WORKS OF FREDERICK H. GETMAN, Ph.D. PUBLISHED BY JOHN WILEY & SONS, Inc. Outlines of Theoretical Chemistry. Second Edition, Thoroughly Revised, xv + 539 pages, 5i X 8. Ill figures. Cloth, $3.50 net. An Introduction to Physical Science. vi + 257 pages, 5 X 7*. 129 figures. Cloth, $1.50 net. Laboratory Exercises in Physical Chemistry. Second Edition, Revised. x + 285 pages, 5 X 7J. 115 figures. Cloth, $2.00 net. OUTLINES OF THEORETICAL CHEMISTRY BY FREDERICK H. GETMAN, Ph.D. (Johns Hopkins) Formerly Associate Professor of Chemistry in Bryn Mawr College SECOND EDITION, THOROUGHLY REVISED AND ENLARGED NEW YORK JOHN WILEY & SONS, INC. LONDON: CHAPMAN & HALL, LIMITED 1918 COPYRIGHT, 1913, 1918, BY FREDERICK H. GETMAN Stanhope jpress F. H.GILSON COMPANY BOSTON. U.S.A. To WHOSE TENDER SYMPATHY HAS HELPED TO LIGHTEN THE DARKNESS OF THE DAYS DURING WHICH THESE PAGES WERE WRITTEN 385126 "I look upon the common operations and practices of chymists, almost as I do on the letters of the alphabet, without whose knowledge 'tis very hard for a man to become a philosopher; and yet that knowledge is very far from being sufficient to make him one." ROBERT BOYLE. (The Sceptical Chymist.) PREFACE TO SECOND EDITION. " The theoretical side of physical chemistry is and will probably remain the dominant one; it is by this peculiarity that it has exerted such a great in- fluence upon the neighboring sciences, pure and applied, and on this ground physical chemistry may be regarded as an excellent school of exact reasoning for all students of natural sciences." Arrhenius. The demand for a second edition of this book has not only afforded the author an opportunity to thoroughly revise the original text, but also has made it possible to include such new material as should properly find a place in an introductory text- book ot theoretical chemistry. The arduousness of the task of revision and amplification has been appreciably lightened by the helpful criticisms and valuable suggestions which have been received from those who have used the first edition with their classes. So numerous were the additional topics suggested that the author found himself confronted with a veritable embarrassment of riches, and not the least difficult part of his task has been the attempt to weave in as many of these suggestions as seemed to be consistent with a well-balanced presentation of the entire subject. The features which distinguish this edition from the preceding edition may be briefly summarized as follows : 1. The necessity of introducing a short chapter on the mod- ern conception of the atom and its structure involved the further necessity of including a preliminary chapter treat- ing of those radioactive phenomena upon which the greater part of our present atomic theory is based. 2. The chapter on solids has been practically rewritten: the space formerly devoted to an outline of crystallog- vii Vlll PREFACE raphy being devoted in the present edition to a discussion of the absorption of heat by crystalline solids and the bear- ing of X-ray spectra on crystalline form. 3. The increasing importance of colloidal phenomena, not only to the chemist but also to the biologist, to the physician, and to the technologist, has made it seem desirable to rewrite the entire chapter devoted to the chemistry of colloids. 4. The Brownian movement and its bearing upon the exist- ence of molecules has been briefly presented in a separate chapter in order to emphasize the importance of the bril- liant experimental work of Perrin and others in confirming the kinetic theory. 5. The chapter treating of electromotive force has been enlarged so as to include a discussion of some of the more valuable methods which have been proposed for deter- mining junction potentials and also to point out several useful applications of concentration cells. 6. An entirely new chapter in which an attempt has been made to present the salient facts and more important theories of photochemistry in succinct form replaces the former chapter treating of the relations between radiant and chemical energy. Among the books to which the author has had frequent re- course in the preparation of this edition should be mentioned Rutherford's " Radioactive Substances and their Radiations," W. H. and W. L. Bragg's "X-Rays and Crystal Structure," Freund- lich's "Kapillarchemie," Perrin's "Les Atomes," and Sheppard's "Photochemistry." It is with a deep sense of gratitude that acknowledgment is made to all of those friends who have offered criticisms of the old edition and suggestions for the new. Special thanks are due to Dr. W. D. Harkins of 'the University of Chicago not only for suggestions but for permission to quote extensively from his papers, to Dr. J. Howard Mathews of the University of Wis- consin for numerous helpful suggestions, to Dr. Walter A. Patrick of Johns Hopkins University for criticisms of the former chapter PREFACE IX on colloids and to Mr. John McGavack for his conscientious work in checking the answers to all of the problems. In the preparation of the indices of names and subjects the author is indebted to his wife and to Miss Mary K. Pease who have given valuable assist- ance in that wearisome and exacting task. To the publishers, Messrs. John Wiley and Sons, Inc., acknowledgment is made of their kindness in permitting the use of Fig. 68 taken from Chamot's "Elementary Chemical Microscopy." FREDERICK H. GETMAN.; STAMFORD, CONN. Aug. 7, 1918. PREFACE. " The last thing that we find in making a book is to know what we must put first." PASCAL. THE present book is designed to meet the requirements of classes beginning the study of theoretical or physical chemistry. A working knowledge of elementary chemistry and physics has been presupposed in the presentation of the subject, the introduc- tory chapter being the only portion of the book in which space is devoted to a review of principles with which the student is assumed to be already fairly familiar. With the exception of a few para- graphs in which the application of the calculus is unavoidable, no use is made of the higher mathematics, so that the book should be intelligible to the student of very moderate mathematical attainments. Wherever the calculus has been employed, the student who is unfamiliar with this useful tool must accept the correctness of the results without attempting to follow the suc- cessive operations by which they are obtained. The contributions to our knowledge in the domain of physical chemistry have increased with such rapidity within recent years, that the prospective author of a general text book finds himself confronted with the vexing problem of what to omit rather than what to include. In selecting material for this book, the author has been guided in large measure by his own experience in teaching theoretical chemistry to beginners and to advanced students. The attempt has been made to present the more difficult portions of the subject, such as the osmotic theory of solutions, the laws of equilibrium and chemical action, and the principles of electro- chemistry, in a clear and logical manner. While the treatment of each topic is necessarily brief yet the effort has been made to avoid the sacrifice of clearness to brevity. CONTENTS CHAP. PAGE PREFACE TO SECOND EDITION vii PREFACE xi I. Fundamental Principles 1 II. Classification of the Elements 20 III. The Electron Theory 31 IV. Radioactivity 42 V. Atomic Structure 57 VI. Gases ; 72 VII. Liquids *. 104 VIII. Solids 153 IX. Solutions 167 X. Dilute Solutions and Osmotic Pressure 187 XI. Association, Dissociation and Solvation 225 XII. Colloids 237 XIII. Molecular Reality 279 XIV. Thermochemistry 286 XV. Homogeneous Equilibrium 312 XVI. Heterogeneous Equilibrium 328 XVII. Chemical Kinetics 359 XVIII. Electrical Conductance 385 XIX. Electrolytic Equilibrium and Hydrolysis 422 XX. Electromotive Force 446 XXI. Electrolysis and Polarization 492 XXII. Photochemistry 504 INDEX OF NAMES 529 INDEX OF SUBJECTS . . 533 THEORETICAL CHEMISTRY. CHAPTER I. FUNDAMENTAL PRINCIPLES. Theoretical Chemistry. That portion of the science of chem- istry which has for its object the study of the laws controlling chemical phenomena is called theoretical or physical chemistry. The first attempt to summarize the more important facts and ideas underlying the science of chemistry was made by Dalton in 1808 in his "New System of Chemical Philosophy." The birth of the science of theoretical chemistry may be considered to be coeval with the appearance of Dalton's epoch-making book. Theoretical chemistry is concerned with the great generaliza- tions of chemical science and bears the same relation to chemistry that philosophy bears to the whole body of scientific truth; it aims to systematize all of the established facts of chemistry and to discover the laws governing the various phenomena of chemical action. Law, Hypothesis and Theory. The science of chemistry is based upon experimentally established facts. When a number of facts have been collected and classified we may proceed to draw inferences as to the behavior of systems under conditions which have not been investigated. This process of reasoning by analogy we term generalization and the conclusion .reached we call a law. It is apparent that a law is not an expression of an infallible truth, but it is rather a condensed statement of facts which have been discovered by experiment. It enables us to predict results with- out recourse to experiment. The fewer the number of cases in which a law has been found to be invalid, the greater becomes our confidence in it, until eventually it may come to be regarded as tantamount to a statement of fact. 1 2 THEORETICAL CHEMISTRY Natural laws may be discovered by the correlation of experi- mentally determined facts, as outlined above, or by means of a speculation as to the probable cause 'of the phenomena in question. Such a speculation in regard to the cause of a phenomenon is called an hypothesis. After an hypothesis has been subjected to the test of experiment and has been shown to apply to a large number of closely related phenomena it is termed a theory. In his address to the British Association (Dundee, 1912), Profes- sor Senier has this to say : " While the method of discovery in chem- istry may be described generally, as inductive, still all the modes of inference which have come down to us from Aristotle, analogical, inductive, and deductive, are freely used. An hypothesis is framed which is then tested, directly or indirectly, by observation and experiment. All the skill, and all the resource the inquirer can command, are brought into his service; his work must be accurate; and with unqualified devotion to truth he abides by the result, and the hypothesis is established, and becomes a part of the theory of science, or is rejected or modified. " Elements and Compounds. All definite chemical substances are divided into two classes, elements and compounds. Robert Boyle was the first to. make this distinction. He de- fined an element as a substance which is incapable of resolution into anything simpler. The substances formed by the chemical combination of two or more elements he termed chemical com- pounds. This definition of an element as given by Boyle was later proposed by Lavoisier and, notwithstanding the vast accumu- lation of scientific knowledge since their time, the definition re- mains very satisfactory today. At the present ]time we have a group of about eighty substances which have resisted all efforts to decompose them into simpler substances. These are the so-called chemical elements. It should be borne in mind, however, that because we have failed to resolve these substances into simpler forms of matter, we are not warranted in maintaining that such resolution may not be effected in the future. Recent investigations of the radioactive elements have shown FUNDAMENTAL PRINCIPLES 3 that they are continuously undergoing a series of transformations, one of the products of which is the inactive element helium. This behavior is contrary to the old view that transformation of one element into another is impossible. At first the attempt was made to explain it by assuming that the radioactive element was a compound of helium with another element, but since the radio- active elements possess all of the properties characteristic of elements as distinguished from compounds, and find appropriate places in the periodic table of Mendeleeff, the "compound theory" must be abandoned. Uranium and thorium, the heaviest ele- ments known, appear to be undergoing a process of spontaneous disintegration over which we have no control. The products of this disintegration have filled the gap in the periodic table between thorium and lead with about thirty new elements, each of which is in turn undergoing transformations similar to those of the parent elements. Professor Soddy * says: "In spite of the existence at one time of a vague belief ( a belief which has no foundation), that all matter may be to a certain extent radioactive, just as all matter is believed to be to a certain extent magnetic, it is recognized today that radioactivity is an exceedingly rare prop- erty of matter." Notwithstanding these remarkable discoveries, we may still hold to the idea of an element as suggested by Boyle and Lavoisier. Professor Walker says: "The elements form a group of substances, singular not only with respect to the resistance which they offer to decomposition, but also with respect to certain regularities dis- played by them and not shared by substances which are designated as compounds." Law of the Conservation of Mass. In 1774, as the result of a series of experiments, Lavoisier established the law of the con- servation of mass which may be stated as follows : In a chemical reaction the total mass of the reacting substances is equal to the total mass of the products of the reaction. It is sometimes stated thus: the total mass of the universe is a constant; but this form of state- ment is open to the objection that we have no means of verification; it is a statement of a fact which transcends our experience. * Chemistry of the Radio-Elements, p. 2. 4 THEORETICAL CHEMISTRY The law of the conservation of mass has been subjected to most rigid investigation by Landolt * in a series of experiments extend- ing over a period of fifteen years. The reacting substances, AB and CD, ^ were placed in the two arms of the in- verted U-tube shown in Fig. 1 which was then sealed at S and the whole weighed upon an extremely sensitive balance. The vessel was then inverted when the following reaction took place : AB + CD-+AD + CB. When the reaction was complete and suf- ficient time had elapsed to allow the vessel AB| I ^zr-nicD to return to its original volume (this some- times required nearly three weeks), it was weighed again using every precaution to avoid errors and any gain or loss in weight noted. Fig. 1. Landolt concluded from the thirty or more reactions which he studied that the gain or loss in weight was less than one ten- millionth of the total weight, f Law of Definite Proportions. The enunciation of the law of the conservation of mass and the introduction of the balance into the chemical laboratory marked the beginning of a new era in the history of chemistry, the era of quantitative chemistry. As the result of painstaking experimental work, Richter and Proust announced the law of definite proportions about the beginning of the nineteenth century. Tnis law may be expressed thus: A definite chemical compound always contains the same elements united in the same proportion by weight. Shortly after the enunciation of this law its truth was questioned by the French chemist Berthollet.| From the results of a series * Zeit. phys. chem., 12, 1 (1893); 55, 589 (1906). f An excellent summary of this important investigation will be found in the Journal de Chimie physique, 6, 625 (1908). J Essai de statique chimique (1803). FUNDAMENTAL PRINCIPLES 5 of brilliant experiments, he became convinced that chemical reac- tions are largely controlled by the relative amounts of the react- ing substances. As we shall see later, he really foreshadowed the work of Guldberg and Waage who were the first to correctly for- mulate the influence of mass on a chemical reaction. Berthollet argued that when two elements unite to form a compound, the proportion of one of the elements in the compound is conditioned solely by the amount of that element which is available. This led to the celebrated controversy between Berthollet and Proust which finally resulted in the establishment of the latter's original statement. Subsequent investigation has only strengthened our faith in the law of definite proportions. Law of Multiple Proportions. Elements are known to unite in more than one proportion by weight. Dalton analyzed the two compounds of carbon and hydrogen, methane and ethylene, and found that the ratio of the weights of carbon to hydrogen in the former was 6 : 2 while in the latter it was 6:1. That is, for the same weight of carbon, the weights of the hydrogen in the two compounds were in the ratio 2:1. A large number of compounds were examined and similar simple ratios between the masses of the constituent elements were found. As a result of these observations, Dalton * formu- lated in 1808 the law of multiple proportions, as follows: When two elements unite in more than one proportion, for a fixed mass of one element the masses of the other element bear to each other a simple ratio. Notwithstanding the fact that Dalton was a careless experimenter the subsequent investigations of Marignac and others have established the validity of his law. Law of Combining Proportions. Dalton pointed out that it is possible to assign to every element a definite relative weight with which it enters into chemical combination. He observed that the weights or simple multiples of the weights of the different elements which unite with a given weight of a definite element, represent the weights of the different elements which combine with each other. The weights of the elements which combine with each other are termed their combining weights. This com- * A New System of Chemical Philosophy (1808). 6 THEORETICAL CHEMISTRY prehensive law of chemical combination may be stated as follows: Elements combine in the ratio of their combining weights or in simple multiples of this ratio. It will be observed that this law really includes the law of definite and the law of multiple pro- portions. If we assume the combining weight of hydrogen to be unity, the combining weights of chlorine, oxygen and sulphur will be 35.5, 8 and 16 respectively. These numbers represent the ratios in which the elements substitute each other in chemical com- pounds. Hydrochloric acid, for example, contains 35.5 parts by weight of chlorine to 1 part by weight of hydrogen and when oxygen is substituted for chlorine, forming water, the new com- pound contains 8 parts by weight of oxygen to 1 part by weight of hydrogen. Similarly, if the oxygen be substituted by sulphur, forming hydrogen sulphide, there will be found 16 parts by weight of sulphur to 1 part by weight of hydrogen. We may say, then, that 35.5 parts of chlorine, 8 parts of oxygen and 16 parts of sul- phur are equivalent. A chemical equivalent may be defined as the weight of an element which is necessary to combine with or displace 1 part by weight of hydrogen. The Atomic Theory. In very early times two different views were entertained by opposing schools of Greek philosophers as to the mechanical constitution of matter. According to the school of Plato and Aristotle, matter was thought to be continuous within the space it appears to fill and to be capable of indefinite subdivision. According to the other school, first taught by Leucippus, and afterwards by Democritus and Epicurus, matter was considered to be made up of primordial, extremely minute particles, distinct and separable from each other but in themselves incapable of division. These ultimate particles were called atoms (iJaTo/uios), signifying something indivisible. While the Aristote- lian doctrine held sway for many centuries yet the notion of atoms was revived at intervals. Xate in the seventeenth century, Boyle seems to have looked upon chemical combination as the result of atomic association. Guided by these early speculations as to the constitution of FUNDAMENTAL PRINCIPLES 7 matter and influenced by his study of the writings of Sir Isaac Newton, Dal ton seems to have formed a mental picture of the part played by atoms in the act of chemical combination. After a few carelessly performed experiments, the results of which ac- corded with his preconceived ideas, he formulated his atomic theory. According to this theory matter is composed of extremely mi- nute, indivisible particles or atoms. Atoms of the same element are all of equal weight, but atoms of different elements have weights proportional to their combining numbers. Chemical compounds are formed by the union of atoms of different kinds. This theory offers a simple, rational explanation of the laws of chemical combination. Since a chemical compound results from the union of atoms, each of which has a definite weight, its composition must be in- variable, which is the law of definite proportions. Again, when atoms combine in more than one proportion, for a fixed weight of atoms of one kind, the weights of the other species of atoms must bear to each other a simple ratio, since the atoms are indivisible units. This is clearly the law of multiple proportions. Finally, the law of combining weights is seen to follow as a necessary consequence of the atomic theory, since the experimen- tally determined combining weights bear a simple relation to the relative weights of the atoms. At the time when Dalton proposed his atomic theory, the number of facts to be explained was comparatively small, but with the enormous growth of the science of chemistry during the past century and with the vast accumulation of data, the theory has proved capable of affording adequate representation of all of the facts, and has opened the way to many important generaliza- tions. While the atomic theory has played a very important part in the development of modern chemistry, and while we recognize that it helps to clarify our thinking and enables us to construct a mental image of tiny spheres uniting to form a chemical compound, yet we must not forget the fact that these atoms are purely hypothetical. Faraday has said: "Whether matter be atomic or not, this 8 THEORETICAL CHEMISTRY much is certain, that granting it to be atomic, it would appear as it now does." Ostwald believes that in the not distant future the atomic theory will be abandoned and chemists will free them- selves from the yoke of this hypothesis, relying solely upon the results of experiment. He says: "It seems as if the adaptabil- ity of the atomic hypothesis is near exhaustion, and it is well to realize that, according to the lesson repeatedly taught by the history of science, such an end is sooner or later inevitable. " Combining Weights and Atomic Weights. The problem of determining the relative atomic weights of the elements would at first sight appear to be a very simple matter. This might appar- ently be accomplished by selecting one element, say hydrogen, it being the lightest known element, as the standard ; a compound of hydrogen and another element may then be analyzed and the amount of the other element in combination with one part by weight of hydrogen determined. This weight will be its atomic weight only when the compound contains but one atom of each element. To determine the relative atomic weight, therefore, we must know in addition to the chemical equivalent of the element, the number of atoms present in the compound. For example, the analysis of water shows it to contain 8 parts by weight of oxygen to 1 part by weight of hydrogen; the chemical equivalent of oxygen is, therefore, 8, and if water contained but one atom of hydrogen the atomic weight of oxygen would be 8. It can be shown, however, that water contains two atoms of hydrogen and one atom of oxygen, therefore, the atomic weight of oxygen must be 16. It is evident, therefore, that neither the analysis nor the synthesis of a compound is sufficient to enable us to determine the number of atoms of an element combined with one atom of hydrogen. We shall proceed to the consideration of the methods by which this problem may be solved. Gay-Lussac's Law of Volumes. Gay-Lussac in 1808, while studying the densities of gases before and after reaction, announced the following law : When gases combine they do so in simple ratios by volume, and the volume of the gaseous product bears a simple ratio to the volumes of the reacting gases when measured under like conditions of temperature and pressure. Thus, one volume of hydro- FUNDAMENTAL PRINCIPLES 9 gen combines with one volume of chlorine to form two volumes of hydrochloric acid; one volume of oxygen combines with two volumes of hydrogen to form two volumes of water (vapor) ; and one volume of nitrogen combines with three volumes of hydrogen to form two volumes of ammonia. In a previous investigation, Gay-Lussac had shown that all gases behave identically when subjected to changes of temperature and pressure. This fact, taken together with the simple volumetric relation just enunciated and the atomic theory, suggested a possible relation between the number of ultimate particles in equal vol- umes of different gases. Berzelius attempted to show that under corresponding condi- tions of temperature and pressure, equal volumes of different gases contain the same number of atoms, but he was compelled to abandon the assumption as untenable. Avogadro' s Hypothesis. It remained for the Italian physicist, Avogadro,* in 1811, to point out the distinction between atoms and molecules, terms which had been used almost synonymously up to his time. He denned the atom as the smallest particle which can enter into chemical combination, whereas the molecule is the smallest portion of matter which can exist in a free state. He then formulated the following hypothesis :f Under the same conditions of temperature and pressure, equal volumes of all gases contain the same number of molecules. This hypothesis has been subjected to such rigid experimental and mathematical tests that its validity cannot be questioned. Avogadro's Hypothesis and Molecular Weights. According to Gay-Lussac when hydrogen and chlorine combine to form hydro- chloric acid, one volume of hydrogen unites with one volume of chlorine yielding two volumes of hydrochloric acid. According to the hypothesis of Avogadro, the number of mole- cules of hydrochloric acid is double the number of molecules of hydrogen or of chlorine, and, consequently, each molecule of the reacting gases must contain at least two atoms. If we take hydrogen as the unit of our system of atomic weights, its molec- * Jour, de Phys., 73, 58 (1811). f Ampere advanced nearly the same hypothesis in 1814. 10 THEORETICAL CHEMISTRY ular weight must be 2. It is convenient to express molecular and atomic weights in terms of the same unit, for then the molec- ular weight of a substance will be simply the sum of the weights of the atoms contained in the molecule. The determination of the approximate molecular weight of a substance, therefore, re- solves itself into ascertaining the mass of its vapor in grams which, under the same conditions of temperature and pressure, will occupy the same volume as 2 grams of hydrogen. This weight is called the gram-molecular weight or the molar weight of the substance, while the corresponding volume is known as the gram-molecular or molar volume. It is nearly the same for all gases and at and 760 mm. it may be taken equal to 22.4 liters. The molecular weights obtained from vapor density meas- urements are approximate only, because of the failure of most gases and vapors to obey the simple gas laws, a condition essen- tial to the strict applicability of Avogadro's hypothesis. Atomic Weights from Molecular Weights. While vapor density determinations as ordinarily carried out do not give exact molecular weights, it is an easy matter to arrive at the true values when we take into consideration the results of chemical analysis. It is apparent that the true molecular weight must be the sum of the weights of the constituent elements, these weights being exact multiples or submultiples of their combining proportions, which proportions have been determined by analysis alone. We select, as the true molecular weight, the value which is nearest to the approximate molecular weight calculated from the vapor density of the substance. For example, the molecular weight of ammonia, as computed from its vapor density, is 17.5 or, in other words, 17.5 grams of ammonia occupy the same volume as 2 grams of hydrogen, measured under the same conditions of temperature and pressure. The analysis of ammonia shows us that for every gram of hydrogen, there are present 4.67 grams of nitrogen. Hence the true molecular weight must contain a multiple of 1 gram of hydrogen and the same multiple of 4.67 grams of nitrogen. The problem is, to find what integral value must be assigned to x in the expression, x (1 + 4.67), in order that it may give the closest approximation to 17.5. Clearly if x = 3 the value of the FUNDAMENTAL PRINCIPLES 11 expression becomes 17, and this we take to be the true molecular weight. This gives 3 X 4.67 = 14 as the probable atomic weight of nitrogen. To decide whether the atomic weight of nitrogen is a multiple or a submultiple of 14, we must determine the molecu- lar weights of a large number of gaseous or vaporizable compounds of nitrogen and select as the atomic weight the smallest quantity of the element which is present in any one of them. The following table gives a list of seven gaseous compounds of nitrogen together with their gram-molecular weights, and the number of grams of the element in the gram-molecule. Compound. Gram-mol. Wt. Grams Nitro- gen. Ammonia 17 14 Nitric oxide 30 14 Nitrogen peroxide 46 14 Methyl nitrate 77 14 Cyanogen chloride 61.5 14 Nitrous oxide 44 28 Cyanogen 52 28 It will be observed that the least weight of nitrogen entering into a gram-molecular weight of any of these compounds is 14 grams, and, therefore, we accept this value as the atomic weight of the element, although there is still a very slight chance that in some other compound of nitrogen a smaller weight of the ele- ment may be found. We shall proceed to point out that there are methods by which the probable values of the atomic weights may be checked. Specific Heat and Atomic Weight. In 1819 the French chem- ists, Dulong and Petit,* pointed out a very simple relation between the specific heats of the elements in the solid state and their atomic weights. This relation, known as the law of Dulong and Petit, is as follows : The product of the specific heat and the atomic weight of the solid elements is constant. The value of this constant, called the atomic heat, is approximately 6.4. A little reflection will show that an alternative statement of this law is that the * Ann. Chim. Phys., 10, 395 (1819). 12 THEORETICAL CHEMISTRY atoms of the elements in the solid state have the same thermal capac- ity. The specific heats, atomic weights and atomic heats of several elements are given in the subjoined table. Element. At. Wt. Sp. Ht. At. Ht. Lithium 7 0.940 6.6 Glucinum 9 0.410 3.7 Boron (amorphous) Carbon (diamond) Sodium Silicon (crystalline) 11 12 23 28 0.250 0.140 0.290 0.160 2.8 1.7 6.7 4.5 Potassium 39 166 6.5 Calcium 40 0.170 6.8 Iron 56 112 6.3 Copper. . 63 0.093 5.9 Zinc 65 0.093 6.1 Silver 108 0.056 6.0 Tin.. 119 0.054 6.5 Gold 197 032 6.3 Mercury 200 032 6.4 It is truly remarkable that elements differing as greatly as lithium and mercury differ, not only in atomic Weight but in other properties as well, should have identical atomic heats. It will be observed that the atomic heats of boron, silicon, carbon and glucinum are too low. This departure from the law of Dulong and Petit is more apparent than real, for in the statement of the law there is no specification as to the temperature at which the specific heat should be determined. The specific heats of all solids, vary with the temperature, this variation being greater in the case of some elements than in that of others. It has been shown that the specific heats of the above four elements increase rapidly with rise of temperature and approach limiting values. As these values are approached the product of specific heat and atomic weight approximates more and more closely to the mean value of the constant, 6.4. The following table gives the values obtained by Weber * for carbon and silicon. Pogg. Ann., 154, 367 (1875). FUNDAMENTAL PRINCIPLES CARBON (DIAMOND). 13 Temperature, degrees. Sp. Ht. At. Ht. -50 0.0635 0.76 +10 0.1128 1.35 85 0.1765 2.12 206 0.2733 3.28 607 0.4408 5.30 806 0.4489 5.40 985 0.4589 5.50 CARBON (GRAPHITE). Temperature, degrees. Sp. Ht. At. Ht. -50 0.1138 1.37 + 10 0.1604 1.93 61 0.1990 2.39 202 0.2966 3.56 642 0.4454 5.35 822 0.4539 5.45 978 0.4670 5.50 SILICON. Temperature, degrees. Sp. Ht. At. Ht. -40 0.136 3.81 +57 0.183 5.13 129 0.196 5.50 232 0.203 5.63 It is evident that this empirical relation can be used to deter- mine the approximate atomic weight of an element when its specific heat is known, thus 6.4 atomic weight = . ' , specific heat The law of Dulong and Petit has been of great service in fixing and checking atomic weights. About twenty years after the law of Dulong and Petit was formulated, Neumann* showed that a similar relation holds for * Pogg. Ann., 23, 1 (1831). 14 THEORETICAL CHEMISTRY compounds of the same general chemical character. Neumann's law may be stated thus: Similarly constituted compounds in the solid state have the same molecular heat. Subsequently Kopp * pointed out that the thermal capacity of the atoms is not appreciably altered when they enter into chemical combination, or in other words, the molecular heat of solid compounds is an additive property, being made up of the atomic heats of the constituent elements. For example, the specific heat of PbBr 2 is 0.054 and its molec- ular weight is 366.8, therefore, the molecular heat is 0.054 X 366.8 = 19.9. Since there are three atoms in the molecule, 19.9 -f- 3 = 6.6 is their average atomic heat, a value in excellent agree- ment with the constant in the law of Dulong and Petit. Neu- mann's law may be used to estimate the atomic heats of elements which cannot be readily investigated in the solid state. The following table gives a list of atomic heats of elements in the solid state derived by means of Neumann's law. Element. At. Ht. Element. At. Ht. Hydrogen 2.3 Carbon 1.8 Oxygen 4.0 Silicon 4.0 Fluorine Nitrogen 5.0 5 5 Phosphorus Sulphur 5.4 5 4 Isomorphism. From a study of the corresponding salts of phosphoric and arsenic acids, Mitscherlich f observed that they crystallize with the same number of molecules of water and are nearly identical in crystalline form, it being possible to obtain mixed crystals from solutions containing both salts. This sug- gested to Mitscherlich a line of investigation which resulted, in 1820, in the establishment of the law of isomorphism which bears his name. This law may be stated as follows: An equal number of atoms combined in the same manner yield the same crystal form, which is independent of the chemical nature of the atoms and dependent upon their number and position. Thus, when one element replaces * Lieb. Ann. (1864), Suppl., 3, 5. f Ann. Chim. Phys. (2), 14, 172 (1820). FUNDAMENTAL PRINCIPLES 15 another in a compound without changing its crystalline form, Mitscherlich assumed that one element has displaced the other, atom for atom. For example, having two isomorphous substances, such as BaCl2.2 H 2 and BaBr 2 .2 H 2 O, we assume that the brom- ine in the second compound has replaced the chlorine in the first and, if the atomic weights of all of the elements in the first com- pound are known, then it is evident that the atomic weight of the bromine in the second compound can be easily calculated. This method was largely used by Berzelius in fixing atomic weights and in checking the values obtained by the volumetric method. It should be remembered that the converse of the law of isomorphism does not hold, since elements may replace each other, atom for atom, without preserving the same form of crystallization. Many exceptions to the law have been pointed out. For example, Mitscherlich himself showed that Na 2 SO 4 and BaMn 2 8 are iso- morphous and yet the two molecules do not contain the same number of atoms. Furthermore, careful measurements of the interfacial angles of crystals have revealed the fact that sub- stances which have been regarded as isomorphous are only approx- imately so, thus the interfacial angles of the apparently isomorph- ous crystalline salts given in the following table differ appreciably. Salt. Interfacial Angle. MgS0 4 .7 H 2 ZnSO 4 .7 H 2 O NiSO 4 .7 H 2 O 89 26' 88 53' 88 56' Ostwald has suggested that the term homeomorphous be applied to designate substances which have nearly identical form. At best the principle of isomorphism is only an approximation and should be employed with caution. Valence. During the latter half of the nineteenth century the usefulness of the atomic theory was greatly enhanced by the introduction of certain assumptions concerning the combining power of the atoms. These assumptions, constituting the so- called doctrine of valence, were forced upon chemists in order that a satisfactory explanation might be offered of the phenomenon 16 THEORETICAL CHEMISTRY of isomerism. A consideration of the following formulas, - HC1, H 2 0, NH 3 , CH 4 , shows that the power to combine with hydrogen increases regularly from chlorine, which combines with hydrogen, atom for atom, to carbon, one atom of which is capable of combining with four atoms of hydrogen. Either hydrogen or chlorine, each of which is capable of combining with but one atom of the other, may be taken as an example of the simplest kind of atom. Any element like hydrogen or chlorine is called a univalent element, whereas elements similar to oxygen, nitrogen and carbon, which are capable of combining with two, three or four atoms of hydrogen, are called bivalent, trivalent and quadri- valent elements respectively. Most elements belong to one or the other of these four classes, although quinquivalent, sexivalent and septivalent elements are known. The familiar bonds or link- ages of structural formulas are graphic representations of the valence of the atoms constituting the molecule. This useful con- ception of valence has made possible the prediction of the prop- erties of many compounds before they have been discovered in nature or in the laboratory. Atomic Weights. Among the first to recognize the importance of Dalton's atomic theory was the Swedish chemist, Berzelius. He foresaw the importance for chemists of a table of exact atomic weights and in 1810 he undertook the task of determining the combining weights of most of the known elements. For nearly six years he was engaged in determining the exact composition of a large number of compounds and calculating the combining weights of their constituent elements, thus compiling the first table of atomic weights. - Numerous investigators since Berzelius have been engaged in this important work, among whom should be mentioned Stas, Marignac, Morley and Richards. On two occasions special stimu- lus was given to such investigations. The first occasion was in 1815 when Prout suggested that the atomic weights of the elements are exact multiples of the atomic weight of hydrogen. The values obtained by Berzelius were incompatible with the hypothesis of Prout, although the atomic weights of several of the elements differed but little from integral values. To test the accuracy of FUNDAMENTAL PRINCIPLES 17 this hypothesis, Stas undertook the determination of the atomic weights of several elements with a degree of accuracy such that his maximum experimental error was less than the difference between the atomic weight found and the nearest whole number. This important series of investigations disproved Prout's hypoth- esis as originally stated. The second occasion when the investi- gation of atomic weights received a special impulse was in 1869 when Mendeleeff brought forward the periodic classification of the elements. When the elements were arranged in the order of their atomic weights, several of them were found to fall in groups with which their chemical and physical properties did not correspond, and Mendeleeff asserted that in these cases the commonly accepted atomic weights, were erroneous. This led to the careful redeter- mination of the atomic weights which Mendeleeff had asserted to be faulty, and in most cases his predictions were confirmed. In connection with the precise determination of atomic weights, the work of Cannizzaro in the latter part of the nineteenth century should be mentioned. He emphasized the importance of Avoga- dro's law as the basis of atomic weight determinations, and drew a sharp distinction between atomic and molecular weights, thus bringing order out of confusion and rendering possible the present system of atomic weights. In recent times, the most noteworthy investigations in this field are those of Morley on the combining ratio of hydrogen and oxygen, and the determination by T. W. Richards and his co-workers of the atomic weights of a large number of elements. International Atomic Weights. Dalton selected hydrogen, the lightest known element, as the unit of his system of combin- ing weights of the elements, but Berzelius pointed out that this was an unwise choice since but relatively few of the elements form stable compounds with hydrogen. He proposed, therefore, that oxygen should be taken as the standard, assigning to it the arbi- trary value 100. This proposal of Berzelius, to substitute oxygen for hydrogen as the unit of atomic weights, did not receive serious consideration until quite recently when the International Com- mittee on Atomic Weights took the matter up and, after careful deliberation, decided in favor of the oxygen standard. The 18 THEORETICAL CHEMISTRY 1917. INTERNATIONAL ATOMIC WEIGHTS. Aluminium A1 O = 16 27.10 Molybdenum Mo = 16 96 00 Antimony Sb 120 20 Neodymium Nd 144 30 Argon. . A 39.88 Neon Ne 20 20 Arsenic As 74.96 Nickel Ni 58 68 Barium Ba 137.37 Niton (radium emanation). Nt 222 40 Bismuth Bi 208.00 Nitrogen N 14 01 Boron B 11.00 Osmium Os 190 90 Bromine Br 79.92 Oxygen o 16 00 Cadmium Cd 112.40 Palladium Pd 106 70 Caesium. Cs 132 81 Phosphorus p 31 04 Calcium Ca, 40.07 Platinum Pt 195 20 Carbon c 12 005 Potassium K 39 10 Cerium Ce 140 25 Praseodymium Pr 140 90 Chlorine Cl 35.46 Radium Ra 226 00 Chromium. . Cr 52 00 Rhodium Rh 102 90 Cobalt Co 58 97 Rubidium Rb 85 45 Columbium Cb 93.10 Ruthenium Ru 101 70 Copper Cu 63.57 Samarium . . . Sa 150 40 Dysprosium Dy 162.50 Scandium . Sc 44 10 Erbium Fr 167.70 Selenium Se 79 20 Europium Eu 152 00 Silicon Si 28 30 Fluorine F 19 00 Silver Ae 107 88 Gadolinium Gd 157.30 Sodium Na 23 00 Gallium. Ga 69 90 Strontium Sr 87 63 Germanium Ge 72 50 Sulphur s 32 06 Glucinum. . Gl 9 10 Tantalum Ta 181 50 Gold Au 197 20 Tellurium Te 127 50 Helium, . . . He 4 00 Terbium Tb 159 20 Holmium Ho 163 50 Thallium. . Tl 204 00 Hydrogen H 1 008 Thorium Th 232 40 Indium. In 114 80 Thulium Tm 168 50 Iodine I 126 92 Tin Sn 118.70 Iridium Iron Ir Fe 193.10 55 84 Titanium Tungsten Ti W 48.10 184.00 Krypton Lanthanum Kr La 82.92 139 00 Uranium Vanadium U v 238.20 51 00 Lead Pb 207 20 Xenon Xe 130 20 Lithium Lutecium Li Lu 6.94 175 00 Ytterbium (Neoytterbium) Yttrium . Yb Yt 173.50 88.70 Magnesium Mg 24 32 Zinc Zn 65.37 Manganese Mercury Mn Hg 54.93 200.60 Zirconium Zr 90.60 FUNDAMENTAL PRINCIPLES 19 atomic weight of oxygen is taken as 16, and the unit to which all atomic weights are referred is one-sixteenth of this weight. The atomic weight of hydrogen on this basis is 1.008. Aside from the fact that most of the elements form compounds with oxygen which are suitable for analysis, the atomic weights of more of the ele- ments approximate to integral values when oxygen instead of hy- drogen is used as the standard. The table on page 18 gives the values of the atomic weights as published by the International Committee on Atomic Weights for 1917. CHAPTER II. CLASSIFICATION OF THE ELEMENTS. Early Attempts at Classification. Many attempts were made to classify the elements according to various properties, such as their acidic or basic characteristics or their valence. In all of these systems the same elements frequently found a place in more than one group, and elements bearing little resemblance to each other were classed together. The early attempts at classifica- tion based upon the atomic weights of the elements were not successful owing to the uncertainty as to the exact numerical values of these constants. Prout's Hypothesis. In 1815, W. Prout, an English physician, observed that the atomic weights of the elements, as then given, did not differ greatly from whole numbers when hydrogen was taken as the standard. Hence he advanced the hypothesis that the different elements are polymers of hydrogen. As has already been pointed out this hypothesis led Stas to undertake his refined determinations of the atomic weights of silver, lithium, sodium, potassium, sulphur, lead, nitrogen and the halogens. As a result of his investigations he says: "I have arrived at the absolute conviction, the complete certainty, so far as is possible for a human being to attain to certainty in such matter, that the law of Prout is nothing but an illusion, a mere speculation definitely contra- dicted by experience." Notwithstanding the fact that Prout 's hypothesis as originally stated was thus disproved by Stas, it still survived in a modified form given to it by J. B. Dumas, who sug- gested that one-half of the atomic weight of hydrogen should be taken as the fundamental unit. When Stas showed that his experiments excluded this possibility, Dumas suggested that the fundamental unit be taken as one-quarter of the atomic weight of hydrogen. Having begun to divide and subdivide, there was no limit to the process, and the hypothesis fell into disfavor, although 20 CLASSIFICATION OF THE ELEMENTS 21 the belief in a primal element, something akin to the protyle (-rrptary) v\rj) of the ancient philosophers, has survived and in modern times has reappeared in the electron theory. Dobereiner's Triads. About 1817 J. W. Dobereiner* observed that groups of three elements could be selected from the list of the elements, all of which are chemically similar, and having atomic weights such that the atomic weight of the middle member is the arithmetical mean of the first and third members of the group. These groups of three elements he termed triads. In the following table a few of these triads are given. Element. At. Wt. Mean atomic weight of triads. Lithium. . 6 94 ) Sodium. . 23 00 23 02 Potassium 39 10 Calcium 40 07 ) Strontium. . 87 63 88 72 Barium 137 37 Chlorine 35.46 Bromine 79.92 > 80 69 Iodine 126 92 Sulphur 32 07 ) Selenium 79 2 78 78 Tellurium * Phosphorus 127.5 31 04 Arsenic . . . . 74 96 75 62 Antimony 120 2 This simple relation, first pointed out by Dobereiner, is clearly a foreshadowing of the periodic law. The Helix of de Chancourtois. The idea of arranging the elements in the order of their atomic weights with a view to emphasizing the relationship of their chemical and physical prop- erties, seems to have first suggested itself to M. A. E. B. de Chan- courtois f in the year 1862. On a right-circular cylinder he traced * Pogg. Ann., 15, 301 (1825). t Vis Tellurique, Classement naturel des Corps Simples. 22 THEORETICAL CHEMISTRY what he termed a " telluric helix" at a constant angle of 45 to the axis. On this curve he laid off lengths corresponding to the atomic weights of the elements, taking as a unit of measure a length equal to one-sixteenth of a complete revolution of the cylinder. He then called attention to the fact that elements with analogous properties fall on vertical lines parallel to the generatrix. Being a mathematician and a geologist he did not express himself in such terms as would attract the attention of chemists and con- sequently his work remained unnoticed until recent times. The Law of Octaves. In 1864 J. A. R. Newlands* pointed out that if the elements are arranged in the order of their atomic weights, the eighth element has properties very similar to the first; the ninth to the second; the tenth to the third; and so on, or to employ Newlands' own words : " The eighth element starting from a given one is a kind of repetition of the first, like the eighth note of an octave in music." This peculiar relationship, termed by New- lands the law of octaves, is brought out in the following table. H Li Gl B C N F Na Mg Al Si P S Cl K Ca Cr Ti Mn Fe Notwithstanding the fact that its author was ridiculed and his paper returned to him as unworthy of publication in the proceed- ings of the Chemical Society, this generalization must be regarded as the immediate forerunner of the periodic law. The Periodic Law. Quite independently of each other and apparently in ignorance of the work of Newlands and de Chan- courtois, Mendeleeff f in Russia and Lothar Meyer in Germany, gamed a far deeper insight into the relations existing between the properties of the elements and their atomic weights. In 1869, Mendeleeff wrote: "When I arranged the elements according to the magnitude of their atomic weights, beginning with the smallest, it became evident that there exists a kind of periodicity * Chem. News, 10, 94 (1864), Ibid., 12, 83 (1865). t Lieb. Ann. Suppl., 8, 133 (1874). CLASSIFICATION OF THE ELEMENTS 23 in their properties. I designate by the name 'periodic law' the mutual relations between the properties of the elements and their atomic weights; these relations are applicable to all the elements and have the nature of a periodic function." This important generalization may be briefly stated thus: The properties of the elements are periodic functions of their atomic weights. The original table of Mendeleeff has been amended and modified as new data has accumulated and new elements have been dis- covered. The accompanying table, though containing several new elements and an entirely new group, is essentially the same as that of Mendeleeff. It consists of nine vertical columns, called groups, and twelve horizontal rows termed series or periods. The second and third periods contain eight elements each, and are known as short periods, while in the fourth series, starting with argon, it is necessary to pass over eighteen elements before another element, krypton, is encountered which bears a close resemblance to argon: such a series of nineteen elements is called a long period. The entire table is composed of two short and five long periods, the last one being incomplete. The positions of the elements are largely determined by their chemical similarity to those in the same group, the hyphens indicating the positions of undiscovered elements. The elements in Group VIII, presented difficulties when Mendeleeff attempted to place them according to their atomic weights and so he was obliged to group them by themselves. This group has wittily been designated as "the hospital for incur- ables." An examination of the table shows that the valence of the elements toward oxygen progresses regularly from Group 0, containing elements which exhibit no combining power, up to Group VIII, where it attains a maximum value of eight in the case of osmium. The valence toward hydrogen on the other hand increases regularly from Group VII to Group IV in which the elements are quadrivalent. 24 THEORETICAL CHEMISTRY V R0 4 . S88888 22 O O t- Ir > Pt > Au. But from analogies existing between osmium, ruthenium and iron and the disposition of the preceding members of Group VIII, Mendeleeff predicted that the atomic weights were in error and that the order of the elements should be Os < Ir < Pt < Au. CLASSIFICATION OF THE ELEMENTS 29 Subsequent atomic weight determinations by Seubert substantiated MendeleefFs prediction. Defects in the Periodic Law. While the arrangement of the elements in the periodic table is on the whole very satisfactory, there are several serious defects in the system which should be pointed out. At the very outset there is difficulty in finding a place for hydrogen in the system. The element is univalent and falls either in Group I, with the alkali metals, or in Group VII with the halogens. While the element is electro-positive it cannot be considered to possess metallic properties. It forms hydrides with some of the metallic elements and can be displaced by the halo- gens from organic compounds. These facts make it extremely difficult to decide whether hydrogen should be placed in Group I or Group VII. The idea has been advanced that hydrogen is the only known member of the first series of the periodic table. These hypothetical elements have been styled proto-elements, the successive members of the series being, proto-glucinum, proto- boron and so on to the last element in the series, proto-fluorine. To find a suitable location for the rare-earth elements in the periodic system is another difficulty which has not been satisfactorily met. Brauner considers that these elements should all be grouped together with cerium (at. wt. = 140.25), but owing to our limited knowledge of the properties of these elements it seems better to defer attempting to place them for the present. In the group of non-valent elements the atomic weight of argon is distinctly higher than that of potassium in the next group. There can be little doubt that the values of the atomic weights are correct and it is evidently impossible to interchange the positions of these two elements in the periodic table, since argon is as much the analogue of the rare gases as potassium is of the alkali metals. A similar discrepancy occurs with the elements, tellurium and iodine. The atomic weight of the former element is appreciably higher than that of the latter and, notwithstanding the attempts of numerous investigators to prove tellurium to be a complex of two or more elements, nothing but failure has attended their efforts. Still another anomaly is encountered in Group VII, where manganese is classed with the halogen family, to which it bears much less 30 THEORETICAL CHEMISTRY resemblance than it does to chromium and iron, its two immediate neighbors. As has already been mentioned, Group VIII is made up of non- conformable elements. If the properties of the elements are dependent upon their atomic weights, it should be impossible for several elements having almost identical atomic weights and different properties to exist, and yet such is the case with the elements of Group VIII. The elements copper, silver and gold, while not closely resembling the other members of Group VIII, are much more closely allied to them than to the alkali metals with which they are also classed. Notwithstanding its imper- fections, the periodic law must be regarded as a truly wonderful generalization which future investigations will undoubtedly show to be but a fragment of a more comprehensive law. CHAPTER III. THE ELECTRON THEORY. Conduction of Electricity through Gases. Within recent years the discovery of new facts relative to the conduction of electricity through gases has led to the development of the so- called electron or corpuscular theory of matter. Under ordinary conditions gases are practically non-conductors of electricity, but when a sufficiently great difference of potential is established between two points within a gas it is no longer able to withstand tne stress, and an electric discharge takes place between the points. The potential necessary to produce such a discharge is quite hign, To pump Fig. 3. several thousand volts being required to produce a spark of one centimeter length in air at ordinary pressures. The pressure of the gas has a marked effect upon the character of the discharge and the potential required to produce it. If we make use of a glass vessel similar to that shown in Fig. 3, the effect of pressure on the nature of the discharge may be studied. This apparatus consists of a straight glass tube about 4 cm. hi diameter and 40 cm. long, into the ends of which platinum elec- trodes are sealed. To the side of the vessel a small tube is sealed so that connection may be established with an air-pump and manometer. If the electrodes are connected with the terminals of an induction coil and the pressure within the tube be gradually diminished, the following changes in the character of the dis- 31 32 THEORETICAL CHEMISTRY charge will be observed. At first the spark becomes more uni- form and then broadens out, assuming a bluish color. When a pressure of about 0.5 mm. is reached, the negative electrode or cathode will appear to be surrounded by a thin luminous layer; next to this will be a dark region, known as the Crookes' dark space; adjoining this will be a luminous portion, called the negative glow, and beyond this will be seen another dark region which is frequently referred to as the Faraday dark space. Between the Faraday dark space and the positive electrode or anode is a luminous portion, called the positive column. By a slight variation of the current and pressure the positive column can be caused to break up into alternate light and dark spaces or strioe, the appearance of which is dependent upon various factors such as Fig. 5. the nature of the gas and the size of the tube. If we use a modi- fication of this tube, such as is shown in Fig. 4, and diminish the pressure to about 0.01 mm., a new phenomenon will be observed. The positive column will vanish and the walls of the tube opposite the cathode will become faintly phosphorescent. The color of the phosphorescence will depend upon the nature of the glass: if the tube is made of soda glass, the glow will be greenish yellow, while with lead glass the phosphorescence will be bluish. The phosphorescence is due to the bombardment of the walls of the THE ELECTRON THEORY 33 tube by very minute particles projected normally from the cathode. These streams of particles are called the cathode rays. Some Properties of Cathode Rays. The following are among the most important properties of the cathode rays : 1. The cathode rays travel in straight lines normal to the cathode: and they cast shadows of opaque objects placed in their path. This property may be demonstrated by means of the apparatus shown in Fig. 5, where a small metallic Maltese cross is interposed in the path of the rays, a distinct shadow being cast on the opposite wall of the tube. The cross may be hinged at the bottom so that it can be dropped out of the path of the rays, when the usual phosphorescence will be obtained. 2. The cathode rays can produce mechanical motion. By means Fig. o. of the apparatus due to Sir William Crookes, Fig. 6, this prop- erty of the cathode rays may be demonstrated. Within the vacuum tube is placed a small paddle wheel which rolls horizon- tally on a pair of glass rails. When the current is applied to the tube, the wheel will revolve, moving away from the cathode. By reversing the current, the wheel will stop and then rotate in the opposite direction owing to the reversal of the direction of the cathode stream. 3. The cathode rays cause a rise of temperature in objects upon which they fall. In the tube shown in Fig. 7, the anode consists of a small piece of platinum : this is placed at the center of curva- ture of the spherical cathode. After pumping down to the proper pressure, if a strong discharge be sent through the tube, 34 THEORETICAL CHEMISTRY To pump the anode will begin to glow, and if the action of the current be continued long enough, the platinum plate may be rendered in- candescent, thus showing the marked heating effect of the cathode rays. 4. Many substances become phosphorescent on exposure to the cathode rays. If the cathode rays be directed upon different substances, such as calc-spar, barium platino-cyanide, willemite, scheelite and various kinds of glass, beautiful phosphorescent effects may be observed. This phosphorescent property is useful in observing and experimenting with the cathode rays. 5. The cathode rays can be deflected from their rectilinear path by a magnetic field. In studying the magnetic deviation of the cathode rays a tube similar to that shown in Fig. 8 has been found very satisfactory. An aluminium diaphragm, A, pierced B Fig. 8. with a 1 mm. hole, is placed in front of the cathode while at the opposite end of tube is placed a phosphorescent screen, D. When the discharge takes place a circular phosphorescent spot will appear on D. If the tube be placed between the poles of an electromagnet, the phosphorescent spot will move at right angles to the direction of the magnetic field. On reversing the polarity of the magnet the spot will move in the opposite direction. Furthermore the direction of the deflection will be found to be THE ELECTRON THEORY 35 similar to that produced by a negative charge of electricity mov- ing in the same direction as the cathode ray. 6. The cathode rays can be deflected from their rectilinear path by an electrostatic field. The same tube which was used in observ- ing the magnetic deflection may be employed in studying the effect of an electrostatic field. .Two insulated metal plates, B and C, are placed on opposite sides of the tube and parallel to each other. When the tube is in action, if a difference of poten- tial of several hundred volts be applied to the plates, the phos- phorescent spot on D will be found to move, the direction of the Elect. To Pump Fig. 9. motion being the same as that of a negatively charged body under the influence of an electrostatic field. Reversal of the field causes the phosphorescent spot to move to the opposite side of the screen. 7. The cathode rays carry a negative charge. Probably the most important characteristic of the cathode rays is their ability to carry a negative charge. While the magnetic and electrostatic deviation of the rays made this fact more than probable, it re- mained for Perrin to demonstrate that a negative electrification accompanies the cathode stream. A modification of Perrin's apparatus due to J. J. Thomson is shown in Fig. 9. It con- 36 THEORETICAL CHEMISTRY sists of a spherical bulb to which is sealed a smaller bulb and a long side tube. The small bulb contains the cathode C and the anode A. The anode consists of a tight-fitting brass plug pierced by a central hole of small diameter. The side tube, which is out of the direct range of the cathode rays, contains two coaxial metallic cylinders insulated from each other, each being perforated with a narrow transverse slit: D is earth-connected and B is connected with an electrometer by means of the rod F. When the tube has been pumped down to the proper pressure for the production of cathode rays, a phosphorescent spot will appear at E directly opposite the cathode C. Upon testing B for possible electrifica- tion by means of the electrometer, it will be found to be uncharged. If the cathode stream be deflected by means of a magnet so that the rays fall upon B, a sudden charging of the electrometer will be observed, proving that B is becoming electrified. Upon deflect- ing the rays still further so that they are no longer incident upon B, the accumulation of charge will immediately cease. If the electrometer be tested for polarity, it will be found to be negatively charged, thus proving the charge carried by the cathode rays to be negative. 8. The cathode rays can penetrate thin sheets of metal. In 1894 Lenard constructed a vacuum tube fitted with an aluminium window opposite the cathode. He showed that the cathode rays passed through the aluminium and are absorbed by different sub- stances outside of the tube, the absorption varying directly with the density of the substance. 9. The cathode rays when directed into moist air cause the forma- tion of fog. This phenomenon has been shown by C. T. R. Wilson to be due to the minute particles in the cathode stream acting as nuclei upon which the water vapor can condense. Velocity of the Cathode Particle. Since the cathode rays consist of minute, negatively-charged particles which can be deflected by a magnetic and an electrostatic field, it is possible to measure their speed and to compute the ratio of the mass of a particle to its charge. The special form of tube shown in Fig. 10 was devised for the purpose by J. J. Thomson. It consists of a glass tube about 60 cm. in length, furnished with a flat cir- THE ELECTRON THEORY 37 cular cathode, C, and an anode, A, in the form of a cylindrical brass plug about 2.5 cm. in length, pierced by a central hole 1 mm. in diameter. Another brass plug, B, is placed about 5 cm. away from A, the two holes being in exactly the same straight line, so that a very narrow bundle of rays may pass along the axis of the tube and fall upon the phosphorescent screen at the opposite end of the tube. Upon this screen is a millimeter scale, SS'. Two parallel plates, D and E, are sealed into the tube for the pur- pose of establishing an electrostatic field. When the tube is connected with an induction coil or other source of high-potential, a phosphorescent spot will appear at F. If a strong magnetic field be applied, the lines of force being at right angles to the plane Fig. 10. of the diagram, the rays will be deflected vertically and the spot on the screen will move from F to G. Let H denote the strength of the magnetic field and let m, e and v represent respectively the mass, charge and velocity of a cathode particle. A magnetic field, H, acting at right angles to the line of flight of the cathode particle will exert a force, Hev, which will tend to deflect the particle from a rectilinear path. This force must be equal to the centrifugal force of the moving particle acting outwards along its radius of curvature. Therefore mv* Hev = - , or (D 38 THEORETICAL CHEMISTRY Since H and r can both be measured, the ratio, , can be deter- 6 mined. Now if a difference of potential be established between D and E, and the lines of force in the electrostatic field have the proper direction, it will be possible to alter the strength of the field so as to just counterbalance the effect of the magnetic field, and bring the phosphorescent spot back to F again. Under these conditions, if X denotes the strength of the electrostatic field, we have Xe = Hev, or -i- (2) Since X and H can both be measured, v can be calculated, and by introducing the value so obtained into equation (1), the ratio e/m can be evaluated. By this method the average value of v has been found to be 2.8 X 10 9 cm. per second, while 1.7 X 10 7 is the mean value of a large number of determinations of the ratio e/m. Comparison of the Ratio of Charge to Mass for the Cathode Particle with that for the Ion in Electrolysis. The ratio of the charge carried by an ion in electrolysis to its mass can be easily computed. Thus it may be shown that the ratio of the charge E, of the hydrogen ion to its mass, M , in electrolysis is about 1 X 10 4 C.G.S. units or ET -JTJ. = 1 X 10 4 approximately. The mass of the hydrogen ion may be considered to be identical with that of the hydrogen atom, the lightest atom known. Com- paring the value of e/m for the cathode particle with the value of E/M for the hydrogen ion in electrolysis, it is evident that the former is about 1700 times greater than the latter. Charge Carried by the Cathode Particle. Until the value of the charge carried by the cathode particle has been determined, it is clearly impossible to compute its mass. Thus, if we consider THE ELECTRON THEORY 39 the last statement of the preceding paragraph, which may be formulated as follows : e/m :E/M :: 1700 : 1, the proportion will remain unaltered whether m = M/1700 and e = E, or e = 1700 and m = M . The method employed to deter- mine the charge carried by a cathode particle is too complicated for a detailed description in this place; merely the general outline will be given. Upon suddenly expanding a volume of saturated water vapor, its temperature is lowered, and a cloud forms, each particle of dust present serving as a nucleus for a fog particle. If sufficient time be allowed for the mist to settle and the vapor to become saturated again, a repetition of the preceding process will result in the formation of less mist, owing to the presence of fewer dust particles. By repeating the operation enough tunes the space may be rendered dust free. As has already been pointed out, cathode particles serve as nuclei for the condensation of water vapor, their function being similar to that of dust particles. It has been shown by Sir George G. Stokes that if a drop of water of radius r, be allowed to fall through a gas of viscosity 77, then the velocity with which the drop falls is given by the equation .-;. where g is the acceleration due to gravity. The viscosity of air at any temperature being known, a cloud can be produced in an appropriate chamber by expansion of water vapor in the presence of cathode particles and the speed, v, with which the cloud falls can be measured, and hence r can be calculated by means of equa- tion (3). If m is the total mass of the cloud and n is the number of drops per cubic centimeter, then m = 4/3 rnrr* (density of water = 1). From a simple application of thermodynamics m may be deter- mined. Knowing the values of m and r, the number of drops in the cloud, n, which is the same as the number of cathode par- ticles, can be calculated. It is a simple matter to measure the total charge in the expansion chamber, and dividing this by the total number of charged particles, gives the charge carried by a 40 THEORETICAL CHEMISTRY single particle. The latest determinations of J. J. Thomson show this to be 3.4 X ~ 10 electrostatic unit. This is practically identical with the calculated value of the charge on the hydro- gen ion in electrolysis, or e = E and therefore m = Af/1700; the mass of the cathode particle is 1/1700 of the mass of the atom of hydrogen. The cathode particle has the smallest mass yet known and has been called the corpuscle or electron. An ingenious modification of the foregoing method devised by Millikan,* has made it possible to determine e with extreme accuracy. Millikan gives as the mean of a large number of deter- minations of e, the following value which he states is in error by less than 0.1 per cent: e = 4.4775 X 10~ 10 electrostatic units. For purposes of comparison, the following values of e obtained by other investigators using different experimental methods are here given : (a) Making use of available data on radiant energy, Planck calculated e = 4.69 X 1Q- 10 . (6) By counting the number of scintillations produced by a known weight of polonium and measuring the total charge, Regener found e = 4.79 X 1Q- 10 . (c) By counting the number of a-particles escaping from a given amount of radium bromide and measuring the total charge, Rutherford and Geiger calculated e = 4.65 X 10~ 10 . The Avogadro Constant. The actual number of molecules in one gram-molecule or the actual number of atoms in one gram- atom of a gas is called the Avogadro Constant. The most accurate method for the calculation oi this constant involves the elementary charge of electricity, e. The quantity of electricity carried by one gram equivalent in electrolysis has been found to be 96,500 coulombs (see p. 390). This quantity, known as the faraday, and commonly designated by F, bears the same relation to e that the actual weight of an atom bears to the number expressing its atomic weight. The relation * Phil. Mag., 6, 19, 209 (1910); Phys. Rev., 39, 349 (1911); Trans. Am. Electrochem. Soc., 21, 185 (1912). THE ELECTRON THEORY 41 between the Avogadro constant N and the ionic and electronic charges F and e, is given by the equation F N = -- e Substituting the above values in this equation and converting coulombs into electrostatic units, we have 96500X3X10* ' 4.4775 X 10- 10 Other Sources of Electrons. Electrons may be produced by other agents than cathode rays. Thus, electrons are emitted by radium and other radioactive substances, by metals and amalgams under the influence of ultra-violet light, and also by gas flames charged with the vapors of salts. It has been shown that from whatever source an electron is derived, the value of the ratio e/m remains constant. This interesting fact has led Thomson to sug- gest that the electron may be regarded as "one of the bricks of which the atom is built up." Before entering upon a discussion of modern views of atomic structure, however, it will be necessary to summarize very briefly some of the salient facts of radiochemistry. CHAPTER IV. RADIOACTIVITY. Discovery of Radioactivity. The first radioactive substance was discovered by Henri Becquerel * in 1896. It had been shown by Roentgen in the previous year that the bombardment of the walls of a vacuum tube by the cathode stream, gives rise to a new type of rays, which, because of their puzzling characteristics, he called X-rays. The portion of the tube where these rays originate was observed to fluoresce brilliantly, and it was at once assumed that this fluorescence might be the cause of the new type of radia- tion. Many substances were known to fluoresce under the stimulus of the sun's rays, and it was natural, in the light of Roentgen's dis- covery, that all substances which exhibit fluorescence should be subjected to careful examination. Among those who became inter- ested in these phenomena was Becquerel. He studied the action" of a number of fluorescent substances, among which was the double sulphate of potassium and uranium. This salt, after exposure to sunlight, was found to emit a radiation capable of affecting a care- fully protected photographic plate. Further investigation proved that the fluorescence had nothing to do with the photographic action, since both uranous and uranic salts were found to exert similar photographic action, notwithstanding the fact that uranous salts are not fluorescent. The photographic activity of both uranous and uranic salts was found to be proportional to their content of uranium. Becquerel also showed that preliminary stimulation by sunlight was wholly unnecessary. Uranium salts which had been kept in the dark for years were found to be just as active as those which had been recently exposed to brilliant sun- light. The properties of the rays emitted by uranium salts differ in many respects from those of the X-rays. The rate of emission of the uranium rays remains unaltered at the highest or the lowest * Compt. rendus, 122, 420 (1896). 42 RADIOACTIVITY 43 obtainable temperatures. The entire behavior of these salts justi- fies the conclusion that the continuous emission of penetrating rays is a specific property of the element uranium itself. This property of spontaneously emitting radiations capable of penetra- ting substances opaque to ordinary light is called radioactivity. Discovery of Radium. Shortly after the discovery of the radioactivity of uranium, the element thorium and its compounds were also found to be radioactive. As a result of a systematic examination by Mme. Curie * of minerals known to contain uranium or thorium, it was learned that many of these were much more radioactive than either uranium or thorium alone. Thus, pitchblende, one of the principal ores of uranium, was found to be four times more active than uranium alone, and chalcolite, a phosphate of copper and uranium, was found to be at least twice as active as uranium. On the other hand, when a specimen of artificial chalcolite, prepared in the laboratory from pure materials, was examined, its activity was found to be proportional to the content of uranium. Mme. Curie concluded from this result that natural chalcolite and pitchblende must contain a minute amount of some substance much more active than uranium. With the assistance of her husband, Mme. Curie undertook the task of separating this unknown substance from pitchblende. Pitchblende is an extremely complex mineral and its systematic chemical analysis calls for skill and patience of a high order. With- out "entering into details as to the analytical procedure, it must suffice here to state the results obtained. Associated with bismuth, a very active substance was discovered, to which Mme. Curie gave the name polonium in honor of her native land, Poland. In like manner, an extremely active substance was found associated with barium in the alkaline earth group. The substance was called radium because of its great radioactivity. While the isolation of pure polonium is extremely difficult and, while sufficient quantities have not been obtained to permit de- terminations of its physical properties, the isolation of radium in relatively large amounts is readily accomplished. The pure bromides of radium and barium are prepared together and the * Compt. rendus, 126, 1101 (1898). 44 THEORETICAL CHEMISTRY two salts are then separated by a series of fractional crystalliza- tions. That the salts of barium and radium are very similar in chemical properties is shown by the fact that they separate to- gether from the same solution. The atomic weight of radium has been determined by several investigators, the accepted value being 226. It is thus, with the exception of uranium, the heaviest known element. In 1910, Mme. Curie * succeeded in obtaining metallic radium. It is a metal possessing a silvery luster, dissolving in water with energetic evolution of hydrogen and tarnishing rapidly in air with the formation of the nitride. It is estimated that one ton of pitchblende contains approxi- mately 0.2 gram of radium. Other Radioactive Substances. Shortly after the discovery of radium and polonium by the Curies, Debierne f discovered another radioactive element in pitchblende. This element, which he named actinium, was found associated with the iron group in the course of the analysis of the mineral. In 1906, Boltwood J discovered in pitchblende, and in several other uranium minerals, the presence of still another radioelement which he named ionium. Ionium is much more active than thorium to which it bears such a close resemblance that the two elements cannot be separated from each other. The lead which is obtained from uranium and thorium minerals is found to be slightly radioactive, the activity being attributed to the presence of a small proportion of a constituent called radio- lead, the chemical properties of which resemble those of ordinary lead. It is interesting to note that recent determinations by Richards of the atomic weight of lead obtained from different sources, reveal differences greater than the possible experimental errors of the determinations. Thus, the values of the atomic weight of lead from pitchblende and from thorite were found to be 206.40 and 208.4 respectively, while the value of the atomic weight of ordinary lead is 207.15. * Compt. rendus, 151, 523 (1910). t Compt. rendus, 129, 593 (1899). J Am. Jour. Sci., 22, 537 (1906). Jour. Am. Chem. Soc., 36, 1329 (1914). RADIOACTIVITY 45 About thirty other radioactive substances have been separated and many of their properties have been determined. All of these radio-elements have been shown to be the lineal descendants of one or the other of the two parent elements, uranium or thorium. lonization of Gases. The radiations emitted by radioactive substances have the power of rendering the air through which they pass conductors of electricity. To account for this action, Thom- son and Rutherford formulated the theory of gaseous ionization. According to this theory, which has since been experimentally confirmed, the radiations break up the components of the gas into positive and negative carriers of electricity called ions. If two parallel metal plates are connected to the terminals of a battery and a radioactive substance is placed between them, the air will be ionized and, owing to the movement of the positive and negative ions toward the plates of opposite sign, an electric current will pass between the plates. If the electric field is weak, the mutual attraction between the positive and negative ions will cause many of them to recombine before reaching the plates and the resulting current will be small. As the strength of the field is increased, the greater will be the speed of the ions toward the plates and the smaller will become the tendency to recombination. Ultimately, with increasing strength of field all of the ions will be swept to the plates as fast as they are formed and the ionization current will attain a maximum value. This limiting or saturation current affords the most accurate method for the measurement of radio- activity. The method is so sensitive that, by means of it alone, it is possible to detect amounts of radioactive products far beyond the reach of the balance or the spectroscope. The theory of gaseous ionization has been confirmed in several different ways, but one of the most striking verifications of the theory is that due to C. T. R. Wilson. Making use of the fact that the ions tend to condense water vapor around themselves as nuclei, Wilson succeeded in actually photo- graphing the path of an ionizing ray in air. Photographic Action of Radiations. It has already been pointed out that the radiations from radioactive substances are capable of affecting a photographic plate. The photographic 46 THEORETICAL CHEMISTRY action of the radiations has been employed quite extensively in studying radioactive phenomena from a purely qualitative stand- point. The method employed, consists in exposing the photo- graphic plate, which has been previously wrapped in opaque black paper, to the action of the radiations. The time of exposure varies with the nature of the substance under examination, a few minutes being required for highly active preparations while sev- eral days or even weeks may be needed for preparations of low activity. Phosphorescence Induced by Radiations. A screen covered with crystals of phosphorescent zinc sulphide is rendered luminous when exposed to fairly intense radiation from a radioactive substance. This phenomenon has been shown to be due to the bombardment of the crystals of zinc sulphide by the so-called ex- rays (see below). When the screen is examined with a lens the phosphorescence is seen to consist of a series of scintillations of very short duration. Nature of the Radiations. The ionizing, photographic, and luminescent properties of the radiations from radioactive sub- stances are not sufficient to differentiate them from cathode rays or X-rays, although each of these properties may be employed in determining their intensity. Evidence of the composite character of the radiations was fur- nished by a study of their penetrating power as well as by investi- gations of the behavior of the rays when subjected to the action of magnetic and electric fields. A thin sheet of aluminum or a few centimeters of air was found sufficient to cut off a large percentage of the rays. The unabsorbed portion of the radiation was found to consist of two distinct types, one of which was cut off by five or six millimeters of lead while the other possessed such great penetrating power that its presence could be readily detected after passing through a layer of lead fifteen centimeters thick. Rutherford named these three distinct types of radiation, the a-, /3-, and 7-rays, respectively. The penetrating powers of the a-, jS-, and 7-rays may be approximately expressed by the propor- tion 1 to 100 to 10,000; i.e., the 0-rays are 100 times more pene- RADIOACTIVITY 47 trating than the a-rays, while the 7-rays are 100 times more penetrating than the /3-rays. The general characteristics of the three kinds of rays may be briefly summarized as follows : (1) a-Rays. The oi-rays consist of positively charged particles moving with speeds approximately one-tenth as great as that of light. These particles have been shown to be identical with helium atoms carrying two positive charges of electricity. They are appreciably deflected from a rectilinear path by magnetic and electric fields. They possess great ionizing power but relatively little penetrating power or photographic action. The depth to which an a-particle penetrates a homogeneous absorbing medium before losing its ionizing power, is known as its " range." The range is proportional to the cube of the initial speed of the a- particle and is one of the characteristic properties of the radio- elements emitting a-rays. (2) (3-Rays. The 0-rays consist of negatively charged particles moving with speeds varying from two-fifths to nine-tenths of the speed of light. They are, in fact, electrons moving with much greater speeds than those shot out from the cathode in a vacuum tube. While the a-particles emitted by a particular radio-element have a definite velocity, the corresponding /3-ray emission consists of a flight of particles having widely different speeds. The pene- trating power of the 0-rays is conditioned by the speed of the particles, those which move most rapidly possessing the greatest penetrating power. The ionizing action of the /3-rays is much weaker than that of the a-rays, while exactly the reverse is true of the photographic action. (3) j-Rays. The 7-rays are identical with X-rays. They con- sist of extremely short waves of light, the wave-length varying from about 1 X 10~ 8 cm. for the rays of low penetrating power to about 1 X 10~ 9 cm. for the most penetrating rays. Obviously the 7-rays cannot be deflected from a rectilinear path by either electric or magnetic fields. Uranium-X and Thorium-X. In 1900, Crookes * precipitated a solution of a uranium salt with ammonium carbonate; when an * Proc. Roy. Soc., 64, 409 (1900). 48 THEORETICAL CHEMISTRY excess of the reagent had been added, all but a minute portion of the precipitate was found to have dissolved. This small insoluble residue, though chemically free from uranium, was found, when tested photographically, to be several hundred times more active, weight for weight, than the original salt. The solution, on the other hand, was found to have lost nearly all of its activity. At the end of a year, however, the solution had entirely regained its original activity, while the insoluble residue had become inactive. The active substance thus separated was called, on account of its unknown nature, uranium-X. Similarly, Rutherford and Soddy,* by precipitating a solution of a thorium salt with ammonium hydroxide, found that a large proportion of the activity remained behind in the thorium-free filtrate. On evaporating the filtrate to dryness and driving off the ammonium salts, a residue was obtained which was, weight for weight, several thousand times more active than the original solution. After standing for a month, this residue was found to have lost its activity, while the precipitate had regained the ac- tivity of the original thorium compound. This active residue was called thorium-X from analogy to Crookes' uranium-X. The fact that uranium-X and thorium-X had each been obtained as the result of specific chemical processes, seemed to warrant the conclusion that they are new substances possessing well-defined properties. The manner in which these substances were obtained led to a variety of speculations as to the mechanism of the process involved in their production. In a subsequent paragraph it will be shown that the so-called disintegration theory offers a most satisfactory explanation of the foregoing experimental results. The Emanations. The element thorium was found by Ruther- ford to give off a radioactive gas or emanation which, when left to itself, rapidly loses its activity in a similar manner to uranium-X and thorium-X. The thorium emanation was found to resemble the inactive gases in its chemical behavior. Thus, it can be sub- jected to the action of lead chromate, metallic magnesium and zinc dust at extremely high temperatures without undergoing change. The only other substances which, at the time of the dis- * Phil. Mag., VI, 4, 370 (1902). RADIOACTIVITY 49 covery of the emanation, were known to resist the action of these reagents under the same conditions were the gaseous elements helium, neon, argon, krypton, and xenon. The most conclusive evidence of the gaseous character of the thorium emanation is the fact that it condenses to a liquid at very low temperatures. Rutherford and Soddy showed that the origin of the thorium emanation is thoriurn-X and not the element thorium itself. Freshly precipitated thorium hydroxide shows only a trace of emanating power, whereas the thorium-X separated from the filtrate possesses this power to a marked degree. As the thorium- X gradually loses its emanating power, the hydroxide shows a corresponding recovery. Radium was found to give off an emanation which behaves similarly to the thorium emanation except that it parts with its activity at a slower rate. The rate at which the radium emanation is produced, together with its longer life, enabled Ramsay and Soddy * not only to measure the volume of the emanation obtained from 60 mg. of radium bromide but also to establish the fact that it obeys Boyle's law. The Active Deposits. It was discovered by Rutherford f for thorium, and by M. and Mme. Curie t for radium, that the emana- tions from these elements are capable of imparting radioactivity to surrounding objects. On the other hand, uranium and polo- nium, which evolve no emanations, have no such influence on their environment. This fact is taken as a proof that induced radio- activity is due to actual contact with the emanations. If a nega- tively charged platinum wire be exposed to the thorium or radium emanation, the whole of its activity will be concentrated on the wire. This so-called active deposit may be transferred from the wire to a piece of sandpaper by rubbing. It may be sublimed from the wire to the walls of the tube in which it is heated, or it may be dissolved from the wire by means of hydrochloric or sul- phuric acid. On evaporating the solution, the activity will be found to reside on the evaporating dish. Microscopic examination of * Proc. Roy. Soc., 73, 346 (1904). f Phil. Mag.. VI, 22, 621 (1911). % Compt. rendus, 129, 714 (1899). 50 THEORETICAL CHEMISTRY the wire fails to reveal any deposit, and the most sensitive balance is incapable of detecting any gain in weight after exposing the wire to the emanation. These experiments leave no room for doubt that the active deposits consist of infinitesimal amounts of solid substances possessing definite chemical properties. The active deposits undergo a gradual loss of activity similar to that observed with the emanations and with uranium-X and thorium-X. The Disintegration Hypothesis. It was soon discovered that the active deposits undergo a series of additional radioactive changes. These subsequent transformations were found to be much more obscure and difficult to follow. In fact, it is only be- cause of the ingenuity and mathematical acumen of those who undertook this difficult research that we are today in possession of such complete knowledge of the succeeding members of the radio- active series of elements. In 1003, Rutherford and Soddy * brought forward their theory of atomic disintegration which affords a perfectly satisfactory in- terpretation of the complicated results already detailed, as well as of the subsequent changes in the active deposits to which reference has just been made. According to this theory, radioactive change is assumed to be due to the spontaneous disintegration of the radio- elements with concomitant formation of new elements. These new elements, which are often less stable than the parent element, are assumed to undergo further disintegration with the production of still other elements, the end of the process ultimately being reached when a stable element is formed. The Radioactive Constant. The activity of a radio-element decays exponentially with time according to the equation It = /oe-", (1) where / is the initial activity, I t the activity at the end of an in- terval of time t, X a constant, and e is the base of the natural system of logarithms. The constant X, known as the radioactive constant, represents the fraction of the total amount of radioactive substance undergoing disintegration in unit of time, provided the latter is so small that the quantity at the end of the time unit is only slightly * Phil. Mag., VI, 5, 576 (1903). RADIOACTIVITY 51 different from the initial quantity. The reciprocal of the radio- active constant is called the average life of the element. Soddy defines the average life of a radio-element as "the sum of the sepa- rate periods of future existence of all the individual atoms divided by the number in existence at the starting point." If n t represents the number of atoms of a radio-element changing in unit time at the end of a time t, and n the corresponding value when t = 0, equation (1) may be written nt = noe~ xt . In order to determine the initial rate of change, let No denote the total number of atoms originally present, and Nt the number re- maining unchanged at time t; we then have ^0 -v / > Al But when t = 0, N = N t and No = - A Hence Nt = On differentiating, we have Or, stated in words, the rate at which the atoms of a radio-element undergo disintegration at any given time is found to be propor- tional to the total number in existence at that time. This law of radioactive change is also peculiar to unimolecular chemical reactions (see p. 361). The velocity of a unimolecular reaction, however, is conditioned by the temperature, whereas the velocity of a radioactive change remains unaltered at the highest and the lowest attainable temperatures. The time required for one-half of a radio-element to undergo transformation is known as the period of half change T, and may be readily calculated from X as follows : log 0.5 = 0.4343 \T T = 52 THEORETICAL CHEMISTRY It has been shown by Geiger and Nuttall * that the radioactive constant X and the range R of the ^-particles shot out from a dis- integrating atom bear the following empirical relation to each other, X = aR b , where a and b are constants, the former constant being character- istic of the particular radioactive series to which the element be- longs. This formula has been found useful in calculating the values of X for the longest and shortest lived elements. Radioactive Equilibrium. It is evident that a state of equi- librium must ultimately be attained among the atoms of a radio- active substance. When the rate of production of a radio-element from its parent element is equal to its rate of disintegration into the next succeeding element of the series, the substance is said to be in radioactive equilibrium. The relative amounts of the suc- cessive members of a series of elements in radioactive equilibrium are inversely proportional to their radioactive constants. In order that measurements of the rate of radioactive change may be strictly comparable, it is necessary to make use of the amounts which are in equilibrium with a fixed amount of the parent element. Thus, the unit adopted for the measurement of the quantity of the radium emanation is the mass of emanation in equilibrium with one gram of radium. This unit is known as the curie. Its mass is Xi/X 2 gram, where Xi and X 2 are the radioactive constants of radium and its emanation respectively. One curie of radium emanation may be shown to occupy 0.63 cu. mm. under standard conditions of temperature and pressure. The Disintegration Series. It appears almost certain that the thirty or more radio-elements are disintegration products of one or the other of the two parent elements, uranium or thorium. One of the most convenient methods of classification of these elements is to arrange them in disintegration series, starting with the parent element and placing the succeeding elements in the order of their production. The accompanying table shows the three series of radio-elements as thus arranged by Soddy. The numbers within the circles are the atomic weights of the elements, while the small * Phil. Mag., VI, 22, 613 (1911). RADIOACTIVITY TABLE SHOWING DISINTEGRATION SERIES 53 54 THEORETICAL CHEMISTRY circles and dots at the right of the larger circles indicate the char- acter of the radiation given out at each stage of the disintegration process. The average life, 1/X, of each element in the series is given below the name of the element. While a detailed account of the properties of the different radio-elements included in this table cannot be given here, attention should be called to the com- plex transformations occurring in the active deposit. It is also of interest to observe that the end-products of the three series bear a striking resemblance to the element lead and that their atomic weights are approximately equal and nearly identical with that of ordinary lead. Counting the a-Particles. It has already been stated that when an a-particle strikes a screen coated with phosphorescent zinc sulphide, a distinct flash of light may be seen when the screen is viewed through a magnifying lens in a dark room. It is obvious that if one could count the number of these scintillations, it would be an easy matter to ascertain the total number of a-particles shot out from a radioactive substance in a given time. By using a phosphorescent screen and a microscope, Regener* has deter- mined in this manner the rate of emission of a-particles from polonium. He found from his different experiments an average emission of 3.94 X 10 5 a-particles per second. The total charge on the a-particles was then measured by collecting them in a suit- able measuring vessel. A charge of 37.7 X 10~ 5 electrostatic units was found to be associated with the total number of a- particles emitted by polonium in one second. Rutherford and Geiger f devised an electrical method for count- ing the a-particles. In their experiment the source of the a- particles was a small disc which had been exposed to the radium emanation for some hours. This disc was placed in an evacuated tube at a measured distance from a small aperture of known cross- section. The aperture was closed with a thin plate of mica through which the a-particles could pass with ease. After passing through the mica plate, the a-particles entered an ionization chamber filled with air at reduced pressure and fitted with two charged * Sitzunsbericht d. K. preuss. Akad., 38, 948 (1909). t Proc. Roy. Soc. A, 81, 141 (1908). RADIOACTIVITY 55 metal plates connected with appropriate apparatus for measuring ionization currents. Whenever an a-particle entered the ioniza- tion chamber, a momentary current passed, producing a sudden deflection of the needle of the electrometer. By counting the number of throws of the needle occurring in a definite interval of time, the total number of a-particles passing through the ionization chamber was determined. Knowing the distance of the source of the radiations from the aperture, together with the area of the aperture, the total number of a-particles emitted by the radio- active disc in a given time could be computed. Rutherford and Geiger thus found that one gram of radium emits very nearly 107 X 10 16 a-particles per year. Having determined the total number of a-particles emitted, it only remained to measure the total charge, in order to calculate the charge carried by a single a-particle. From a series of very consistent measurements, the charge carried by a single a-particle was found to be 9.3 X 10~ 10 electrostatic units. Since the fundamental charge e has been shown to be 4.48 X 10~ 10 electrostatic units, it follows that the o:-particle carries two ionic charges of electricity. Helium Atoms and a-Particles Identical. In 1909, Ruther- ford and Royds * performed a crucial experiment to determine the nature of the a-particle. A glass bulb was blown with walls thin enough to permit the passage of the a-particles but sufficiently strong to withstand atmospheric pressure. The bulb was filled with radium emanation and then enclosed in an outer glass tube to which a spectrum tube had been sealed. On exhausting the outer tube and examining the spectrum of the residual gas, no evidence of helium was obtained until after an interval of twenty- four hours. After four days the characteristic yellow and green lines were plainly visible and at the end of the sixth day, the com- plete spectrum of helium was obtained. The unavoidable con- clusion from this experiment is, that the presence of helium in the outer tube must have been due to the a-particles which were projected through the thin walls of the inner tube. In another experiment, the inner tube was filled with pure helium under pressure while the exhausted outer tube was examined for helium. * Phil. Mag., VI, 17, 281 (1909). 56 THEORETICAL CHEMISTRY No trace of helium could be detected spectroscopically even after an interval of several days, thus proving that the helium detected in the first experiment must have resulted from the a-particles which had been shot out from the radium emanation with sufficient energy to penetrate the thin walls of the inner tube. These experi- ments leave no room for doubt that an a-particle becomes a helium atom when its positive charge is neutralized. Rate of Production of Helium. The rate of production of helium from the series in equilibrium with one gram of radium has been determined experimentally by Rutherford and Bolt- wood* to be 156 cu. mm. per year. This result agrees closely with the calculated rate of production, viz., 158 cu. mm. per gram of radium per year. Energy Evolved by Radium. Curie and Laborde f were the first to call attention to the interesting fact that the temperature of radium compounds was uniformly higher than that of their environment. Careful measurements have shown that one gram of radium evolves heat at the rate of approximately 135 gram- calories per hour. That the greater part of this heat energy is due to the a-particles may be proven by a direct calculation of their mean kinetic energy. The magnitude of the store of energy con- tained in radium may be realized from the statement, that one gram of radium, before it entirely disappears, evolves an amount of heat energy nearly one million times greater than that evolved in the formation of one gram of water from its elements. * Phil. Mag., VI, 22, 586 (1911). t Compt. rendus, 136, 673 (1904). CHAPTER V. ATOMIC STRUCTURE. The Modern Conception of Atomic Structure. As a result of the investigations of Thomson,* Rutherford,f Nicholson,! Bohr, and others, a theory of atomic structure has been developed which affords a satisfactory interpretation of many of the important relationships among the chemical elements. Briefly stated, this theory assumes that the atom consists of a central, positively charged nucleus, surrounded by a miniature solar system of electrons. The investigations of Rutherford and Geiger|| show that the character of the deflection of a-particles shot out from radioactive atoms at speeds approximating 20,000 miles per second, and consequently completely penetrating other atoms, is such as to indicate an extremely high concentration of positive electricity on the central nucleus. The central nucleus which is supposed to represent nearly the entire mass of the atom, is thought to be very small in comparison with the size of the atom as a whole. Recent investigations make it appear probable that the maximum diameter of the nucleus of the hydrogen atom is about one one-hundred-thousandth of the diameter commonly attributed to the atom. In commenting on this statement, Har- kins says: 1f "On this basis the atom would have a volume a million-billion times larger than that of its nucleus, and thus the nucleus of the atom is much smaller in comparison with the size of the atom than is the sun when compared with the dimensions of its planetary system." It is highly probable that the central nucleus is itself made up of a definite number of units of positive electricity together with a small number of attendant electrons. * Phil. Mag., 7, 237 (1904). t Popular Science Monthly, 87, 105 (1915). { Phil. Mag., 22, 864 (1911). Phil. Mag., 26, 476, 857 (1913). H Phil. Mag, 21, 669 (1911). 1f Science, 66, 419 (1917). 57 58 THEORETICAL CHEMISTRY It is further assumed that the units of positive electricity are hydrogen atoms, each of which has been deprived of one electron. If the mass of an atom is largely due to the presence of hydrogen nuclei, then we should expect Prout's hypothesis to hold and the atomic weights of the elements to be exact multiples of the atomic weight of hydrogen. When we consider, however, that according to electromagnetic theory the total mass of a body composed of positive and negative units is dependent upon the relative posi- tions of these units when packed together, it is evident that the mass of the atom will not necessarily be an exact multiple of the mass of the hydrogen atom. It has already been pointed out that helium is a product of many radioactive transformations. This fact may be taken as an indication of the extraordinary stability of the helium atom. Be- cause of its stability, the nucleus of this atom has come to be con- sidered as a secondary unit of positive electricity. The nucleus of the helium atom, or the nucleus of an ex-particle is assumed to consist of four hydrogen nuclei with two nuclear electrons. The Atomic Number. Since the algebraic sum of the positive and negative electrification on an atom must be zero, it follows that the charge resident upon the nucleus must be equal to the number of electrons outside the nucleus. This number, which has come to be recognized as more important and characteristic than the atomic weight, is known as the atomic number. X-Rays and Atomic Structure. The discovery by W. L. Bragg in 1912 that X-rays undergo reflection at crystal surfaces and the subsequent development by Mr. Bragg and his father, W. H. Bragg, of the X-rays pectrometer, has led to a series of investigations of the utmost importance to both the chemist and the physicist. In order that the significance of these investigations may be understood it may be well to summarize very briefly some of the more important properties of the X-ray. The bombardment of metal plates, usually of platinum, by elec- trons gives rise to X-rays. The radiation issuing from an X-ray tube is very far from homogeneous. When screens of different materials and varying thicknesses are interposed in the path of the rays, the degree of absorption is irregular. It has been found, ATOMIC STRUCTURE 59 however, that every substance when properly stimulated is capable of emitting a homogeneous and characteristic X-radiation, the penetrating power of which is wholly determined by the nature of the elements of which the substance is composed. The pene- trating power of this typical X-radiation increases with the atomic weight of the radiating element. With elements whose atomic weights are less than 24, the radiation is too feeble to be measured. It is important to note that this property of the elements is not a periodic function of the atomic weight. This type of X-radiation is entirely independent of external conditions, indicating that it is closely connected with the internal structure of the atoms from which it emanates. The rays possess the power of affecting the photographic plate and also of rendering gases through which they pass conductors of electricity. It is estimated that the wave-length of an X-ray is about 1 X 10~ 8 to 1 X 10~ 9 cm., or about one ten-thousandth of the wave-length of sodium light. It is obvious that the spacing of the lines of a grating capable of diffracting such short waves must be of the order of magnitude of intermolecular distances. It is well known that a grating owes its power of analyzing a complex system of light waves into its component wave-trains, to the series of paral- lel lines which are ruled upon its surface at exactly equal intervals. When a train of waves is incident upon a grating, each line acts as a center from which a diffracted train of waves emerges. Such a grating is relatively simple in its action since it consists of a single series of centers of diffraction lying in one plane. The power of a crystal surface to reflect X-rays, however, is due to the fact that the crystal is in reality a three-dimensional diffraction grating, the atoms or molecules of which the crystal is built up, acting as the centers of diffraction. It must be borne in mind that the reflection of X-rays is in no way dependent upon the existence of a polished surface on the outside of the crystal, but rather upon the regularly spaced atoms or molecules within the crystal. To ordinary waves of light the atomic structure is so fine grained as to behave as a continuous medium, whereas to the short X-ray waves, the crystal acts as a discontinuous structure of regularly arranged particles, each of which functions as a diffraction center. 60 THEORETICAL CHEMISTRY Spectra. By making use of the reflecting power of one of the cleavage planes of a crystal, and employing different metals as anti-cathodes in an X-ray tube, Moseley * succeeded in photo- graphing the X-ray spectra of the characteristic radiations of a number of the elements. He showed that the X-ray spectrum of an element is extremely simple and consists of two groups of lines known as the "K " and "L " radiations. As a result of careful study of the "K " radiations of thirty-nine elements from alumin- ium to gold, Moseley discovered that these radiations are char- acterized by two well-defined lines whose vibration frequency v is connected with the atomic number of the element AT, by the simple relation v = A (N I) 2 , where A is a constant. When the square roots of the frequencies of the elements are plotted as abscissae against their atomic num- bers as ordinates, the points are found to lie on a straight 50 1 1 1 1 *j T^ I loo line as shown in Fig. 11. On the other hand, if the square roots of the frequen- cies are plotted against the atomic weights of the ele- ments, the relation- 20 1 ^4 1 1 1 1 40 ship is no longer rec- tilinear. When the elements are ar- ranged in the order of their atomic num- bers instead of in the order of their atomic weights, the irregularities f hitherto noted in connection with argon, cobalt, and tellurium entirely disappear. In reviewing Moseley's work on X-ray spectra, Soddy { says : * Phil. Mag., 26, 210 (1913); 27, 703 (1914). t See p. 29. J Ann. Reports on the Prog, of Chemistry, p. 278 (1914). 80 10 14 18 22 Square Root of Frequency X 10" 8 Fig. 11. -ATOMIC STRUCTURE 61 "A veritable roll-call of the elements has been made by this method. Thirty-nine elements, with atomic weights between those of aluminium and gold, have been examined in this way, and in every case the lines of the X-ray spectrum have been found to be simply connected with the integer that represents the place assigned to it by chemists in the periodic table." One of the most interesting results of this " roll-call " of the elements is the fixing of the number of possible rare-earth elements. Between barium and tantalum there are places for only fifteen rare-earth elements and fourteen of these places are filled. While future investigation may necessitate some rearrangement in the order of tabulation, the total number of these elements is limited to fifteen. Periodicity among the Radio-elements. The problem of plac- ing the newly discovered radio-elements in the periodic table re- mained unsolved until 1913, when Fajans * and Soddy,f working independently, discovered an important generalization concerning the changes in chemical properties resulting from the expulsion of a- and /3-particles during radioactive transformation. This impor- tant generalization may be stated as follows: The expulsion of an a-particle causes a radioactive element to shift its position in the periodic table two places in the direction of decreasing atomic weight, whereas the emission of a (3-particle causes a shift of one place in the opposite direction. This generalization not only agrees with our present theory of atomic structure, but may be shown to be a necessary consequence of this theory. The loss of an a-particle or helium atom involves a loss of 4 units in atomic weight and of 2 units of positive electricity from the nucleus of the atom. In consequence of this loss, the atomic number is diminished by 2 units and the resulting new element will find a place in the periodic table two groups to the left of that occupied by the parent element. On the contrary, while the expulsion of a /3-particle, or electron, involves practically no change mass, the nucleus of the parent atom suffers a loss of 1 unit of egative electricity. This loss is equivalent to a gain of 1 unit of * Physikal. Zeit., 14, 49 (1913). f Chem. News, 107, 97 (1913). 62 THEORETICAL CHEMISTRY positive electricity, or to an increase of 1 unit in the atomic number, and in consequence, the position of the new element in the periodic table will be shifted one group to the right of that occupied by the parent element. Soddy's arrangement of all of the radio-elements in accordance with this generalization is shown in Fig. 12, Thus, starting with the element uranium in Group VIA, we may follow the successive RADIO-ELEMENTS AND PERIODIC LAW ALL ELEMENTS IN THE SAME PLACE IN THE PERIODIC TABLE ARE CHEMICALLY NON-SEPARABLE AND (PROBABLY) 'ECTROSCOPICALLY INDISTINGUISHABLE Relative No. of Negative Electrons 543 1 240 Fig. 12. steps in the radium disintegration series which was discussed in the preceding chapter. The element UXi resulting from U by the loss of an a-particle is placed in Group IVA. This element in turn undergoes a /3-ray change producing the element UX 2 which is accordingly placed in Group VA. The element UII in Group VIA is formed from UX 2 by 0-ray disintegration, while the ele- ment lo results from the loss of an a-particle by UII with a con- sequent shifting of two places to the left in the table. A similar loss of an a-particle by lo brings us to the element Ra in Group ATOMIC STRUCTURE 63 IIA. In the successive steps of this disintegration from U to Ra, three a-particles or 12 units of atomic mass are lost, and the atomic weight of Ra, as calculated from that of U, agrees with the atomic weight found by direct experiment. In like manner the remain- ing stages of the disintegration may be followed to the end-product in Group IVB. Isotopes. Perhaps the most striking feature in the table is the occurrence of several different elements in the same place, as for example in Group IVB, where in the place occupied by the element Pb, we also find RaB, RaD, ThB, and AcB, together with four other elements to which no names have been assigned, but which are none the less stable end-products. The individual members of such a group of elements occupying the same place in the periodic table, and being in consequence chemically identical, are known as isotopes. Isotopic elements have identical arc and spark spectra and, except for differences in atomic weight, are chemically indistinguishable. Making use of the fact that two gaseous elements having differ- ent atomic weights diffuse at different rates, Thomson and Aston have recently succeeded in separating neon into two isotopes having atomic weights 20 and 22 respectively. This is the only method which has thus far given promise of success in effecting isotopic separations. It is interesting to note in Soddy's table (Fig. 12), that "the ten occupied spaces (groups) contain nearly forty distinct elements, whereas if chemical analysis alone had been available for their recognition, only ten elements could have been distinguished." The Hydrogen-Helium System of Atomic Structure. A generalization similar to that just outlined for the radio-elements has been found by Harkins and Wilson * to hold true for the lighter elements which apparently do not undergo appreciable a-ray dis- integration. Beginning with helium and adding 4 units of atomic weight for each increase of 2 units in the atomic number, gives the atomic weights of the elements in the even-numbered groups of the periodic table, neglecting small changes in mass due to nuclear packing. This rule has been found to hold very closely for all of the elements having atomic weights below 60. * Proc. Nat. Acad., Vol. I, p. 276 (1915). 64 THEORETICAL CHEMISTRY The atomic weights of the elements of the odd-numbered groups can be calculated by a similar rule, provided that the atom of lithium, the first member of the odd-numbered groups, be assumed to be made up of 1 hydrogen and 3 helium nuclei. The following table gives the results as calculated by Harkins and Wilson for the first three series of the periodic table. The so-called theoretical atomic weights are calculated on the basis H = 1, while the experimentally determined values are on the basis O = 16 or H = 1.0078. The remarkably close agree- ment between the two sets of values is taken as an indication that the packing effect, resulting from the formation of the elements from hydrogen nuclei and attendant electrons, is very small. This packing effect has been estimated to involve a decrease in atomic mass of about 0.77 per cent, and is believed to be due almost entirely to the formation of the helium atom. The hydro- gen-helium hypothesis of atomic structure offers a rational ex- planation of many interesting but hitherto obscure facts concern- ing the nature of the elements. Relation between Atomic Weights and Atomic Numbers. For all elements whose atomic weights are less than that of nickel, Harkins finds the following simple mathematical relation to hold, w = 2# + 4 + J(-i)"- 1 , where W is the atomic weight and N is the atomic number. In other words, the atomic weights are a linear function of the atomic numbers. The Periodic Law. In the light of recent discoveries the periodic law acquires new significance; in fact to-day the periodic law may be regarded as the most comprehensive generalization in the whole science of chemistry. Attention has already been directed in an earlier chapter to the most apparent of the imperfections in Mendeleeff's system of classification of the elements. While the later tables are more complete than the original, owing in part to the discovery of new elements, it must be admitted nevertheless that relatively little real progress has been made until recently toward removing the seemingly inherent defects of the system. ATOMIC STRUCTURE 65 02 O ! ^ o I fe w I +88 Cirv. ' ^ o O5 Oi 8t^ c^d co co ojoco W Wo^ SI a bo ' ffi + VjH 10 3 lea or in words, the ratio of the molecular heat of vaporization to the absolute boiling temperature of a liquid is constant, the numerical * Phil. Mag. (5), 18, 54 (1884). LIQUIDS 119 value of the ratio being approximately 21. This is known as Trouton's law. While this relation holds quite well for many liquids, Nernst has pointed out that the constant varies with the temperature, and has proposed two other forms of the equation. Bingham has simplified the equations of Nernst to the following form : 17+0.011 T. While this modification of the Trouton equation has been found to hold for a large number of substances, there are other substances for which the left side of the equation has a value greater than that of the right side. Bingham infers that where this occurs, the substance in the liquid state has a greater molecular weight than it has in the gaseous state, or in other words, the liquid is associated. It is evident that an associated liquid will require an expenditure of energy over and above that required for vapori- zation, to break down the molecular complex. The difference between the values of the two sides of the equation may be taken as a rough measure of the degree of association. Boiling-Point and Critical Temperature. An interesting rela- tion has been pointed out by Guldberg * and Guye.f These two investigators have shown that the absolute boiling temper- ature of a liquid is about two-thirds of its critical temperature. That this empirical relation holds for a variety of different sub- stances is shown in the accompanying table. RELATION OF BOILING-POINT TO CRITICAL TEMPERATURE. Substance. T b T e T b /T e Oxygen 90 155 0.58 Chlorine . 240 414 0.58 Sulphur dioxide Ethyl ether Ethyl alcohol 263 308 351 429 467 516 0.61 0.66 0.68 Benzene 353 562 0.63 Water 373 637 0.59 Phenol 454 691 0.66 * Zeit. phys. chem., 5, 376 (1890). t Bull. Soc. Chim., (3), 4, 262 (1890). 120 THEORETICAL CHEMISTRY Molecular Volume. In dealing with the volume relations of liquids it is customary to employ the molecular volume, i.e., the volume occupied by the molecular weight of the liquid in grams. The justification of this procedure is that when we compare the gram-molecular weights of liquids, the comparison involves equal numbers of molecules of the different substances. Since , mass volume = j r ) density we may write and similarly, molecular weight molecular volume = - 3 r > density atomic weight atomic volume = -= r density Relations between the molecular volumes of liquids were first pointed out by Kopp.* On comparing the molecular volumes of different liquids at their boiling-points, he found that constant differences in composition correspond to constant differences in the molecular volumes. Thus the molecular volumes of the successive members of an homologous series differ by the same number of units, this difference corresponding, for example, to a CH 2 group. In like manner the molecular volumes of various groups have been determined, and from these in turn the atomic volumes of the constituent elements have been worked out. The atomic volumes assigned by Kopp to some of the elements com- monly entering into organic compounds are as follows : C = 11 Cl=22.8 1=37.5 Hydroxyl O = 7.8 # = 5.5 Br = 27.8 S = 22.6 Carbonyl O = 12.2 The value of the atomic volume is found to be dependent upon the manner of linkage ; thus oxygen in the hydroxyl group has the atomic volume, 7.8, while oxygen in the doubly linked condi- tion, as in the carbonyl group, has the atomic volume, 12.2. By means of such a table of experimentally determined atomic * Lieb. Ann., 41, 79 (1842); 96, 153, 303 (1855); 96, 171 (1855). LIQUIDS 121 volumes, Kopp showed that it is possible to calculate the molec- ular volume of a liquid with a fair degree of accuracy. For example, the molecular volume of acetic acid C 2 H 4 O 2 , may be calculated from the atomic volumes of its constituent atoms as follows : 2 C = 2 X 11 = 22 4H = 4 X 5.5 = 22 IHydroxylO = 1 X 7.8 = 7.8 1 Carbonyl = 1 X 12.2 = 12.2 Molecular volume = 64 . The density of acetic acid at its boiling-point is 0.942, and its molecular weight is 60, therefore the observed value of the molec- ular volume is 60 -T- 0.942 = 63.7, a result which is in excellent agreement with that calculated from the atomic volumes of the constituents. The more recent investigations of Thorpe, Lessen, Schiff and Buff afford a confirmation of the conclusion reached by Kopp, that the molecular volumes of liquids are in general additive. While Kopp found that his results were most regular when the molecular volumes were determined at the boiling temperatures of the respective liquids, the reason for this did not appear until after van der Waals had developed his theory of corresponding states. As has been pointed out in the preceding paragraph the boiling-points of most liquids are approximately two-thirds of their respective critical temperatures, and therefore the boiling- points are corresponding temperatures. Co-volume. By studying various series of hydrocarbons, alcohols and ethers, Traube * has been led to suggest that the molecular volume of a liquid be looked upon as made up of the atomic volumes of its constituent elements and a magnitude which he terms the co-volume. This latter he defines as the space surrounding a molecule within which it is free to vibrate and from which other molecules are excluded. The co-volume appears to be nearly constant for a large number of substances, its mean value at a temperature of 15 C. being 25.9 cc. The values * Uber den Raum der Atorae. J. Traube. Ahrens' Sammlung Chemischer und chemisch-technischer Vortraege, 4, 255 (1899). 122 THEORETICAL CHEMISTRY assigned by Traube to the atomic volumes of some of the elements are as follows : C = 9.9 = 5.5 Br = 17.7 N (trivalent) = 1.5 H = 3.1 Cl = 13.2 1=21.4 N (pentavalent) = 10.7 Traube has worked out a series of constants which must be de- ducted to allow for ring formation and for double and treble link- ing. By means of these values, it is possible to calculate the molecular volume of a substance by adding together the respective atomic volumes of the constituents of the liquid and the co- volume, 25.9. It is of course necessary to know the molecular weight of the substance together with its constitution, so that due allowance may be made for unsaturation. For example, the molecular volume of ethyl ether, C^ioO, may be calculated by Traube's method as follows : 4C = 4X9.9 = 39.6 10 H = 10 X 3.1 = 31 10= 1 X5.5 = 5.5 76.1 Co- volume 25 . 9 Molecular volume 102.0 The molecular volume, as determined from the molecular weight and density at 15 C., is 74 -r- 0.7201 = 102.7. The method of Traube may be employed in roughly checking the accepted value of the molecular weight of a liquid provided its density at 15 C. is known, since in the equation M/d = 2 atomic volumes + 25.9, expressing Traube's relation, M is the only unknown quantity. It is apparent that the liquid must be non-associated, since for an associated substance the normal co-volume must necessarily accompany the polymerized molecule. In this case the formula becomes M/d = 2 atomic volumes -f 25.9/X where n denotes the number of simple molecules in the polymer. Obviously when the molecular weight of a liquid is known, the experimental determination of the co-volume, (M/d 2 atomic vols.) may be used to estimate the degree of association. The LIQUIDS 123 values thus obtained are not in satisfactory agreement with the factors of association derived by means of other methods. Refractive Power of Liquids. The velocity of transmission of light through any medium depends upon its nature, especially Fig. 27. upon its density. When a ray of light passes from one medium into another it is refracted, the degree of refraction being such that the ratio of the sines of the angles of incidence and refrac- tion is constant and characteristic for the two media. This 124 THEORETICAL CHEMISTRY fundamental law of refraction was discovered by Snell about 1621. According to the wave theory of light, the ratio of the sines of the angles of incidence and refraction is identical with the ratio of the velocities of light in the two media. The ratio is termed the index of refraction and is usually denoted by the letter n. Representing by i and r, the angles of incidence and refraction, and by vi and v 2 , the respective velocities of light in the two media, we have _ sin i _ Vi sin r v 2 Various forms of apparatus have been devised for the determina- tion of the refractive index of liquids. Of these the best known and most satisfactory is the refractometer of Pulfrich, an improved form of which is shown in Fig. 27. While the limits of this book prohibit a detailed description of the apparatus, the fundamental principles involved in its construction will be readily understood from the accompanying diagram, Fig. 28. The liquid or fused solid is placed in a small glass cell, C, which is cemented to a rec- tangular prism of dense optical glass, P, the refractive index of which is generally 1.61. A beam of monochromatic light, from a sodium flame or a spectrum-tube containing hydrogen, is allowed to enter the prism in a direction par- allel to the horizontal surface of separation between the glass and the liquid. After passing through Fig. 28. the liquid and the prism, the beam emerges making an angle i with its original direction. By means of a telescope, the emergent beam can be observed and its position noted, the angle of emergence being read on a divided circle attached to the telescope. From the angle of emergence thus determined, the index of refraction of the liquid can be calculated in the following manner. The value of the index of refraction, N, for air/glass being known, we have N = smi sinr (1) LIQUIDS 125 The angle of incidence of the last ray entering the prism from the liquid is 90, or sin ii = 1. The index of refraction, n\ y for liquid/ glass may be calculated thus, sm ri sin 7*1 But sn 7*1 = cos r = V 1 sin 2 r. (3) Transposing equation (1) and substituting in equation (3), we have or sin ri*** VN 2 -sin?i. (4) Therefore, substituting equation (4) in equation (2), we have N sn Remembering that n = N/rii, we have for the index of refraction, n, for air/liquid, by substitution in equation (5), n = v sin 2 z, or if N = 1.61, _ n = V2.5921 -sin 2 !. The values of VAT 2 sin 2 i are generally given, for different values of i, in tables supplied with the refractometer, thus saving the experimenter a somewhat laborious calculation. The value of n thus obtained is the index of refraction from air into the liquid; if the index from vacuum into the liquid, the so-called absolute index, is required, the value of n must be multiplied by 1.00029. The index of refraction is dependent upon temperature, pressure and in general upon all conditions which affect the density of the medium. Furthermore, it is dependent upon the wave-length of the light employed, the index for the red rays being greater than that for the violet rays. It is therefore necessary in making 126 THEORETICAL CHEMISTRY measurements of refractive indices to use light of a definite wave- length, or what is termed monochromatic light. The sodium flame is most frequently used for this purpose, the wave-length being represented by the letter D. Measurements of the refrac- tive index referred to the D-line of sodium are commonly desig- nated by the symbol UD- When incandescent hydrogen is employed as a source of light, the refractive index may be determined for the C-, F- and G-lines, the respective values being represented by nc, np, and no- Specific and Molecular Refraction. Various attempts have been made to express the refractive power of a liquid by a formula which is independent of variations of temperature and pressure. Of the different formulas proposed but two need be mentioned. The first, due to Gladstone and Dale,* is as follows: n- 1 in which d denotes the density of the liquid and r x is the so-called specific refraction. The other formula, proposed by Lorenz f and Lorentz,J has the following form: = This formula is superior to that of Gladstone and Dale which is purely empirical. It is based upon the electromagnetic theory of light and gives values of r 2 which are quite independent of the temperature. In order that we may compare the refractive powers of different liquids, the specific refractions are multiplied by their respective molecular weights, the resulting products being termed their molecular refractions. As the result of a large number of experiments, it has been shown that the molecular refraction of a compound is made up of the sum of the refractive constants of the constituent atoms, or in other words refractive power is an additive property. The values of the refractive con- stants of the elements and commonly-occurring groups have been * Phil. Trans. (1858). t Wied. Ann., n, 70 (1880). t Ibid., 9, 641 (1880). LIQUIDS 127 determined with great care by Briihl and others, the method employed being similar to that used by Kopp in connection with his investigations on molecular volumes. Thus, Briihl found in the homologous series of aliphatic compounds that a difference of CH 2 in composition corresponds to a constant difference of 4.57 in molecular refraction. Then, having determined the molecular refraction of a ketone or an aldehyde of the composition, C n H 2n O, he subtracted n times the value of CH 2 and obtained the atomic refraction of carbonyl oxygen. By deducting the molecular refraction of the hydrocarbon, C n H 2n +2, from that of the corre- sponding alcohol, CnHtn+zO, he obtained the atomic refraction of hydroxyl oxygen. By subtracting six times the value of CH 2 from the molecular refraction of hexane, C 6 Hi 4 , he obtained tne refractive constant for hydrogen or 2 H = 2.08. In like manner the refractions of other elements and groups of elements were determined. Just as in the case of molecular volumes so with molecular refractions, the arrangement of the atoms in the molecule must be taken into consideration. Bruhl,* who has devoted much time to the investigation of the effect of constitution upon refraction, has pointed out that the molecular refraction of compounds con- taining double and triple bonds is greater than the calculated value, and he has assigned to these bonds definite constants of refrac- tion. The values of the atomic refractions for a few of the elements as given by Briihl are as follows : C= 2.48 Hydroxyl = 1.58 H= 1.04 Carbonyl O =2.34 Cl = 6.02 Double bond = 1.78 I = 13.99 Triple bond = 2.18 More recent investigations bring out the fact that, when double or triple bonds occupy adjacent positions in the molecule, the simple additive relations no longer obtain. The determination of the molecular refraction of a liquid affords a means of ascertain- ing or confirming its chemical constitution. For example, geraniol has the formula CioHi 8 0, and its chemical behavior is such as to * Proc. Roy. Inst., 18, 122 (1906). 128 THEORETICAL CHEMISTRY warrant the conclusion that it is a primary alcohol. The value of UD is 1.4745, from which the molecular refraction is calcu- lated to be 48.71. The molecular refraction calculated from the atomic refractions given in the preceding table is: - IOC = 10X2.48 = 24.80 18H = 18 X 1.04 = 18.72 1 Hydroxyl = 1 X 1.58 = 1.58 Molecular refraction 45. 10 The difference between the theoretical and experimental values of the molecular refraction is 48.71 45.10 = 3.61, which is approximately twice the value of a double bond, 1.78 X 2 = 3.56. From this we conclude that the molecule of geraniol contains two double bonds. Furthermore an alcohol of the formula, Ci Hi 8 O, containing two double bonds cannot possess a ring structure and therefore must be a member of the aliphatic group of compounds. This conclusion is supported by the chemical properties of the substance.* In a similar manner the Kekul6 formula for benzene has been confirmed, the difference between the theoretical and experimental values of the molecular refraction indicating the presence of three double bonds in the molecule. Specific Refraction of Mixtures. The specific refraction of an homogeneous mixture or solution is the mean of the specific refractions of its constituents. Thus, if the specific refractions of the mixture and its two components are represented by ri, r 2 , and r 3 , then p , (100 -p) riS= I66 rH ~~~ where p denotes the percentage of the constituent whose specific refraction is r 2 . Hence it is possible to determine the specific refraction of a substance in solution by measuring the refractive indices and densities of the solution and solvent. If the refrac- tive indices of the solvent, solution and dissolved substance are * The accepted structural formula of geraniol is H - C - CH 2 OH (CH 3 ) 2 C = CH-CH 2 .CH 2 - C - CH 8 LIQUIDS 129 represented by wi, HZ, and n s respectively, and if d\ t ck, and ds denote the corresponding densities, then we have g2 - 1 100 - p nf - 1 where p is the percentage of the dissolved substance. As has already been mentioned, the formula of Lorenz-Lorentz is based upon the electromagnetic theory of light. According to this theory n 2 l/n 2 + 2 expresses the fraction of the unit of volume of the substance which is actually occupied by it. From this it M n 2 1 follows that the molecular refraction, -r -^ r, is an expression a n^ + 2. of the volume actually occupied by the atomic nuclei of the molecule. It is interesting to note that the ratio of the sum of the atomic volumes, calculated by the method of Traube, to the corrected molecular volume, as determined by the Lorenz-Lorentz formula, is approximately constant, or atomic volumes - TT 2 _ - =3.45 approximately. This may be considered as the ratio of the volume within which the atoms execute their vibrations to their actual material volume. Rotation of the Plane of Polarized Light. Some liquids when placed in the path of a beam of polarized light possess the prop- erty of rotating the .plane of polarization to the right or to the left. Such liquids are said to be optically active. Those substances which rotate the plane of polarization to the right are termed dextro-rotatory, while those which cause an opposite rotation are called levo-rotatory. The determination of the rotatory power of a liquid is made by means of an instrument known as a polarimeter, a convenient form of which is shown in Fig. 29. The essential parts of this instrument are two similar Nicol prisms placed one behind the other with their axes in the same straight line. The light after passing through the forward prism, P, known as the polarizer, has its vibrations reduced to a single plane; it is said to be plane polarized. On entering the rear Nicol prism, A, 130 THEORETICAL CHEMISTRY known as the analyzer, the light will either pass through or be completely stopped, depending upon the position of the prism. If the analyzer be slowly rotated, it will be observed that the positions of maximum transmission and extinction occur at points 90 apart. If the analyzer be rotated, so that its axis is at right angles to the axis of the polarizer, the field observed will be dark, no light being transmitted. If now a tube similar to that shown in Fig. 30 be filled with an optically active liquid and placed between the polarizer and analyzer, the field will Fig. 29. become light again, due to the rotation of the plane of polariza- tion by the optically-active substance. The extent to which the plane of polarization has been rotated can be determined by turning the analyzer until the field becomes dark again, and read- ing on the divided circle, K, the number of degrees through which it has been moved. When it is necessary to turn the analyzer to the right, the substance is dextro-rotatory, and when it is neces- sary to turn it to the left, the substance is levo-rotatory. Various LIQUIDS 131 optical accessories have been added to the simple polarimeter described above to render the instrument more sensitive, but for these details the student must consult some special treatise.* The angle of rotation is dependent upon the nature of the liquid, the length of the column of substance through which the light passes, the wave-length of the light used, and the temperature at which the measurement is made. It is customary in polarimetric Fig. 30. work to employ sodium light and, unless otherwise specified, it may be assumed that a given rotation corresponds to the D-line. Specific and Molecular Rotation. The results of polarimetric measurements are expressed either as specific rotations or as molecular rotations, the latter being preferable since the optical activities of different substances may then be compared. The specific rotation is obtained by dividing the observed rota- tion by the product of the length of the column of liquid and its density, or r i a Wi = w' where [a] t is the specific rotation at the temperature, t, a the observed angle, I, the length of the column of liquid in decimeters and d, its density. If the specific rotation is multiplied by the molecular weight of the substance, the molecular rotation is ob- tained, but owing to the fact that the resulting numbers are too large, it is customary to express the molecular rotation as one one-hundredth of this value, thus Ma The specific and molecular rotations of solutions of optically active substances may also be determined, if we assume that the * See for example, " The Optical Rotatory Power of Organic Substances and its Practical Applications." H. Landolt, trans, by J. H. Long. 132 THEORETICAL CHEMISTRY solvent is without effect. While this assumption is justifiable with aqueous solutions, it is not so when non-aqueous solvents are used. If g grams of an optically active substance be dissolved in v cc. of solvent, then av M av [*!-, and [M]< = JOO-^> or if the composition of the solution is expressed in terms of weight instead of volume, g grams of substance being dissolved in 100 grams of solution of density d } then 100 gdl and [a M ]t = Ma gdl Optical Activity and Chemical Constitution. The fact that some substances have the power of rotating the plane of polarized light was first discovered by Biot, but the credit for recognizing the chemical significance of this fact belongs to Pasteur.* He discovered that ordinary recemic acid can be separated into two optically active modifications, one of which is dextro- and the other levo-rotatory, the numerical values of the two rotations Fig. 31. being identical. If a solution of sodium ammonium racemate be allowed to evaporate at a low temperature, crystals of the composition NaNH^EUOe 4 H 2 O will separate. On close in- spection it will be found that the crystals are not all alike, but that they may be divided into two classes, one class showing some unsymmetrical crystal surfaces which are oppositely placed in the crystals of the other class. The crystals of one class may * Ann. Chim. Phys. (3), 24, 442 (1848); 28, 56 (1850); 31, 67 (1851). LIQUIDS 133 be regarded as the mirror images of those of the other class: such crystals are said to be enantiomorphous. The forms usually assumed by the two enantiomorphous modifications of sodium ammonium reacemate are shown in Fig. 31. After separating the two forms Pasteur dissolved each in water, making the solu- tions of the same strength. The solution of the crystals with the " right-handed faces" was found to be dextro-rotatory, while that of the crystals with the " left-handed faces" was found to be levo- rotatory. Pasteur then decomposed the two salts obtained from sodium ammonium racemate and obtained the corresponding acids, which he called dextro- and levo-racemic acids. It was subsequently shown that the two acids were identical with dextro- and levo-tartaric acids. Finally, when Pasteur mixed equiv- alent amounts of concentrated solutions of dextro-and levo- tartaric acids, an appreciable evolution of heat was observed, indicating that a chemical reaction had taken place. After allowing the solution to stand for some time, crystals of ordinary racemic acid were obtained. Thus it wus clearly proven that an optically inactive substance may be separated into two opti- cally active modifications, possessing equal and opposite rotatory powers, and that by mixing equivalent quantities of the two optically active forms, the optically inactive substance may be recovered. Pasteur discovered and applied three other methods in addi- tion to the mechanical method already described, for the separation of a substance into its optically active modifications. These are as follows: (a) Method of Crystallization; (b) Method of Forma- tion of Derivatives ; and (c) Method of Ferments. Method of Crystallization. To a supersaturated solution of the racemic modification a very small crystal of one of the active forms is added. This will induce the separation of crystals of the same form, inoculation with a dextro-crystal producing the dextro-form and inoculation with a levo-crystal producing the levo-form. Method of Formation of Derivatives. In this method an optic- ally active substance, generally an alkaloid, is added to the racemic modification, producing optically active derivatives having differ- 134 THEORETICAL CHEMISTRY ent solubilities. Thus if cinchonine, an optically active alkaloid having the formula, CuK^O, be added to the racemic modifica- tion of tartaric acid, the cinchonine salt of the levo-acid will crystallize first. The crystals of the cinchonine salt are then removed and after adding ammonia to displace the alkaloid, dilute sulphuric acid is added and the pure levo-tartaric acid is obtained. Method of Ferments. Notwithstanding the fact that optical antipodes resemble each other so closely in most of their properties, Pasteur found that certain micro-organisms have the power of distinguishing sharply between these forms. For example, if penicillium glaucum be introduced into a solution of racemic tartaric acid, it thrives at the expense of the dextro-acid and eventually leaves the pure levo-form. In this method one of the active modifications is always lost. Pasteur was the first to point out that there must be some inti- mate connection between optical activity and the constitution of the molecule. It remained for Le Bel * and van't Hoff f to for- mulate independently and almost simultaneously an hypothesis to account for optical activity on the basis of molecular constitu- tion. Their important work laid the foundation of spatial chemis- try, commonly termed stereochemistry (derived from the Greek 0-re/oeos = a solid). Le Bel accepted Pasteur's view that optical activity is dependent upon a condition of asymmetry, but whether this asymmetry is a property of the crystal alone or whether it belongs to the molecule of the optically active substance, was the question he set himself to answer. He found, on dissolving certain optically active crystals in an inactive solvent, that the optical activity is imparted to the solution and therefore he concluded that the condition of asymmetry must exist in the chemical mole- cule. All of the optically active substances known to Le Bel were compounds of carbon. An examination of the formulas of these compounds led him to ascribe the cause of their optical activity to the presence of an asymmetric carbon atom, that is, a carbon atom combined with four different atoms or groups of atoms. * Bull. Soc. Chim. (2), 22, 337 (1874). t Ibid. (2), 23, 295 (1875). LIQUIDS 135 One of the simplest examples is afforded by lactic acid, the struc- tural formula of which is H I CH 3 - C - COOH I OH In this formula the asymmetric carbon atom is placed at the center and is in combination with hydrogen, hydroxyl, methyl and carboxyl. In connection with his work on the relation between optical activity and asymmetry, Le Bel pointed out that active forms never result from laboratory syntheses, the racemic modification being invariably obtained. Van't Hoff reached con- clusions similar to those of Le Bel and proposed the additional COOH COOH Fig. 32. theory of the asymmetric tetrahedral carbon atom. Since the four valences of the carbon atom are equivalent, as the work of Henry on methane has shown them to be, van't Hoff pointed out that the only possible geometrical arrangement of the atoms in the molecule of methane, must be that in which the carbon atom is placed at the center of a regular tetrahedron with the four hydrogen atoms at the four apices. He then pointed out that when the four valences of the tetrahedral carbon atom are satis- fied with different atoms or groups, no plane of symmetry can be passed through the figure, the carbon atom being asymmetric. This conception of Le Bel and van't Hoff forms the basis of all stereochemistry, and has proved of inestimable value to the organic chemist in enabling him to explain the existence of many isomeric compounds. Thus, ordinary lactic acid can be split into 136 THEORETICAL CHEMISTRY two optically active isomers. Aside from the fact that one acid is dextro- and the other is levo-rotatory, the properties of the two acids are practically identical. If the formulas are written spatially, the different groups can be arranged about the asymmetric carbon atom in such a way that the two tetrahedra shall be mirror images of each other, as shown in Fig. 32. It will be observed that these two tetrahedra can in no way be super- posed so that the same groups fall over each other, that is to say, they are enantiomorphous forms. In tartaric acid there are two asymmetric carbon atoms as is evident when its structural formula is written as follows: H H I I HOOC - C - C - COOH I I OH OH If the stereochemical formulas of the dextro- and levo-acids be represented as in Fig. 33, (a) and (b), it will be apparent that the theory admits of the existence of another isomer with the atoms and groups arranged as in Fig. 33 (c). Racemic Acid A COOH d-Tartarte Ae *) COOH Z-Tartaric Acid ib) Fig. 33. OH COOH Meao-Tartaric Acid (C) In this arrangement the asymmetry of the upper tetrahedron is the reverse of that of the lower, and consequently the optical activity of one-half of the molecule exactly "compensates the optical LIQUIDS 137 activity of the other half, and the molecule as a whole is inactive. It is evident that such a tartaric acid could not be split into two active forms. Actually there are four tartaric acids known, viz., (1) inactive racemic acid which is separable into (2) dextro-tartaric acid and (3) levo-tartaric acid; and (4) meso-tartaric acid, an in- active substance which has never been separated into two active forms, but which has the same formula, the same, molecular weight and in general the same properties as the dextro- or levo- tartaric acids. Inactive forms, such as meso-tartaric acid, are said to be inactive by internal compensation. This constitutes one of many beautiful confirmations of the van't Hoff theory of the asymmetric tetrahedral carbon atom. Meso-tartaric acid furnishes an illustration of the fact that asymmetric carbon atoms may be present in the molecule with- out imparting optical activity to the substance. The converse of this proposition, however, that optical activity is dependent upon asymmetric carbon atoms, is generally true. Quite recently some substances apparently containing no asymmetric carbon atoms have been discovered which are optically active. An example of such a substance is 1-methyl cyclohexylidene-4 acetic acid, t& which the following formula has been assigned: CH 3 CH )C:CH-COOH Other atoms aside from carbon may be asymmetric; thus certain compounds of nitrogen, sulphur and tin have been shown to be optically active. The theory also furnishes an explanation of the fact, pointed out by Le Bel, that optically active forms are never obtained by direct synthesis. Since the rotatory power is dependent upon the arrangement of the atoms and groups in the molecule, it follows from the doctrine of probability that as many dextro as levo configurations will be formed and consequently the racemic modification will be obtained. Up to the present time no satisfactory generalization has been discovered as to the factors determining the molecular rotation in any particular case. An 138 THEORETICAL CHEMISTRY attempt in this direction has been made by Guye,* in which he ascribes the magnitude of the observed rotation to the relative masses of the atoms or groups which are in combination with the tetrahedral carbon atom. But it cannot be mass alone which conditions optical activity, since substances are known which rotate the plane of polarization notwithstanding the fact that their molecules have two groups of equal mass in combination with the asymmetric carbon atom. The molecular rotations of the members of homologous series exhibit some regularities, but on the other hand many exceptions occur which cannot be satis- factorily explained. About all that can be said at the present time is, that optical activity is a constitutive property. Magnetic Rotation. That many substances acquire the power of rotating the plane of polarized light when placed in an intense magnetic field was first observed by Faraday f in 1846. The relation between chemical composition and magnetic rotatory power has since been investigated very exhaustively by W. H. Per kin, | his experiments in this field having been continued for more than fifteen years. In brief, Perkin's method of investi- gating magnetic rotatory power consisted in introducing the liquid to be examined into a polarimeter tube 1 decimeter in length and then placing the tube axially betweeen the perforated poles of a powerful electromagnet, as shown in Fig. 34. Upon exciting the magnet it was found that the plane of polarization had been rotated, either to the right or the left, the direction of rotation depending upon the direction of the current, the intensity of the magnetic field and the nature of the liquid. Perkin used the sodium flame as his source of light and carried out all of his experiments at 15 C. He expressed his results by means of the formula, * Compt. rend., no, 714 (1890). t Phil. Trans., 136, 1 (1846). t Jour, prakt. Chem. [2], 31, 481 (1885); Jour. Chem. Soc., 49, 777; 41, 808; 53, 561, 695; 59, 981; 61, 287, 800; 63, 57; 65, 402, 815; 67, 255; 60, 1025 (1886-1896). LIQUIDS 139 Ma/d, a being the observed angle of rotation, d, the density of the liquid and M its molecular weight. All measurements were expressed in terms of water as a standard: thus if Ma/d is the rotation for any substance and M'a'/d' is the corre- sponding rotation for water, then, according to Perkin, the molecular magnetic rotation' will be given by the ratio, Ma/d : M'a'/d' or Mad'/M'a'd. The molecular magnetic rotation for a large number of organic compounds has been determined by Perkin, who has shown it to be an additive property. In any one homologous series the value of the molecular magnetic rotation is given by the formula mol. mag. rotation = a + rib, where a is a constant characteristic of the series, b is a constant corresponding to a difference of CH 2 in composition, its value being 1.023, and n is the number of carbon atoms contained in the molecule. This formula is applicable only to compounds which are strictly homologous, isomeric substances in two differ- ent series having quite different rotations. The constitution of the molecule exerts as great an influence on magnetic rotation as it does on refraction, a double bond causing an appreciable increase in the value of a. The results of experiments on mag- netic rotation show that nothing like the same regularities exist as have been discovered for molecular refraction and molecular volume. The rotatory powers of various inorganic substances have been determined, but the results are too irregular to admit of any satisfactory interpretation. Absorption Spectra. When a beam of white light is passed through a colored liquid or solution and the emergent beam is examined with a spectroscope, a continuous spectrum crossed by a number of dark bands is obtained. A portion of the light has been absorbed by the liquid. Such a spectrum is known as an absorption spectrum. If instead of passing the light through a liquid it is passed through an incandescent gas, a spectrum will be obtained which is crossed by numerous fine lines, termed Fraunhofer lines. Such lines occupy the same positions as the corresponding colored lines in the emission spectrum of the gas. 140 THEORETICAL CHEMISTRY It follows, therefore, that the absorption spectrum is quite as char- acteristic of a substance as its emission spectrum, and from a careful study of the absorption spectra of liquids we may expect to gain some insight into their molecular constitution. The pioneer workers in this field were Hartley and Baly * and it is largely to them that we owe our present experimental methods. The instrument employed for photographing spectra is called a Fig. 35. spedrograph, a very satisfactory form being shown in Fig. 35. It differs from an ordinary spectroscope in that the eye-piece is replaced by a photographic camera. This attachment is clearly shown in the illustration. The plateholder is so constructed that only a narrow horizontal strip of the plate is exposed at any one time, thus making it possible to take a series of photographs on the same plate by simply lowering the holder. By means of a millimeter scale, also shown in the illustration, the plateholder can be moved through the same distance each time before expos- * See numerous papers in the Jour. Chem. Soc., since 1880. LIQUIDS 141 ing a fresh portion of the plate, thus. insuring an equally-spaced series of spectrum photographs. In order that spectra in the ultra- violet region may be photographed, it is customary to equip the in- strument with quartz lenses and a quartz prism, ordinary glass not being transparent to the ultra-violet rays. Using a spectrograph furnished with a quartz optical system, it is possible to photograph on a single plate the entire spectrum from 2000 to 8000 Angstrom units. A scale of wave-lengths photographed on glass is provided with the instrument so that the wave-lengths of lines or bands can be read off directly by laying the scale over the photographs. The source of light to be used depends upon the character of the investigation. If a source rich in ultra-violet rays is desired, the light from the electric spark obtained between electrodes pre- pared from an alloy of cadmium, lead and tin is very satisfactory; or the light from an arc burning between iron electrodes may be used. For investigations in the visible region of the spectrum the Nernst lamp is unsurpassed. In using the spectrograph for the purpose of studying the constitution of a dissolved substance, it is necessary to determine not only the number and position of the absorption bands, but also the persistence of these bands as the solution is diluted. According to Beer's law the product of the thickness, t, of an absorbing layer of solution of molecular concentration, ra, is con- stant, or mt = k. If then the thickness of a given layer of solu- tion is diminished n times, its absorption will be the same as that of a solution whose concentration is only I/ nth of that of the original solution. Thus, by varying the thickness of the absorbing layer we can produce the same effect as by changing the concentration. The convenient device of Baly for altering the length of the absorbing column of liquid is shown in Fig. 36 attached to the collimator of the spectrograph. It consists of two closely-fitting tubes, one end of each tube being closed by a plane, quartz disc. The outer tube is fitted with a small bulbed-f unnel and is graduated in millimeters. The two tubes are joined by means of a piece of rubber tubing which prevents leakage of the contents, and at the same time admits of the adjustment of the column of liquid to the desired length by simply sliding the smaller tube in or out. 142 THEORETICAL CHEMISTRY Molecular Vibration and Chemical Constitution. There are two systems of graphic representation of the results of spectro- scopic investigations. In the first system, due to Hartley, the wave-lengths or their reciprocals, the frequencies, are plotted as abscissae and the thicknesses of the absorbing layers, in milli- meters, are plotted as ordinates. Such curves are known as curves of molecular vibration. The second system, due to Baly and Desch, is a modification of that developed by Hartley. Fig. 36. Baly and Desch suggested that for various reasons it would be more advantageous, if instead of plotting the thicknesses of the absorbing layers as ordinates, the logarithms of these thicknesses be plotted. Both methods have their advantages and both are used. As an illustration of the value of curves of molecular vibration in connection with questions of chemical constitution, we will take the case of o-hydroxy-carbanil. The constitution of this substance was known to be represented by one of the two following formulas: /\ /\ C 6 H/ ^>CO, or C 6 HC - O.C 2 H 6 N - C 2 H 5 Lactam ether Lactim ether 144 THEORETICAL CHEMISTRY Hartley concluded, therefore, that formula (a) represents the structure of the molecule of o-hydroxy-carbanil. It is beyond the scope of this book to discuss at greater length the bearing of absorption spectra upon chemical constitution; but the student is earnestly advised to consult some book * treating of this important subject or to read some of the original papers. Surface Tension. The attraction between the molecules of a liquid manifests itself near the surface where the molecules are subject to an unbalanced internal force. The condition of a liquid near its surface is roughly depicted in Fig. 38, where the dots A, B, and C represent molecules and the circles represent the spheres within which lie all of the other molecules which exert an appreciable attraction upon A, B, and C. The shaded portions rep- Fig. 38. resent those molecules whose attractions are unbalanced. These unbalanced forces will evidently tend to diminish the surface to a minimum value. That is, the contraction of the surface of a liquid involves the expenditure of energy by the liquid. The surface film of a liquid is consequently in a state of tension. Some liquids wet the walls of a glass capillary tube while others do not. When the liquid wets the tube, the surface is concave and the liquid rises in the tube; on the other hand, when the liquid does not adhere, the surface is convex and the liquid is depressed in the tube. The law governing the elevation or depression of a liquid in a capillary tube was discovered by Jurin and may be stated thus : The elevation or depression of a liquid in a capillary tube is inversely proportional to the diameter of the tube. Let Fig. 39 represent a * Relation between Chem. Constitution and Phys. Properties. Samuel Smiles. LIQUIDS 145 capillary tube of radius r, immersed in a vessel of liquid whose density is d, and let the elevation of the liquid in the capil- lary be denoted by h. Then the weight of the column of liquid in the capillary will be irr 2 hdg, where g is the acceleration due to gravity. The force sustaining this weight is 2 irry cos 0, the Fig. 39. vertical component, of the force due to the tension of the liquid surface at the walls of the tube, 7 being the surface tension and the angle of contact of the liquid surface with the walls of the tube. Therefore or Trr 2 hdg = 2 irry cos 0, hdgr 7 = 2cos0 In the case of water and many other liquids is so small that we may write = 0, the foregoing expression becoming 7 = 1/2 hdgr. Thus the surface tension of a liquid can be calculated provided its density and the height to which it rises in a previously calibrated tube is known. When h and r are expressed in centimeters, y will be expressed in dynes per centimeter or ergs per square centi- meter. A simple form of apparatus for the determination of surface tension used by the author is shown in Fig. 40. A capil- lary ,tube, A, of uniform bore is sealed to a glass rod, E, which is held in position in the test tube, B, by means of a cork stopper. A short right-angled tube, D, and a thermometer, F, are also 146 THEORETICAL CHEMISTRY passed through the same cork stopper. The liquid whose surface tension is to be measured is introduced into the tube, B, the cork inserted and the tube placed inside of the larger tube, C, containing a liquid of known boiling-point. When the thermometer, F, has become stationary, the capillary elevation of the liquid is measured ( with a cathetometer. The tube, D, permits the escape of vapor from the liquid in B and at the same time insures equality of pres- sure inside and outside of the ap- paratus. The spiral tube, G, serves as an air condenser, preventing loss of vapor from the liquid in the outer tube. The surface tension of a liquid has been found to depend upon the nature of the liquid and also upon its temperature. Surface Tension and Molecular Weight. In 1886, Eotvos * showed that the surface tension multiplied by the two-thirds power of the mo- lecular weight and specific volume is a function of the absolute tem- perature, or y(Mv)l = f(T), where 7 is the surface tension, M Fig. 40. the molecular weight, v the specific volume or reciprocal of the density, and T the absolute temperature. Ramsay and Shields f modified the equation of Eotvos as follows : 7 (Mv)l = k(t c -t- 6), (1) t c being the critical temperature of the liquid, t the temperature of the experiment, and k a constant independent of the nature of * Wied. Ann., 27, 448 (1886). t Zeit. phys. Chem., 12, 431 (1893). LIQUIDS 147 the liquid. The physical significance of the two-thirds power of the molecular volume has been explained by Ostwald in the follow- ing manner : Assuming the molecules to be spherical, we shall have for two different liquids, the proportion Vi:F 2 ::ri 8 :r 2 , where Vi and Vz represent the volumes and ri and r 2 the radii of their respective molecules. Similarly the ratio of the surfaces, Si and >S 2 , of the molecules in terms of their respective radii, will be Sn&iirtirf. From these two proportions it follows that the ratio of the molec- ular surfaces in terms of the molecular volumes, will be Si:&::Fi*:F s *. Making use of the value of M as determined in the gaseous state, Ramsay and Shields found the value of k for a large number of liquids to be equal to 2.12 ergs. Among the liquids for which this value of k was found were benzene, carbon tetrachloride, carbon disulphide and phosphorus trichloride. For certain other liquids such as water, methyl and ethyl alcohols and acetic acid, k was found to have values much smaller than 2.12. Ramsay and Shields attributed these abnormalities to an increase in molec- ular weight due to association, and suggested that the degree of association might be calculated from the equation xi = 2.12/fc', or where x denotes the factor of association, and k' is the value of the constant for the associated liquid in equation (1). It was further pointed out by Ramsay and Shields that equation (1) affords a means of calculating the molecular weight of a pure liquid, provided we assume that for a non-associated liquid the mean value of k is 2.12. Since it is not an easy matter to deter- mine the critical temperature with accuracy, Ramsay and Shields made use of a differential method, and thus eliminated t e from equation (1). If the surface tension of a liquid be measured at 148 THEORETICAL CHEMISTRY two temperatures h~and k, and the corresponding densities are di and dz, we shall have 71 (M/di)i = k(t e -ti- 6), (3) and 72 (M/d,)i = k(t c -U- 6). (4) Subtracting equation (4) from equation (3), we obtain or solving equation (5) for M, we have The method of Ramsay and Shields is the best known method for the determination of the molecular weight of a pure liquid. If M is known to be the same in the liquid and gaseous states, orj in other words, if k is independent of the temperature, even though its value is not exactly 2.12, the critical temperature of the liquid can be calculated by means of equation (1). In order that the^correct value of the critical temperature may be obtained, Ramsay and Shields found it necessary to use the specific value of k for the liquid whose critical temperature is sought. As an illustration of the method of calculation, the following example is taken from the work of Ramsay and Shields. For carbon disulphide, 7 at 19.4 = 33.58 7 at 46. 1 = 29.41 d at 19.4 = 1.264 d at 46. 1 = 1.223. We have then for 7 (M/d)l, at the two temperatures, (76/1.264)1 X 33.58 = 515.4, and (76/1.223)1 X 29.41 = 461.4. Substituting in the equation 71 (MM)! - 72(M/d 2 )f *> tz ti we have, 515.4 - 461.4 46.1 - 19.4 = 2.022. LIQUIDS 149 This value of k is so nearly equal to the mean value, 2.12, that we assume M to be the same in the liquid and gaseous states, and therefore we may substitute in equation (1) and calculate the critical temperature of carbon disulphide thus, 7 (M/d)l = k (t c - t- 6), or solving for t c) we have t c = fa - + t + 0. Substituting the data given above, in the preceding equation, we obtain t c = 515.4/2.022 + 6 + 19.4, or t c = 280.3 C. Surface Tension and Drop-Weight. Morgan and his co- workers,* from measurements of the volumes of a single drop falling from the carefully-ground tip of a capillary tube, have shown that the weight of the falling drop, from such a tip can be used in place of the surface tension in the equation of Ramsay and Shields for the calculation of molecular weights and critical temperatures. The modified equation may be written thus : _ ln where wi amd w z are the respective weights of the falling drop at the temperatures, ti and t 2 . The value of k obviously depends upon the tip employed. The results obtained by the drop-weight method have been shown to be more trustworthy than those obtained by the method of capillary elevation. Morgan has further pointed out that when the experimental data are substituted in the preceding formula, the magnification of the experimental errors is appreci- ably greater than when use is made of the original formula, w (M/d)t = k(t c -t- 6). Morgan recommends therefore that this formula be used for the determination of molecular weights. After having calibrated * Jour. Am. Chem, Soc., 30, 360 (1908); 30, 1055 (1908). 150 THEORETICAL CHEMISTRY a particular tip with pure benzene (a liquid which is known to be non-associated), and thus ascertaining the value of k, the drop- weights at several different temperatures are determined. If we assume M to have the same value in the liquid and gaseous states, the value of t c can be computed by substituting the experi- mental data in the preceding equation. If at the different tem- peratures at which drop-weights are determined, the same value of t c is obtained, then we may infer that the liquid is non-asso- ciated and, therefore, that the assumption made as to the value of M is confirmed. It is a singular fact that the calculated value of t c for some liquids does not agree with the experimental value, although it remains constant throughout an extended range of temperatures. Morgan considers a constant value of t c to be an indication of non-association, even if the value is fictitious. In this method the constancy of the calculated value of the critical temperature becomes the criterion of molecular association, and thus affords a means of determining whether the molecular weight in the liquid state is identical with that in the gaseous state. The values of t c calculated from the drop-weights of an associated liquid become steadily smaller as the temperature increases. A large number of liquids have been studied by this method, and the results indicate that many of the substances which were con- sidered to be associated by Ramsay and Shields are in reality non-associated ; in fact, it appears from the work of Morgan that association is much less common among liquids than has hitherto been supposed. Dielectric Constants. In 1837, Faraday discovered that the attraction or repulsion between two electric charges varies with the nature of the intervening medium or dielectric. If qi and q 2 represent two charges which are separated by a distance r, the force of attraction or repulsion, /, is given by the equation D' r* where D is a specific property of the medium known as the dielectric constant. The dielectric constant of air is taken as unity. Var- ious methods have been devised for the experimental determination LIQUIDS 151 of the dielectric constant, but the scope of this book forbids even a brief description of the apparatus or an outline of the processes of measurement. For a description of these methods the student is referred to any one of the more complete physico- chemical laboratory manuals, or to the original communications of Nernst * and Drude.f The values of the dielectric constants for some of the more common solvents are given in the accompanying table. DIELECTRIC CONSTANTS AT 18 C. Substance. D Hydrogen dioxide 92 8 Water 77 Formic acid 63 Methyl alcohol 33 7 Ethyl alcohol 25.9 Ammonia, liquid 22 Chloroform 5 Ether 4.4 Carbon disulphide 2.6 Benzene 2.3 The importance of this property of liquids will become more apparent in subsequent chapters, especially in those devoted to 3lectrochemistry. PROBLEMS. 1. It is desired to compare the molecular volumes of alcohol and ether. If the molecular volume of ether is determined at 20 C., at what temper- ature must the molecular volume of alcohol be determined? The boiling points of alcohol and ether are 78 and 35 respectively. Ans. 61 ^C. 2. A volume of 50 liters of air in passing through a liquid at 22 C. causes the evaporation of 5 grams of substance, the molecular weight of which is 100. What is the vapor pressure of the liquid in grams per square centimeter? Ans. 25. * Zeit. phys. Chem., 14, 622 (1894). t Ibid., 23, 267 (1897). 152 THEORETICAL CHEMISTRY 3. The boiling-point of ethyl propionate is 98.7 C. and its heat of vaporization is 77.1 calories. Calculate its molecular weight. 4. The heat of vaporization of liquid ammonia at its boiling-point, under atmospheric pressure ( 33. 5 C.) is 341 calories. Is liquid am- monia associated? 5. Calculate the molecular volume of ethyl butyrate. The molecular volume determined by experiment is 149.1. 6. For propionic acid, d = 1.0158 and no = 1.3953. Calculate the molecular refraction by the formula of Lorenz-Lorentz and compare the value so obtained with that derived from the atomic refractions of the constituent elements. 7. The density of ether is 0.7208, of ethyl alcohol, 0.7935 and of a mixture of ether and alcohol containing p per cent of the latter, 0.7389. At 20 C. the refractive indices for sodium light are, for ether, 1.3536, for alcohol, 1.3619, and for the mixture, 1.3572. Calculate the value of p, using the Gladstone and Dale formula. Ans. 20.81. 8. At 20 C. the density of chloroform is 1.4823 and the refractive index for the D-line is 1.4472. Given the atomic refractivities of carbon and hydrogen, calculate that of chlorine, using the Lorenz-Lorentz formula. Ans. 5.999. 9. Calculate the surface tension of benzene in dynes per centimeter t the radius of the capillary tube being 0.01843 cm., the density of the liquid, 0.85, and the height to which it rises in the capillary, 3.213 cm. Ans. 24.71 dynes/cm. 10. Find the molecular weight of benzene, the surface tension at 46 C. being 24.71 dynes per centimeter, its critical temperature, 288.5 C., its density, 0.85 and the value of k = 2.12. Ans. 77.7. 11. At 14.8 C. acetyl chloride (density = 1.124) ascends to a height of 3.28 cm. in a capillary tube the radius of which is 0.01425 cm. At 46.2 C. in the same tube the elevation is 2.85 cm. and the density = 1.064. Calculate the critical temperature of acetyl chloride. Ans. 234.6C. 12. From a certain tip the weights of a falling drop of benzene are 35.329 milligrams (temp. = 11.4, density = 0.888) and 26.530 milli- grams (temp. = 68.5, density = 0.827). The molecular weight is the same in the liquid and gaseous states. Calculate the critical temper- ature of benzene. Ans. 286. 1C. CHAPTER VIII. SOLIDS. General Properties of Solids. Solids differ from gases and liquids in possessing definite, individual forms. Matter in the solid state is capable of resisting considerable shearing and tensile stresses. In terms of the kinetic theory of matter, the mutual attractive forces exerted by the molecules of solids must be re- garded as superior to the attractive forces between the molecules of gases and liquids. With one or two exceptions all solids ex- pand when heated, but there is no simple law expressing the relation between the increment of volume and the temperature. Rigidity is another characteristic property of solids, it being much more ap- parent in some than in others. Many solids are constantly under- going a process of transformation into the gaseous state at their free surfaces, such a change being known as sublimation. Just as when a gas is sufficiently cooled it passes into the liquid state, so on cooling a liquid below a certain temperature, it passes into the solid state. The reverse transformations are also possible, a solid being liquefied when sufficiently heated, and the resulting liquid completely vaporized if the heating be continued. Heat energy is required to effect transition from the solid to the liquid state, just as heat energy is required to effect transition from the liquid to the gaseous state. Obviously a substance in the solid state contains less energy than it does in the liquid state. The number of calories required to melt 1 gram of a solid substance is called its heat of fusion. It is often difficult to decide whether a substance should be classi- fied as a solid or as a liquid. For example the behavior of certain amorphous substances such as pitch, amber and glass, is similar to that of a very viscous, inelastic liquid. Solids are generally classified as crystalline or amorphous. In crystalline solids the molecules are supposed to be arranged in some definite order, this 153 154 THEORETICAL CHEMISTRY arrangement manifesting itself in the crystal form. An amor- phous solid on the other hand may be considered as a liquid possessing great viscosity and small elasticity. The physical properties of amorphous solids have the same values in all direc- tions, whereas in crystalline solids the values of these properties may be different in different directions. When an amorphous solid is heated it gradually softens and eventually acquires the properties characteristic of a liquid, but during the process of heat- ing there is no definite point of transition from the solid to the liquid state. On the other hand, when a crystalline solid is heated there is a sharp change from one state to the other at a definite temperature, this temperature being termed the melting- point. Crystallography. The study of the definite geometrical forms assumed by crystalline solids is termed crystallography. The number of crystalline forms known is exceedingly large, but it is possible to reduce the many varieties to a few classes or systems by referring their principal elements the planes to definite lines called axes. These axes are so drawn within the crystal that the crystal surfaces are symmetrically arranged about them. This system of classification was proposed by Weiss in 1809. He showed that notwithstanding the multiplicity of crystal forms encountered in nature, it is possible to consider them as belonging to one of six systems of crystallization. The six systems of Weiss are as follows: 1. The Regular System. Three axes of equal length, inter- secting each other at right angles (Fig. 41). 2. The Tetragonal System. Two axes of equal length and the third axis either longer or shorter, all three axes intersecting at right angles (Fig. 42). 3. The Hexagonal System. Three axes of equal length, all in the same plane and intersecting at angles of 60, and a fourth axis, either longer or shorter and perpendicular to the plane of the other three (Fig. 43). 4. The Rhombic System. Three axes of unequal length, all intersecting each other at right angles (Fig. 44). SOLIDS 155 5. The Monodinic System. Three axes of unequal length, two of which intersect at right angles, while the third axis is per- pendicular to one and not to the other (Fig. 45). 6. The Tridinic System. Three axes of unequal length no two of which intersect at right angles (Fig. 46). The position of a plane in space is determined by three points in a system of coordinates, and consequently the position of the a Fig. 41. Fig. 42. Fig. 43. Fig. 44. Fig. 45. Fig. 46. face of a crystal is likewise determined by its points of inter- section with the three axes, or by the distances from the origin of the system of coordinates at which the plane of the crystal face intersects the three axes. These distances are called the parameters of the plane. The fundamental law of crystallography discovered by Steno 156 THEORETICAL CHEMISTRY in 1669 may be stated thus: The angle between two given crystal faces is always the same for the same substance. The fact that every crystalline substance is characterized by a constant inter- facial angle, affords a valuable means of identification which is used by both chemists and mineralogists. The instrument em- ployed for the measurement of the interfacial angles of crystals is called a goniometer. The crystal to be measured is mounted at the center of the graduated circular table of the goniometer, and the image of an illuminated slit, reflected from one surface of the crystal, is brought into coincidence with the cross-wires in the eye- piece of the telescope. The table is then turned until the image of the slit, reflected from the adjacent face of the crystal, coincides with the cross-wires. The interfacial angle of the crystal is de- termined by the number of degrees through which the table has been turned. Properties of Crystals. The properties of all crystals, except those belonging to the regular system, exhibit differences depend- ent upon the direction in which the particular measurements are made. Thus the elasticity, the thermal and electrical con- ductivities, and in fact all of the physical properties of crystals which do not belong to the regular system, have different values in different directions. Crystals whose physical properties have the same values in all directions are termed isotropic, while those in which the values are dependent upon the direction in which the measurements are made, are called anisotropic. Crystals belonging to the regular system, and amorphous substances are isotropic. Certain amorphous substances, such as glass, which are normally isotropic, may become anisotropic when subjected to tension or compression. The phenomenon of double refrac- tion observed in all crystals, except those belonging to the regular system, is due to their anisotropic character. Crystals belonging to the tetragonal and hexagonal systems resemble each other in one respect, viz.: that in all of them there is one direction, called the optic axis, or axis of double refraction (coincident with the prin- cipal crystallographic axis), along which a ray of light is singly refracted, while in all other directions it is doubly refracted. In crystals belonging to the rhombic, monoclinic, and triclinic systems, SOLIDS 157 there are always two directions along which a ray of light is singly refracted. A crystal of Iceland spar (CaCO 3 ) affords a beauti- ful illustration of double refraction. On placing a rhomb of this substance over a piece of white paper on which there is an ink spot, two spots will be seen. On turning the crystal, one spot will remain stationary while the other spot will revolve about it. This property of Iceland spar is utilized in the construction of Nicol prisms for polariscopes. The examination of sections of anisotropic crystals in a polari- scope between crossed Nicol prisms, reveals something as to their crystal form. As has been stated, crystals of the tetragonal and hexagonal systems are uniaxial. If a section is cut from such a crystal perpendicular to the optic axis, and this is placed between the crossed Nicol prisms of a polariscope, in a convergent beam of white light, a dark cross and concentric, spectral-colored circles will be observed, Fig. 47. Upon turning the analyzer through 90 the colors of the circles will change to the respective comple- mentary colors and the dark cross will become light. Crystals of the rhombic, monoclinic, and triclinic systems are biaxial. If a section of a biaxial crystal, cut perpendicular to the line bisect- ing the angle between the two axes, be placed in the polariscope Fig. 46. Fig. 47. and examined as in the preceding case, a series of concentric spectral-colored lemniscates surrounding two dark centers and pierced by dark, hyperbolic brushes, will be observed, as shown in Fig. 48. On rotating the analyzer, the colors will change to the corresponding complementary colors, as in the case of uniaxial crystals. The appearance of these figures is so varied and cha- racteristic as to furnish, in many cases, a very satisfactory means of identifying anisotropic crystals. 158 THEORETICAL CHEMISTRY x Etch Figures. The solubility of crystals has been shown to be different in different directions. Thus, if the surface of a crys- talline substance be highly polished and then treated for a short time with a suitable solvent, faint patterns, known as etch figures, will appear as a result of the inequality of the rate of solution in different directions. When these figures are examined under the microscope the crystal-form can generally be determined. The examination of etch figures has come to be of prime importance to the metallographer. Thus, when an appropriate solvent is applied to the polished surface of an alloy, not only is the crystal form revealed by the etch figures, but also the presence of various chemical compounds may be recognized. By a careful study of the etch figures developed on the surface of highly polished steel, the metallographer may gather important information as to its previous history, especially its heat treatment. Crystal Form and Chemical Composition. From the pre- ceding paragraphs it might be inferred that the same substance always assumes the same crystal form. While this is true in general, there are some substances which appear in several different crystal forms. This phenomenon is termed polymor- phism. Calcium carbonate is an example of a substance crystallizing in more than one form. As calcite, it crystallizes in the hexagonal system, while as aragonite, it crystallizes in the rhombic system. Such a substance is said to be dimorphous. Of the several factors controlling polymorphism, temperature is the most important. Thus sulphur crystallizes at temperatures above 95.6 in the mon- oclinic system, while at lower temperatures it assumes the rhombic form. The temperature at which it changes from one form into the other is termed its transition temperature. As has been mentioned in an earlier chapter (p. 14), some substances may crystallize in the same form, the characteristic interfacial angles being nearly identical. Such substances are said to be iso- morphous. This phenomenon, discovered by Mitscherlich, has been of great use in connection with the earlier investigations on atomic weights, as has already been pointed out. There can be little doubt as to the existence of an intimate SOLIDS 159 connection between crystalline form and chemical composition. Ever since the early part of the nineteenth century, when Haiiy established the science of crystallography, various attempts have been made by chemists and crystallographers to connect crystal- line form with chemical constitution. In 1906, Barlow and Pope * made a most notable contribution to the theories concerning the relation between crystalline form and chemical constitution. Their ideas may be summarized as follows : If each atom be considered as appropriating a certain space, called its sphere of atomic influence, then (1) the spheres of atomic influence are so arranged as to occupy the smallest possible volume in every crystal; (2) the volumes of the spheres of atomic influence in any substance are proportional to the valences of the constituent atoms; (3) the volumes of the spheres of influence of the atoms of different elements of the same valence are nearly equal, any variation being in har- mony with their relations in the periodic system. Barlow and Pope have shown that the general agreement between theory and observation is most satisfactory, a particularly strong argument in favor of this theory being the very plausible explanation which it furnishes for a large number of crystallographic facts. It is without doubt the best working hypothesis which has yet been offered for the investigation of the dependence of crystalline form upon a definite chemical constitution. Compressibilities of the Solid Elements. A series of careful measurements of the compressibilities of the elements by T. W. Richards and his collaborators,! has revealed the fact that com- pressibility is a periodic function of atomic weight. Richards has advanced some interesting suggestions as to the importance of compressibility in connection with intermolecular cohesion and atomic volume. In fact, Richards' theory of compressible atoms may be regarded as a valuable supplement to the the- ory of Barlow and Pope and, taken together, these two theories constitute a rational basis for the science of chemical crystallog- raphy. X-Rays and Crystal Structure. In 1912, Laue pointed out * Jour. Chem. Soc., 91, 1150 (1907). f Zeit. phys. Chem., 61, 77, 100, 171, 183 (1908). 160 THEORETICAL CHEMISTRY that the regularly arranged atoms or molecules of a crystal should act as a three-dimensional diffraction grating toward the X-rays. He showed mathematically that on traversing a thin section of a crystal, a pencil of X-rays should give rise to a diffraction pattern arranged symmetrically round the primary beam as a center. A photographic plate placed perpendicular to the path of the rays and behind the crystal should reveal, on development, a central spot due to the action of the primary rays, and a series of symmetri- cally grouped spots due to the diffracted rays. Laue's predictions were verified experimentally by Friedrich and Knipping, who obtained numerous plates showing a vari- ety of geometrical patterns corresponding to the structural dif- ferences of the crystals examined. The analysis of the Laue diffraction patterns, while furnishing valuable information as to the internal structure of crystals, is nevertheless extremely complex. W. H. Bragg and his son W. L. Bragg * have devised an X-ray spectrometer in which use is made of the fact that the regularly spaced atoms of a crystal reflect the X-rays in much the same way that light is reflected (diffracted) by a plane grating. By observing the angles of reflection from the different faces of a crystal for an incident radiation of known wave-length, it is an easy matter to calculate the distances between the atoms of the crystal which function as diffraction centers. By means of the X-ray spectrometer the internal structure of a number of crystals has been determined. One of the most in- teresting results to the chemist is that obtained with the diamond. The X-ray spectra of the diamond reveal the fact that each carbon atom is situated at the center of a regular tetrahedron formed by four other carbon atoms. In commenting on this method of studying crystal structure, W. H. Bragg says: " Instead of guessing the internal arrange- ment of the atoms from the outward form assumed by the crystal, we find ourselves able to measure the actual distances from atom to atom and to draw a diagram as if we were making a plan of a building." * See "X-Rays and Crystal Structure" by W. H. Bragg and W. L. Bragg. SOLIDS 161 Heat Capacity of Solids. Recent investigations of specific heats of solids at extremely low temperatures have resulted in the formulation of several interesting relationships between heat capacity and temperature. At ordinary temperatures the molecules of a crystalline solid may be assumed to be in a state of violent, unordered motion. As the temperature is lowered, the amplitude of the molecular oscil- lations steadily diminishes until finally, at the absolute zero, there is in all probability a complete cessation of motion. In the neigh- borhood of the absolute zero, where the amplitude of the molecular oscillations is negligible, a crystalline solid may be assumed to possess the properties characteristic of a perfectly elastic body. In other words, the crystalline forces holding the molecules to- gether would preponderate over the feeble thermal forces tending to initiate molecular oscillations within the solid. Under these conditions the solid as a whole would exhibit the same behavior as a single molecule, that is to say, the solid would function as a perfectly elastic body. On this assumption Debye * has derived the following equation expressing the heat capacity of a solid, C v , in terms of its absolute temperature T, TS C v = 71.903 In this equation 6 is a constant characteristic of each solid and has the same dimensions as T. The value of 6 varies between the limits 6 = 50 for calcium and = 1840 for carbon. The agree- ment between the observed and calculated values of C v has been found to be excellent up to T = 9/12. When this latter temperature is exceeded, the molecules of the solid begin to absorb more and more heat energy and to vibrate independently about their centers of oscillation. The failure of Debye's equation is to be expected under these conditions since the solid is no longer behaving as one large molecule. Obviously the lighter the molecules and the greater the crystalline forces within the solid, the higher must the temperature become before * Ann. Physik., 39, 789 (1912). 162 THEORETICAL CHEMISTRY the individual molecules can acquire appreciable kinetic energy. This is apparent from the familiar dynamical principle, that the kinetic energy of a vibrating particle is proportional to its mass and to the square of its vibration frequency. In the case of lead, which is a soft, malleable solid with a relatively low melting point, it is reasonable to infer that the crystalline forces are feeble, and consequently we should expect that molecular and atomic vibra- tions would be set up at quite low temperatures. Furthermore, since the atoms of lead are extremely heavy, their kinetic energy must be great. The correctness of these conclusions is confirmed by the fact that the Debye equation when applied to lead has been found to hold only over a very short range of temperature. On the other hand, the equation has been found to hold for the diamond up to a temperature of about 200 absolute. In this case we have a solid in which the crystalline forces are extremely powerful and in which the atoms are relatively light. A fairly high temperature must be attained before the energy absorbed by the individual atoms of the diamond acquires appreciable magnitude. The absorption of energy by the vibrating molecules continues to increase as the temperature is raised until ultimately, at the melting point of the solid, the crystalline forces become negligible. As this temperature is approached therefore, the intermolecu- lar restraint becomes less and less and the mean kinetic energy of the molecules approaches that of the molecules of the molten solid. As has already been stated in Chapter I (p. 11), Dulong and Petit, in 1819, discovered the interesting fact that the atomic heats of the solid elements have a constant value of 6.5. The importance of this generalization in connection with the verification of atomic weights has already been pointed out. Quite recently, Lewis * has directed attention to the fact that it is much more rational to calculate the atomic heat of an element from the specific heat at constant volume rather than from the specific heat at constant pressure. While it is impossible to measure the specific heat at constant volume, its value may be derived from the specific heat * Jour. Am. Chem. Soc., 29, 1165 (1907). SOLIDS 163 at constant pressure by an application of the laws of thermo- dynamics. Thus, Lewis has obtained the formula TaW Lp ~ "41.780' where T denotes the absolute temperature, a the coefficient of expansion, the coefficient of compressibility, C p and C v the atomic specific heats at constant pressure and constant volume respectively and V the atomic volume. By means of this equa- tion, Lewis has established the following generalization: Within the limits of experimental error, the atomic heat at constant volume, at 20 C., is the same for all solid elements whose atomic weights are greater than that of potassium, and is equal to 5.9. In the case of a solid having a high melting point, the violent agitation of its con- stituent molecules and atoms as the temperature is raised, will undoubtedly produce a corresponding increase in the amplitude of vibration of its electrons together with an increase in their trans- lational velocity among the molecules. Under these conditions, the specific heat at constant volume should be greater than 5.9 calories. This conclusion cannot be verified experimentally until the values of a and /3 in the foregoing equation have been deter- mined at high temperatures. The complete heat capacity curves for three typical solid ele- ments, lead, aluminium, and carbon, are given in Fig. 49. It is apparent from these curves that the absorption of heat energy by a crystalline solid may be considered as taking place in three dis- tinct stages, as follows: (1) In the neighborhood of the abso- lute zero, the heat capacity remains practically zero; (2) the heat capacity increases rapidly with the temperature; and (3) the heat capacity increases slowly, approaching asymptotically the limiting value 5.9 for T = oo. For a malleable, low melting element of high atomic weight, such as lead, the first two stages are very short and the final stage commences at a low temperature. On the contrary, with a hard, high melting element of low atomic weight, such as carbon in the form of diamond, the final stage is not reached at any temperature within the range covered by the experiments. 164 THEORETICAL CHEMISTRY The Nernst-Lindemann Equation. Recently, several equa- tions have been derived expressing the heat capacity of a solid in terms of temperature and arbitrary constants. All of these equations are based upon the so-called " quantum theory " accord- ing to which the absorption of heat energy by matter is supposed to take place in a discontinuous manner, the discrete units of energy being termed quanta. While the discussion of the quantum theory and the equations connecting heat capacity and tempera- ture lies outside of the scope of this book, mention should never- theless be made of the empirical equation derived by Nernst and Lindemann.* This equation gives values of C v which are in re- 100 200 300 T Fig. 49. 400 500 markably close agreement with the values determined by direct experiment. The equation may be written in the following form : In this expression, R is the molecular gas constant R = 2 calories, e is the base of the natural system of logarithms and is a constant depending upon the nature of the solid. * Zeit, Elektrochem., 17, 817 (1911). SOLIDS 165 The value of 6 may be calculated with a fair degree of accuracy by means of the equation fd\i (A) in which T/ denotes the absolute melting point of the substance, A its atomic weight and d its density. Liquid Crystals. In addition to possessing well-defined geo- metrical forms, crystalline substances are characterized by their resistance to deformation when subjected to mechanical stress, and by the property of melting sharply at definite temperatures with the production of transparent liquids. In 1888, two substances, cholesteryl acetate and cholesteryl benzoate, were found by Reinitzer * to behave in an anomalous manner when heated. At definite temperatures these substances melted to turbid liquids which, in turn, became clear on further heating, the latter change also taking place at definite tempera- tures. On cooling the clear liquids, the reverse series of changes was found to occur. Examination of the turbid liquids revealed the fact that they resembled ordinary liquids in their general behavior, such as assum- ing the spherical shape when suspended in a medium of the same density, or of rising in a capillary tube under the influence of sur- face tension. But in addition to possessing the properties char- acteristic of the liquid state, Lehmann discovered that they possessed optical properties which had hitherto been observed only with solid, crystalline substances. Their behavior towards polarized light was such as to warrant the conclusion that these turbid liquids are anisotropic. In view of these facts, Lehmann proposed that liquids possessing these properties should be called liquid crystals, the term implying that under ordinary condi- tions, the crystalline forces in these substances are so feeble that the crystals readily undergo deformation and actually flow like liquids. That these turbid liquids are not emulsions, is proven by the fact that when they are examined under the micro- scope, the turbidity is found to be due to the aggregation of a * Monatshefte, 9, 435 (1888). 166 THEORETICAL CHEMISTRY myriad of differently oriented transparent crystals. All subse- quent investigation of liquid crystals has failed to show any lack of homogeneity. The number of such substances known at the present time is fairly large. CHAPTER IX. SOLUTIONS. Classification of Solutions. Having dealt with the properties of pure substances in the gaseous, liquid and solid states we now proceed to the consideration of the properties of mixtures of two or more pure substances. When such a mixture is chemically and physically homogeneous, and no abrupt change in its prop- erties results from an alteration of the proportions of the com- ponents of the mixture, it is termed a solution. When one substance is dissolved in another, it is customary to designate as the solvent that component which is present in the larger proportion, the other component being termed the solute. When not more than one-tenth mol of solute is present in one liter of solution, the solution is said to be dilute. The detailed study of dilute solu- tions will be deferred until the next chapter. There are nine possible classes of solutions, as follows : (1) Solution of gas in gas; (2) Solution of liquid in gas; (3) Solution of solid in gas; (4) Solution of gas in liquid; (5) Solution of liquid in liquid; (6) Solution of solid in liquid; (7) Solution of gas in solid; (8) Solution of liquid in solid; (9) Solution of solid in solid. While examples of all of these different types of solutions are^known, only the more important classes will be considered here. Solutions of Gases in Gases. In solutions of this class the components may be present in any proportions, since gases are completely miscible. In a mixture of gases where no chemical action occurs, each gas behaves independently, the properties of 167 168 THEORETICAL CHEMISTRY the gaseous mixture being the sum of the properties of the con- stituents. Thus, the total pressure of a mixture of several gases is equal to the sum of the pressures which each gas would exert were it alone present in the volume occupied by the mixture. This law was discovered by Dalton * and is known as Dalton's law of partial pressures. If the partial pressures of the constituent gases be denoted by pi, p 2 , p 3 , etc., and P and V represent the total pres- sure and the total volume of the gaseous mixture, then PV = V (Pl + P2 + P3 + . . . ) Dalton's law holds when the partial pressures are not too great, its order of validity being the same as that of the other gas laws. Dal- ton's law can be tested experimen- tally by comparing the total pressure of the gases with the sum of the pres- sures exerted by each gas before mixture. Van't Hoff pointed out the possibility of measuring the par- tial pressure of one of the two com- ponents of a gas mixture, provided a diaphragm could be found which would be pervious to one of the gases but not to the other. It was shown shortly afterward by Ram- say,! that the walls of a vessel of palladium, when sufficiently heated, permit the free passage of hydrogen but not of nitrogen. The walls are A sketch of the apparatus used by HytLnogen Fig. 50. said to be semi-permeable. Ramsay in the verification of Dalton's law is shown in Fig. 50. A small vessel of palladium, P, containing nitrogen, is connected with a manometer AB, which serves to measure the pressure of the gas in P. The vessel P is enclosed within a larger vessel C, which can be filled with hydrogen at known pressure. On heating P and * Gilb. Ann., 12, 385 (1802). t Phil. Mag. (5), 38, 206 (1894). SOLUTIONS 169 passing a current of hydrogen at a definite pressure through C, the hydrogen enters P until the pressures due to hydrogen inside and outside are equal. The total pressure in P, measured on the manometer, is greater than the pressure in C. The difference between the two pressures is very nearly equal to the partial pres- sure of the nitrogen. Conversely, if a mixture of the two gases be introduced into P, which is then heated and maintained at suf- ficiently high temperature to insure its permeability to hydrogen, the partial pressure of the nitrogen can be determined by passing a current of hydrogen at known pressure through C until equilib- rium is attained, as shown by the manometer. The difference between the external and internal pressures is the partial pressure of the nitrogen. This experiment has a very important bearing upon the modern theory of solution. Solutions of Gases in Liquids. The solubility of gases in liquids is limited, the extent to which they dissolve depending upon the pressure, the temperature, the nature of the gas, and the nature of the solvent. When a liquid cannot absorb any more of a gas at a definite temperature, it is said to be saturated, and the solution is called a saturated solution. The solubility of a gas in a liquid is defined by Ostwald as the ratio of the volume of the gas absorbed to the volume of the absorbing liquid at a specified temperature and pressure, or if the solubility of the gas be represented by S, we have S = v/V, where v is the volume of gas absorbed and V is the volume of the absorbing liquid. The " absorption coefficient" of Bunsen in terms of which he expressed the results of his measurements of the solubility of gases, may be defined as the volume of a gas, reduced to C. and 76 cm. pressure which is absorbed by unit volume of a liquid at a certain temperature and under a pressure of 76 cm. of mercury. In certain cases the volume of the gas absorbed is found to be independent of the pressure, so that if a is the coefficient of gaseous expansion, and Bunsen's coefficient of absorption, then S = j8 (1 + at). 170 THEORETICAL CHEMISTRY The solubilities of a few gases in water and alcohol as determined by Bunsen are given in the following table : Gas. Water. Alcohol. 15 15 Hydrogen 0.0215 0.0489 1.797 0.0190 0.0342 0.1002 0.0693 0.2337 4.330 0.0673 0.2232 3.199 Oxygen Carbon dioxide The solubility of gases in water is appreciably diminished by the presence of dissolved solids or liquids, especially electrolytes. Various theories have been proposed to account for the diminished solubility of gases in salt solutions but the most satisfactory is that due to Philip,* who suggests that the phenomenon is caused by the hydration of the dissolved salt. A portion of the water in the salt solution is supposed to be in combination with the salt, the water which is thus removed from the role of solvent, being no longer free to absorb gas. The solubility of a gas increases with increase in pressure. For gases which do not react chem- ically with the solvent, there exists a simple relation between pressure and solubility, discovered by Henry, f This relation, known as Henry 1 s law may be stated as follows : When a gas is absorbed in a liquid, the weight dissolved is proportional to the pressure of the gas. Since pressure and volume, at constant temperature, are inversely proportional (Boyle's law), the law of Henry may be stated thus : The volume of a gas absorbed by a given volume of liquid is independent of the pressure. There is yet another form in which the law may be stated which is instructive in connection with the modern theory of solution. When a definite volume of liquid is saturated with a gas at constant temperature and pres- sure, a condition of equilibrium is established between the gas in solution and that in the free space over the solution, therefore, Henry's law may be stated as follows: The concentration of the dissolved gas is directly proportional to that in the free space above * Trans. Faraday Soc., 3, 140 (1907). f Gilb. Ann., 20, 147 (1805). SOLUTIONS 171 the liquid. If c\ represents the concentration of the gas in the liquid and Cz the concentration in the free space above the liquid, Henry's law may be expressed thus : ci/cz = k, where k is known as the solubility coefficient. Dalton showed that the solubility of the individual gases in a mixture of gases is directly proportional to their partial pressures, the solubility of each gas being nearly independent of the presence of the others. As will be seen from the foregoing table, the solubility of a gas in a liquid diminishes with increase in temperature. Concerning the influence of the nature of the gas on its solubility, it may be said that those gases which exhibit acid or basic reactions are the most soluble, the solubilities of neutral gases being small. In the case of many of the very soluble gases Henry's law does not hold. For example, ammonia, a gas having marked basic properties and a large coefficient of solubility, does not obey Henry's law at ordinary temperatures, the mass of ammonia absorbed not being proportional to the pressure. The curve showing the variation in solubility with pressure at C. has two marked discontinuities. At temperatures above 100 C. the gas obeys Henry's law. Sulphur dioxide behaves similarly, the law holding only for temperatures exceeding 40 C. With regard to the connection between the solvent power of a liquid and its nature but little is known. About all that can be said is, that the order of solubility of gases in different liquids is the same. Thus in the preceding table the solubilities of hydro- gen, oxygen and carbon dioxide in water and in alcohol will be seen to be approximately proportional. A slight change in volume always results when a gas is dissolved in a liquid. In general it may be said that the less compressible a gas is, the greater is the increase in volume produced when it is absorbed by a liquid. It is of interest to note that the increase in volume caused by the solution of a gas is nearly equal to the value of b for the gas in the equation of van der Waals. This is shown in the following table : 172 THEORETICAL CHEMISTRY Gas. Increase in Vol. 6 Oxygen 0.00115 0.00145 0.00106 0.00125 0.000890 0.001359 0.000887 0.000866 Nitrogen Hydrogen . Carbon dio xide Solutions of Liquids in Liquids. Solutions of liquids in liquids can be divided into three classes as follows: (1) Liquids which are miscible in all proportions; (2) Liquids which are partially miscible; and (3) Liquids which are immiscible. Examples of these three classes in the order mentioned are, alcohol and water, ether and water, and benzene and water. As to the cause of miscibility and non-miscibility of liquids very little is known. Partial Miscibility. If a small amount of ether is added to a large volume of water in a separatory funnel and the mixture vigorously shaken, a perfectly homogeneous solution will be obtained. On gradually increasing the amount of ether, shaking after each addition, a concentration will eventually be reached at which a separation into two layers will take place. The upper layer is a saturated solution of water in ether and the lower layer is a saturated solution of ether in water. So long as the relative amounts of the two liquids is such that the mixture does not become homogeneous on standing, the composition of the two layers will be independent of the relative amounts of the two components. Measurements of the mutual solubility of liquids have been made by Alexieeff * by placing weighed amounts in sealed tubes and observing the temperature at which the mixture became homogeneous. In general the solubility of a pair of partially miscible liquids increases with the temperature, and therefore it may be inferred that at a sufficiently high temperature the mixture will become perfectly homogeneous. An example of this type of binary mixture is furnished by phenol and water, the solubility curve of which is shown in Fig. 51. In this diagram temperature is plotted on the axis of ordinates and percentage composition of the solution on the axis of abscissae. Starting * Jour, prakt. Chem., 133, 518 (1882); Bull. Soc. Chem., 38, 145 (1882). SOLUTIONS 173 with a small amount of phenol and adding it in increasing quan- tities to a large volume of water, a concentration will eventually be reached at which the solution will separate into two layers. This concentration is represented by the point A. On raising the 100 # Percentage Water in Phenol Percentage Phenol in Water Fig. 51. 100 * temperature, the solubility of phenol in water increases, as shown by the curve AB. In like manner, starting with pure phenol and adding increasing amounts of water, separation into two layers will occur at a concentration represented by the point C. As the temperature is raised the solubility of water in phenol increases, as shown by the curve CB. When the temperature is raised above 68.4 C., corresponding to the point B, phenol and water become miscible in all proportions. If we start with a solution whose temperature and composition is represented by the point a, the addition of increasing amounts of phenol, at constant temperature will be represented by the dotted line afed. When the point / is reached, the solution will separate into two layers the composition of which will be inde- pendent of the relative amounts of phenol and water. At e the solution will again become homogeneous. If the solution repre- 174 THEORETICAL CHEMISTRY sented by the point a be again chosen as the starting point, and its composition be kept unaltered while the temperature is raised to a value above 68.4 C., the change will be represented by the dotted line ab. If now the temperature be maintained constant and the percentage of phenol increased, the alteration in composition will be effected without discontinuity, as represented by the dotted line be. On cooling the solution represented by the point c to the initial temperature of a, the point d will be reached. Thus it is possible to pass from a to d by the path abed without causing a separation of the components into two layers. There is an analogy between the solubility curve of a pair of partially mis- cible liquids and the dotted, parabolic curve in the diagram of the isothermals of carbon dioxide, shown in Fig. 23. In both cases there is but one phase outside of the curves, while two phases are coexistent within the area enclosed by the curves. The analogy may be traced further, since in each case only one phase can exist above a certain temperature. The temperature corre- sponding to the apex of the parabolic curve in Fig. 23, is termed the critical temperature of carbon dioxide and by analogy the temperature corresponding to the point, B, in Fig. 51 is called the critical solution temperature. The mutual solubilities of some pairs of partially miscible liquids were found by Alexieeff to di- minish with increasing temperature. Thus a mixture of ether and water, which is perfectly homogeneous at ordinary temperatures, becomes turbid on warming. A specially interesting pair of liquids is nicotine and water. At ordinary temperatures these liquids are miscible in all proportions. If the temperature is raised above 60 C., the solution becomes turbid owing to incom- plete miscibility. On continuing to heat the mixture the mutual solubility of the liquids begins to increase, until at 210 C. they become completely soluble again. The solubility relations of this binary mixture are shown in Fig. 52. The closed solubility curve defines the limits of the coexistence of two layers, all points outside of the curve representing homogeneous solu- tions. Complete Miscibility. The study of the vapor pressures of binary mixtures of completely miscible liquids is of great im- SOLUTIONS 175 portance in connection with the possibility of separating them by the process of distillation. The experimental investigations of Konowalow * on homogeneous binary mixtures of liquids have shown that such pairs of liquids may be divided into three classes 100 i Percentage Water in.Nicotine 210 # Percentage Nicotine in Water Fig. 52. 100* as follows: (1) Mixtures having a maximum vapor pressure corresponding to a certain composition, e.g., propyl alcohol and water; (2) Mixtures having a minimum vapor pressure corre- sponding to a certain composition, e.g., formic acid and water; and (3) Mixtures having vapor pressures intermediate between the vapor pressures of the pure components, e.g., methyl alcohol and water. In considering the possibility of separating binary mixtures of liquids belonging to these three classes, it is essential to determine the composition of both solution and escaping vapor. When a pure liquid is boiled the composition of the escaping vapor is the same as that of the liquid itself, but this is, in general, * Wied. Ann., 14, 34 (1881). 176 THEORETICAL CHEMISTRY not the case when a binary mixture is distilled. The composition of the liquid mixture in the distilling flask generally alters contin- uously when such a mixture is distilled. (1) The relation between the vapor pressure and composition of all possible mixtures of propyl alcohol and water is represented graphically in Fig. 53. In this diagram the compositions of the mixtures are plotted as abscissae and vapor pressures as ordinates. The vapor pressures of the pure components, water and propyl alcohol, at a definite temperature are represented by A and C. The maximum in the vapor-pressure curve corresponds to a mix- 100* Water Propyl Alcohol 8Q# 100* Fig. 53. ture containing 80 per cent of propyl alcohol. The dotted curve represents the boiling-points of the various mixtures under normal atmospheric pressure. Konowalow has shown that the vapor of a binary mixture with a minimum or maximum boiling-point has the same composition as that of the liquid. The vapor of all mixtures containing less than 80 per cent of propyl alcohol will be relatively richer in alcohol than the liquid mixture, since the vapor of propyl alcohol is quite insoluble in water. If the amount of alcohol in the mixture exceeds 80 per cent, then the vapor will be relatively richer in water. Thus, whatever may be the com- position of the mixture in the distilling flask, the distillate will approximate to the composition of the mixture having the mini- mum boiling-point. The residue in the flask will gradually SOLUTIONS ,177 change to pure water if the original concentration were below 80 per cent, or to pure alcohol if the original concentration were above 80 per cent. (2) The second type of binary mixture of liquids is illustrated by formic acid and water, the vapor pressure and boiling-point curves for which are shown in Fig. 54, A mixture containing 73 per cent of formic acid has a minimum vapor pressure and a maximum boiling-point. At this concentration the vapor and the liquid have the same composition. The vapor of mixtures con- oo f Water 760 totofl Porffiic Acid- Fig. 54. 100 1 taining less than 73 per cent of acid is relatively richer in water than the liquid, while the vapor of mixtures containing more than 73 per cent of acid contains relatively less water than the liquid. Any mixture of formic acid and water when distilled will thus leave a residue in the distilling flask containing 73 per cent of acid; this residue will distil at constant temperature like a homo- geneous liquid. It was thought for a long time that such constant boiling mixtures were definite chemical compounds of the two liquids. Thus a mixture of hydrochloric acid and water contain- ing 20.2 per cent of acid boils at 110 C. under atmospheric pres- sure. The composition of such a mixture corresponds very nearly to the formula, HC1.8 H 2 0. Roscoe * showed that these mix- * Lieb. Ann. 116, 203 (1860). 178 THEORETICAL CHEMISTRY tures are not definite chemical compounds since the composition of the distillate changes when the distillation is carried out under different pressures. (3) The vapor-pressure and boiling-point curves for methyl alcohol and water, a mixture typical of the third class of com- pletely miscible liquids, are shown in Fig. 55, the heavy line representing vapor pressures at 65.2 C. and the dotted line the boiling points under normal atmospheric pressure. In this case 100 # Water Methyl Alcohol Fig. 55. 100* the composition of both vapor and liquid alter continuously on distillation. The .distillate will contain a relatively larger amount of alcohol and the residue in the distilling flask, an excess of water. If this distillate be redistilled from a clean flask, a second dis- tillate still richer in alcohol will be obtained. By repeating this process a sufficient number of times, a more or less complete separation of the two components of the mixture can be effected. This process is termed fractional distillation. Immistibility. When two immiscible liquids are brought together, the total vapor pressure is equal to the sum of the vapor pressures of the components; hence when such a mixture is dis- tilled, the two liquids will pass over in the ratio of their respective SOLUTIONS 179 vapor pressures, the boiling-point of the mixture being the temper- ature at which the sum of the vapor pressures of the two liquids is equal to the pressure of the atmosphere. The relation between vapor pressure and composition in this case will be represented by a horizontal line drawn at a distance above the axis of abscissae equal to the sum of the vapor pressures of the components. Nitrobenzene and water may be chosen as an example of a pair of liquids which are practically immiscible. Under a pressure of 760 mm. the mixture boils at 99 C. The vapor pressure of water at this temperature is 733 mm. ; the vapor pressure of nitro- benzene must be 760 733 = 27 mm. Notwithstanding the relatively small vapor pressure of nitrobenzene in the mixture, considerable quantities of it distil over with the water. It is this fact that makes possible separations of liquids by the process of steam distillation so frequently employed by the organic chem- ist. The relative weights of water and nitrobenzene passing over in a steam distillation may be calculated as follows : The relative volumes of steam and vapor of nitrobenzene which distil over will be in the ratio of their respective vapor pressures at the temperature of the experiment, and consequently the relative weights of the two liquids which pass over will be in the ratio, Pidi : pzdz, where pi and p 2 denote the respective vapor pressures of water and nitrobenzene, and di and d 2 the corresponding vapor densities. If Wi and w 2 denote the weights of the two liquids in the state of vapor, then wi : w 2 :: pii or, since vapor density is proportional to molecular weight, we may write w ii w 2 :: piMi : p 2 M 2 . Substituting in this proportion the values given above for the vapor pressures of steam and nitrobenzene, we have W L : w 2 :: 733 X 18 : 27 X 123 or, it* :*:: 13,194: 3321. Thus the weights of water and nitrobenzene in the distillate are approximately in the ratio of 4 to 1 notwithstanding the fact that 180 THEORETICAL CHEMISTRY the ratio of their vapor pressures at the boiling-point of the mix- ture is 27 to 1. If an organic substance is not decomposed by steam, it is possible to effect an appreciable purification by steam distillation, even though its vapor pressure be relatively small. As will be seen from the above example, it is the high molecular weight of the nitrobenzene which compensates for its low vapor pressure. It is the small molecular weight of water which renders it so suitable for steam distillation. Finally, the vapor-pressure and boiling-point relations of binary mixtures of partially miscible liquids must be considered. In 100 # Water / B c\ 'A \/ \ X 1 \ \ \ / D \ \ \ / \ \ 760 mm.- / B 7 " c Of Isobutyl Alcohol 10( Fig. 56. general when two liquids are mixed, each lowers the vapor pressure of the other, so that the vapor pressure of the mixture is less than the sum of the vapor pressures of the components. As has al- ready been pointed out, the composition of the two layers in a binary mixture of partially miscible liquids is independent of the relative amounts of the components present; hence the vapor pressure remains constant so long as the solution remains hetero- geneous. The vapor-pressure and boiling-point curves for a binary mixture of partially miscible liquids, (isobutyl alcohol and water), are shown in Fig. 56. The horizontal portion BC, repre- sents the vapor pressures, at 88.5 C., of mixtures of isobutyl alcohol and water where two layers are present. -The vapor SOLUTIONS 181 pressure of the homogeneous mixtures are represented by AB and CD, AB corresponding to solutions of isobutyl alcohol in water, and CD to solutions of water in isobutyl alcohol. The dotted line A'B'C'D' represents the boiling-points of all possible mix- tures of isobutyl alcohol and water, under normal atmospheric pressure. Solutions of Solids in Liquids. The solubility of a solid in a liquid is limited and is dependent upon the temperature, the nature of the solute and the nature of the solvent. When a solvent has taken up as much of a solute as it is capable of dis- solving at a definite temperature, the solution is said to be satu- rated. There are two general methods for the preparation of saturated solutions: (1) An excess of the finely-divided solute is agitated with a known amount of the solvent, at a definite temperature, until equilibrium is attained; (2) the solvent is" heated with an excess of the solute to a temperature higher than that at which saturation is required, and then cooled in contact with the solid solute to the desired temperature. Both of these methods give equally satisfactory results provided sufficient time is allowed for the establishment of equilibrium, and provided the solid substance is always present in excess. The solubility of a solid in a liquid may be expressed as the number of grams of the solute in a given mass or volume of solvent or solution, but it is usually expressed as the number of grams of solute in 100 grams of solution. . The solubility of solids has recently been shown to be somewhat dependent upon their state of division. Thus, Hulett * has found that a saturated solution of gypsum at 25 C. contains 2.080 grams of CaS0 4 per liter, whereas when very finely divided gypsum is shaken with this solution, it is possible to increase the content of dissolved CaSCX to 2.542 grams per liter. When a saturated solution is cooled, every trace of solid solute being excluded, the excess of dissolved solid may not separate. Such a solution is said to be supersaturated. As a general rule the solubility of solids in liquids increases with the temperature, as shown in Fig. 57. Several exceptions to this rule are known, among which may be mentioned calcium * Jour. Am. Chem. Soc., 27, 49 (1905). 182 THEORETICAL CHEMISTRY hydroxide, calcium sulphate above 40 C., and sodium sulphate between the temperatures of 33 C. and 100 C. Solubility curves are usually continuous, but exceptions to this rule are common: the solubility curve of sodium sulphate fur- 100 40 60 Temperature Fig. 57. nishes an illustration. The discontinuity in the solubility curve of sodium sulphate is due to the fact that we are not dealing with one solubility curve, but with two solubility curves. At temper- atures below 33 C., the dissolved salt is in equilibrium with the decahydrate, Na 2 S04.10 H 2 0, whereas at temperatures above 33 C. the dissolved salt is in equilibrium with the anhydrous salt, Na 2 S0 4 . The solubility of Na 2 S0 4 .10H 2 O increases with the temperature, while the solubility of Na 2 S0 4 diminishes. That we are actually dealing with two solubility curves, is proved by the fact that the solubility curves of the hydrated and anhydrous salts in supersaturated solutions are continuations of the corre- sponding curves for saturated solutions, as shown by the dotted SOLUTIONS 183 curves in Fig. 57. If we select any point, such as p, lying between a dotted curve and a full curve, it is apparent that it represents a solution supersaturated with respect to Na 2 SO4.10 H 2 O, but un- saturated with respect to NaaSO^ If pure anhydrous sodium sulphate be shaken with this solution it will slowly dissolve, where- as if a trace of the hydrated salt be added, the solution will deposit NasSC^.lO H 2 0, until the amount remaining in solution corre- sponds to the solubility of the hydrate at that temperature. Supersaturated solutions of some substances can be preserved indefinitely, provided all traces of the solid phase are excluded. Such solutions are called metastable. On the other hand there are some supersaturated solutions which deposit the excess of solid solute even when all traces of it are excluded. These solu- tions are termed labile. The distinction between metastable and labile solutions is not sharp. If a metastable solution is suffi- ciently cooled, or if its concentration is sufficiently increased, it may be made to pass over into the labile condition. The concen- tration at which this transition occurs is termed the metastable limit. The stability of supersaturated solutions has recently been shown by Young * to be greatly influenced by vibrations or sudden shocks within the solution. He has been able to control the amount of overcooling in a supersaturated solution, by alter- ing the intensity of the vibrations due to the friction between glass or metal surfaces within the solution. Very little is known concerning the relation between solubility and the specific properties of solute and solvent. Owing to the fact that the change in volume resulting from the solution of a solid in a liquid is very small, the effect of pressure on the solution is almost negligible. The chief factors condition- ing the change in solubility due to increasing pressure, are the heat of solution of the solute in the nearly saturated solution, and the change in volume on solidification. Very few experiments have been made to determine the effect of pressure on solubility. Van't Hoff states that the solubility of a solution of ammonium chloride, a salt which expands when dissolved, decreases by 1 per cent for 160 atmospheres, while the solubility of copper sulphate, * Jour. Am. Chem. Soc., 33, 148 (1911). 184 THEORETICAL CHEMISTRY a salt which contracts when dissolved, increases by 3.2 per cent for 60 atmospheres. Solid Solutions. In general, when a dilute solution is suffi- ciently cooled the solvent separates in the form of crystals which are almost entirely free from the solute. When, however, the temperature of a solution of iodine in benzene is reduced to the freezing-point, the crystals which separate are found to contain iodine. Furthermore, the depression of the freezing-point of the solvent is found to be less than that calculated on the assumption that the solvent crystallizes uncontaminated with the solute. Such solutions were first studied by van't Hoff.* He found that when the concentration of such abnormal solutions is varied, the ratio of the amount of solute in the liquid solvent to the amount of solute in the solidified solvent remains constant. Thus, in solu- tions of iodine in benzene, the ratio of the concentration of iodine in the liquid to its concentration in the crystallized benzene is constant. In the following table Ci is the concentration of iodine in the liquid benzene, and Cz is the concentration of iodine in the solid benzene. Cl C 2 Cl/Cj 3.39 2.587 0.945 1.279 0.925 0.317 0.377 0.358 0.336 Van't Hoff pointed out the analogy between the distribution of the solute between the solid and liquid solvent, and the distribution of a gas between a liquid and the free space above it. In other words, the distribution follows Henry's law for the solution of a gas in a liquid. Shice the crystals containing both solute and solvent are perfectly homogeneous, van't Hoff suggested that they be regarded as solid solutions. The mixed crystals which separate from solutions of isomorphous substances being chem- ically and physically homogeneous, are to be considered as solid solutions. Many alloys possess the properties characteristic of * Zeit. phys. Chem., 5, 322 (1890). SOLUTIONS 185 solid solutions; hardened steel, for example, being regarded as a homogeneous solid solution of carbon in iron. One of the char- acteristic properties of a dissolved substance is its tendency to diffuse into the pure solvent. Interesting experiments performed by Roberts- Austen * have shown that even solids have the prop- erty of mixing by diffusion. Thus, by keeping gold and lead in contact at constant temperature for four years, he was able to detect the presence of gold in the layer of lead at a distance of 7 mm. from the surface of separation. Many other instances of diffusion in solids have been observed. f Instances of gases and liquids dissolving in solids are also known. Thus platinum, palladium, charcoal and other sub- stances have the property of taking up large volumes of hydrogen. This phenomenon, known as occlusion, is but little understood. Van't Hoff has suggested that when hydrogen dissolves in palla- dium we are really dealing with two solid solutions: one a solu- tion of hydrogen in palladium and the other a solution of palladium in solid hydrogen, the system being analogous to that of two partially miscible liquids. Certain natural silicates, the so-called zeolites, are transparent and homogeneous. Since they contain varying quantities of water they may be regarded as examples of solutions of liquids in solids. This classification is further justified by the fact that portions of the water may be removed and replaced by other substances, such as alcohol, with apparently no change in the transparency or homogeneity of the mineral. PROBLEMS. 1. 2.3 liters of hydrogen under a pressure of 78 cm. of mercury, and 5.4 liters of nitrogen at a pressure of 46 cm. were introduced into a vessel containing 3.8 liters of carbon dioxide under a pressure of 27 cm. What was the pressure of the mixture? Ans. 140 cm. of mercury. 2. Air is composed of 20.9 volumes of oxygen and 79.1 volumes of nitrogen. At 15 C. water absorbs 0.0299 volumes of oxygen and 0.0148 * Proc. Roy. Soc., 67, 101 (1900). t See Report on Diffusion in Solids, by C. H. Desch, Chem. Newg, 106, 153 (1912). 186 THEORETICAL CHEMISTRY volumes of nitrogen, the pressure of each being that of the atmosphere. Calculate the composition of the mixture of gases absorbed by the water. Ans. 34.8% by vol. of O, and 65.2% by vol. of N. 3. The vapor pressure of the immiscible liquid system, aniline-water, is 760 mm. at 98 C. The vapor pressure of water at that temperature is 707 mm. What fraction of the total weight of the distillate is aniline. Ans. 0.28. 4. The boiling-point of the immiscible liquid system, naphthalene-water, is 98 C. under a pressure of 733 mm. The vapor pressure of water at 98 C. is 707 mm. Calculate the proportion of naphthalene in the dis- tillate. Ans. 0.207. CHAPTER X. DILUTE SOLUTIONS AND OSMOTIC PRESSURE. Osmotic Pressure. In the preceding chapter reference was made to the fact that diffusion is a characteristic property of solu- tions. If a few cubic centimeters of a concentrated solution of cane sugar are placed at the bottom of a tall cylinder, and water is added, care being taken to prevent mixture, the sugar immedi- ately begins to diffuse into the water, the process continuing until the concentration of sugar is the same throughout the liquid. The sugar molecules move from a region of high concentration to a region of low concentration, the rate of diffusion being rela- tively slow owing to the viscosity of the medium. A similar process is encountered in the study of gases, but the rate of gas- eous diffusion is extremely rapid. In terms of the kinetic theory, the movement of the molecules of a gas from regions of high concentration to regions of low concentration, is to be considered as due to the pressure of the gas. By analogy, we may regard the process of diffusion in solutions as a manifestation of a driving force, known as the osmotic pressure. Semi-permeable Membranes. The use of a semi-permeable membrane for the measurement of the partial pressure of nitrogen in a mixture of nitrogen and hydrogen, has already been explained. A similar method may be employed for the measurement of the osmotic pressure of a solution, provided a suitable semi-perme- able membrane can be found. Such a membrane must prevent the passage of the molecules of solute and must be readily perme- able to the molecules of solvent; it must exert a selective action on solute and solvent. If a solution is separated from the pure solvent by a semi-permeable membrane, diffusion of the solute is no longer possible. Since equilibrium of the system can only be attained when the concentrations on both sides of the mem- brane are equal, it follows that the solvent must pass through 187 188 THEORETICAL CHEMISTRY the membrane and dilute the more concentrated solution. A number of semi-permeable membranes have been discovered which are readily permeable to water and nearly, if not entirely, impermeable to various solutes. About the middle of the eight- eenth century Abbe Nollet discovered that certain animal mem- branes are permeable to water but not to alcohol. Artificial semi-permeable membranes were first prepared by M. Traube.* If a glass tube, provided with a rubber tube and pinch-cock, be partially filled, by suction, with a solution of copper sulphate, and then immersed in a solution of potassium ferrocyanide, a thin film of copper ferrocyanide will be formed at the junction of the two solutions. When the film has once been formed, further precipitation of copper ferrocyanide will cease, the solutions on either side of the film remaining clear. Traube showed that this membrane is semi-permeable. He also showed that a number of other gelatinous precipitates possess the property of semi-permeability. A membrane formed in the above manner is easily ruptured and is wholly inadequate for quantitative or even qualitative experiments. Pfeffer f devised a method for strengthening the membrane. By depositing the precipitate in the walls of a porous clay cup, the area of unsup- ported membrane is greatly diminished and its resisting power correspondingly increased. Pfeffer directs that the cup to be used for this purpose must be throughly washed, and its walls allowed to become completely permeated with water. The cup is then filled to the top with a solution of copper sulphate, con- taining 2.5 grams per liter, and allowed to stand for several hours in a solution of potassium ferrocyanide, containing 2.1 grams per liter. The two solutions diffuse through the walls of the cup and on meeting, deposit a thin membrane of copper ferrocyanide. When precipitation is complete, the cup is thoroughly washed and soaked in water. The cup is then filled to the top with a solution of cane sugar, and a rubber stopper, fitted with a long glass tube of narrow bore, is inserted, care being taken to exclude air-bubbles. The stopper is then made fast with a suitable * Archiv. fur Anat. und Physiol., p. 87 (1867). f Osmotische Untersuchungen, Leipzig, 1877. DILUTE SOLUTIONS AND OSMOTIC PRESSURE 189 cement, and the cup completely immersed in a beaker of water. The completed apparatus is shown in Fig. 58. If the formation of the membrane has been successful, the level of the liquid in the vertical glass tube will slowly rise and will eventually attain a height of several meters. If the mem- brane is sufficiently strong and no leaks develop, the passage of water through the membrane will continue until the hydro- static pressure of the column of liquid in the tube is great enough to overcome the tendency of the water to force its way into the sugar solution. As a general rule, the membrane becomes ruptured be- fore equilibrium is attained. Measurement of Osmotic Pressure. The first direct measurements of osmotic pressure were made by Pfeffer. His ex- periments deserve brief consideration, since the results obtained furnish the basis of the modern theory of solution. The cell used was similar to that described above, but instead of employing a ver- tical glass tube as a manometer, the cup was connected, as shown in Fig. 59, with a closed mercury manometer. The sub- stitution of the closed for the open man- ometer is necessitated by the fact, that with an open manometer so much water entered the cell that the concentration of the solution became appreciably diminished, and the pressure actually measured corresponded to a solu- tion of smaller concentration than that introduced into the cell. With the closed manometer, when a trace of water has entered the cell, sufficient pressure is developed to prevent the further en- trance of more water. Pfeffer calculated that with a cell, the capacity of which was 16 cc., the volume of water entering before equilibrium was attained, did not exceed 0.14 cc. In his experi- ments, Pfeffer determined the density of the cell contents before and Fig. 58. 190 THEORETICAL CHEMISTRY after measurement of the osmotic pressure, and corrected for any change in concentration. With this apparatus he made numer- ous measurements of th osmotic pressures of different solutions, the entire apparatus being immersed in a constant-temperature bath. With solutions of cane sugar he obtained the results given in the accompany- ing table, where C denotes the per- centage concentration of the solution, and P the corresponding osmotic pressure, expressed in centimeters of mercury. The temperature varied from 13.5 C. to 14.7 C. C P P/C 1 53.5 53.5 2 101.6 50.8 4 208.2 52.0 6 307.5 51.2 It is evident from these results, that the osmotic pressure is propor- tional to the concentration of the solution, since P/C is approximately constant. The deviations from con- stancy in the ratio of pressure to concentration may be ascribed to experimental errors, since the dif- ficulties involved in these measure- ments are very great. Pfeffer also Fig. 59. studied the influence of temperature on osmotic pressure, and showed that as the temperature is raised the pressure increases. The following table gives his results for a 1 per cent solution of cane sugar. DILUTE SOLUTIONS AND OSMOTIC PRESSURE 191 Temperature. Osmotic Pressure. 6. 8 cm. 50.5 13. 2 52.1 14. 2 53.1 22. 54.8 36. 56.7 Osmotic Pressure and the Nature of the Membrane. Pfeffer also studied the effect of the nature of the membrane on osmotic pressure. In addition to copper ferrocyanide, he used membranes of calcium phosphate and Prussian blue. His results seemed to indicate that the magnitude of the osmotic pressure developed, was dependent upon the nature of the membrane used. The variations observed have since been shown to have been due to leakage of the calcium phosphate and Prussian blue mem- branes, the copper ferrocyanide membrane being the only one which was capable of withstanding the pressure. Ostwald * has devised an ingenious theoretical demonstration of the fact that osmotic pressure must be independent of the nature of the mem- brane employed in measuring it. Let A and B, in Fig. 60, repre- A B Fig. 60. sent two different semi-permeable membranes placed in a glass tube of wide bore. Let us imagine the space between the two membranes to be filled with a solution, and the tube immersed in a vessel of water. If the osmotic pressures developed at A and B are pi and p 2 respectively, and p 2 is less than pi, then water will pass through A until the pressure pi is reached. Since the pressure at B only reaches the value p 2 , however, the pres- sure pi can never be attained, and a steady stream of water from A to B, under the pressure p\ pz, will result. This, however, would be a perpetual motion, and since this is impossible, the osmotic pressures at the two membranes must be the same. * Lehrb. d. allg. Chem., I., p. 662.. 192 THEORETICAL CHEMISTRY Theoretical Value of Osmotic Pressure. The physico-chem- ical significance of Pfeffer's results was first perceived by van't Hoff.* In a remarkably brilliant paper, he pointed out the existence of a striking parallelism between the properties of gases and the properties of dissolved substances. We have already called attention to the analogy between osmotic pressure and gas pressure: we now proceed to trace the connec- tion between osmotic pressure, volume and temperature, as first pointed out by van't Hoff. Pfeffer's experiments showed that at constant temperature, the ratio, P/C, is constant for any one solute. Since the concentration varies inversely as the volume in which a definite amount of solute is dissolved, we obtain, by substituting l/V for C, the equation, PV = constant, which is plainly the analogue of the familiar equation of Boyle for gases. An examination of Pfeffer's data for osmotic pressures at differ- ent temperatures, convinced van't Hoff that the law of Gay- Lussac is also applicable to solutions. In the following table, the osmotic pressures in atmospheres for a 1 per cent solution of cane sugar at different temperatures are recorded, together with the pressures calculated on the assump- tion that the osmotic pressure is directly proportional to the absolute temperature. Temperature. P (obs.). P (calc.). 6. 8 0.664 0.665 13. 7 0.691 0.681 15. 5 0.684 0.686 22. 0.721 0.701 32. 0.716 0.725 36. 0.746 0.735 Since the laws of Boyle and Gay-Lussac are both applicable, we may write an equation for dilute solutions corresponding to that already derived for gases, or PV = R'T, in which P is the osmotic pressure of a solution containing a defi- * Zeit. phys. Chem., i, 481 (1887). DILUTE SOLUTIONS AND OSMOTIC PRESSURE 193 nite weight of solute in the volume, V, of solution, T being the absolute temperature of the solution and R r a constant corre- sponding to the molecular gas constant. The molecular gas constant R has already been evaluated and has been found to be equal to 0.0821 liter-atmosphere. Making use of Pfeffer's data, van't Hoff calculated the value of R' in the above equation, in the following manner : the osmotic pressure of a 1 per cent solution of cane sugar at C. is 0.649 atmosphere, and since the concentration of the solution is 1 per cent, the volume of solution containing 1 mol of sugar, will be 34,200 cc. or 34.2 liters. Substituting these values in the equa- tion, we have D , PV 0.649 X 34.2 R' = -- = 273 = 0-0 81 3 liter-atmos., a value which is nearly the same as that of the molecular gas constant, R. The equality of R and R' leads to a conclusion of the greatest importance, as was pointed out by van't Hoff, viz., " the osmotic pressure exerted by any substance in solution is the same as it would exert if present as a gas in the same volume as that occupied by the solution, provided that the solution is so dilute that the volume occupied by the solute is negligible in comparison with that occupied by the solvent. " It should be remembered that we are not justified in concluding from this proposition of van't Hoff, that osmotic pressure and gaseous pressure have a common origin. While the origin of osmotic pressure may be kinetic, it is also conceivable that it may result from the mutual attraction of solvent and solute, or that it may bear some relation to the surface ten- sion of the solution. Up to the present time no wholly satis- factory explanation of the cause of osmotic pressure has been advanced. Just as 1 mol of gas at C. and 760 mm. pressure occupies a volume of 22.4 liters, so when 1 mol of a substance is dissolved and the solution diluted to 22.4 liters at C., it will exert an osmotic pressure of 1 atmosphere. In other words, molar weights, or quantities proportional to molar weights, of different substances, when dissolved in equal volumes of the same solvent exert the same 194 THEORETICAL CHEMISTRY osmotic pressure. If we deal with n mols of solute instead of 1 mol, the general equation becomes PV = nRT. But n = g/M, where g is the number of grams of solute per liter, and M is its molecular weight. Substituting in the preceding equation, we have PV = g/M-RT, or, M- gET "PV Since P, V, g, R and T are all known, M can be calculated. The direct measurement of the osmotic pressure of a solution does not afford a practical method for the determination of the molecular weight of dissolved substances, because of the experimental difficulties involved and the time required for the establishment of equilibrium. There are other and simpler methods for deter- mining molecular weights in solution, based upon certain proper- ties of solutions which are proportional to their respective osmotic pressures. Recent Work on the Direct Measurement of Osmotic Pressure. It is only within the past decade that the investigations of Pfeffer have been confirmed and extended by elaborate and systematic experiments on the direct measurement of osmotic pressure. Morse and his co-workers,* while employing a method essentially the same as that of Pfeffer, have, as the result of much patient labor, brought the apparatus to such a high state of perfection, that the experimental errors are now estimated to affect only the second place of decimals in the numerical data expressing osmotic pressures in atmospheres. The most important of the improvements introduced by Morse are the following: (1) the improvement of the quality of the membrane; (2) the improve- ment of the connection between the cell and the manometer, and (3) the improvement of the means of accurately measuring * Am. Chem. Jour., 26, 80 (1901); 34, 1 (1905); 36, 1, 39 (1906); 37, 324, 425, 558 (1907); 38, 175 (1907); 39, 667 (1908); 40, 1, 194 (1908); 41, 1, 257 (1909). DILUTE SOLUTIONS AND OSMOTIC PRESSURE 195 the pressure. The membrane of copper ferrocyanide is deposited electrolyti- cally. After thorough washing and soaking in water, the porous cup, made from specially prepared clay, is filled with a solution of potassium ferrocy- anide and immersed in a solution of cop- per sulphate. An electric current is then passed from a copper electrode in the solution of copper sulphate, to a platinum electrode immersed in the solution of potassium ferrocyanide. This drives the copper and ferrocy- anide ions toward each other, and the membrane of copper ferrocyanide is thus formed in the walls of the cup. The passage of the current is continued until the electrical resistance reaches a value of about 100,000 ohms. The cell is then rinsed, and soaked in water for several hours, and then the electro- lytic treatment is repeated until the electrical resistance attains a maximum value. A solution of cane sugar is now introduced into the cell, which is con- nected with the manometer and im- mersed in water. When the pressure has attained its maximum value, the apparatus is dismantled and the cell, after thorough washing and soaking in water, is again subjected to the electro- lytic process of membrane forming. In this way the weak places in the mem- brane which may have yielded to the high pressure, can be repaired, and by Fig. 61. continued repetition of this treatment the membrane can ultimately be brought to its maximum power 196 THEORETICAL CHEMISTRY of resistance. A sketch of the Morse apparatus is shown in Fig. 61. A description of the details of this apparatus lies beyond the scope of this book. The results of the work of Morse and his students are of the highest importance. The osmotic pressures of solutions of cane sugar and dextrose have been shown to be proportional to the respective concentrations, provided the con- centration is referred to unit volume of solvent instead of unit volume of solution. Thus in their experiments, the solutions were made up containing from 0.1 to 1.0 mol of solute in 1000 grams of water. Morse calls such solutions weight-normal solutions in con- trast to volume-normal solutions, in which 1 mol or a fraction of a mol of solute is dissolved in water and the solution diluted to 1 liter. The following data taken from the work of Morse, shows that when concentration is expressed on the weight-normal basis, there is direct proportionality between osmotic pressure and con- centration. The figures refer to solutions of dextrose at 10 C. Molar Concentration. Osmotic Pressure. Per 1000 gm. Water. Per Liter of Solution. In Atmos. Relative to First as Unity. 0.1 0.099 2.39 1.00 0.2 0.196 4.76 1.99 0.5 0.474 11.91 4.98 1.0 0.901 23.80 9.96 Morse and his co-workers also conclude from their experiments at temperatures ranging from C. to 25 C., that the temper- ature coefficients of osmotic pressure and gas pressure are prac- tically identical. In other words, their results confirm the conclusions of van't Hoff, that the law of Gay-Lussac is applicable to solutions. The results of the experiments of Morse are of special interest in connection with the proposition of van't Hoff, that the osmotic pressure of a dilute solution is the same as that which the solute would exert if it were gasified at the same temper- ature and occupied the same volume as the solution. The data in the following table is taken from the work of Morse on solutions DILUTE SOLUTIONS AND OSMOTIC PRESSURE 197 of cane sugar at 15 C. In addition to the observed osmotic pressures, the table contains the corresponding gas pressures, calculated (1) on the assumption that the solute when gasified occupies the same volume as the solution (proposition of Van't Hoff), and (2) on the assumption that it occupies the same volume as the solvent alone. Molar Concentration. Osmotic Pressure in Atmos. Per 1000 gm. of Water. Per Liter of Solution. Obs. Calc. (a). Calc. (b). 0.1 0.098 2.48 2.30 2.35 0.2 0.192 4.91 4.51 4.70 0.4 0.369 9.78 8.67 9.40 0.6 0.533 14.86 12.51 14.08 0.8 0.684 20.07 16.07 18.79 1.0 0.825 25.40 19.38 23.49 The calculated pressures, recorded in the last column, are in much closer agreement with the observed osmotic pressures, than are the calculated pressures, recorded in the fourth column of the table. The proposition of van't Hoff should then be modified to read as follows : A dissolved substance in dilute solution exerts an osmotic pressure equal to that which it would exert if it were gas- ified at the same temperature, and the volume of the gas were reduced to that of the solvent in the pure state. The investigations of Morse and his co-workers may be summarized thus: (1) the law of Boyle is applicable to dilute solutions, provided the concentration is referred to 1000 grams of solvent and not to 1 liter of solution; (2) the law of Gay-Lussac is also applicable to dilute solutions, that is, the temperature coefficients of osmotic pressure and gas pressure are equal, and (3) the small departures from the theo- retical values of the osmotic pressures may be traced to hydration of the solute. Direct measurements of the osmotic pressure of concentrated solutions of cane sugar, dextrose and mannite have been made by the Earl of Berkeley and E. G. J. Hartley.* The method em- * Proc. Roy. Soc., 73, 436 (1904); Trans. Roy. Soc. A., 206, 481 (1906). 198 THEORETICAL CHEMISTRY ployed by these investigators is slightly different from that of Pfeffer or Morse; the tendency of water to pass through the semi-permeable membrane is offset by the application of a counter pressure to the solution. A membrane of copper ferrocyanide is deposited electrolytically very near the outer surface of a tube of porous porcelain. This tube is placed co-axially within a large cylindrical vessel of gun metal, an absolutely tight joint between the two being secured by an ingenious system of dermatine rings and clamps. The open ends of the porcelain tube are closed by rubber stoppers fitted with capillary tubes bent at right angles, one of the latter being provided with a glass stop-cock. When a determination of osmotic pressure is to be made, the apparatus is placed in a horizontal position and water is introduced into the porcelain tube, completely filling it and the connecting capillary tubes up to a certain level. The gun metal vessel is then filled with the solution, and connected with an auxiliary apparatus by means of which a gradually increasing hydrostatic pressure can be applied. If no pressure is applied to the solution, water will pass through the semi-permeable membrane into the solution, and the level of the water in the capillary tubes will fall. In carrying out a measurement, therefore, as soon as the solution is introduced into the gun metal vessel, hydrostatic pressure is applied, the mag- nitude of the pressure being so adjusted as to counterbalance the osmotic pressure of the solution. The level of the water in the capillary tubes serves to indicate the relative magnitudes of the osmotic and hydrostatic pressures. When the level of the water in the capillary tubes remains constant, the two pressures are in equilibrium. The following are the values of the equilibrium pressures of solutions of cane sugar, dextrose and mannite at C. It must be remembered that when the two pressures are in equi- librium, there is always a pressure of one atmosphere on the solvent. As will be seen the pressures developed in the more concen- trated solutions are enormous and it is a surprising fact, that even in cases where the highest pressures were measured, hardly a trace of sugar was found in the pure solvent, the membrane retaining its property of semi-permeability throughout the entire range of pressures. The figures in the third column are calculated DILUTE SOLUTIONS AND OSMOTIC PRESSURE 199 CANE SUGAR. Osmotic Pressure in Atmospheres. Cono. gm. per Liter. Obs. Calc. 180.1 13.95 13.95 300.2 26.77 28.74 420.3 43.97 32.55 540.4 67.51 41.85 660.5 100.78 51.16 750.6 133.74 58.14 DEXTROSE. Osmotic Pressure in Atmospheres. Cone. gm. per Liter. Obs. Calc. 99.8 199.5 319.2 448.6 548.6 13.21 29.17 53.19 87.87 121 . 18 13.21 26.41 42.25 59.28 72.61 MANNITE. Osmotic Pressure in Atmospheres. Cone. gm. per Liter. Obs. Calc. 100 110 125 13.1 14.6 16.7 13.1 14.4 16.4 on the assumption that there is direct proportionality between osmotic pressure and concentration. It is apparent that in every case the observed osmotic pressure is greater than the calculated. Even when the concentrations are expressed on the weight-normal 200 THEORETICAL CHEMISTRY basis, as recommended by Morse, the osmotic pressure increases more rapidly than the concentration. This is well shown in the accompanying diagram, Fig. 62, due to the Earl of Berkeley. In this diagram, the osmotic pressures of solutions of cane sugar are plotted against concentrations, curve A representing the actually observed osmotic pressures; curve C 120 100 200 300 400 500 600 700 Grams Cane Sugar per liter of Solution Fig. 62. being traced on the assumption that osmotic pressure may be calculated from the equation, PV = RT, where V denotes the volume of solvent containing 1 mol of cane sugar; and curve B, a straight line, being drawn on the assumption that osmotic pressure may be calculated from the equation, PV = RT, where V represents the volume of solution containing 1 mol. While the theoretical and observed values of the osmotic pres- DILUTE SOLUTIONS AND OSMOTIC PRESSURE 201 sure are approximately equal in the more dilute solutions, it is obvious that the observed values of the osmotic pressure of the concentrated solutions are always greater than the calculated values, even when the calculation is made on the assumption that V in the equation, PV = RT, is the volume of the solvent. The abnormally high osmotic pressures observed by the Earl of Berkeley have been discussed by Callendar * who suggests hydra- tion of the solute as a probable cause. He shows, that if 5 molecules of water are assumed to be asso- ciated with each molecule of cane sugar in the most concentrated solutions studied by the Earl of Berkeley, the discrepancy between the observed and calculated values of the osmotic pressure dis- appears. Comparison of Osmotic Pressures. Although the difficulties involved in the direct determination of osmotic pressure are many, these can be avoided by the employment of one of several indirect methods which have been devised for the comparison of osmotic pressures. All of these methods depend upon the exchange of water which occurs when two solutions are separated by a semi-permeable membrane. The movement of the water will always be in such a direction as to tend to equalize the osmotic pressures on opposite sides of the membrane, or, in other words, the transfer of water will take place from the solution with the lesser osmotic pressure to the solution with the greater osmotic pressure. The Plasmolytic Method. In this method, solutions of various substances are prepared, the concentration of each being such that its osmotic pressure is the same as that of a particular plant cell. Obviously the osmotic pressures of all of these solutions must be equal : such solutions are said to be isotonic or isosmotic. The plasmolytic method for the comparison of osmotic pressures was developed by the Dutch botanist, De Vries.f This method depends upon the shrinking or swelling of the protoplasmic sac of plant cells when they are immersed in a solution whose osmotic * Proc. Roy. Soc. A., 80, 466 (1908). f Jahrb. wiss. Botanik., 14, 427 (1884); Zeit. phys. Chem., 2, 415 (1888); 3, 103 (1889). 202 THEORETICAL CHEMISTRY pressure differs from that of their own sap. De Vries found that the cells of Tradescantia discolor, Curcuma rubricaulis, and Begonia manicata fulfil the necessary conditions, viz.; the cell walls are strong and resist alteration when immersed in solutions, the cells are readily permeable to water, and the cell contents are colored, thus enabling the slightest contraction or expansion to be de- tected. The "cell walls are lined on the inside with a thin, elastic, semi-permeable membrane which encloses the colored contents of the cell. The content of the cell consists of an aqueous solu- tion of several substances, among which may be mentioned glucose, potassium and calcium malate, together with coloring matter. The osmotic pressure of the cell contents ranges from four to six atmospheres. The semi-permeable membrane expands when the contents of the cell increases and contracts when the contents diminishes. In making a comparison of osmotic pres- sures by this method, tangential sections are cut from the under side of the mid-rib of the leaf of one of the above plants, e.g., Tradescantia discolor, and are placed in the solution whose osmotic pressure it is desired to compare with that of the cell contents. The cells are then observed under the microscope, any decrease in pressure below the normal resulting in a detachment of the semi-permeable membrane from one or more points of the cell wall. This contraction always occurs when the cells are im- mersed in a concentrated solution, the phenomenon being termed plasmolysis. When the solution in which the cells are placed has a lower osmotic pressure than the cell contents, no visible effect is produced, the increased pressure within the cell simply forcing the membrane closer to the rigid cell walls. By starting with a concentrated solution, the osmotic pressure of which is greater than that of the cell, and gradually diluting it, a concen- tration will ultimately be reached at which the elastic membrane will just completely fill the cell. This solution is isotonic with the cell contents. In this method the very reasonable assump- tion is made that all of the cells have the same osmotic pressure, any differences which might have existed having equalized them- selves in the living plant. The microscopic appearance of cells of Tradescantia discolor when immersed in different solutions is DILUTE SOLUTIONS AND OSMOTIC PRESSURE 203 shown in Fig. 63. The appearance of the normal cell when im- mersed in water or in a solution whose osmotic pressure is less than that of the cell contents, is shown in A . When the cell is immersed in a 0.22 molar solution of cane sugar it appears as in B, this solution having a greater osmotic pressure than the cell contents. When the cell is immersed in a molar solution of Fig. 63. potassium nitrate, there is marked plasmolysis, as shown in C. De Vries determined the concentrations of a large number of solutions which were isotonic with the cell contents. He ex- pressed his results in terms of the isotonic coefficient, which he defined as the reciprocal of the molar concentration. The iso- tonic coefficient of potassium nitrate was taken equal to 3. A few of De Vries' results are given in the following table. Substance. Formula. Isotonic Coefficient. Glycerol . C 3 H 8 O 8 1.78 Glucose C 6 H 12 O 6 1.81 Cane sugar 1.88 Malic acid C 4 H 6 O 5 1.98 Tartaric acid C 4 H 6 O 6 2.02 Citric acid C 6 H 8 O 7 2.02 Potassium nitrate KNO 3 3.00 Magnesium chloride MgCl 2 4.33 204 THEORETICAL CHEMISTRY De Vries applied the plasmolytic method to the determination of the molecular weight of raffinose. At that time there was considerable uncertainty as to the correct formula of crystallized ramnose, three different formulas, all consistent with the results of analysis, having been proposed as follows: CisH^Oie- 5 H 2 O, Ci2H 22 Oii.3 H 2 O, and C 3 6H 64 O 3 2. De Vries found that a 3.42 per cent solution of cane sugar was isotonic with a 5.96 per cent solution of ramnose. Letting the unknown molecular weight of ramnose be represented by M, then 3.42 :5.96 ::342:M. Solving the proportion we have, M = 596. This result has since been confirmed by purely chemical methods, and the formula, Ci8H 32 Oi 6 .5 H 2 0, the molecular weight, of which is 594, is thus established. The Blood Corpuscle Method. The red blood corpuscle is a cell, the contents of which is enclosed by a thin elastic semi- permeable membrane. Unlike the plant cells, there is no resistant cell wall to give support to the membrane, so that when red blood corpuscles are immersed in water they at first swell, owing to the osmotic pressure developed, and finally burst. When the mem- brane is ruptured, the coloring matter of the cell, the haemoglobin, escapes and the water acquires a deep red color. Advantage of this behavior of red blood corpuscles was taken by Hamburger * for the comparison of osmotic pressures. He found that when a 1.04 per cent solution of potassium nitrate is added to the defibrinated blood of a bullock, the corpuscles will settle completely to the bottom, while the supernatant liquid will remain clear. On the other hand, if a 0.96 per cent solution of potassium nitrate is used, the corpuscles will not settle and the supernatant liquid becomes colored. If more dilute solutions of potassium nitrate are used, the solution acquires a still deeper color. By careful adjustment, a concentration of potassium nitrate can be found in which the red blood corpuscles will just settle. In like manner, the concentration of solutions of other substances can be so adjusted as to cause the precipitation of the * Zeit. phys. Chem., 6, 319 (1890). DILUTE SOLUTIONS AND OSMOTIC PRESSURE 205 corpuscles. These solutions are isotonic. Without going into details, it may be said that the isotonic coefficients obtained by Hamburger, agree well with those obtained by the plasmolytic method. The Haematocrit Method. In this method developed by Hedin,* advantage is again taken of the properties of red blood corpuscles. As has already been stated, when red blood cor- puscles are immersed in solutions of gradually diminishing concentration of the same solute, they continue to swell and ulti- mately the semi-permeable envelope bursts. On the other hand, when the corpuscles are immersed in solutions of gradually increas- ing concentration, they shrink, owing to the transfer of water from the corpuscles. It is apparent that there must be a certain concentration for each solute which will cause no change in the volume of the corpuscles. To determine this concentration, use is made of an instrument known as an hcematocrit. This is simply a graduated thermometer-tube which may be attached to the spindle of a centrifugal machine. When the spindle is re- volved at high speed, the corpuscles collect in the bottom of the graduated tube. A measured volume of blood is centrifuged until no further shrinkage in volume of the corpuscles can be detected in the hsematocrit. The same volume of blood is then added to each of a series of solutions whose concentration dimin- ishes progressively, and the volume of the corpuscles is determined as in pure blood. In this way the concentration of the solution is found, in which the volume of the corpuscles is the same as in the undiluted blood. By proceeding in a similar manner with solutions of different substances, a series of isotonic coefficients can be determined. The following table gives a comparison of the isotonic coefficients of various substances obtained by the plasmolytic, blood corpuscle and hsematocrit methods. The iso- tonic coefficients are referred to that of cane sugar as unity. There are other methods which may be used for the comparison of osmotic pressures, among which may be mentioned that due to Wladimiroff,f involving the use of bacteria, and the interesting * Ibid., 17, 164 (1895). * t Zeit. phys. Chem., 7, 529 (1891). 206 THEORETICAL CHEMISTRY Substance. Plasmolytic Method. Corpuscle Method. Hsematocrit Method. 1.00 1.00 1.00 MgSO 4 1.09 1.27 1.10 KNO 3 1 67 1 74 1 84 NaCl 1.69 1.75 1 74 CHs.COOK 1.67 1.66 1.67 CaCl 2 2.40 2.36 2.33 method developed by Tammann,* in which artificially prepared membranes are employed. Osmotic Pressure and Diffusion. That there is a very close connection between osmotic pressure and diffusion, has already been pointed out. In fact the osmotic pressure of a solution may be regarded as the driving force which causes the molecules of a dissolved substance to distribute themselves uniformly through- out the solution. The process of diffusion was first systematically investigated by Graham f in 1850, but it was not until five years later that the general law of diffusion was enunciated by Fick.| He proved theoretically and experimentally that the quantity of solute, ds, which diffuses through an area A, in a time dt, when the concen- tration changes by an amount dc, in a distance dx, at right angles to the plane of A, is given by the equation ds= -DA^-dt, dx in which D is a constant, known as the coefficient of diffusion. Interpreting the equation of Fick in words, we see that the coeffi- cient, of diffusion is the amount of solute which will cross 1 square centimeter in 1 second, if the change of concentration per centi- meter is unity. The phenomena of diffusion have also been investigated by * Wied. Ann., 34, 299 (1888). t Phil. Trans. (1850), p. 1, 805; (1851), p. 483. J Pogg. Ann., 94, 59 (1855). DILUTE SOLUTIONS AND OSMOTIC PRESSURE 207 Nernst * and Planck, f If we have a tall cylindrical vessel con- taining a solution of a non-electrolyte in its lower part, and pure water at the top, the solute will slowly diffuse upward into the water. Assuming the osmotic pressure at a height x, to be P, and letting A denote the area of cross-section of the cylinder, the solute in the layer whose volume is A dx, will be subjected to a force equal to A dP, the negative sign indicating that the force acts in the direction of diminishing pressure. If c is the concentration in mols per cubic centimeter, the force acting on each molecule in this layer will be - A.m = __ l.^L cA dx c dx Let F denote the force necessary to drive a single molecule through the solution with the velocity of one centimeter per second. Since the velocity is constant, the resistance due to the viscosity of the medium must also be denoted by F. The velocity attained will be JL &P_ cF ' dx ' If dN represents the number of molecules crossing each layer in a time dt, then, since the number crossing unit area per second is proportional to the concentration and to the mean velocity of the molecules, we shall have dN = - ~-~Acdt y cF dx or,. , A7 1 , dP ,. dN= ~F A Tx dL When the solution is dilute, we may apply the general equation, PV = RT, remembering that V = 1/c. Substituting in the pre- ceding equation, we have , A , RT dc .. dN = - -=r A -j- dt. F dx * Zeit. phys. Chem., 2, 40, 615 (1888); 4,129 (1889). t Wied. Ann., 40, 561 (1890). 208 THEORETICAL CHEMISTRY Comparing this equation with that of Fick, we see that the coefficient of diffusion D, corresponds to the factor, RT/F. From the equation of Nernst it is possible to calculate the force required to drive a molecule of solute through the solution with unit veloc- ity. Thus, solving the above equation for F, we have A , F = - A-j-dt. dN dx By means of this equation, it has been calculated that the force necessary to drive one molecule of formic acid through water with a velocity of one centimeter per second at C. is equal to the weight of 4,340,000,000 kilograms. It is difficult at first to realize that such enormous forces are operative in solutions, but when one considers the minute size of the molecules and the great resistance offered by the medium, it becomes evident that a very large driving force must be applied to produce an appreciable movement of the solute through the solution. Principle of Soret. If a solution is maintained at a uniform temperature it will ultimately become homogeneous; if, on the other hand, two parts of a homogeneous solution are kept at different temperatures for some time, the solution will become more concentrated in the colder portion. This phenomenon was first investigated by Soret.* The experiments of Soret are of special interest, since they furnish a means of determining the influence of temperature on osmotic pressure. Thus, if the law of Gay-Lussac holds for osmotic pressure, the colder portion of a solution should increase in concentration by 1/273 for each degree of difference in temperature. The experimental results are in satisfactory agreement with the requirements of theory, and con- stitute another proof of the applicability of the gas laws to dilute solutions. Lowering of Vapor Pressure. It has long been known that the vapor pressure of a solution is less than that of the pure sol- vent, provided the solute is non-volatile. The investigations of von Babo and Wiillner f on the lowering of vapor pressure of * Ann. Chem. Phys. (5), 22, 293 (1881). t Pogg. Ann., 103, 529 (1858). DILUTE SOLUTIONS AND OSMOTIC PRESSURE 209 various liquids when non-volatile substances are dissolved in them, resulted in the following generalizations: (1) The lower- ing of the vapor pressure of a solution is proportional to the amount of solute present ; and (2) For the same solution, the lowering of the vapor pressure of the solvent by a non-volatile solute is at all tempera- tures a constant fraction of the vapor pressure of the pure solvent. In 1887, Raoult,* as the result of an exhaustive experimental investigation, enunciated the following laws: (1) When equi- molecular quantities of different non-volatile solutes are dissolved in equal volumes of the same solvent, the vapor pressure of the solvent is lowered by a constant amount] and (2) The ratio of the observed lowering of the vapor pressure to the vapor pressure of the pure sol- vent is equal to the ratio of the number of mols of solute to the total number of mols in the solution. The ratio of the observed lowering to the original vapor pressure is called the relative lowering of the vapor pressure. Letting p\ amd pz denote the vapor pressures of solvent and solution, Raoult's second law may be put in the form Pi- P2 = n pi N + n' in which n and N represent the number of mols of solute and solvent respectively. Some of Raoult's results for ethereal solu- tions are given in the accompanying table. Mols of Solute PI Pt PI Substance. per 100 mols of Solution. for Solution. for 1 molar per cent Solution. Turpentine 8 95 0885 0099 Methyl salicylic acid 2 91 026 0089 Methyl benzoic acid Benzoic acid 9.60 7 175 0.091 070 0.0095 0097 Trichloracetic acid 11 41 120 0105 Aniline . 7 66 081 0106 The results given in the fourth column of the table are nearly * Compt. rend., 104, 1430 (1887); Zeit. phys. Chem., 2, 372 (1888); Ann. Chem. Phys. (6), 15, 375 (1888). 210 THEORETICAL CHEMISTRY constant, and are in close agreement with the theoretical value of the relative lowering of a 1 molar per cent solution calculated as follows: ^ - = Tnff4T7 = - 0099 - pi When the solution is very dilute, the number of mols of solute is negligible in comparison with the number of mols of solvent, and the equation of Raoult may be written Pi - P2 = _rc Pi N' Since n = g/m, and N = W/M, where g and W are the weights of solute and solvent respectively, and m and M are the correspond- ing molecular weights, the above equation becomes Pi- P2 _ gM pi Wm' This equation enables us to calculate the molecular weight of a dissolved substance from the relative lowering of the vapor pressure produced by the solution of a known weight of solute in a known weight of solvent. Solving the equation for m, we have Pl W PI- Pz As an illustration of the application of this equation, we may take the determination of the molecular weight of ethyl benzoate from the following experimental data: The vapor pressure at 80 C. of a solution of 2.47 grams of ethyl benzoate in 100 grams of benzene is 742.6 mm.: the vapor pressure of pure benzene at 80 C. is 751.86 mm. Substituting in the equation, we have = 2.47 X 78 751.86 100 ' 751.86 - 742.6 ~ The molecular weight calculated from the formula, C6H 5 COO.C 2 H5, is 150. The difficulties which attend the accurate measurement of the vapor pressure of a solution by the static method have already DILUTE SOLUTIONS AND OSMOTIC PRESSURE 211 been mentioned. While there are other methods which are pref- erable for the determination of the molecular weight of dissolved substances, the vapor pressure method has one marked advan- tage, it can be used for the same solution at widely divergent temperatures. The method devised by Walker and already described in connection with the determination of the vapor pressure of pure liquids (p. 92) is well adapted to the measure- ment of the vapor pressure of solutions. Connection between Lowering of Vapor Pressure and Osmotic Pressure. The relation between osmotic pressure and the lowering of vapor pressure has been derived in the following manner by Arrhenius.* Imagine a very dilute solution contained in the wide glass tube A, Fig. 64. The tube, A, is closed at its lower end with a semi- permeable membrane, and dips into a vessel, B, which contains the pure solvent. The entire apparatus is covered by a bell-jar C, and the enclosed space exhausted. Let h be the difference in level between the solvent and solution when equilibrium is established, that is, when the hydrostatic pressure of the column of liquid is equal to the osmotic pressure. When equilibrium is _ attained, the vapor pressure of the solution at .h will be equal to the vapor pressure of the solvent at this height. If the vapor pressure of the pure sol- vent is pi, and p 2 is the vapor pressure of the solution at h, we shall have Pi Pz = hd, (1) where d denotes the density of the vapor. Let v be the volume of 1 mol of solvent in the state of vapor, then Piv = RT, and RT v f T^J^^-- Fig. 64. Zeit. phys. Chem., 3, 115 (1889). 212 THEORETICAL CHEMISTRY If the molecular weight of the solvent is M, we may replace v by M/d, when the preceding equation becomes pi The solution being very dilute the osmotic pressure may be cal- culated from the equation PV = nRT, where P is the osmotic pressure of the solution, V the volume of the solution containing 1 mol of solute, and n the number of mols of solute present. If s represents the density of the solvent and also of the solution, since it is very dilute, we may write P = hs, and F-2, S where g is the number of grams of the solvent in which the n mols of solute are dissolved. Substituting these values of P and V in the general equation, we have PV = nRT = hg, and solving for h, .--?. Substituting the values of d and h, given hi equations (2) and (3), in equation (1) we have Rearranging equation (4), and remembering that N = g/M, we obtain Pi "ft _ n Pi ~N' This equation it will be seen, is identical with that derived experi- mentally by Raoult for very dilute solutions. DILUTE SOLUTIONS AND OSMOTIC PRESSURE 213 Van't Hoff showed, by an application of thermodynamics to dilute solutions, that the relation between osmotic pressure and the relative lowering of the vapor pressure is expressed by the equation Pi - Pz = MP pi sRT' in which the symbols have the same significance as above. This equation may be reconciled easily with the equation of Raoult. If n in equation (5) be replaced by its equal, PV/RT, the equation becomes Pi - pa pi NET' But V = NM/s, hence Pi - Pz = MP f pi sRT* This equation shows that the relative lowering of the vapor pressure is directly proportional to the osmotic pressure. Elevation of the Boiling-Point. Just as the vapor pressure of a solution is less than that of the pure solvent, so the boiling-point of a solution is correspondingly higher than the boiling-point of the solvent. It follows that when equimolecular quantities of different substances are dissolved hi equal volumes of the same solvent, the elevation of the boiling-point is constant. Thus, the molecular weight of any soluble substance may be determined by comparing its effect on the boiling-point of a particular solvent, with that of a solute of known molecular weight. The elevation in boiling-point produced by dissolving 1 mol of a solute in 100 grams, or 100 cubic centimeters, of a solvent is termed the moke- ular elevation, or boiling-point constant of the solvent. In deter- mining the boiling-point constant of a solvent, a fairly dilute solution is employed and the elevation in the boiling-point is observed; the value of the constant is then calculated on the .assumption that the elevation in boiling-point is proportional to the concentration. If g grams of a substance of unknown molecular weight m, are dissolved in W grams of solvent, and the boiling-point is raised 214 THEORETICAL CHEMISTRY A degrees, then, since m grams of the substance when dissolved in 100 grams of solvent, produce an elevation of K degrees (the molecular elevation), it follows that IQOg.A. . K W therefore, Q The accompanying table gives the boiling-point constants for 100 grams and 100 cubic centimeters of some of the more com- mon solvents. Solvent. Molecular Elevation. 100 gr. 100 cc. Water 5.2 11.5 21.0 16.7 26.7 35.6 30.1 5.4 15.6 30.3 22.2 32.8 Ethyl alcohol Ether Acetone Benzene Chloroform Pyridine As an example of the calculation of the molecular weight of a dissolved substance by the above formula, we may take the calculation of the molecular weight of camphor in acetone from the following data : When 0.674 gram of camphor is dissolved in 6.81 grams of acetone, the boiling-point of the solvent is raised 1.09. Substitut- ing in the formula, we have 0.674 m = 100 X 16.7 X 6.81 X 1.09 = 151. The molecular weight of camphor according to the formula Cio H 16 0, is 152. The molecular elevation of the boiling-point can be calculated by means of the formula, 0.02 T 2 K = w DILUTE SOLUTIONS AND OSMOTIC PRESSURE 215 in which T is the absolute boiling-point of the solvent, and w is the heat of vaporization for 1 gram of the solvent at its boiling- point. This formula will be derived in a subsequent paragraph of this chapter. The calculated values of K are in close agreement with the values obtained experimentally by Raoult and others. As an example, the calcu- lated value of the molecular elevation for water, the heat of vaporization of which at 100 C. is 537 calories, is K 0.02 X (373) 2 537 = 5.2, a value in exact agreement with the exper- imental value given in the table. Experimental Determination of Mo- lecular Weight by the Boiling-Point Method. One of the simplest and most convenient of the various forms of appa- ratus which have been devised for the determination of the boiling-point of solutions, is that developed by Jones,* and shown hi Fig. 65. The liquid whose boiling-point is to be determined is intro- duced into the vessel A, which already contains a platinum cylinder P, em- bedded in some glass beads. Sufficient liquid is introduced to insure the com- plete covering of the bulb of the ther- mometer, as shown in the sketch. Fig. 65. The side tube of A is connected with a condenser, C. The vessel A, is surrounded by a thick jacket of asbestos J, and rests on a piece of asbestos board in which a circular hole is cut, and over which a piece of wire gauze is placed. The liquid is heated by means of a burner, B. The platinum cylinder is the feature which differentiates this apparatus from the various other forms * Am. Chem. Jour., 19, 581 (1897). 216 THEORETICAL CHEMISTRY of boiling-point apparatus. It has the two-fold object of pre- venting the condensed solvent from coming in direct contact with the bulb of the thermometer, and of reducing the effect of radia- tion to a minimum. The liquid in A is boiled, using a very small flame, until the thermometer remains constant; this temperature is taken as the boiling-point of the liquid. The apparatus is now emptied and dried. A weighed amount of the liquid is then intro- duced into A, and to this is added a known weight of solute; the thermometer is replaced and the boiling-point of the solution is determined. The difference between the readings of the thermom- eter when immersed in the solution, and in the solvent alone, gives the boiling point elevation. For further details concerning the boiling-point method as applied to the determination of molecular weights, the student is referred to any one of the standard labo- ratory manuals. Osmotic Pressure and Boiling-Point Elevation. Imagine a dilute solution containing n mols of solute in G grams of solvent, and let dT be the elevation in the boiling-point. Suppose a large quantity of the solution to be introduced into a cylinder, fitted with a frictionless piston, and closed at the bottom by a semi- permeable membrane. Let the cylinder and contents be raised to the absolute temperature T, the boiling-point of the solvent, and then let pressure be exerted on the piston just sufficient to overcome the osmotic pressure of the solution. In this way, let a quantity of solvent corresponding to 1 mol of solute be forced through the semi-permeable membrane. The volume V, thus expelled is the volume corresponding to G/n grams of solvent. If the osmotic pressure of the solution is P, then the work done in moving the piston and expelling the solvent is PV. Now let the portion of the solvent which has been forced through the semi- permeable membrane be vaporized. For this operation G/n . w calories will be required, w being the heat of vaporization for 1 gram of solvent at its boiling-point. Then let the entire system be raised to the boiling-point of the solution (T + dT), the pre- viously expelled G/n grams of vapor being allowed to mix with the solution. The heat of vaporization lost at T is thus recov- ered at the slightly higher temperature, (T + dT). Finally, the DILUTE SOLUTIONS AND OSMOTIC PRESSURE 217 entire system is cooled to T, and is thus restored to its original state. Applying the well-known thermodynamic relation, that the ratio of the work done to the heat absorbed, is the same as the ratio of the difference in temperature to the absolute initial temperature of the system, we have PV dT ~ = "T' w n therefore, -* But, since PV = RT, equation (1) may be written .. w G If n = 1 and G = 100 grams, then dT = K, (the molecular eleva- tion of the boiling-point), or RT* 100 w' Or putting R = 2 calories, we have X-*S*H. (2) W Equation (1) shows that the osmotic pressure of a solution is directly proportional to the elevation of the boiling-point. Equation (2) was originally derived by van't Hoff at about the time when Raoult determined the values of K experimentally. Lowering of the Freezing-Point. Of all the methods employed for the determination of molecular weights in solution, the freez- ing-point method is the most accurate and the most widely used. It was pointed out by Blagden * over a century ago, that the de- pression of the freezing-point of a solvent by a dissolved substance is directly proportional to the concentration of the solution. When equimolecular quantities of different substances are dissolved in equal volumes of the same solvent, the lowering of the freezing- point is constant. The molecular weight of any soluble sub- * Phil. Trans., 78, 277 (1788). 218 THEORETICAL CHEMISTRY stance can be found, as in the boiling-point method, by comparing its effect on the freezing-point of a solvent with that of a solute of known molecular weight. The molecular lowering of the freezing- point, or the freezing-point constant, of a solvent is defined as the de- pression of the freezing-point produced by dissolving 1 mol of solute in 100 grams or 100 cubic centimeters of solvent. The freezing-point constants of a few common solvents are given in the following table. Molecular I )epression. 100 gr. 100 cc. Water 18.5 18.5 Benzene . . . . . . x 50 56 Phenol 74 Naphthalene 69 Acetic acid 39 41 Van't Hoff showed that the molecular lowering of the freezing- point of a solvent K, can be calculated from the absolute freez- ing-point T, and the heat of fusion w, for 1 gram of solvent at the temperature T, by means of the formula K _ 0.02 T 2 w This expression is analogous to that which applies to the molec- ular elevation of the boiling-point. The agreement between the observed and the calculated values of K is very satisfactory, as the following calculation for water shows : 0.02) oU It is of interest to note that the calculated value of K for water is lower than the experimental values originally obtained by Raoult and others. Subsequent experiments, carried out with greater care and better apparatus, by Raoult, Abegg and Loomis gave values in close agreement with that derived theoretically. A formula analogous to that employed for the calculation of the molecular weight of a dissolved substance from the elevation it DILUTE SOLUTIONS AND OSMOTIC PRESSURE 219 produces in the boiling-point of a solvent, may be used for the calculation of molecular weight from freezing-point depression. Thus, if g grams of solute when dissolved in W grams of solvent produce a depression A of the freezing-point of the solvent, the molecular weight m, is given by the formula where K is the molecular lowering of the freezing-point. EXAMPLE. When 1.458 grams of acetone are dissolved in 100 grams of benzene, the freezing-point of the solvent is de- pressed 1.22, therefore the molecular weight of acetone is 1 The molecular weight of acetone, calculated from the formula C 3 H 6 O, is 58. In order to obtain trustworthy results with the freezing-point method, it is necessary that only the pure solvent separate out when the solution freezes, and that excessive overcooling be avoided. When too great overcooling occurs, the subsequent freezing of the solution results in the separation of so large an amount of solvent in the solid state, that the observed freezing point corresponds to the equilibrium temperature of a more concentrated solution than that originally prepared. A formula for the correction of the concentration, due to excessive overcooling, has been derived by Jones.* If the overcooling of the solution in degrees be represented by u, the heat of fusion of 1 gram of sol- vent at the freezing-point by w, and the specific heat of the sol- vent by c, then the fraction of the solvent which will solidify, /, may be calculated by the formula, . _ cu w' When water is used as the solvent, c = 1 and w = 80. There- fore, for every degree of overcooling, the fraction of the solvent separating as ice will be 1/80, and the concentration of the original * Zeit. phys. Chem., 12, 624 (1893). 220 THEORETICAL CHEMISTRY solution is increased by just so much. It is simpler, however, to apply the correction directly to the freezing-point depression in- stead of to the concentration. Experimental Determination of Molecular Weight by the Freezing-Point Method. The apparatus almost universally em- ployed for the determination of molecular weights by the freezing- point method is that devised by Beckmann,* and shown in Fig. 66. It consists of a thick- walled test tube A, provided with a side tube, and fitted into a wider tube AI, thus surround- ing A with an air space. The whole is fitted into the metal cover of a large battery jar, which is filled with a freez- ing mixture whose temperature is several de- grees below the freezing-point of the solvent. The tube A is closed by a cork stopper, through which passes the thermometer and stirrer. The thermometer is generally of the Beckmann differential type. This instrument has a scale about 6 in length, each degree being divided into hundredths; the quantity of mercury in the bulb can be varied by means of a small reservoir at the top of the scale, so that the zero of the instrument can be adjusted for use with solvents of widely different freez- ing-points. In carrying out a determination with the Beckmann apparatus, a weighed quantity of solvent is placed in A, and the temperature of the refrigerating mixture regulated so as to be not more than 5 below the freezing-point of the solvent. The tube A is removed from its jacket, and is immersed in the freezing mixture until the solvent begins to freeze. It is then replaced in the jacket AI, and the solvent is vigorously stirred. The thermometer rises during the stirring until the true freezing-point is reached, after which it remains constant. This temperature is taken as the freezing temperature of the solvent. * Zeit. phys. Chem., 2, 683 (1888) . Fig. 66. DILUTE SOLUTIONS AND OSMOTIC PRESSURE 221 The tube A is now removed from the freezing mixture, and a weighed amount of the substance whose molecular weight is to be determined is introduced. When the substance has dissolved, the tube is replaced in AI and the solution cooled not more than a degree below its freezing-point. A small fragment of the solid solvent is dropped into the solution, which is then stirred vigorously until the thermometer remains constant. The maxi- mum, temperature is taken as the freezing-point of the solution. The difference between the freezing-points of solution and solvent is the depression sought. For further details con- cerning the determination of molecular weights by the freezing- point method, the student is referred to a physico-chemical laboratory manual. Osmotic Pressure and Freezing-Point Depression. Let dT be the freezing-point depression produced by n mols of solute in G grams of solvent, the solution being dilute. Imagine a large quantity of this solution to be confined within a cylinder fitted with a frictionless piston, the bottom of the cylinder being closed by a semi-permeable membrane. Let the cylinder and contents be cooled to the freezing temperature of the solvent T, and then let pressure be applied to the piston until a quantity of solvent corresponding to 1 mol of solute is forced through the semi-perme- able membrane. This requires an expenditure of energy equiva- lent to PV, where P is the osmotic pressure of the solution and V is the volume of solvent expelled. The volume V is clearly the volume of G/n grams of solvent. Now let the expelled portion C 1 of solvent be frozen and the system deprived of w calories of heat, where w is the heat of fusion of 1 gram of the solvent at the temperature T. The temperature of the solution is then lowered to its freezing- point (T dT), and the G/n grams of solidified solvent dropped into it. The solidified solvent melts, thereby restoring to the system at the temperature (T dT), the heat of fusion formerly taken from it. Finally, the temperature of the system is raised to T, the initial temperature of the cycle. Applying the familiar thermodynamic relation, that the ratio of the work done to the 222 THEORETICAL CHEMISTRY heat absorbed, is the same as the ratio of the difference in temper- ature to the initial absolute temperature, we have PV AT rn w n From which we obtain w G But PV = RT, hence equation (1) becomes w G If n = 1 and G = 100 grams, then dT = K, the molecular lower- ing of the freezing-point, and RT* ~ 100 W' Or putting R = 2 calories, we have An equation to which reference has already been made. It is evident from equation (1) that the osmotic pressure of a solution is directly proportional to the freezing point depression. Molecular Weight in Solution. As has been pointed out, the molecular weight of a dissolved substance can be readily calcu- lated, provided that the osmotic pressure of a dilute solution of known concentration at known temperature is determined. But the experimental difficulties attending the direct measurement of osmotic pressure are so great, that it is customary to employ other methods based upon properties of dilute solutions which are proportional to osmotic pressure. We have shown that in dilute solutions osmotic pressure is directly proportional (1) to the relative lowering of the vapor pressure, (2) to the elevation of the boiling-point, and (3) to the depression of the freezing-point. From this it follows, that equimolecular quantities of different substances dissolved in equal volumes of the same solvent, exert the DILUTE SOLUTIONS AND OSMOTIC PRESSURE 223 same osmotic pressure, and produce the same relative lowering of vapor pressure, the same elevation of boiling-point, and the same depression 'of freezing-point. Since equimolecular quantities of different substances contain the same number of molecules, it is evident that the magnitude of osmotic pressure, relative lowering of vapor pressure, elevation of boiling-point and depression of freezing- point, is dependent upon the number of particles present in the solu- tion and is independent of their nature. It has been pointed out by Nernst that any process which involves the separation of solvent from solute, may be employed for the determination of molecular weights. A little reflection will convince the reader that the four methods discussed in this chapter involve such separ- ation. Both van't Hoff and Raoult emphasized the fact that the formulas derived for the determination of molecular weights in so- lution depend upon assumptions which are valid only for dilute solutions. It follows, therefore, that we are not justified in apply- ing these formulas to concentrated solutions. Up to the present time we have no satisfactory theory of concentrated solutions, neither can we state up to what concentration the gas laws apply. PROBLEMS. 1. At 10 C. the osmotic pressure of a solution of urea is 500 mm. of mercury. If the solution is diluted to ten times its original volume, what is the osmotic pressure at 15 C.? Ans. 50.89 mm. 2. The osmotic pressure of a solution of 0.184 gram of urea in 100 cc. of water was 56 cm. of mercury at 30 C. Calculate the molecular weight of urea. Ans. 62.12. 3. At 24 C. the osmotic pressure of a cane sugar solution is 2.51 atmos- pheres. What is the concentration of the solution in mols per liter? Ans. 0.103. 4. At 25. 1 C. the osmotic pressure of solution of glucose containing 18 grams per liter was 2.43 atmospheres. Calculate the numerical value of the constant R, when the unit of energy is the gram-centimeter. Ans. 84,231. 5. The vapor pressure of ether at 20 C. is 442 mm. and that of a solu- tion of 6.1 grams of benzoic acid in 50 grams of ether is 410 mm. at the same temperature. Calculate the molecular weight of benzoic acid in ether. Ans. 124. 224 THEORETICAL CHEMISTRY 6. At 10 C. the vapor pressure of ether is 291.8 mm. and that of a solution containing 5.3 grams of benzaldehyde in 50 grams of ether is 271.8 mm. What is the molecular weight of benzaldehyde? Ans. 106.6. 7. A solution containing 0.5042 gram of a substance dissolved in 42.02 grams of benzene boils at 80. 175 C. Find the molecular weight of the solute, having given that the boiling-point of benzene is 80.00 C., and its heat of vaporization is 94 calories per gram. Ans. 181.9. 8. A solution containing 0.7269 gram of camphor (mol. wt. = 152) in 32.08 grams of acetone (boiling-point = 56.30 C.) boiled at 56.55 C. What is the molecular elevation of the boiling-point of acetone? What is its heat of vaporization? Ans. K = 16.74; w = 129.5 cals. per gm. 9. A solution of 9.472 grams of CdI 2 in 44.69 grams of water boiled at 100.303 C. The heat of vaporization of water is 536 calories per gram. What is the molecular weight of CdI 2 in the solution? What conclusion as to the state of CdI 2 in solution may be drawn from the result? Ans. 363.2. 10. The freezing-point of pure benzene is 5.440 C. and that of a solu- tion containing 2.093 grams of benzaldehyde in 100 grams of benzene is 4.440 C. Calculate the molecular weight of benzaldehyde in the solu- tion. K for benzene is 50. Ans. 104.6. 11. A solution of 0.502 gram of acetone in 100 grams of glacial acetic acid gave a depression of the freezing-point of 0.339 C. Calculate the molecular depression for glacial acetic acid. Ans. 39. 12. By dissolving 0.0821 gram of m-hydroxybenzaldehyde (C 7 H 6 2 ) in 20 grams of naphthalene (melting point 80. 1 C.) the freezing-point is lowered by 0.232 C. Assuming that the molecular weight of the solute is normal in the solution, calculate the molecular depression for naphtha- lene and the heat of fusion per gram. Ans. K = 68.96; w = 36.2 cals. per gm. CHAPTER XI. ASSOCIATION, DISSOCIATION AND SOLVATION. Abnormal Solutes. As has already been pointed out the acceptance of Avogadro's hypothesis was greatly retarded by the discovery of certain substances whose vapor densities were ab- normal. Thus, the vapor density of ammonium chloride is approx- imately one-half of that required by the formula NH 4 C1, while the vapor density of acetic acid corresponds to a formula whose molecular weight is greater than that calculated from the formula, C2H 4 O2. The anomalous behavior of ammonium chloride and kindred substances has been shown to be due, not to a failure of Avogadro's law, but to a breaking down of the molecules, a process known as dissociation. The abnormally large molecular weight of acetic acid on the other hand, has been ascribed to a process of aggregation of the normal molecules, known as asso- ciation. In extending the gas laws to dilute solutions similar phenomena have been encountered. Association in Solution. When the molecular weight of acetic acid in benzene is determined by the freezing-point method, the depression of the freezing-point is abnormally small and conse- quently, as the formula shows, the molecular weight will be greater than that correspond- ing to the formula, C2H 4 O 2 . Acetic acid in benzene solution is thus shown to be associated. Almost all compounds containing the hydroxyl and cyanogen groups when dissolved in benzene are found to be associated. Solvents, such as benzene and chloro- form, are frequently termed associating solvents, although it is doubtful whether they exert any associating action. There is considerable experimental evidence to show that those substances 225 226 THEORETICAL CHEMISTRY whose molecules are associated in benzene and chloroform solu- tion, are also associated in the free condition. Just as the depres- sion of the freezing-point of a solution of an associated substance is abnormally small, so its osmotic pressure and other related properties will be less than the calculated values. Dissociation in Solution. Van't Hoff * pointed out that the osmotic pressure of solutions of most salts, of all strong acids, and of all strong bases is much greater for all concentrations than would be expected from the osmotic pressure of solutions of sub- stances, like cane sugar or urea, for corresponding concentrations. He was unable to account for this abnormal behavior, and in order to render the general gas equation applicable, he introduced a factor i, the modified equation being PV = iRT. If the osmotic pressure of some substance, like cane sugar, which behaves normally, be represented by P > the factor i is given by the expression . P Since the osmotic pressure of a solution is proportional to the relative lowering of its vapor pressure, to the elevation of its boiling-point, and to the lowering of its freezing-point, we may write Po where the symbols have their usual significance. The subscript refers in each case to a substance which behaves normally, and A denotes either boiling-point elevation or freezing-point depression. A more definite conception of the abnormal behavior of salts will be gained by an inspection of the accompanying tables. In the first column is recorded the molar concentration of the solu- tion; the second column gives the observed depressions of the * Zeit. phys. Chem., i, 501 (1887). ASSOCIATION, DISSOCIATION AND SOLVATION 227 freezing-point and the third column contains the values of the ratio of the observed depression to the normal depression, or i. POTASSIUM CHLORIDE. POTASSIUM SULPHATE. TO A i m A i 0.05 0.1750 1.88 0.05 0.2270 2.33 0.10 0.3445 1.85 0.10 0.4317 2.32 0.20 0.6808 1.83 0.20 0.8134 2.18 0.40 1.3412 1.80 0.30 1 . 1673 2.09 ALUMINIUM CHLORIDE. SODIUM CHROMATE. TO A i TO A i 0.046 0.076 0.102 0.276 0.446 0.578 3.22 3.15 3.04 0.1 0.2 0.5 1.0 0.450 0.850 1.960 3.800 2.42 2.28 2.11 2.04 It is apparent from the above data that the depression of the freezing-point of water caused by these salts is abnormally large, a fact which points to an increase in the number of dissolved units over that corresponding to the initial concentration. The Theory of Electrolytic Dissociation. In 1887, Arrhenius * advanced an hypothesis to account for the abnormal osmotic activity of solutions of acids, bases and salts. He pointed out that just as the exceptional behavior of certain gases has been completely reconciled with the law of Avogadro, by assuming a dissociation of the vaporized molecule into two or more simpler molecules, so the enhanced osmotic pressure and the abnormally great freezing-point depression of solutions of acids, bases and salts can be explained, if we assume a similar process of dissoci- ation. He proposed, therefore, that aqueous solutions of acids, bases and salts be considered as dissociated, to a greater or less extent, into positively- and negatively-charged particles or ions, and that the increase in the number of dissolved units due to this * Zeit. phys. Chem., i, 631 (1887). 228 THEORETICAL CHEMISTRY dissociation is the cause of the enhanced osmotic activity. Accord- ing to this hypothesis, hydrochloric acid, potassium hydroxide and potassium chloride, when dissolved in water, dissociate in the following manner: TT/^l _v TI* I OT XlV^l r XI -J- \jL KC1 ->K' + C1', where the dots indicate positively-charged ions and the dashes negatively-charged ions. In each of the above cases, one molecule yields two ions, so that, if dissociation is complete, the maximum osmotic effect should not be greater than twice that produced by an equimolecu- lar quantity of a substance which behaves normally. Reference to the preceding table shows that the value of i for potassium chloride approaches the limiting value of 2 as the solution is diluted. The other salts given in the table dissociate, according to Arrhenius, in the following way : - K 2 S0 4 -^2K--hS0 4 " NasCr0 4 -> 2 Na* + Cr0 4 " Aid. -> A!"' + 3 Cl'. If these equations correctly represent the process of dissociation, then when dissociation is complete, the osmotic effect of infinitely dilute solutions of potassium sulphate and sodium chromate should be three times the effect produced by an equimolecular quantity of a normal solute, and in the case of aluminium chloride, the maximum effect should be four times the effect due to a normal substance. A glance at the table shows that the values of i for solutions of the three salts approach these respective limits. If this hypothesis of ionic dissociation be accepted, then it becomes possible to calculate the degree of ionization in any solution by comparing its freezing-point depression with the freezing-point lowering of an equimolecular solution of a normal substance. Let us suppose that the degree of dissociation of 1 molecule of a dissolved substance is a, each molecule yielding n ions. Then ASSOCIATION, DISSOCIATION AND SOLVATION 229 1 a. will be the undissociated portion of the molecule, and the total number of dissolved units will be 1 a. -f- na. If A is the depression of the freezing-point produced by the sub- stance, and Ao the depression produced by an equimolecular quantity of an undissociated substance, then 1 a + na A or 1 a i- 1 n - 1 It will be observed that this formula is identical with that derived for the degree of dissociation of a gas (p. 91). If this formula be applied to the freezing-point data for solutions of potassium chloride given in the preceding table we find the following per- centages of dissociation corresponding to the different concentra- tions : POTASSIUM CHLORIDE. m A i a 0.05 0.1750 1.88 88 0.10 0.3445 1.85 85 0.20 0.6808 1.83 83 0.40 1.3412 1.80 80 The figures in the last column show that the degree of dissocia- tion increases as the concentration diminishes. It was further pointed out by Arrhenius that all of the substances which exhibit abnormal osmotic effects, when dissolved in water, yield solutions which conduct the electric current, whereas, aqueous solutions of such substances as cane sugar, urea and alcohol, exert normal osmotic pressures, but do not conduct electricity any better than the pure solvent. In other words, only electrolytes * are capable of undergoing ionic dissociation ; hence Arrhenius termed his hypoth- esis the electrolytic dissociation theory. As we have seen, when potassium chloride is dissolved in water, it is supposed to dissoci- * The term electrolyte strictly refers to the solution of an ionized substance, although it is often applied to acids, bases and salts because, when dissolved, they produce electrolytes. To avoid confusion, the term "ionogen" (ion former) has been proposed for those substances which give conducting solu- tions. 230 THEORETICAL CHEMISTRY ate into positively-charged potassium ions and negatively-charged chlorine ions. Accordingly when two platinum electrodes, one charged positively and the other negatively, are introduced into the solution, the potassium ions move toward the negative elec- trode and the chlorine ions move toward the positive electrode, the passage of a current through the solution consisting in the ionic transfer of electric charges. Since the undissociated molecules, being electrically neutral, do not participate in the transfer of electric charges, it follows that the conductance of a solution of an electrolyte is dependent upon the degree of dissociation. The relation between electrical conductance and the degree of ioniza- tion will be discussed in a subsequent chapter. It may be stated at this point, however, that the values of a based upon measure- ments of electrical conductance, while showing some discrepancies in individual cases, are in general in good agreement with the values obtained by the freezing-point method. Furthermore, the values of a obtained from freezing-point measurements are in har- mony with those based upon De Vries' isotonic coefficients. It will be seen, on referring to the table of isotonic coefficients (p. 181), that solutions of electrolytes show enhanced osmotic activity. Thus, the osmotic pressures of equi-molecular solutions of cane sugar, potassium nitrate and calcium chloride are to each other as 1 : 1.67 : 2.40. The following table illustrates the general agreement between the values of i calculated, (a) from electrical conductance, (b) from freezing-point depression, and (c) from De Vries' isotonic coefficients. Substance. Molar Cone. (a) (b) (c) KC1.. 14 1 86 1 82 1 81 LiCl 13 1 84 1 94 1 92 Ca(NO 3 ) 2 18 2 46 2 47 2 48 MgCl 2 19 2 48 2 68 2 79 CaCl 2 184 2 42 2 67 2 78 It must be remembered that the values of i derived from freez- ing-point measurements correspond to temperatures in the ASSOCIATION, DISSOCIATION AND SOLVATION 231 neighborhood of C., while those derived from the other methods correspond to temperatures ranging from 17 C. to 25 C. Chemical Properties of Completely Ionized Solutions. The chemical properties of an ion are very different from the properties of the atom or radical when deprived of its electrical charge. For example, the sodium ion is present in an aqueous solution of sodium chloride, but there is no evidence of chemical reaction with the solvent; whereas, the element in the electrically-neutral con- dition reacts violently with water, evolving hydrogen and form- ing a solution of potassium hydroxide. Again, take the element chlorine: when chlorine in the molecular condition, either as gas or in solution, is added to a solution of silver nitrate, no precipitate of silver chloride is formed. Further, chlorine in such compounds as CHC1 3 , CC1 4 , etc., is not precipitated by silver nitrate, since these compounds are not dissociated by water. Or, chlorine may be present in a compound which is dissociated by water and yet not exhibit its characteristic reactions, because it is present in a complex ion. Thus, potassium chlorate dissociates in the follow- ing manner : KClOa - K' + On adding silver nitrate, there is no precipitation, because the chlorine forms a complex ion with oxygen. Physical Properties of Completely Ionized Solutions. The physical properties of completely ionized solutions are, in general, additive. This is well illustrated by a series of solutions of colored salts, the color of which is due to the presence of a par- ticular ion. It is found, when the solutions are sufficiently dilute to insure complete dissociation, that they all have the same color. The additive character of the colors of solutions of electrolytes is brought out in a striking manner by a comparison of their absorption spectra. Ostwald * photographed the absorp- tion spectra of solutions of the permanganates of lithium, cadmium, ammonium, zinc, potassium, nickel, magnesium, copper, hydrogen and aluminium, each solution containing 0.002 gram-equivalents * Zeit. phys. Chem., 9, 579 (1892). 232 THEORETICAL CHEMISTRY of salt per liter. The absorption spectra, as shown in Fig. 67, will be seen to be practically identical, the bands occupying the same position in each spectrum. This affords a strong con- firmation of the theory of electrolytic dissociation, according to which a dilute solution is to be regarded as a mixture of elec- trically equivalent quantities of oppositely charged ions, each of which contributes its specific properties to the solution. The permanganate ion being colored, and common to all of the salts examined, and the positive ions of the various substances being colorless, it follows that when dis- sociation is complete, the absorp- tion spectra of all of the solutions must be identical. A number of other properties of completely dissociated solutions have been shown to be additive. Among these may be mentioned density, specific refraction, surface ten- sion, thermal expansion, and magnetic rotatory power. Addi- tional evidence in favor of the theory of electrolytic dissociation will be furnished in forthcoming chapters. Notwithstanding the large number of facts which can be satisfactorily interpreted by the theory, there are directions in which it requires amplification and modification. Of the various objections which have been urged against the theory of electrolytic dissociation, one is of sufficient weight to call for brief consideration here. When two elements, such as potassium and chlorine, combine to form potassium chloride, the reaction is violent and a large amount of heat is developed. Nevertheless, Fig. 67. ASSOCIATION, DISSOCIATION AND SOLVATION 233 according to the ionization theory, the strong, mutual affinity of these two elements is overcome by the act of solution in water, the molecule being split into two oppositely-charged atoms. Obvi- ously such a separation calls for the expenditure of a large amount of energy, and the question naturally arises : What is the source of this energy? While this question cannot be fully an- swered here, it may be pointed out that we have abundant evidence to show that the ions are hydrated, each being surrounded by an " atmosphere" of solvent. In view of this fact, it has been suggested * that dissociation in aqueous solution is caused by the mutual attraction between the ions and the molecules of the sol- vent, the heat of ionic hydration furnishing the energy necessary for the separation of the ions. Freezing-Point Depressions Produced by Concentrated Solu- tions of Electrolytes. As has already been mentioned, the dissociation of electrolytes in aqueous solution increases with the dilution, becoming complete at a concentration of about 0.001 molar. We should expect the dissociation to diminish with in- creasing concentration, until, if the electrolyte is sufficiently solu- ble, the depression of the freezing-point becomes normal. Recent investigations by Jones and his co-workers f have shown that the facts are contradictory to this expectation. They found that the value of the molecular depression of the freezing-point of water produced by a number of chlorides and bromides, diminished with increasing concentration up to a certain point, as would be ex- pected, and then increased again. The increase in the molecular depression became very marked at great concentrations; in fact, the molecular depression in a molar solution was frequently greater than the molecular depression corresponding to a completely dissociated salt. This phenomenon was systematically studied by Jones and the author J and the fact was established that it is quite general. * Trans. Faraday Soc., i, 197 (1905); 3, 123 (1907). f Am. Chem. Jour., 22, 5, 110 (1899); 23, 89 (1900). j Zeit. phys. Chem., 46, 244 (1903); Phys. Rev., 18, 146 (1904): Am. Chem. Jour., 31, 303 (1904); 32, 308 (1904); 33, 534 (1905); 34, 291 (1905); Zeit. phys. Chem., 49, 385 (1904); Monograph No. 60, Carnegie Institution of Washington. 234 THEORETICAL CHEMISTRY This abnormal depression of the freezing-point may be accounted for by assuming that the dissolved substance has entered into com- bination with a portion of the water, thus removing it from the role of solvent. The formation of a loose molecular complex be- tween one molecule of the solute and a large number of molecules of water, acts as a single dissolved unit in depressing the freezing- point of the pure solvent. Evidently the total amount of water present, which functions as solvent, is diminished by the amount of water which has been appropriated by the solute. The abnor- malities observed in the depression of the freezing-point of con- centrated solutions of electrolytes can be explained by assuming that the molecules of solute, or the resulting ions, are in combina- tion with a number of molecules of solvent. This hypothesis is termed the solvate theory, and the loose molecular complexes are called solvates. Since the solvate theory was first proposed, con- siderable evidence has been accumulated to confirm its correct- ness. Reference has already been made to the work of Philip on the solubility of gases in saline solutions, from which he concludes that the dissolved salts enter into combination with a portion of the solvent. The experiments of Morse and the Earl of Berkeley on osmotic pressure, also seem to point to the solvation of the dissolved substance. PROBLEMS. 1. At 18 C. a 0.5 molar solution of XaCl is 74.3 per cent dissociated. What would be the osmotic pressure of the solution in atmospheres at 18 C.? Ans. 20.79. 2. A solution containing 3 mols of cane sugar per liter was found by the plasmolytic method to be isotonic with a solution of potassium nitrate containing 1.8 mols per liter. What is the degree of ionization of the potassium nitrate? Ans. 67 per cent. 3. The vapor pressure of water at 20 C. is 17.406 mm. and that of a 0.2 molar solution of potassium chloride u 17.296 mm. at the same tem- perature. Calculate the degree of dissociation of the salt. Ans.. 75.38 per cent. 4. The degree of dissociation of a 0.5 molar solution of sodium chloride at 25 is 74.3 per cent. Calculate the osmotic pressure of the solution at the same temperature. Ans. 21.32 atmos. ASSOCIATION, DISSOCIATION AND SOLVATION 235 5. A solution containing 1.9 mols of calcium chloride per liter is isotonic with a solution of glucose containing 4.05 mols per liter. What is the degree of ionization of the calcium chloride? Ans. 56.6 per cent. 6. At C. the vapor pressure of water is 4.620 mm. and of a solution of 8.49 grams of NaNO 3 in 100 grams of water 4.483 mm. Calculate the degree of ionization of NaN0 3 . Ans. 64.9 per cent. 7. At C. the vapor pressure of water is 4.620 mm. and that of a solution of 2.21 grams of CaCl 2 in 100 grams of water is 4.583 mm. Cal- culate the apparent molecular weight and the degree of ionization of CaCl 2 . Ans. M = 49.66, a = 62 per cent. 8. The boiling-point of a solution of 0.4388 gram of sodium chloride in 100 grams of water is 100. 074 C. Calculate the apparent molecular weight of the sodium chloride and its degree of ionization. K = 5.2. Ans. M = 30.84, a = 89.7 per cent. 9. The boiling-point of a solution of 3.40 grams of BaCk in 100 grams of water is 100.208 C. K = 5.2. What is the degree of ionization of theBaCl 2 ? Ans. 72.5 per cent. 10. At 100 C. the vapor pressure of a solution of 6.48 grams of ammo- nium chloride in 100 grams of water is 731.4 mm. K = 5.2. What is the boiling-point of the solution? Ans. 101.086 C. 11. A solution of 1 gram of silver nitrate in 50 grams of water freezes at 0.348 C. Calculate to what extent the salt is ionized in solution. K = 18.6. Ans. 59 per cent. 12. A solution of NaCl containing 3.668 grams per 1000 grams of water freezes at 0.2207 C. Calculate the degree of ionization of the salt. K = 18.6. Ans. 89.2 per cent. 13. The freezing-point of a solution of barium hydroxide containing 1 mol in 64 liters is 0.0833 C. What is the concentration of hydroxyl ions in the solution? Take K = 18.9 for concentrations in mols per liter. Ans. 0.0284 gm.-ion per liter. 14. The vapor pressure of water at C. is 4.620 mm., and the lower- ing of the vapor pressure produced by dissolving 5.64 grams of sodium chloride in 100 grams of water is 0.142 mm. What is the freezing-point of the solution? K = 18.6. Ans. - 3.177 C. 15. A solution containing 8.34 grams Na 2 S0 4 per 1000 grams of water freezes at 0.280 C. Assuming dissociation into 3 ions, calculate the 236 THEORETICAL CHEMISTRY degree of ionization and the concentrations of the Na* and SO/' ions. K = 18.6. Ans. a = 78.2 per cent; cone. Na" = 0.0918 gm.-ion per liter; cone. S0 4 " = 0.0459 gm.-ion per liter. CHAPTER XII. COLLOIDS. Crystalloids and Colloids. In the course of his investigations on diffusion in solutions, Thomas Graham * drew a distinction between two classes of solutes, which he termed crystalloids and colloids. Crystalloids, as the name implies, can be obtained in the crystalline form: to this class belong nearly all of the acids, bases and salts. Colloids, on the other hand, are generally amorphous, such substances as albumin, starch and caramel being typical of the class. Because of the gelatinous character of many of the sub- stances in this class, Graham termed them colloids (xoXXa = glue, and erdos= form). The differences between the two classes are most apparent in the physical properties of their solutions. Thus, crystalloids diffuse much more rapidly than colloids; the velocity of diffusion of caramel being nearly 100 times slower than that of hydrochloric acid at the same temperature. While crystalloids exert osmotic pressure, lower the vapor pressure and depress the freezing-point of the solvent, colloids have very little effect upon the properties of the solvent. The marked differences in the rates of diffusion of crystalloids and colloids render their separation comparatively easy. If a solution containing both crystalloids and colloids be placed in a vessel over the bottom of which is stretched a colloidal membrane, such as parchment, and the whole is immersed in pure water, the crystalloids will pass through the membrane, while the colloids will be left behind. This process was termed by Graham, dialysis, while the apparatus employed to effect such a separation was called a dialyzer. When a solution of sodium silicate is added to an excess of hydrochloric acid, the products of the reaction, silicic acid and sodium chloride, remain in solution. When the mixture is placed in a dialyzer, the sodium chloride and the hydrochloric acid, being crystalloids, diffuse * Lieb. Ann., 121, 1 (1862). 237 238 THEORETICAL CHEMISTRY through the membrane of the dialyzer, leaving behind the colloidal silicic acid. The terms crystalloid and colloid, as used at the present time, have acquired different meanings from those assigned to them by Graham. The terms are now considered to refer, not so much to different classes of substances, as to different states which almost all substances can assume under certain conditions. Colloidal Solutions. A colloidal solution is one in which the solute is a colloid, although the latter may not be included among the substances classified as such by Graham. For example, arsenious sulphide, ferric hydroxide or finely-divided gold may form colloidal solutions. In bringing such substances into the colloidal state, mere agitation with water will not suffice, but some indirect method must be employed. Nomenclature. Graham distinguished between two condi- tions in which colloids were obtainable, the term sol being applied to forms in which the system resembled a liquid, while the term gel was used to designate those forms which were solid and jelly- like. When one of the components of the solution was water, the two forms were called a hydrosol and a hydrogel. In like manner, when alcohol was one of the components, the terms alcosol and alcogel were applied to the two forms. As the knowledge of colloids has developed it has become neces- sary to supplement Graham's nomenclature by the introduction of various other terms. It is known to-day that the essential difference between colloidal suspensions and solutions on the one hand, and true solutions on the other, is due to the difference in the degree of subdivision or degree of dispersity of the dissolved sub- stance. In a true solution the dissolved substance is generally present either in the molecular or ionic condition, as may be shown by means of the familiar osmotic methods for molecular weight de- termination. In colloidal solutions, however, the degree of dis- persity is not so great and has been found to vary from above the limit of microscopic visibility (1 X 10~ 5 cm.) to that of molecular dimensions (1 X 10~ 8 cm.). When the degree of dispersity varies from 1 X 10~ 3 cm. to 1 X 10~ 5 cm. the particles are termed microns. The properties of the disperse phase at this degree of COLLOIDS 239 dispersity differ appreciably from the properties of the same sub- stance when present in large masses. When the degree of dis- persity lies between 1 X 10~ 5 cm. and 5 X 10~ 7 cm. the particles are known as submicrons. The existence of particles whose diam- eters are approximately 1 'X 10~ 7 cm. has been demonstrated by Zsigmondy with the ultramicroscope; these minute particles are termed amicrons. When the degree of dispersity is increased beyond this limit all heterogeneity apparently vanishes and we enter the realm of true solutions. When the dispersion is not too great, colloidal solutions may be divided into suspensions and emulsions according to whether the disperse phase is a solid or a liquid. As the dispersion is increased we obtain suspension and emulsion colloids which may be con- veniently called suspensoids and emulsoids. Suspensoids and emulsoids are included under the general term dispersoids. In certain cases, although the disperse phase is unquestionably liquid, the systems resemble suspensoids in their behavior, while in other cases, where the disperse phase is solid, the systems exhibit proper- ties characteristic of emulsoids. For this reason the classification of sols as suspensoids and emulsoids is not entirely satisfactory. A better system is that in which the presence or the absence of affinity between the disperse phase and the dispersion medium is made the basis of classification. Where there is marked affinity between the two phases, the system is termed lyophile, and where such affinity is absent, the system is termed lyophobe. When the dispersion medium is water, the terms hydrophile and hydrophobe are employed. In the reversible transformation of a sol into a gel, we are not warranted in referring to the change from gel to sol as an act of solution, for if the gel really dissolved, a solution and not a sol would result. Various terms have been proposed for these reversible transformations but perhaps the most satisfac- tory are the terms gelation and solation, the former designating the formation of a gel from a sol and the latter the reverse process. Lyotrope Series. The differences between lyophile and lyo- phobe sols are frequently very marked, this being especially true 240 THEORETICAL CHEMISTRY of their behavior toward chemical reagents. The action of chemi- cal reagents on lyophobe sols is almost wholly confined to the disperse phase, while the addition of reagents to lyophile sols fre- quently produces a more marked effect on the dispersion medium than on the disperse phase. It should be observed that the physi- cal properties of a lyophobe sol and of the pure dispersion medium are practically identical, while exactly the reverse is true of lyo- phile sols. It is well known that the addition of a foreign sub- stance to a reaction-mixture frequently exerts a marked influence on the speed of the reaction, notwithstanding the fact that the nature of the added substance may be such as to render its partici- pation in the reaction highly improbable. If a series of reagents are arranged in the order of their influence on a particular reaction, it has been found that the same sequence is preserved when the same reagents are added to other reactions of widely different character. In some reactions the reagents may produce effects directly opposite to those which they produce in others, but the sequence remains unchanged. For example, when the same re- action takes place either in an acid or in an alkaline medium, the substances which promote the reaction when the medium is acid, retard it when the medium is alkaline, but the sequence of the added substances remains the same under both conditions. These facts make it appear highly probable that the effects produced by the addition of foreign substances to a chemical reaction are to be ascribed to the changes which they produce in the pure solvent. It is not without significance that the same sequence of reagents is maintained whether we observe their influence on different chemi- cal reactions or on certain physical properties of the solvent, such as its density, viscosity, and surface tension. The following examples * afford a striking illustration of the persistence of the sequence of reagents, generally known as the lyotrope series. The ions which precede the formula (H 2 0) reduce the velocity of reaction or cause a diminution in the magnitude of the particular physical property tabulated. The ions which follow the formula (H 2 0) exert the opposite effect. * Freundlich, Kapillarchemie, p. 411. COLLOIDS 241 1. The hydrolysis of esters by acids. Anions: S0 4 (H 2 0) Cl < Br. Cations: (H 2 O) Li < Na < K < Rb < Cs. 2. The hydrolysis of esters by bases. Anions: I > NO 3 > Br > Cl (H 2 0) S 2 3 < S0 4 . Cations: Cs > Rb > K > Li (H 2 0). 3. The viscosity of aqueous solutions. Anions: NO 3 > Cl (H 2 O) S0 4 (potassium salts). Cations: Cs > Rb > K (H 2 O) Na < Li (chlorides), 4. The surface tension of aqueous solutions. (H 2 0) K N0 3 < Cl< S0 4 < C0 3 . It will be observed that although the ions which accelerate the acid hydrolysis retard the basic hydrolysis, the sequence of the ions nevertheless remains the same. The Ultramicroscope. When a narrow beam of sunlight is admitted into a darkened room, the dust particles in its path are rendered visible by the scattering of the light at the surface of the particles. If the air of the room is free from dust, no shining particles will be seen and the space is said to be " optically void." When the particles of dust are very minute, the beam of light acquires a bluish tint. The blue color of the sky is thus attrib- uted to the presence of extremely fine particles of dust in the air together with minute drops of condensed gases in the upper regions of the atmosphere. The visibility of a beam of light due to the scattering effect of minute particles, is known as the Tyndall phenomenon. Almost all colloidal solutions exhibit this phenomenon when a powerful beam of light is passed through them, thus proving the presence of discrete particles in the solutions. The ultramicroscope is an instrument devised by Siedentopf and Zsigmondy * for the detection of colloidal particles much too small to be seen by the naked eye. A powerful beam of light issuing from a horizontal slit is brought to a focus within the colloidal solution under examination by means of a microscope objective, and this image is viewed through a second micro- * " Colloids and the Ultramicroscope " by R. Zsigmondy. Trans, by Alexander. John Wiley & Sons, Inc. 242 THEORETICAL CHEMISTRY scope, the axis of which is at right angles to the path of the beam. When examined in this way a colloidal solution appears to be swarming with brilliantly colored particles, moving rapidly in a dark field; whereas a true solution, if properly prepared, appears optically void. With the ultramicroscope it is possible to count the number of particles present in a given volume of a col- loidal solution. By means of a chemical analysis, the mass of colloid per unit of volume can be determined and from this the average mass of each particle can be calculated. If the particles be assumed to be spherical in shape and to have the same density as larger masses of the same substance, we can calculate the volume of a single particle and from this its diameter. Thus, Burton * in his experiments on gold, silver and platinum sols, found the average diameter of the colloidal particles to range from 0.2 to 0.6 micron. Zsigmondy's latest ultramicroscope, Fig. 68, consisting of two compound microscopes placed at right angles and haying their objectives so cut away as to permit them to be brought together in focus, enables the observer to discern particles whose diameters range from 1 to 2 milli-microns. The ultramicroscopic character of emulsoids is by no means sharply defined, notwithstanding the fact that they exhibit the Tyndall phenomenon. It has been suggested by Zsigmondy that the lack of sharpness in definition observed with emulsoids is probably due to the relatively small difference between the re- fractive indices of the disperse phase and the dispersion medium. Where the difference between the refractive indices of the two phases is very great, as in the case of the metallic sols, excellent definition is obtained. It is of interest to note that although the basic hydroxides are apparently suspensoids, yet in their ultra- microscopic characteristics they closely resemble emulsoids. Ultrafiltration. Almost all sols can be filtered through ordinary filter paper without undergoing more than a slight change in con- centration due to initial adsorption. The rate of filtration varies widely, depending upon the viscosity of the sol. As a general rule, * Phil. Mag., ii, 425 (1906) . COLLOIDS 243 emulsoids filter more slowly than suspensoids owing to the high viscosity of the former. Fig. 68. By filtering an arsenious sulphide sol through a porous earthen- ware filter, Linder and Picton * succeeded in obtaining four differ- * Jour. Chem. Soc., 61, 148 (1892). 244 THEORETICAL CHEMISTRY ent sizes of particles which they described as follows: (1) visible under the microscope, (2) exhibited the Tyndall phenomenon, (3) retained by porous plate, and (4) passed through porous plate un- changed. By employing plates of different degrees of porosity and determining the average size of the pores which just permit nitra- tion, it is possible to determine the size of the particles which con- stitute the disperse phase of a sol. If we make use of a series of graduated filters, prepared by impregnating filter paper with a solution of collodion in acetic acid, it is not only possible to sepa- rate suspensoids from their dispersion media, but also to effect the concentration of emulsoids. Furthermore such filters are useful in removing impurities from sols, the impurity passing through the filter in a manner similar to the passage of the solvent through the membrane in the process of dialysis. Ultrafiltration is an exceed- ingly complex process involving the phenomena of adsorption and dialysis in addition to the ordinary process of mechanical separation. The complexity of the process is well illustrated by the phenomena attendant upon the filtration of almost any positive hydrosol. Thus, if we attempt to filter a ferric hydroxide hydrosol through a porous plate, or even through an ordinary filter paper, we shall find that the colloid will be partially retained by the filter. This is due to the fact that the filter becomes negatively charged in con- tact with water and, on the entrance of the positively charged sol into the pores of the filter, the colloid is immediately discharged and the disperse phase precipitated. After the pores of the filter become partially stopped with particles of the colloid, the sol will then pass through unchanged. Classification of Dispersoids. There is abundant evidence in favor of the view that colloidal solutions and simple suspensions are closely related. Suspensions of all grades exist, from those in which the suspended particles are coarse-grained and visible to the naked eye, down to those in which a high-power microscope is required to render the suspended particles visible. Colloidal solutions have also been shown to be non-homogeneous, the presence of discrete particles being revealed by means of the ultramicroscope. It follows, therefore, that the size of the particles in solution determines whether a substance is to be considered as COLLOIDS 245 a colloid or not. At one extreme we have true solutions in which no lack of homogeneity can be detected, even by the ultramicro- scope, and at the other extreme we have coarse-grained suspen- sions, in which the particles are visible to the naked eye. Between these two limits all possible degrees of subdivision are possible and it is a very difficult matter to draw sharp lines of distinction between true solutions and colloidal solutions on the one hand, and between colloidal solutions and suspensions on the other. One of the most satisfactory schemes of classification is that of von Weimarn and Wo. Ostwald.* Because of the fact that sus- pensions, colloidal solutions, and true solutions represent varying degrees of dispersion of the solute, all three types of system are termed by these authors, dispersoids. The dispersoids are classi- fied as shown in the accompanying diagram. DISPERSOIDS. Coarse dispersions (sus- pensions, emulsions, etc.). Magnitude of particles greater than 0.1 M-* Colloidal solutions. Molecular Ionic Magnitude of particles dispersoids. dispersoids. between 0.1 M and 1 MM- Decreasing degree of Magnitude of particles, "colloidity." about 1 MM and smaller. Increasing degree of dis- persion. * 1 M = 1 micron = 0.001 mm. Density of Colloidal Solutions. As we have seen, suspensoids are commonly regarded as sols in which the disperse phase is solid, while emulsoids are considered to be sols in which the disperse phase is liquid. While this distinction between the two classes of sols is generally well defined, it should be borne in mind that there * Koll. Zeitschrift, 3, 26 (1908). 246 THEORETICAL CHEMISTRY are colloidal solutions in which it is extremely difficult to determine the physical state of the disperse phase. The fundamental difference between suspensoids and emulsoids manifests itself most clearly in those properties which undergo appreciable change in consequence of solution. Among these may be mentioned density, viscosity and surface tension. It was shown by Linder and Picton * that the density of suspensoids follows the law of mixtures. This is clearly shown by the following table in which are given the observed and calculated values of the density of a series of arsenious sulphide sols. DENSITY OF ARSENIOUS SULPHIDE SOLS. As 2 S 3 (grams per liter). Density (obs.). Density (calc.). 44 1.033810 1.033810 22 1.016880 1.016905 11 1.008435 1.008440 2.45 1.002110 1.002100 0.1719 1.000137 1.000134 The density of emulsoids, on the other hand, cannot be calcu- lated from the composition of the sol. This fact may be taken as evidence in favor of the view that a closer relation exists between the disperse phase and the dispersion medium in emulsoids than in suspensoids. Viscosity of Colloidal Solutions. Owing to the fact that the concentrations of most suspensoids are relatively small, it follows that their viscosities differ but little from the viscosity of the pure dispersion medium. In general, it may be said that the viscosity of suspensoid sols is slightly greater than that of the dispersion medium. On the other hand, the viscosity of emulsoid sols is frequently much greater than that of the pure dispersion medium. The vis- cosity of emulsoids also increases with increasing concentration, as is shown by the data of the following table. * Jour. Chem. Soc., 67, 71 (1895). COLLOIDS 247 VISCOSITY OF EMULSOIDS. Sol. Temp. 20 Concentration. Viscosity. Per cent. Gelatine 1 021 Gelatine 2 037 Silicic acid. . . . 0.81 0.012 Silicic acid. . 99 0.016 Silicic acid. . . 1.96 0.032 Silicic acid 3.67 0.165 Viscosity of water at 20 = 0.0120. Surface Tension of Colloidal Solutions. The surface tension of suspensoid sols has been shown by Linder and Picton to be practically identical with that of the dispersion medium. As a general rule, the surface tension of emulsoid sols is appre- ciably smaller than that of the dispersion medium. According to Quincke * the surface tensions of gelatine sols are appreciably less than the surface tension of the dispersion medium. The differ- ence between suspensoids and emulsoids in respect to surface tension undoubtedly accounts for the fact that, in general, the former are not adsorbed while the latter are. Osmotic Pressure of Colloidal Solutions. The osmotic pres- sure of colloidal solutions is very small. This is what we should expect with solutions of substances which exhibit a slow rate of diffusion. As has been pointed out, diffusion is closely connected with osmotic pressure; hence, if the rate of diffusion is slow, the osmotic pressure exerted by the solution should be small. In some eases the osmotic pressure is so small as to escape detection. The experimental determination of the osmotic pressure of col- loidal solutions is complicated by the difficulty of removing the last traces of electrolytes from the colloid. Owing to their great osmotic activity, the presence of the merest trace of electrolytes may mask the true osmotic effect of the colloid. It should be borne in mind, however, that semipermeable membranes are much less permeable by colloids than by electrolytes and, in consequence of this fact, the impurities in the colloid would be gradually removed by prolonged dialysis. If the total osmotic pressure were due to * Wied. Ann., Ill, 35, 582 (1885). 248 THEORETICAL CHEMISTRY the presence of small amounts of impurities in the colloid, then as these are removed, the pressure should steadily diminish and ulti- mately become zero. As a matter of fact, the final value of the osmotic pressure of a colloidal solution, although generally very small is never zero. This final, positive value of the osmotic pressure has been shown to be wholly independent of the method of preparation of the sol. Although differences in the method of preparation may introduce different impurities which give rise to different initial values of the osmotic pressure, in each case the same final value is obtained. Of course it must be admitted that the possibility exists that a minute portion of electrolyte which cannot be removed by dialysis is retained by the colloid, but even then it is difficult to account for the constancy of the final value of the osmotic pressure irrespective of the method of preparation of the sol. The values of the osmotic pressure of suspenoids are in- variably small and by no means concordant. The following table gives the results obtained by Duclaux * with colloidal solutions of ferric hydroxide. OSMOTIC PRESSURE OF COLLOIDAL FERRIC HYDROXIDE. Cone. Fe(OH) 3 Per cent. Pressure in cm. of Water. 1.08 0.8 2.04 2.8 3.05 5.6 5.35 12.5 8.86 22.6 Inspection of the table shows that, even in the most concen- trated solution, the osmotic pressure is very small. Furthermore, it is apparent that although the osmotic pressure increases with the concentration, the variables are not proportional. Observa- tions on the variation of the osmotic pressure of colloidal solu- tions with temperature, show that, in general, as the temperature is raised the pressure increases at a more rapid rate than that required by the law of Gay-Lussac. Employing membranes of collodion and parchment paper, * Compt. rend., 140, 1544 (1905); Jour. Chim. Phys., 7, 405 (1909). COLLOIDS 249 Lillie * and others have demonstrated that 'the values of the os- motic pressure of emulsoids are, in general, considerably greater than the corresponding values obtained with suspensoids. The osmotic pressures of several typical emulsoids are given in the following table. OSMOTIC PRESSURES OF EMULSOIDS. Sol. Concentration (grams per liter) . Osmotic Pressure (mm. of mercury). Egg albumin 12.5 20 Gelatine 12 5 6 Starch iodide 30 15 Dextrin 10 165 It will be observed that the values of the osmotic pressure in the preceding table are appreciably greater than those given for ferric hydroxide hydrosols. This is in agreement with the well-estab- lished fact that emulsoids diffuse more rapidly than suspensoids. It has been observed, that the value of the osmotic pressure of gelatine solutions at ordinary temperatures can be increased by maintaining the solutions at a higher temperature for a short time and then cooling to the initial temperature. After standing for several days at the original temperature, however, the osmotic pressure of the solution returns to its former value. This phenom- enon would seem to indicate th^t the osmotic pressure of colloidal solutions is not completely defined by the two variables, tempera- ture and concentration. It has been suggested that the degree of aggregation of the colloid is partially dependent upon the tempera- ture; the molecular aggregates tending to break up as the tempera- ture is raised, thus increasing the number of dissolved units and therefore causing a corresponding increase in the osmotic pressure. Molecular Weight of Colloids. We have already learned that the knowledge of the osmotic pressure of a solution enables us to calculate the molecular weight of the solute, provided the solu- tion is dilute and obeys the gas laws. As we have seen, other * Am. Jour. Physiol., 20, 127 (1907), 250 THEORETICAL CHEMISTRY factors than concentration and temperature determine the osmotic pressure of colloidal solutions, so that we are not justified in attempting to calculate the molecular weight of a colloid from the observed value of the osmotic pressure of its solution. Values for the molecular weight of colloids calculated from their effect on the vapor pressure, the boiling-point, and the freezing-point of the solvent are also untrustworthy, since the same factors which influence the osmotic pressure necessarily affect these related properties. This becomes evident when we reflect that an osmotic pressure of 1 mm. of mercury corresponds to a depression of the freezing-point of about 0.0001. Owing to the difficulty of ob- taining absolutely pure emulsoid sols, all determinations of their freezing-point depressions must be affected with an experimental error appreciably larger than the observed depression. Hydrosols of albumin, gelatine, etc., prepared with extreme care by Bruni and Pappada * failed to produce any detectable depression of the freezing-point of water. Electroendosmosis. The movement of a liquid through a porous diaphragm, due to the passage of an electric current be- tween two electrodes placed on opposite sides of the diaphragm, is known as eledroendosmosis. This phenomenon, which was first observed by Reuss in 1807, has since been made the subject of numerous investigations by Wiedemann,f Quincke J and Perrin, the latter having worked out a satisfactory theoretical interpreta- tion of the phenomenon. If a porous partition be placed in the horizontal portion of a U-tube and an electrode be inserted in each arm of the tube, it will be found, on filling the tube with a feebly conducting liquid and passing a current, that the liquid will com- mence to rise in one arm of the tube and will continue to rise until a definite equilibrium is established. For a given difference of potential between the two electrodes, there will be a definite differ- ence in the level of the liquid in the two arms of the tube. The majority of substances acquire a negative electric charge when * Rend. R. Accad. dei Lincei, (5), 9, 354 (1900). t Pogg. Ann., 87, 321 (1852). t Ibid., 113, 513 (1861). Jour. Chim. Phys., 2, 601 (1904). COLLOIDS 251 immersed in water. The water, under these conditions, becomes positively charged and will, in consequence, migrate toward the cathode. On the other hand, certain substances acquire a positive charge on immersion in water and in these cases the direction of migration will obviously be reversed. It has been found that acids cause negative diaphragms to be- come less negative and positive diaphragms to become more posi- tive. The action of alkalies is, as we should expect, the reverse of that of acids. There is an interesting connection between the valence of the ions resulting from the dissociation of dissolved salts, and the difference of potential existing between the liquid and the diaphragm. When the diaphragm is positively charged, the difference of potential is found to be 'conditioned by the valence of the anion, and when the diaphragm is negatively charged, the difference of potential is determined by the valence of the cation. Cataphoresis. When a difference of potential is established between two electrodes immersed in a suspension of finely-divided quartz or shellac, the suspended particles move toward the positive electrode. This phenomenon is called cataphoresis and was first observed by Linder and Picton.* They showed that when the terminals of an electric battery are connected to two platinum electrodes dipping into a colloidal solution of arsenious sulphide, there is a gradual migration of the colloid to the positive pole. A similar experiment with a solution of colloidal ferric hydroxide resulted in the transport of the dissolved colloid to the negative pole. It follows, therefore, that the particles of colloidal arsenious sulphide are negatively charged, while those of colloidal ferric hydroxide carry a positive charge. It has been found that all colloids carry an electric charge. In the table on page 252, some typical colloids are classified according to the character of their electrification in aqueous solution. The nature of the charge varies with the dispersion medium used, colloidal solutions in turpentine, for example, having charges oppo- site to those in water. Direct measurements of the velocity with which the particles * Jour. Chem. Soc., 61, 148 (1892). 252 THEORETICAL CHEMISTRY move in cataphoresis have been made by Cotton and Mouton.* By observing with a microscope the distance over which a single ELECTRICAL CHARGES OF HYDROSOLS. Electro-positive. Electro-negative. Metallic hydroxides Methyl violet Methylene blue Magdala red Bismarck brown Haemoglobin All the metals Metallic sulphides Aniline blue Indigo Eosine Starch particle traveled in a given interval of time under a definite po- tential gradient, they calculated the average velocity of migration of a number of suspensoids. The following table gives the velocity of migration of a few typical suspensoids. VELOCITY OF MIGRATION OF SUSPENSOIDS. f Suspensoid. Average Diameter of Particles. Velocity cm. /sec. for Unit Potential Gradient. Arsenic trisulphide. ... ... . 50M 22X10" 5 Quartz 1 MM 30X10- 5 Gold (colloidal) < 100 MM 40X10- 5 Platinum (colloidal) <100 MM 30X1Q-* Silver (colloidal) < 100 MM 23 6X10- 5 Bismuth (colloidal) . . < 100 MM 11 OX10- 5 Lead (colloidal) <100 uu 12 OX1Q-* Iron (colloidal) <100 U.LL 19 OxlO~ 6 Ferric hydroxide (colloidal) 100 MM 30 OX 10-* t Freundlich, Kapillarchemie, p. 234. It will be observed that not only are the velocities of migration nearly constant, but also that they are apparently independent of the size and nature of the particles. The presence of electrolytes, especially acids and bases, exercises a marked effect upon the electrical behavior of suspensoids. Owing to the comparative instability of suspensoids, the addition of electrolytes usually results in the complete precipitation of the colloid. * Jour. Chim. Phys., 4, 365 (1906). COLLOIDS 253 Emulsoids also show the phenomenon of cataphoresis, but their velocities of migration are appreciably less than the corresponding velocities of suspensoids, and their behavior in an electric field is such as to make it appear quite probable that the character of their electric charge is entirely fortuitous. Furthermore, emulsoids are much more susceptible to the influence of electrolytes than are suspensoids. W. B. Hardy * has found that the direction of migration of albumin, modified by heating to 100 C., is dependent upon the reaction of the dispersion medium. A very small quantity of free base causes the particles of albumin to move toward the positive electrode, while the addition of an equally small amount of acid results in a reversal of the direction of migration. Similar reversals of charge have been observed by Burton f in colloidal solutions of gold and silver. When small amounts of aluminium sulphate are added to colloidal solutions of these metals, the charge is gradually neutralized and eventually the colloidal particles acquire a reversed charge. Electrical Conductance of Colloidal Solutions. The electrical conductance of suspensoids differs so slightly from that of the pure dispersion medium that it is difficult to decide whether the small increase in conductance may not be due to the presence of traces of adsorbed electrolytes. In order to ascertain to what extent the conductance of suspensoids is dependent upon the presence of adsorbed electrolytes, Whitney and Blake J investigated the effect of successive electrolyses upon the conductance of a gold hydrosol. If the pure sol is incapable of enhancing the conductance of the dispersion medium, then as the sol is subjected to successive elec- trolyses, the conductance should steadily diminish and ultimately become identical with that of the dispersion medium. Whitney and Blake found that the conductance converged to a definite limiting value which was slightly greater than the conductance of the dispersion medium. From these experiments we seem to be warranted in concluding that suspensoids conduct the electric current very feebly. * Jour. Physiol., 24, 288 (1899). t Phil. Mag., 12, 472 (1906). J Jour. Am. Chem. Soc., 26, 1339 (1904). 254 THEORETICAL CHEMISTRY Emulsoids appear to conduct rather better than suspensoids. Whitney and Blake measured the conductance of silicic acid and gelatine sols and found the specific conductance of the former to be 100 X 10~ 6 reciprocal ohms and that of the latter to be 68 X 10~ 6 reciprocal ohms. On the other hand, Pauli * found that an albumin sol which had been prepared with extreme care was virtually a non-conductor. It should be remembered, however, that the albumins are closely related to the simple amino-acids which are known to be exceedingly poor conductors. Precipitation of Colloids by Electrolytes. One of the most important and interesting divisions of the chemistry of colloids is that which treats of the precipitation of suspensoids and emulsoids by electrolytes. In general, x it may be said that the precipitation of colloids by electrolytes is an irreversible process. Colloidal solutions are more or less unstable systems irrespective of the methods employed in their preparation, and the addition of a small amount of an electrolyte is usually found to be sufficient to cause the sol immediately to become opalescent, and ultimately to pre- cipitate, leaving the dispersion medium perfectly clear and free from the disperse phase. Some exceptions to this general state- ment as to the behavior of colloids are known. For example, Whitney and Blake f found that precipitated gold could be caused to return to the colloidal state by treatment with ammonia, while Linder and Picton { discovered that a ferric hydroxide hydrosol, which had been precipitated with sodium chloride, could be re- stored to the colloidal condition by simply removing the electrolyte with water. The sedimentation of suspensions, such as kaolin in water, is also promoted by the addition of electrolytes. On the other hand, non-electrolytes have no effect upon the stability of suspensoids or simple suspensions. Precipitation of Suspensoids. The phenomenon of the pre- cipitation of suspensoids has been carefully investigated by Freundlich. He has found that an amount of electrolyte which * Beitrag. Chem. Phys. Path., 7, 531 (1906). t Jour. Am. Chem. Soc., 26, 1341 (1904). t Jour. Chem. Soc., 61, 114 (1892); 87, 1924 (1905). Zeit. phys. Chem., 44, 131 (1903). COLLOIDS 255 is incapable of bringing about an instantaneous precipitation, may become effective after an interval of time. He has also shown that the total quantity of electrolyte required to precipitate a suspensoid completely depends upon whether the electrolyte is added all at one time or in successive portions. In order to com- pare the precipitating action of various electrolytes, Freundlich proposed the following procedure, which prevents the possibility of irregularities due to the time factor: To 20 cc. of a solution of a suspensoid, 2 cc. of the solution of the electrolyte are added, the resulting solution being shaken vigorously; the mixture is then set aside for two hours, after which a small portion is filtered off, and the filtrate is examined for the suspensoid. In the following table some of the results obtained by Freundlich with colloidal solu- tions of ferric hydroxide are given. The data represent the mini- mum concentration for each electrolyte which produced precipita- tion in two hours. It will be seen that very small amounts of the electrolytes are required to precipitate the suspensoid, and further, that the pre- cipitating power of an electrolyte is dependent upon the charge of the negative ion. The greater the charge, the smaller is the quantity of electrolyte required to produce precipitation. PRECIPITATING ACTION OF ELECTROLYTES ON FERRIC HYDROXIDE HYDROSOL. (16 milli-mols Fe(OH) 2 per liter.) Electrolyte. Concentration (milli-mols per liter). NaCl 9.25 KC1 9.03 BaCl 2 9.64 KN0 3 11.9 KBr 12.5 Ba(N0 3 ) 2 14.0 KI 16.2 HC1 400 ca. Ba(OH) 2 0.42 H 3 (SO) 4 0.204 MgS0 4 0.217 K 2 Cr 2 O 7 0.194 H 2 SO 4 0.5ca. 256 THEORETICAL CHEMISTRY The significance of the relation between ionic charge and pre- cipitating power was first pointed out by Hardy,* who formulated the following rule : The precipitation of a colloidal solution is de- termined by that ion of an added electrolyte which has an electric charge opposite in sign to that of the colloidal particles. It has already been pointed out that colloidal particles of arse- nious sulphide are negatively charged, hence, according to Hardy's rule, the positive ions of the added electrolyte will condition the precipitation of the suspensoid. The experiments of Freundlich confirm this prediction, as is shown by the following table: PRECIPITATING ACTION OF ELECTROLYTES ON ARSENIOUS SULPHIDE HYDROSOL. . (7.54 milli-mols As 2 S 3 per liter.) t Electrolyte. Concentration (milli-mols per liter). KC1 49.5 KNO 3 50.0 KC 2 H 3 O 2 110.0 NaCl 51.0 LiCl 58.4 MgCl 2 0.717 MgS0 4 0.810 . CaCl 2 0.649 SrCl 2 0.635 BaCl 2 0.691 Ba(N0 3 ) 2 0.687 ZnCl 2 0.685 Aids 0.093 A1(N0 3 ) 3 0.095 Precipitation and Valence. An examination of the preceding tables reveals the fact that although the ionic concentration neces- sary to bring about precipitation, in accordance with Hardy's rule, decreases with increasing valence, the diminution in concentration is not, as we might expect, inversely proportional to the valence of the precipitating ion. The absence of any simple quantitative relation between the valence of an ion and its precipitating con- * Zeit. phys. Chem., 33, 385 (1900). COLLOIDS 257 centration is undoubtedly due to the influence of several potent factors, such as adsorption and the protective action of ions whose electric charge is of the same sign as that of the colloidal substance. Precipitation of Emulsoids. The action of electrolytes on emulsoids is much more obscure than the action of electrolytes on suspensoids. Nothing approaching a generalization similar to Hardy's rule for the precipitation of suspensoids has been found to apply to the precipitation phenomena manifested by emulsoids. Owing to the fact that emulsoids are liquids, and in consequence of their greater degree of dispersity, it has been suggested that emulsoids probably resemble true solutions more closely than sus- pensoids. In fact there is reason for assuming that a portion of the colloid is actually dissolved in the dispersion medium. This may account for the fact that the precipitation of emulsoids is sometimes reversible and sometimes irreversible. It is to be re- gretted that, up to the present time, so many of the investigations on emulsoids have been carried out with materials of questionable purity and of insufficient uniformity. The addition of salts to gelatine generally causes irreversible precipitation, provided the concentration of the salt is not too low. The precipitation of albumin by some salts is reversible while by others it is irreversible. Those transformations which are initially reversible gradually become irreversible on standing. Although relatively concentrated solutions of salts of the alkalies and alkaline earths are required to precipitate albumin, very dilute solutions of the salts of the heavy metals are found to be sufficient to bring about complete precipitation. Action of Heat on Emulsoids. When an albumin hydrosol is gradually heated, a temperature is ultimately reached at which coagulation occurs. The exact nature of this transformation is not understood, but it is believed to be largely chemical. This belief is based upon the fact that the reaction of the natural albu- mins toward litmus is altered by heating. Slight acidity of the sol is essential to complete coagulation, while an excess or a de- ficiency of acids causes a portion of the albumin to remain in the sol. The presence of various salts has been found to exert a marked influence on the temperature of coagulation of albumin. 258 THEORETICAL CHEMISTRY The coagulation temperature is invariably raised at first, attaining a constant value in some cases, while in others it decreases after reaching a maximum temperature. It is especially interesting to note that the anions of the salts follow the usual lytropic sequence. The effect of heat on a gelatine sol is very different from its effect on an albumin sol. If a fairly concentrated gelatine sol is heated and then permitted to cool, it sets into a jelly which is not reconverted into a sol when the temperature is again raised. Fur- thermore, the change does not take place at a definite temperature. In studying the phenomenon of gelation, either the temperature or the time of gelation may be determined. The melting-point of pure gelatine ranges from 26 to 29 while the solidifying tempera- ture lies between 25 and 18. The melting and solidifying tem- perature of gelatine sols vary with the concentration; a 5 per cent sol melts at 26. 1 and solidifies at 17.8, while a 15 per cent sol melts at 29.4 and solidifies at 25.5. The temperature of the gel- sol transformation is affected by the addition of salts, some tending to raise the temperature of gelation and others to lower it. The order of the anions arranged according to their influence on the gel-sol transformation is as follows : Raising temperature : SO 4 > Citrate > Tartrate > Acetate (H 2 0). Lowering temperature: (H 2 0) Cl < C10 3 < NO 3 < Br < I. The same lytropic order was found by Schroeder * in an investi- gation of the viscosity of gelatine sols. It is noteworthy that the influence of salts on the temperature of gelation of agar-agar and other similar substances is analogous to their effect on gelatine, the same lytropic sequence being main- tained. Protective Colloids. The precipitating action of electrolytes on suspensoids may be inhibited by adding to the solution of the suspensoid a reversible colloid. The protective action of a re- versible colloid is not due, as might be supposed, to the increased viscosity of the medium and the resultant resistance to sedimenta- tion, since amounts of a reversible colloid, too minute to produce any appreciable increase in the viscosity of the medium, can pre- * Zeit. phys. Chem., 45, 75 (1903). COLLOIDS 259 vent precipitation. Thus, Bechhold * has shown that while a mixture of 1 cc. of a suspension of mastic and 1 cc. of a 0.1 molar solution of MgS0 4 diluted to 3 cc. with water, is completely pre- cipitated in 15 minutes, no precipitation will occur within 24 hours, if two drops of a 1 per cent solution of gelatine be added before diluting to 3 cc. Gum arabic and ox-blood serum exert a similar protective action when added to a suspension of mastic. The protective power of reversible colloids differs widely and Zsigmondy f has attempted to make this the basis of a method of classification of colloidal sub- stances. A red solution of colloidal gold becomes blue on the addition of a small amount of sodium chloride, owing to the in- crease in the size of the colloidal particles. Various colloidal sub- stances when added to a red colored gold sol protect the colloidal particles from precipitation by a solution of sodium chloride, no change in color following the addition of the electrolyte. A definite amount of each colloidal substance is required to prevent the change from red to blue in the color of the gold sol. In employing this color change as a means of differentiating colloidal substances, Zsigmondy introduced the "gold number," which may be defined as the weight in milligrams of a colloidal substance which is just insufficient to prevent the change from red to blue in 10 cc. of a gold sol after the addition of 1 cc. of a 10 per cent solution of sodium chloride. The following table gives the gold numbers of a few colloids. GOLD NUMBERS OF COLLOIDS. Colloid. Gold Number. Gelatine 005-0 01 Casein (in ammonia) 01 Egg-albumin Gum arabic Dextrin Starch, wheat Starch, potato 0.15-0.25 0.15-0.25)0.5-4 6-12: 10-20 4-6 (about) 25 (about) Sodium stearate . . 10 (at 60 )- 01 (at 100) Sodium oleate Cane sugar Urea 0.4-1 8 8 * Zeit. phys. Chem., 48, 408 (1904). f Zeit. analyt. Chem., 40, 697 (1901). 260 THEORETICAL CHEMISTRY The gold number has proven useful in differentiating the various kinds of albumin, as is shown in the following table. GOLD NUMBERS OF ALBUMINS. Albumin. Gold Number. Egg white (fresh) Globulin 0.08 0.02-0.05 Ovomucoid . 0.04-0.08 Albumin (Merck) 0.1-0.3 Albumin (cryst.) Albumin (alkaline) 2-8 0.006-0.04 The addition of alkali to any one of the first five albumins of the above table reduces the gold number to that of alkaline albumin. Sulphide sols may be protected as well as metallic sols, and further- more the ability to exert protective action is not confined to organic colloids alone. In general, it may be said that when a suspensoid sol is mixed with an emulsoid sol in the proper proportions, the suspensoid sol acquires most of the characteristic properties of the protecting colloid. The masking of the properties of a suspensoid sol by a protecting colloid is probably to be ascribed to the formation of a thin film of adsorbed emulsoid over the suspensoid. Reciprocal Precipitation. A further deduction of the elec- trical theory of precipitation is, that when two oppositely-charged colloids are mixed, they should precipitate each other, and the resulting precipitate should contain both colloids. Experiments carried out by Biltz* have confirmed these predictions. He showed that when a solution of a positively-charged colloid is added to a solution of a negatively-charged colloid, precipitation occurs, unless the quantity of the added colloid is either relatively very large or very small. He also showed that when two colloids of the same electrical sign are mixed no precipitation occurs. Just as the amount of precipitation caused by the addition of an elec- trolyte to a sol is conditioned by the rate at which the electrolyte is added, so also the precipitation of one colloid by another is de- * Berichte, 37, 1095 (1904). COLLOIDS 261 pendent upon the manner in which the two sols are mixed. The extent to which one sol is precipitated by another sol of opposite sign is largely determined by the amount of one that is added to a definite amount of the other. This is clearly shown by the data of the following table which give the results obtained by Biltz on adding ferric hydroxide sol to 2 cc. of an antimony trisulphide sol containing 2.8 mg. per cc. PRECIPITATION OF COLLOIDAL ANTIMONY TRISULPHIDE BY COLLOIDAL FERRIC HYDROXIDE. Fe2O 3 (mg.). Immediate Result. Result after One Hour. 0.8 Cloudy Almost homogeneous 3.2 Small flakes Unchanged 4.8 Flakes Yellow liquid 6.4 8.0 12.8 Complete precipitation Slow precipitation Cloudy Complete precipitation Complete precipitation Slight precipitation 20.8 Cloudy Homogeneous It will be seen that the addition of a small amount of ferric hydroxide produces hardly any precipitation. With the addition of larger amounts of ferric hydroxide, the amount of precipitation increases until finally complete precipitation is attained. The addition of larger quantities of ferric hydroxide produces either little or no precipitation. It has been found that at the concen- tration which just produces complete precipitation, the electrical charges on the two sols are equivalent. When the amount of ferric hydroxide exceeds that required for complete precipitation, it is more than probable that the particles of colloidal antimony tri- sulphide are completely enveloped by the particles of ferric hy- droxide and thereby rendered inactive. When we come to study the action of one emulsoid on another, we find, as might be expected from the general behavior of emul- soids toward electrolytes, that the phenomena are more complex and very much less well-defined than with suspensoids. Although mutual precipitation does take place with emulsoids, the close resemblance between emulsoids and true solutions renders the phenomenon more or less indistinct. 262 THEORETICAL CHEMISTRY When a suspensoid sol is added to an emulsoid sol having an opposite electric charge, precipitation may or may not occur according to the relative amounts of the two colloids in the mix- ture. When the two colloids are present in electrically equivalent quantities precipitation occurs, otherwise one colloid exerts a protective action on the other. Characteristics of Gels. Gels are generally obtained by cooling or evaporating emulsoid sols and, since the latter are known to be two-phase liquid systems, it is natural to infer that gels may also be two-phase systems. According to this conception, the only difference between an emulsoid sol and a gel is, that in the latter the concentration of at least one of the phases is greatly increased and thereby imparts greater viscosity and rigidity to the system. It has been suggested that the more concentrated of the two phases forms the walls of an assemblage of cells within which the more dilute phase is enclosed. The view that gels possess a distinct cellular structure is fully confirmed by microscopic examination. The extreme sensitiveness of gels to changes in temperature and to the presence of extraneous substances renders their investi- gation exceedingly difficult. Notwithstanding the experimental difficulties involved in the study of gels, sufficient knowledge has been gained of their properties to make a brief account of these necessary in any treatment of the subject of colloids. Physical Properties of Gels. The process of gel formation from a dry gelatinous colloid and water invariably involves con- traction. This statement should not be confused with the fact that a gel on immersion in water undergoes appreciable increase in volume. Gels have been shown by Barus * to be considerably more com- pressible than solids. The compressibility increases as the tem- perature is raised until, when the gel is transformed into the sol, the compressibility becomes equal to that of pure water. The temperature of some gels, such as rubber and gelatine, is lowered by compression and raised by tension. The thermal expansion of gels is nearly identical with that of the more fluid component of the gel. * Am. Jour. Sci., 6, 285 (1898). COLLOIDS 263 The rate at which pure substances diffuse in gels differs only slightly from the rate of diffusion in pure water, provided the con- centration of the gel is not too great: The slight resistance offered by gels to diffusion of dissolved substances may be regarded as further evidence in favor of their cellular structure. The modulus of elasticity of a gel, cast in a cylindrical mold, is given by the formula PI where P is the tension which produces the increase in length AZ in a cylinder whose length is I and whose radius is r. It has been found that the modulus of elasticity in gelatine gels increases as the square of the concentration of the gel. The time of recovery, after releasing the tension, increases as the concentration of the gel increases. The shearing modulus for a gel is given by the formula 2(1+;*) where /* denotes the ratio of the relative contraction of the diameter to the relative change in length. The viscosity of a gel may be calculated from the shearing modulus by means of the equation n = E s r, where r denotes the time of recovery. Since both E s and r increase with the concentration of the gel it is apparent that the viscosity of the gel must also increase with the concentration. As is well known, when glass is subjected to pressure or is un- equally strained, it exhibits the phenomenon of double refraction. Since glass bears some resemblance to gels in being a highly viscous, supercooled liquid, it might reasonably be inferred that gels should also show double refraction. Experiments with collodion and gelatine have shown that these substances, when subjected to pressure, behave similarly to glass. Hydration and Dehydration of Gels. The complementary processes of hydration and dehydration of gels are extremely inter- 264 THEORETICAL CHEMISTRY esting. Following Freundlich * we will consider the subject very briefly under the two following heads: (a) Non-elastic Gels and (b) Elastic Gels. (a) Nvn-elastic Gels. When freshly prepared aluminium or fer- ric hydroxide gels were placed in a desiccator, it was found by van Bemmelen f that the rate at which these substances lost water was continuous. Furthermore, on removing the dried gels from the desiccator it was found that the process of recovery of moisture was also continuous. It will be shown in a subsequent chapter (p. 320) that a definite hydrate in the presence of its products of dissociation, possesses a constant vapor pressure so long as any of that particular hydrate is present. The fact that the vapor pressure of the gels investigated by van Bemmelen did not remain constant but decreased continuously as the water was removed, proved conclusively that no chemical compounds were formed in the process of gel hydration. Another gel which has been made the subject of much careful investigation is silicic acid. The dehydration curve of silicic acid is continuous, with the exception of a short portion where an ap- preciable amount of water is lost without much of any change in the vapor pressure. This portion of the curve corresponds to a marked change in the appearance of the gel. The gel which had hitherto been clear and transparent became opalescent 'soon after the vapor pressure had attained a temporarily constant value. The opalescence gradually permeated the entire mass, until the gel acquired a yellow color by transmitted light, and a bluish color by reflected light. These colors suggest a marked increase in the degree of dispersity of the gel, an inference the correctness of which subsequent investigation has fully confirmed. The curve of hydration, while resembling the curve of dehydra- tion in many respects, departs quite widely from it in others. The change in dispersity, as indicated by the appearance in reverse order of the color and opalescent phenomena mentioned above, also was manifest. * Kapillarchemie, p. 486. t Zeit. anorg. Chemie, 5, 466 (1894); 13, 233 (1897); 18, 14, 98 (1898); 30, 265 (1902). COLLOIDS 265 (b) Elastic Gels. The absence in elastic gels of a horizontal portion in the dehydration curves, and the fact that although elastic gels may have become saturated with water vapor, they still retain the power of taking up large amounts of liquid water when immersed in that medium, constitute the chief differences between elastic and non-elastic gels. The amount of water which can be taken up by an elastic gel is exceedingly large. Thus, on exposing a plate of gelatine weighing 0.904 gram for eight days in an atmosphere saturated with water vapor, Schroeder * found that it had taken up 0.37 gram of water. On exposing for a longer period of time under the same conditions, he found no further gain in weight, and on removing the gelatine plate from the moist atmosphere and placing it in a desiccator, the plate slowly gave up the absorbed moisture and regained its origi- nal weight. On the other hand, when the plate, after having absorbed the maximum weight of moisture from the air, was im- mersed in water, it was found to increase in weight very rap- idly. Thus, on immersing the above plate which weighed 1.274 grams when saturated in moist air, and allowing it to remain in water for one hour, it was found to have taken up 5.63 grams of water. After an immersion of twenty-four hours, the plate was found to have taken up the maximum weight of water it was capable of absorbing at that temperature. On remov- ing the plate, it was found to part with the absorbed water very readily, even in moist air, the greater part of the absorbed water being so loosely held that the vapor pressure of the gel remained the same as that of pure water at the temperature of the experiment. It is impossible to measure directly the pressure produced by gels when they take up water, but some idea of the magnitude of these pressures may be obtained by coating a glass plate with gelatine, which has imbibed the maximum amount of water, and observing the degree to which the glass plate is bent by the drying gelatine film. Frequently the elastic limit of the glass is exceeded and the plate breaks under the stress produced by the dehydration of the gel. * Zeit. phys. Chem., 45, 75 (1903). 266 THEORETICAL CHEMISTRY Velocity of Imbibition. The rate at which gels imbibe water has been studied by Hofmeister.* The velocity of imbibition of water by thin plates of gelatine and agar-agar was measured by removing the plates at definite intervals and determining their increase in weight. Owing to the time consumed in making the weighings, the time intervals are affected by an appreciable error. The following table gives the data of a single experiment with gelatine. VELOCITY OF IMBIBITION IN GELATINE. (Thickness of plate 0.5 mm.) Time (min.). Water Imbibed (grams). I 5 3.08 0.090 10 3.88 0.084 15 4.26 0.084 20 4.58 0.064 25 4.67 0.075 00 4.96 If the weight of water imbibed in t minutes is w t , and w^ is the maximum weight of water which a gel can take up under the con- ditions, then the velocity of imbibition should be given by the equation dw , -~jj = k(w M - w t ), which on integration becomes t (w x - w t ) The figures in the third column of the preceding table were cal- culated by means of this equation and, although the values are not strictly constant, the variation is no greater than might be expected where the experimental error is so large. Heat of Imbibition. The process of imbibition is accom- panied by an evolution of heat. Quantitative measurements * Arch. exp. Pathol. u. Pharmakol., 27, 395 (1890). COLLOIDS 267 of the heat evolved when gels take up water have been made by Wiedemann and Ludeking * and also by Rodewald.f The following table gives Rodewald's data on the heat of imbibition of starch. HEAT OF IMBIBITION OF STARCH. (Weight of dry starch 100 grams.) Per Cent, Water. Heat in Calories per Gram of Starch. 0.23 28.11 2.39 22.60 6.27 15.17 11.65 8.43 15.68 5.21 19.52 2.91 It will be observed that the greatest development of heat ac- companies the initial stages of imbibition where very small amounts of water are taken up. This is what we might expect when we remember that it is the last remaining portion of water which is most difficult to remove from a gel, and that it is only through the application of heat that its complete removal can be effected. Imbibition in Solutions. When a gel is immersed in a saline solution, the salt distributes itself between the solvent and the gel. The rate of imbibition is found to vary greatly according to the salt which is present in the solution. Thus, the velocity of imbibition has been found to be accelerated by the presence of the chlorides of ammonium, sodium and potassium and by the nitrate and bromide of sodium ; on the other hand, the pres- ence of the nitrate, sulphate and tartrate of sodium retard im- bibition. The effect of acids and bases on imbibition appears to be similar to the influence of salts. Adsorption. The change in concentration which occurs at the boundary between two heterogeneous phases is termed adsorp- * Wied. Ann., 25, 145 (1885). t Zeit. phys. Chem., 24, 206 (1897). 268 THEORETICAL CHEMISTRY tim. At the surface of a solid surrounded by a gas or vapor, the phenomenon is generally known as gaseous adsorption, since any difference which may occur in the concentration of the solid phase is much too small to be detected. At the boundary between liquid and gaseous phases, the concentration of each phase un- doubtedly undergoes alteration. In the case of the boundary, be- tween solid and liquid phases, the only apparent inequality in concentration occurs on the liquid side of the boundary, notwith- standing the fact that the adsorbed substance is quite commonly regarded as being bound to the surface of the solid phase. The cause of this erroneous conception is, that the extremely thin layer of liquid in which the alteration in concentration actually occurs, is the layer which wets the surface of the solid and hence is the layer which adheres to the solid when it is removed from the liquid. The retention of gases by charcoal is a typical example of gaseous adsorption, while the removal of coloring matter by charcoal in the purification of various organic substances may be cited as an example of adsorption of a liquid by a solid. If the adsorbed substance increases in concentration in the vicinity of the boundary, the adsorption is said to be positive; if it decreases, the adsorption is said to be negative. Adsorption of Gases. In gases, adsorption-equilibrium is attained with remarkable rapidity. Thus, if a gas is admitted into a vessel containing some freshly prepared cocoanut charcoal, the pressure will fall immediately to a value which corresponds to the removal of the entire adsorbed volume of gas. The concentration of adsorbed gas on the surface of a solid, when equilibrium is attained, has been shown to be approximately 1 X 10~ 7 gram per square centimeter. This value is of the same order of magnitude as the strength of the limiting capillary layer of a liquid and, therefore, lends support to the suggestion put for- ward some years ago by Faraday that an adsorbed film of gas may be present in the liquid state. The amount of gas adsorbed by a solid increases with the pres- sure and diminishes with increasing temperature. The following empirical equation, expressing the relation between the amount of COLLOIDS 269 gas adsorbed by a solid and the pressure, has been proposed by Freundlich * x - 1 = apn , m where x is the total mass of gas adsorbed on a surface of m sq. cm. under a pressure p, and where a and n are constants. This equation, known as the adsorption isotherm, has been found to hold quite generally. Adsorption in Solutions. The adsorption phenomena occur- ring at the surface of contact of a solid with a solution are similar to the phenomena which have just been discussed. Because of the frequency of its occurrence in many of the more common operations of both laboratory and factory, the subject of adsorption in solu- tions deserves fuller treatment. The general characteristics of adsorption in solutions may be briefly summarized as follows : (1) Adsorption in solutions is generally positive, i.e., on shaking a solution with a finely-divided adsorbent, the volume concentra- tion of the solution will diminish. (2) The amount of positive adsorption may be sufficient to re- move almost all of the solute from a solution, especially if the solu- tion is dilute. On the other hand, negative adsorption is always very small, and frequently is immeasurable. (3) Adsorption is directly proportional to the so-called " specific surface," the latter term being defined as the ratio of the total. sur- face of the adsorbent to its volume. (4) On shaking a definite weight of an adsorbent with a given volume of solution of known concentration, a definite equilibrium will be established. If the solution is then diluted with a known amount of solvent, the adsorption will decrease until it acquires the same value which it would have attained, had the same weight of adsorbent been introduced directly into the more dilute solution. For example, if 1 gram of charcoal is agitated with 100 cc. of a 0.0688 molar solution of acetic acid for 20 hours, adsorption is found to reduce the original concentration of the acid to 0.0678 molar. * Zeit. phys. Chem., 57, 385 (1906). 270 THEORETICAL CHEMISTRY In a second experiment, if 1 gram of charcoal is shaken for the same period of time with 50 cc. of 2 X 0.0688 = 0.1376 molar acetic acid, and then, after adding 50 cc. of water, the shaking is continued for an additional period of 3 hours, the final concentration of the acid will be found to be the same as in the first experiment. (5) It is impossible to determine the specific surface of an ad- sorbent directly owing to its porosity. However, according to (3) adsorption is directly proportional to the specific surface and there- fore the weights of different adsorbents which produce the same amount of adsorption may be assumed to possess equal specific surfaces. (6) Adsorption in solution is largely dependent upon the surface tension of the solvent. In solutions of the same substance in different solvents, the greatest adsorption occurs in that solution whose solvent possesses the highest surface tension. (7) The order of efficiency of adsorption is not only independent of the nature of the solvent but also of the nature of the adsorbed substance. The Adsorption Isotherm. The empirical equation of Freund- lich for gaseous adsorption has been found to apply equally well to adsorption equilibria in solutions. The equation may be written as follows : x 1 = ac n , ra where x is the weight of substance adsorbed by a weight m of ad- sorbent from a solution whose volume-concentration at equilibrium is c, and where, as before, a and n are constants. The constant n varies in different cases from n = 2 to n = 10; within these limits the value of n is independent of the temperature and also of the natures of the adsorbed substances and the adsorbent. Although the value of the constant a varies over a wide range, the ratio of its values for two adsorbents in different solutions is practically con- stant. The following table contains the data given by Freundlich * on the adsorption of acetic acid by charcoal. * Kapillarchemie, p. 147. COLLOIDS 271 ADSORPTION OF AQUEOUS ACETIC ACID BY CHARCOAL. t = 25; a = 2.606; l/n = 0.425. Concentration (mols per liter). x/m (obs.). x/m (calc.). 0.0181 0.467 0.474 0.0309 0.624 0.596 0.0616 0.801 0.798 0.1259 1.11 1.08 0.2677 1.55 1.49 0.4711 2.04 1.89 0.8817 2.48 2.47 2.785 3.76 4.01 The validity of the adsorption isotherm is best tested graphically, by plotting the logarithms of the experimentally determined values of x/m against the logarithms of the corresponding concentrations. If the equation holds, a straight line should be obtained. The curves shown in Fig. 69 represent x/m as a function of c, and log x/m as a function of log c : it will be observed that the loga- rithmic plot is practically rectilinear. log C c Fig. 69. Surface Energy of Colloids. In almost all colloidal solutions there exists a difference of potential between the particles of the col- loid and the surrounding medium. The importance of this factor 272 THEORETICAL CHEMISTRY in interpreting the behavior of colloids has already been empha- sized. Another factor of equal importance in connection with colloidal phenomena, is that which depends upon the enormous surface of contact between the colloid and the surrounding medium. There is an abundance of evidence showing that a colloidal solution is non-homogeneous, or in other words, that it is essentially a sus- pension of finely-divided particles in a fluid medium. An immense increase in superficial area results from the division and sub- division of matter. To bring about this comminution requires a large expenditure of energy. In a colloidal solution this energy is stored up in the colloidal particles in the form of surface energy, which may be defined as the product of surface area and surface tension. For example, suppose 1 cc. of a substance to be reduced to cubical particles measuring 0.1 /i on each edge, and let the particles be suspended in water at 17 C. The total energy involved can be calculated as follows: The volume of a single particle is 0.1 ij? or [1 X 10~ 15 cc. ; hence the total number of particles is 1 X 10 15 . The surface of a single particle is 6 X (0.1 ju) 2 , or 6 X 10~ 10 sq. cm., and the total surface is 6 X 10 5 sq. cm. The surface tension of water at 17 C. is 71 dynes; hence the total surface energy is 71 X 6 X 10 5 = 4.32 X 10 7 ergs. This enormous figure shows that where the surface of the disperse phase is highly developed, as it is in colloidal solutions, the surface energy becomes a very important factor in determining the behavior of the system. This is especially the case when the degree of aggregation of the colloidal particles is changed, since a relatively small change in the amount of aggregation may involve a great change in the surface exposed and a corresponding change in the surface energy. A very close connection exists between the electrical and surface factors in a colloidal solution. Surface Concentration. It has been pointed out in an earlier chapter (p. 144) that as the result of unbalanced molecular attrac- tion, the surface of a liquid behaves like a tightly stretched mem- brane. In consequence of this contractile force, or surface tension, the pressure at the surface of a liquid is greater than the pressure within the liquid. COLLOIDS 273 The experiments of Soret (p. 208) and the theoretical deductions of van't Hoff have shown that when a dilute solution is unequally heated, the solute distributes itself in accordance with the gas laws, the solution becoming more concentrated in the cooler portion. Just as the homogeneity of a dilute solution has been shown to be disturbed by inequality of temperature, so also inequality of pressure may be assumed to cause differences in concentration in the solution. Although direct experimental verification is difficult, there is abundant evidence for the view that the concentration at the surface of solution differs from the volume-concentration of the solution in consequence of the greater pressure in the surface layer. The mathematical relation between surface concentration and surface tension was first deduced by J. Willard Gibbs * in 1876. The following simplified derivation of this important equation is due to Ostwald. Let s be the surface of a solution whose surface tension is 7, and let it be assumed that the surface contains 1 mol of the solute. If a very small portion of the solute enters the sur- face layer from the solution, thereby causing a diminution dy in the surface tension, the corresponding change in energy will be s dy. But this gain in energy must be equivalent to the osmotic work involved in effecting the removal of the same weight of solute from the solution. Let v be the volume of solution containing unit weight of solute, and let dp be the difference in the osmotic pressures of the solution before and after its removal: the osmotic work will be v dp. Since the gain in surface energy and the osmotic work are equal, we have s dy = v dp. The solutions being dilute, we may assume that the gas laws hold, and since v = RT/p, we may write RT, sdy = - dp, dy RT or - dp sp Trans. Conn. Acad., Vol. Ill, 439 (1876). 274 THEORETICAL CHEMISTRY Since pressure is directly proportional to concentration, the pre- ceding equation becomes dry = _RT dc sc But s has already been defined as the surface which contains 1 mol of solute in excess, from which it follows that the excess of solute in unit surface is 1/s. Writing u = 1/s, we have c dy -RTW which is the equation of Gibbs. From this equation it is evident that if the surface tension, 7, increases with the concentration, then u is negative and the surface concentration is less than the concentration of the bulk of the solution. This is clearly negative adsorption. On tha other hand, if 7 decreases as the concentration increases, u is positive and the surface concentration is greater than the concentration of the bulk of the solution, or the adsorption is positive. Finally, if the surface tension is independent of the concentration, then the con- centration of the solute in both the surface layer and the bulk of the solution will be the same. Preparation of Colloidal Solutions. Since 1861, when Graham published his first paper on colloids, numerous investigators have devised methods for the preparation of colloidal solutions. Within recent years our knowledge of this class of solutions has been greatly increased, many crystalloidal substances having been obtained in the colloidal condition. As a result of these investi- gations, we no longer speak of crystalloidal and colloidal matter, but use the terms crystalloid and colloid to distinguish two dif- ferent states. In fact it is now recognized that it is simply a matter of overcoming certain experimental difficulties, before it will be possible to obtain all forms of matter in the colloidal state. The scope of this book forbids a detailed account of the various methods which have been devised for the preparation of colloidal solutions.* We must content ourselves with a general classification of these methods into two groups as follows : * See "Die Methoden zur Herstellung Kolloider Losungen anorganischer Stoffe," by Theodore Svedberg, Dresden, 1909. COLLOIDS 275 (1) Crystallization Methods, and (2) Solution Methods. These two divisions are sufficiently comprehensive to include all of the known methods for the preparation of colloidal solutions, with the possible exception of the electrical methods which may be con- sidered as forming a separate group. Crystallization Methods. The crystallization methods include the following subdivisions : - (1) Methods involving cooling of a liquid or solution. Example: On cooling an alcoholic solution of sulphur in liquid air, a transparent, highly dispersed, solid sol is obtained. (2) Methods involving change of medium. Example: On gradually adding a solution of mastic in alcohol to a large volume of water, the mastic is precipitated in a finely divided condition and a colloidal mastic hydrosol results. (3) Reduction methods. Example: On adding a cold, dilute solution of hydrazine hydrate to a dilute, neutral solution of auric chloride, a dark blue gold sol is obtained. In. addition to hydrazine, numerous other reducing agents may be employed, such as phosphorus, carbon monoxide, hydrogen, acetylene, formaldehyde, acrolein, various carbohydrates, hy- droxylamine, phenylhydrazine, and metallic ions. (4) Oxidation methods. Example: On oxidizing a solution of hydrogen sulphide by air or sulphur dioxide, a colloidal solution of sulphur is obtained. (5) Hydrolysis methods. Example: When a solution of ferric chloride is slowly added to a large volume of boiling water, the salt undergoes hydrolysis, and on cooling the dilute solution, a reddish-brown ferric hydroxide sol is obtained. (6) Methods involving metathesis. Example: A colloidal solution of silver may be prepared by adding a few drops of a dilute solution of sodium chloride to a dilute solution of silver nitrate, provided the resulting solution of sodium nitrate is below the precipitating concentration. 276 THEORETICAL CHEMISTRY Solution Methods. Under this heading are to be grouped the so-called " peptization" methods. The term, peptization, was intro- duced bv Graham to express the transformation of a gel into a sol. To-day we understand a peptizer to be a substance which, if sufficiently concentrated, is capable of effecting the solution of a solid which is insoluble in its dispersion medium. A typical example of peptization is afforded by silver chloride which forms a sol on prolonged digestion with a solution which contains either Ag* or Cl'. It is apparent that the rate of peptization can be con- trolled by dilution of the peptizer, and that when the sol stage is attained, the peptizer may be readily removed by dialysis. Numerous reactions are known in which the conversion of an insoluble precipitate into a sol can only be effected through the removal of the excess of electrolyte by prolonged washing or dial- ysis. A familiar example of this type of peptization is furnished by the tendency of many precipitates to run through the filter after too prolonged washing with water. Electrical Methods. These methods of preparing colloidal solu- tions depend upon the dispersive action of a powerful electric discharge upon compact metals. In 1897 Bredig * discovered, while studying the action of the electric current on different liquids, that if an arc be established between two metallic wires immersed in a liquid, minute particles of metal are torn off from the negative terminal and remain suspended in the liquid indefi- nitely. In order to prepare a colloidal solution by the method of electrical dispersion, Bredig recommends that a direct current arc be established between wires of the metal of which a colloidal solution is desired, the ends of the wires being submerged in water in a well-cooled vessel, as shown in Fig. 70. The current employed ranges in strength from 5 to 10 amperes, and the voltage lies be- tween 30 and 110 volts. A rheostat and an ammeter are included in the circuit. The wires are brought in contact for an instant in order to establish the arc, after which they are separated about 2 mm. During the gentle hissing of the arc, clouds of colloidal metal are projected out into the water from the negative wire, a portion of * Zeit. Elektrochem., 4, 514 (1897); Zeit. phys. Chem., 31, 258 (1899). COLLOIDS 277 the metal torn off being distributed through the water as a coarse suspension. The size of the particles disrupted from the negative terminal is dependent upon the strength of the current, a current Ammeter Fig. 70. of 10 amperes producing a greater proportion of colloidal metal than a current of 5 amperes. The addition of a trace of potassium hydroxide to the water has been shown to facilitate the process of dispersion. When gold wires are used, deep red colloidal solu- tions are obtained, which after standing for several weeks, acquire a bluish-violet color. With extra precautions, the red colloidal gold solutions may be preserved for two years. These solutions have been shown by Bredig to contain about 14 mg. of gold per 100 cc. In this manner Bredig prepared colloidal solutions of platinum, palladium, iridium, and silver. The method of Bredig has been improved and extended by Svedberg. A diagram of Svedberg's apparatus is shown in Fig. 71. The secondary terminals of an induction coil, capable of giving a spark ranging from 12 to 15 cm. in length, are connected in parallel with the electrodes and a glass plate-condenser having a surface of approximately 225 sq. cm. Minute fragments or grains of the metal of which a sol is desired are placed on the bottom of the vessel containing the dispersion medium. The electrodes, which need not necessarily be of the same metal, are immersed as shown in the diagram, and during the process of electrical dispersion, the contents of the vessel are gently stirred with one or the other of the 278 THEORETICAL CHEMISTRY electrodes. With this apparatus Svedberg has succeeded in pre- paring colloidal solutions of tin, gold, silver, copper, lead, zinc, cadmium, carbon, silicon, selenium, and tellurium. He has also Induction Coil Fig. 71. obtained all of the alkali metals in the colloidal state, ethyl ether being used as the dispersion medium. An interesting observation made by Svedberg in the course of his experiments is that the color of a metal is the same in both the colloidal and gaseous states. CHAPTER XIII. MOLECULAR REALITY. The Brownian Movement. If a liquid in which fine particles of matter are suspended, such as an aqueous suspension of gam- boge, be examined under the microscope, the suspended particles will be seen to be in a state of ceaseless, erratic motion. This phenomenon was first observed in 1827 by the English botanist, Robert Brown, while examining a suspension of pollen grains, and has been called the Brownian Movement in honor of its discoverer. Ever since its discovery, the Brownian Movement has been the subject of numerous investigations. It was not until 1863, how- ever, that Wiener suggested that the cause of the phenomenon was the actual bombardment of the suspended particles by the mole- cules of the suspending medium. Twenty-five years later a similar conclusion was reached independently by Gouy, who showed that neither light nor convection currents within the liquid could possibly give rise to the motion. Furthermore, Gouy showed the movement to be independent of external vibration and only slightly influenced by the nature of the suspended particles. The smaller the particles and the less viscous the suspending medium, the more rapid the motion was found to be. By far the most striking feature of the phenomenon, however, is the fact that the motion is cease- Perrin's Experiments. The first quantitative investigation of the Brownian Movement was undertaken by Perrin in 1909. It has been shown (p. 100) that the mean kinetic energy Ek of one mol of a perfect gas is given by the expression E k = t pv. (1) Since pv = RT, we may write tf* = 7p (2) 279 280 THEORETICAL CHEMISTRY where N denotes the Avogadro Constant, i.e., the number of molecules contained in one mol of any gas. It is evident that if Ek can be measured, equation (2) affords a means of calculating N, provided we are warranted in applying an equation which has been derived for the gaseous state to a suspension of fine particles in a liquid medium. It has already been shown that the simple gas laws hold for dilute solutions and therefore we may assume that, at the same temperature, the mean kinetic energy of the dissolved molecules is equal to that of the gaseous molecules. In other words, at the same temperature, the mean kinetic energy of all the molecules of all fluids is the same, and is directly proportional to the absolute temperature. Since the gas laws apply equally well to dilute solutions containing either large or small molecules, Perrin held that there was no a priori reason for assuming that the grains of a suspension should not conform to the same laws. If this assumption be correct, the grains of a uniform suspension should so distribute themselves under the influence of gravity . that, when equilibrium is attained, the lower layers will have a higher concentration than the upper layers. In other words, the distribution should be strictly analogous to the distribution of the air over the surface of the earth, the density being greatest at the surface and diminishing as the altitude increases. Let us imagine a suspension to be confined within a tall vertical cylinder whose cross-sectional area is s sq. cm. As- suming that the suspension has come to equilibrium under the influence of gravitation, let n be the number of grains per unit of volume at a height h from the base of the cylinder. Since the concentration diminishes as the height increases, the number of grains at a height h + dh will be n dn. The osmotic pressure of the grains at the height h will be f nE k , where E k is the mean kinetic energy of each grain. In like manner, the osmotic pressure at the height h + dh will be f (n - dn) E k . The difference in osmotic pressure between the two levels is -f dnE k and since the pressure acts over a surface of s sq. cm., the difference of osmotic forces acting over the cross-sectional area of the cylinder is -f s dnE k . Since the system is in equilibrium, this difference in osmotic forces must be balanced by the difference in the attraction MOLECULAR REALITY 281 of gravitation at the two levels. Let (j> be the volume of a single grain, D its density, and 5 the density of the suspending medium. The resultant downward pull upon a single grain will be < (D 5) g, where g is the acceleration due to gravity. The volume of liquid between the two levels being s dh, it follows that the total down- ward pull upon all the grains included between the two levels must be nsh(j> (D 5) g. It is this force which opposes the tendency of the grains to distribute themselves uniformly throughout the entire volume of the suspending medium, or, in other words, it is the force which acts in opposition to the osmotic force s dnEk- When equilibrium is established, these two forces must be equal, and we may then write -f s dnE k = ns dh(j> (D - 5) g. (3) If n and n denote the number of grains per unit of volume at each of two planes h units apart, we obtain, on integrating equation (3), E k log e n /n = (D - 5) gh. (4) On substituting in equation (4) the value of Ek in equation (2), and transforming to Briggsian logarithms, we have 2.303 RT/N log wo/n = | irr*g (D - 5) h, (5) < being expressed in terms of the mean radius, r, of a single grain. It is evident that if we can measure n, n , D, and r in equation (5), the calculation of the Avogadro Constant, N, be- comes possible. The determination of the density of the grains, D, was carried out in two different ways with suspensions of gamboge and mastic which had been rendered uniform by a process of centrifuging. In the first method, the grains were dried to constant weight at 110, and then by heating to a higher temperature, a viscous liquid was obtained which, on cooling, formed a glassy solid. The density of this solid was determined by suspending it in a solution of potassium bromide of known density. In the second method for the determination of D, Perrin measured the masses mi and my Zeit. Elektrochem., 14, 235 (1908). 284 THEORETICAL CHEMISTRY \ 5 MOLECULAR REALITY 285 where y is the viscosity of the suspending medium and where the other symbols have their usual significance. This equation was tested by Perrin, using suspensions of gam- boge and mastic. Some of the results obtained are given in the following table: VALUES OF N CALCULATED BY EINSTEIN'S EQUATION. Suspension. rin microns.* TO X 1015. No. of dis- placements. JVxlO-. Gamboge in water 0.367 246 1500 69 Gamboge in 10% solution of glycerine 0.385 290 100 64 Mastic in water. 0.52 650 1000 73 Mastic in 27% solution of urea . . 5.50 750,000 100 78 * The micron is one-millionth of a meter or one ten-thousandth of a centimeter. It will be seen that notwithstanding the large variations in the granular masses of the different suspensions recorded in the table, the values of N, calculated by means of Einstein's equation, are quite concordant. Perrin gives as the mean value of all of his experiments, N = 68.5 X 10 22 . Recent Investigations of the Brownian Movement. Nord- lund * has recently repeated Perrin 's experiments, employing a colloidal solution of mercury and an arrangement of apparatus whereby the movements of the particles could be recorded photo- graphically. The mean value of N derived from twelve carefully executed experiments was 59 X 10 22 , the average deviation of the results of the individual experiments from the mean being approxi- mately 10 per cent. The Brownian Movement in gases has been studied by Millikan f and by Fletcher t employing a minute drop of oil as the suspended particle. In the gaseous state, where the intermolecular distances are greater than in the liquid state, not only are the collisions less frequent but the mean free paths are appreciably longer. These conditions are favorable to the study of the Brownian Movement and offer an opportunity for the determination of the Avogadro Constant with a high degree of accuracy. As the mean of nearly six thousand measurements, Fletcher gives N = 60.3 X 10 22 , this value being accurate to within 1.2 per cent. * Zeit. phys. Chem., 87, 60 (1914). f Phys. Rev., i, 220 (1913). $ Ibid., 4, 453 (1914). CHAPTER XIV. THERMOCHEMISTRY. General Introduction. A chemical reaction is almost invari- ably accompanied by a thermal change. In the majority of cases heat is evolved; a violent reaction developing a large amount of heat, while a feeble reaction develops a comparatively small amount. Such reactions are said to be exothermic. A relatively small number of chemical reactions are known which take place with an absorption of heat. These are termed endoihermic reac- tions. Instances of chemical reactions unaccompanied by any thermal change are very rare and are almost wholly confined to the reciprocal transformations of optical isomers. These facts, which were first observed by Boyle and Lavoisier, led to the view that the amount of heat evolved in a chemical reaction might be taken as a measure of the chemical affinity of the reacting sub- stances. However, with the advance of our theoretical knowledge, it is now known that this is not true, although a parallelism between heat evolution and chemical affinity frequently exists. Thermochemistry is concerned with the thermal changes which accompany chemical reactions. Thermal Units. Heat is a form of energy, and like other forms of energy it may be resolved into two factors; an intensity factor, the temperature, and a capacity factor, which may be measured in any one of several units. Among these units those defined below are the most frequently employed. The small calorie (cal.) is the quantity of heat required to raise the temperature of 1 gram of water from 15 C. to 16 C. The temperature interval is specified because the specific heat of water varies with the temperature.* The large or kilogram calorie (Cal.) is the quantity of heat required to raise the temperature of 1000 grams of water from 15 C. to 16 C. The Ostwald or average 286 THERMOCHEMISTRY 287 calorie (K), is the quantity of heat required to raise the temper- ature of 1 gram of water from the melting point of ice to the boiling point of water under a pressure of 760 mm. of mercury. It is approximately equal to 100 cal. or to 0.1 Cal. The joule (j), a unit based on the C.G.S. system, is equal to 10 7 ergs. This being inconveniently small is generally multipled by 1000, giving the kilojoule (J), which is therefore equal to 10 10 ergs. The last two units are open to the objection that their values are depend- ent upon the mechanical equivalent of heat, any change in the accepted value of which would involve a correction of the unit of heat. The different capacity factors of heat energy are related as follows : 1 cal. = 0.001 Cal. = 0.01 K (approx.) = 4.183 j = 0.004183 J. Thermochemical Equations. In order to represent the changes in energy which accompany chemical reactions, an additional meaning has been assigned to the chemical symbols. As ordina- rily used, these symbols represent only the molecular or formula weights of the reacting substances. In a thermochemical or energy equation the symbols represent not only the weight in grams expressed by the formula weights of the substances, but also the amount of heat energy contained in the formula weight in one state as compared with the energy contained in a standard state. For example, the energy equation, C + 2 O = CO 2 + 94,300 cal., indicates that the energy contained in 12 grams of carbon and 32 grams of oxygen exceeds the energy contained in 44 grams of carbon dioxide, at the same temperature, by 94,300 calories. In writing energy equations it is very essential that we have some means of distinguishing between the different states of aggrega- tion of the reacting substances, since the energy content of a substance is not the same hi the gaseous, liquid, and solid states. In the system proposed by Ostwald, ordinary type is used for liquids, heavy type for solids, and italics for gases. Another and more convenient system has been proposed, in which solids are designated by enclosing the symbol or formula within square brackets; 'liquids by the simple, unbracketed symbol or formula; 288 THEORETICAL CHEMISTRY and gases by enclosing the symbol or formula within parentheses. The above equation should, therefore, be written in the following manner : [C] + (2 0) = (C0 2 ) + 94,300 cal. Thermochemical Measurements. In order to measure the number of calories evolved or ab- sorbed when substances react, it is necessary that the reaction should proceed rapidly to completion. This condition is fulfilled by two classes of processes. In the first class we may mention the processes of solu- tion, hydration, and neutralization; and in the second class, the process of combustion. The apparatus used for measur- ing the capacity factor of heat energy is a calorimeter. This instrument may be given a variety of forms, depending upon the particular use to which it is to be put. A simple form of calorimeter is shown in Fig. 73. It consists of two concentric metal cylinders, A and B, insulated from each other by an air jacket, the inner vessel being supported on vulcanite points. Through a vulcanite cover passes a thin walled test tube, in which the reaction is Fig. 73. allowed to take place. An accurate thermometer and a ring-stirrer also pass through the cover of the calorimeter. In order to deter- mine the thermal capacity of the calorimeter, B is nearly filled with water, and a known mass of water, m, at a temperature t\ is intro- duced intoC. Let the initial temperature of the water in B be tz. The water in B is stirred until the contents of both B and C have acquired the same temperature, 3 . When thermal equilibrium has been estab- lished, it is evident that m (ti U) calories are required to raise THERMOCHEMISTRY 289 the temperature of the apparatus and the water in B, (^ $) degrees. From this data it is an easy matter to calculate the number of calories required to raise the temperature of the appar- atus and water in B I degree, this being the thermal capacity of the apparatus. The calorimeter may now be used to determine the heat evolved or absorbed in a reaction. Suppose, for exam- ple, that it is desired to measure the heat of neutralization of an acid by a base. Equivalent quantities of both acid and base are dissolved in equal volumes of water, care being taken to make the solutions dilute. A definite volume of one solution is introduced into C and an equal volume of the other solution is placed in a vessel from which it can be quickly and completely transferred to C. When both solutions have acquired the same temperature, the thermometer hi B is read and then the two solutions are mixed. When the reaction is complete, the temperature of the water in B is again noted. If the thermal capacity of the calori- meter is Q, and the rise in temperature produced by the reaction is 6, then Qd is the amount of heat evolved by the reaction. To this quantity of heat must be added the number of calories re- quired to raise the temperature of the products of the reaction 8 degrees. The solutions of the products being dilute, their specific heats may be assumed to be equal to unity. From the total quan- tity of heat so obtained, the number of calories evolved when mo- lecular quantities react can be readily calculated. The chief source of error in calorimetric measurements is loss by radiation. This may be reduced to a minimum (1) by making the thermal capac- ity of the calorimeter large, and (2) by so arranging matters that the initial temperature of the water in the calorimeter is as much below the temperature of the room as the final temperature is above it. The Combustion Calorimeter. The combustion of many sub- stances, such as organic compounds, proceeds very slowly in air under ordinary pressures. Such reactions can be accelerated, if they are caused to take place in an atmosphere of compressed oxygen. For this purpose the combustion calorimeter was de- vised by Berthelot.* In this apparatus the essential feature is * Ann. Chim. Phys., (5), 23, 160 (1881); (6), 10, 433 (1887). 290 THEORETICAL CHEMISTRY the so-called combustion bomb, shown in Fig. 74. This consists of a strong steel cylinder lined with platinum or gold, and fur- nished with a heavy threaded cover. The substance to be burned is placed in a platinum capsule fastened to the support R, and a short piece of fine iron wire of known mass is connected with the electric terminals Z, Z, the middle portion of the wire dipping into the substance. The cover is then screwed down tight, and the bomb is filled with oxygen under a pressure of from 20 to 25 atmospheres. The bomb is then submerged in the calorimeter, as shown in Fig. 75. The mass of water in the calorim- eter being known and its temperature having I been read, an electric current is passed through g S the iron wire in the bomb causing it to burn and thus ignite the substance. The rise in temperature due to the combustion is ob- served, and the quantity of heat evolved is calculated. Corrections must be applied for loss by radiation, for the heat evolved from the combustion of the iron, and for the heat evolved from the oxidation of the nitrogen of the residual air in the bomb. For the details of ,the method of determin- ing heats of combustion the student must consult a laboratory manual. Fig. 74. Law of Lavoisier and Laplace. In 1 780, Lavoisier and Laplace,* as a result of their thermochemical investigations, enunciated the following law : The quantity of heat which is required to decompose a chemical compound is precisely equal to that which was evolved in the formation of the compound from its elements. This first law of thermochemistry will be seen to be a direct corollary of the law of the conservation of energy which was first clearly stated by Mayer in 1842. Law of Constant Heat Summation. A generalization of funda- mental importance to the science of thermochemistry was discov- ered in 1840 by Hess.f He pointed out that the heat evolved in a t * Oeuvres de Lavoisier, Vol. II, p. 283. t Fogg. Ann., 50, 385 (1840). THERMOCHEMISTRY 291 chemical process is the same whether it takes place in one or in several steps. This is known as the law of constant heat summation. The Fig. 75. truth of the law may be illustrated by the equality of the heat of formation of ammonium chloride in aqueous solution, when pre- pared in two different ways. 292 THEORETICAL CHEMISTRY Thus, (A) (NH 3 ) + (HC1) = [NEUC1] +42,100 cal. [NH 4 C1] + aq. = NH 4 C1, aq. - 3,900 cal. 38,200 cal. (B) (NH 3 ) + aq. = NH 3 , aq. + 8,400 cal. (HC1) + aq. = HC1, aq. +17,300 cal. NH 3 , aq. + HC1, aq. = NH 4 Cl,aq. +12,300 cal. 38,000 cal. It will be observed that the total amount of heat evolved in the formation and solution of ammonium chloride is the same within the limits of experimental error, whether gaseous ammonia and hydrochloric acid are allowed to react and the resulting prod- uct is dissolved in water, or whether the gases are each dissolved separately and then allowed to react. It should be noted that when a substance is dissolved in so much water that the addition of more water or the removal of a small portion of water produces no thermal effect, it is customary to denote it by the symbol aq. (Latin aqua = water). Thus, NH 4 C1, aq. + nH 2 = NH 4 C1, aq., NH 4 C1, aq. - nH 2 O = NH 4 C1, aq. By means of the law of constant heat summation it is possible to find indirectly the amount of heat developed or absorbed by any reaction, even though it is impossible to carry it out experimen- tally. For example, it is impossible to measure the heat evolved when carbon burns to carbon monoxide. But the heat evolved when carbon monoxide burns to carbon dioxide, and also the heat evolved when carbon burns to carbon dioxide, can be accurately determined. The energy equations are as follows :- [C] + 2(0) = (C0 2 ) + 94,300 cal. (1) (CO) + (0) = (C0 2 ) + 67,700 cal. (2) THERMOCHEMISTRY 293 Treating these equations algebraically, and subtracting equation (2) from equation (1), we have [C] + (0) = (CO) + 26,600 caL, or, the heat of combustion of carbon to carbon monoxide is 26,600 calories. Again, as a further illustration of the applicability of the law of Hess, we may take the calculation of the heat of forma- tion of hydriodic acid from its elements, making use of the follow- ing energy equations : 2 KI, aq. + 2 (Cl) = 2 KC1, aq. + 2 [I] + 524 K (1) 2 HI, aq. + 2 KOH, aq. = 2 KI, aq. + 2 H 2 + 274 K (2) 2 HC1, aq. + 2 KOH, aq. = 2 KC1, aq. + 2 H 2 + 274 K (3) 2 (HI) + aq. = 2 HI, aq. + 384 K, (4) 2 (HC1) + aq. = 2 HC1, aq. + 346 K, (5) 2 (H) + 2 (Cl) = 2 (HC1) + 440 K, (6) adding equations (1) and (2), 2 (Cl) + 2 HI, aq. + 2 KOH, aq. = 2 KC1, aq. + 2 [I] + 2 H 2 + 798 K. (7) Subtracting equation (3) from equation (7), 2 (Cl) + 2 HI, aq. - 2 HC1, aq. = 2 [I] + 524 K, or 2 (Cl) + 2 HI, aq. = 2 [I] + 2 HC1, aq. + 524 K, (8) adding equations (4) and (8), 2 (HI) + aq. + 2 (Cl) = 2 [I] + 2 HC1, aq. + 908 K, (9) subtracting equation (5) from equation (9), 2 (HI) + 2 (Cl) - 2 (HC1) = 2 [I] + 562 K, or 2 (HI) + 2 (Cl) = 2 [I] + 2 (HC1) + 562 K, (10) subtracting equation (6) from equation (10), 2(HI)-2(H) =2[I] + 122K, or 2(H) + 2(I) = 2 (HI)- 122 K. In a similar manner, practically any heat of formation may be calculated, provided the proper energy equations are combined. 294 THEORETICAL CHEMISTRY Heat of Formation. The intrinsic energy of the substances entering into chemical reaction is unknown, the amount of heat evolved or absorbed in the process being simply a measure of the difference between the energy of the reacting substances and the energy of the products of the reaction. Thus, in the equation [C] + 2 (0) = (C0 2 ) + 94,300 cal., the difference between the energy of a mixture of 12 grams of carbon and 32 grams of oxygen, and the energy of 44 grams of carbon dioxide is seen to be 94,300 calories. The equation is clearly incomplete since we have no means of determining the intrinsic energies of free carbon and oxygen. Furthermore, since the elements are not mutually convertible, we have no means of determining the difference in energy between them. It is cus- tomary, therefore, in view of this lack of knowledge, to put the intrinsic energies of the elements equal to zero. If the heats of formation of the substances present in a reac- tion are known, it is much simpler to substitute these in the energy equation and solve for the unknown term. This method avoids the laborious process of elimination from a large number of energy equations, as in the preceding pages. If all of the sub- stances involved in a reaction are considered as decomposed into their elements, it is evident that the final result of the reaction will be the difference in the sums of the heats of formation on the two sides of the equation. This leads to the following rule: To find the quantity of heat evolved or absorbed in a chemical reac- tion, subtract the sum of the heats of formation of the substances initially present from the sum of the heats of formation of the products of the reaction, placing the heat of formation of all elements equal to zero. The energy equation for the formation of carbon dioxide from its elements may then be written as follows : + = (CO 2 ) + 94,300 cal., or (CO 2 ) = - 94,300 cal. That is, the energy of 1 mol of carbon dioxide is 94,300 calories. Therefore in writing an energy equation we make use of the fol- THERMOCHEMISTRY 295 lowing rule : Replace the formulas of each compound in the equa- tion representing the reaction by the negative values of their respective heats of formation and solve for the unknown term. This unknown term may be either the heat of a reaction or the heat of formation of one of the reacting substances. The following examples will serve to illustrate the application of the above rules : (1) Let it be required to find the heat of the following reaction [MgClJ + 2 [Na] = 2 [NaCl] + [Mg] + x, where x is the heat of the reaction. The heat of formation of MgCl 2 is 151 Cal., and that of NaCl is 97.9 Gal, therefore, - 151 + = - (2 X 97.9) + + z, or x = 44.8 Cal. (2) The heat of combustion of 1 mol of methane is 213.8 Cal., and the heats of formation of the products, carbon dioxide and water, are 94.3 Cal. and 68.3 Cal., respectively. Let it be required to find the heat of formation of methane. Representing the heat of formation of methane by x, we have (CH 4 ) + 2 (0 2 ) = (C0 2 ) + 2 (H 2 0) + 213.8 Cal., - x + o - - 94.3 - 2 X 69.3 + 213.8, or x = 17.1 Cal. (3) The heat of combustion of 1 mol of carbon disulphide is 265.1 Cal., the thermochemical equation being CS 2 + 3 (0 2 ) = (COa) + 2 (S0 2 ) + 265.1 Cal. The heats of formation of carbon dioxide and sulphur dioxide are 94.3 Cal. and 71 Cal. respectively. The heat of formation of carbon disulphide x, may then be calculated as follows : - x + = - 94.3 - 2 X 71 + 265.1, or x = -28.8 Cal. Carbon disulphide is thus seen to be an endothermic compound. Heat of Solution. The thermal change accompanying the solution of 1 mol of a substance in so large a volume of solvent that subsequent dilution of the solution causes no further thermal 296 THEORETICAL CHEMISTRY change is termed the heat of solution. The solution of neutral salts is generally an endothermic process. This fact may be readily accounted for on the hypothesis that considerable heat HEATS OF FORMATION AND SOLUTION. Substance. Heatpf Formation. Heat of Solution. Water, vapor 58 . 7 Water, liquid 68.4 Hydrochloric acid 22 . Sulphuric acid 193 . 1 Ammonia 12 . Nitric acid 41 . 9 Phosphoric acid 302 . 9 Potassium hydroxide 103 . 2 Potassium chloride 104.3 Potassium bromide 95 . 1 Potassium iodide 80 . 1 Potassium nitrate 119.5 Sodium hydroxide 101 . 9 Sodium chloride 97 . 6 Sodium bromide 85 . 6 Sodium sulphate 328.8 Sodium nitrate 111.3 Sodium carbonate 272 . 6 Ammonium chloride 75 . 8 Ammonium nitrate 88.0 Calcium hydroxide 215.0 Calcium chloride 170.0 Magnesium sulphate 502 . Ferrous chloride 82 . Ferric chloride 96 . 1 Zinc chloride 97.0 Zinc sulphate 30.0 Cadmium chloride 293 . 2 Cupric chloride 51 . 6 Cupric sulphate 182 . 6 Mercuric chloride 53 . 2 Silver nitrate 28 . 7 Stannous chloride 80 . 8 Stannic chloride 127 . 3 Lead chloride 82.8 Lead nitrate.. 105.5 20.3 17.8 8.4 7.2 2.7 13.3 -3.1 -5.1 -5.1 -8.5 10.9 1.2 -0.2 0.2 -5.0 5.6 -4.0 -6.2 3.0 17.4 20.3 17.9 63.3 15.6 18.5 3.0 11.1 15.8 -3.3 -5.4 0.3 29.9 -6.8 -7.6 must be absorbed as heat of fusion and heat of vaporization before the solid salt can assume a condition in solution which closely resembles that of a gas. The heat of solution of hydrated salts is less than the heat of solution of the corresponding anhydrous THERMOCHEMISTRY 297 salts. For example, the heat of solution of 1 mol of anhydrous calcium nitrate is 4000 calories, while the heat of solution of 1 mol of the tetrahydrate is 7600 calories. The difference between the heats of solution of the anhydrous and hydrated salts is termed the heat of hydration. The heats of formation and heats of solu- tion in water of some of the more common compounds are given in the preceding table, the values being expressed in large calories. Heat of Dilution. The heat of dilution of a solution is the quantity of heat per mol of solute which is evolved or absorbed when the solution is greatly diluted. Beyond a certain dilution, further addition of solvent produces no thermal change. While there is a definite heat of solution for a particular solute in a par- ticular solvent, the heat of dilution remains indefinite, since the latter is dependent upon the degree of dilution. Those gases which obey Henry's law are practically the only substances which have no appreciable heats of solution or dilution. The following tables give the heats of dilution of hydrochloric and nitric acids. HEAT OF DILUTION OF SOLUTIONS OF HYDROCHLORIC ACID. Heat of solution = 20.3 cal. HC1+H 2 O 5.37 HC1+2 H 2 O 11.36 HC1 + 10H 2 O ; 16.16 HC1+50H 2 O HCl-f 300 H 2 O 17.1 17 3 HEAT OF DILUTION OF SOLUTIONS OF NITRIC ACID. Heat of solution = 7.15 cal. HNO 3 +H 2 O 3.84 HNOs+2 H 2 O 2.32 HNOs+4 H 2 O 1.42 HNOs+6 H 2 O 0.2 HNOa+ 8 H 2 O . . . -0.04 HNOa+100 H 2 O -0.03 298 THEORETICAL CHEMISTRY Reactions at Constant Volume. When a chemical reaction takes place without any change in volume, or when the external work resulting from a change in volume is not included in the heat of the reaction, the process is said to take place at constant volume. That is to say, the condition of constant volume is a condition which involves no external work, either positive or negative. Under these conditions the total energy of the react- ing substances is equal to the total energy of the products of the reaction, plus the quantity of heat developed by the reaction. Reactions at Constant Pressure. When a chemical reaction is accompanied by a change in volume, the system suffers a loss of heat equivalent to the work done against the atmosphere, if the volume increases; or the system gains an amount of heat equivalent to the work done on the system by the atmosphere, if the volume decreases. Under these conditions the reaction is said to take place at constant pressure. The difference between constant volume and constant pressure conditions, then, is that under the former, the heat equivalent of the work corresponding to any change in volume which may occur is not considered as having any effect upon the energy of the system; whereas under the latter, due account is taken of the change in energy resulting from change in volume. Suppose that in a reaction, 1 mol of gas is formed. Under standard conditions of temperature and pressure the volume of the system will be increased by 22.4 liters. The formation of gas involves the performance of work against the atmosphere, this work being done at the expense of the heat energy of the system. To calculate the heat equivalent of the work done, let us imagine the gas enclosed in a cylinder fitted with a piston whose area is 1 square centimeter. The normal pressure of the atmosphere on the piston is 76 cm. of mercury or 1033.3 grams per square centimeter. If the increase in the volume of the gas is 22.4 liters, the piston will be raised through 22,400 cm. and the work done will be 1033.3 X 22,400 gram-centimeters. The heat equivalent of this change in volume will be (1033.3 X 22,400) -J- 42,600 = 542.3 calories or 0.5423 large calories. This amount of heat must be added to the heat of the reaction. It should be observed that this correction is independent of the actual THERMOCHEMISTRY 299 value of the pressure upon the system. Thus, if the pressure is increased n times, the volume of the gas will be reduced to l/n of its former value, and the work done will involve moving the piston through l/n of the distance against an n-fold pressure, which is plainly equivalent to the former amount of work. While the correction is independent of the pressure it is not independent of the temperature. The familiar equation, pv = RT, shows us that the work done by a gas is directly proportional to its absolute temperature. Thus, if a gas is evolved at 273 absolute, it will occupy -double the volume it would occupy at 0, and the work done at 273 will involve moving the piston through twice the distance that it would have to be moved at 0. Theoretically, a gas evolved at the absolute zero would occupy no volume and hence no work would be done. Introducing the correction for temperature, we see that - X T = 1.986 T cal., must be added to the heat of the reaction, where T is the absolute temperature at which the change in volume occurs. For all ordinary purposes it is sufficiently accurate to take 2 T calories as the correction. Thus, suppose n mols of gas to be formed in a reaction at 17 C., the amount of heat absorbed will be nX 2(273 + 17) = 580 n cal. Under constant pressure conditions, the symbols, in addition to their usual significance, represent the energy plus or minus the term, 2 T per mol, the positive or negative sign being used according as the gas is absorbed or formed. Since the constant volume condition is a condition in which no account is taken of the external work, even if a change in volume does occur during the reaction, and the constant pressure condition is one in which the external work is taken into consideration, it is apparent that the relation of the heat energy of a reaction at constant volume, Q V) to the heat energy at constant pressure Q p , can be represented by the equation cal., 300 THEORETICAL CHEMISTRY where n denotes the number of mols of gas formed in excess of those initially present. This equation is of great importance in connection with the determination of heats of combustion in the bomb-calorimeter in which the reactions necessarily take place under constant volume conditions. Since it is customary to state heats of reaction under constant pressure conditions, the foregoing equation makes it possible to convert heats of combustion deter- mined under constant volume conditions into heats of combustion under constant pressure conditions. For example, the combustion of naphthalene takes place in accordance with the equation CioH 8 + 12 (0 2 ) = 10 (C0 2 ) + 4 (H 2 0) + 1242.95 Cal. It is apparent that the combustion is accompanied by the forma- tion of 2 mols of gas, and at 15 C. the correction will be Q p = 1242.95 - 2 X 0.002 (273 + 15), or Q p = 1241.8 Cal. The volume occupied by solids or liquids is so small as to be negligible and does not enter into these calculations. Variation of Heat of Reaction with Temperature. If a chem- ical reaction be allowed to take place first at the temperature t\, and then at the temperature ^, the amounts of heat developed in the two cases will be found to be quite different. Let Qi and Q 2 represent the quantities of heat evolved at the temperatures t\ and k, respectively. Let us imagine that the reaction takes place at the temperature t\, Qi units of heat being evolved; and then let the products of the reaction be heated to the temperature ^. If c' represents the total thermal capacity of the products of the reaction, then the quantity of heat necessary to produce this rise in temperature will be c' (k ti). Now let us imagine the reacting substances, at the temperature t\ t to be heated to the temperature k, and then allowed to react with the evolution of Q 2 units of heat. The heat necessary to produce this rise in temperature in the reacting substances is c (^ 1), where c is the total thermal capacity of the original substances. Having started with the same substances at the same initial temperature, and having obtained the same products at the same final temper- THERMOCHEMISTRY 301 ature, we have, according to the law of the conservation of energy, Qi - c' (fe - *0 = Q 2 - c & - *0, or or, where the change in temperature is very small, ^-c-c' dt " C ' If c' is greater than c then the sign of dQ/dt will be negative, or, in other words, an increase in temperature will cause a decrease in the heat of reaction. On the other hand, if c is greater than c', dQ/dt will be positive and the heat of reaction will increase with the temperature. EXAMPLE. The reaction between hydrogen and oxygen at 18 C. is represented by the following equation^ 2 (H 2 ) + (O 2 ) = 2 (H 2 0) + 1367.1 K. Suppose it is required to find how much heat will be evolved when equal masses of the two gases react at 110 C., the product of the reaction being maintained at this temperature, and the pressure remaining constant. The specific heats per gram of the different substances involved are as follows : Hydrogen = 3.409; Oxygen = 0.2175; Water (between 18 and 100) = 1; Water (between 100 and 110) =0.5. The heat of vaporization of water is 537 calories per gram. For liquid water per degree we have, dQ/dt = (4 X 3.409 + 32 X 0.2175) - (36 X 1) = - 15.404 cal. and for (100- 18) =82, we have, 82 X (-15.404) =-1263 cal. The heat of formation of liquid water at 100 is, therefore, 1367.1 - 12.63 = 1354.47 K. When the liquid water is vaporized at 100, (36 X 537) calories of this heat is absorbed, or the formation of steam at 100 from hydrogen and oxygen, evolves 1354.47 - 193.32 = 1161.15 K. 302 THEORETICAL CHEMISTRY For steam per degree, we have, dQ/dt = (4 X 3.409 + 32 X 0.2175) - (36 X 0.5) = 2.596 cal., and for the interval (110 ~- 100) = 10, 10 X 2.596 = 25.96 cal. Or for the total heat evolved, we have 1161.15 + 0.2596 - 1161.41 K. Heat of Combustion. The heat evolved during the complete oxidation of unit mass of a substance is termed its heat of combustion. The unit of mass commonly chosen in all physico- chemical calculations is the mol. An enormous amount of experimental work has been done by Thomsen,* Berthelot,t and Langbein t on the determination of the heats of combustion of a large number of organic compounds. A few of their results are given in the accompanying tables. SATURATED HYDROCARBONS. Hydrocarbon. Heat of Combustion. Difference. Methane, CH 4 Cal. 211 9 Cal. Ethane, C 2 H 6 370 4 158.5 Propane CsHg 529 2 158.8 Butane, C4Hio 687.2 158.0 Pentane, CsHi2 847 1 159.9 UNSATURATED HYDROCARBONS. Hydrocarbon. Heat of Combustion. Difference. Ethylene, C 2 H 4 Cal. 333 4 Cal. Propylene, C 3 H 6 492 7 159.3 Isobutylene, C 4 H8 650.6 157.9 Amylene CsHio 807 6 157.0 Acetylene, CaH2 310.1 Allylene, C 3 H 4 467.6 157.5 * Thermochemische Untersuchungen, 4 Vols. t Essai de Mecanique Chimique, Thermochimie, Donees et Lois Numeri- ques. $ Jour, prakt. Chem., 1885 to 1895. THERMOCHEMISTRY 303 ALCOHOLS. Alcohol. Heat of Combustion. Difference. Methyl alcohol, CH 4 O Cal. 182 2 Cal. Ethyl alcohol, C 2 H 6 O . . 340 5 158.3 Propyl alcohol, C 3 H 8 O 498 6 158.1 Isobutyl alcohol, C 4 Hi O 658 5 159.9 It will be observed that a very nearly constant difference in the heat of combustion corresponds to a constant difference of a CH 2 group in composition. A number of interesting relations between heats of combustion of compounds and their differences in composition have been discovered, but these cannot be taken up at this time. It has also been pointed out that the heat of com- bustion of organic compounds is conditioned not only by their composition, but also by their molecular constitution. Some exceedingly interesting and important results have been obtained with the different allotropic forms of the elements. For example, when equal masses of the three common allotropic forms of carbon are burned in oxygen, the amounts of heat evolved are found to be quite different, as is shown by the following energy equations : [C] diamond + 2 (0) = (C0 2 ) + 94.3 Cal. [C] graphite + 2 (O) = (C0 2 ) + 94.8 Cal. [C] amorphous + 2 (0) = (C0 2 ) + 97.65 Cal. It is apparent that amorphous carbon contains the greatest amount of energy of any one of the three allotropic modifications, and, .therefore, when amorphous carbon is changed into diamond, the reaction must be accompanied by the evolution of (97.65 94.3) = 3.35 Cal. In like manner, the allotropic forms of sulphur and phosphorus have different heats of combustion. The follow- ing equations show the heat equivalents of the differences in intrinsic energy between the allotropic forms : S (monoclinic) = S (rhombic) + 2.3 Cal. P (white) = P (red) + 3.71 Cal. 304 THEORETICAL CHEMISTRY When the same substance is burned in oxygen and then in ozone, it is found that more heat is evolved in ozone than in oxygen. The energy equation expressing the change of ozone into oxygen may be written thus, (0 8 ) = 1 J (0 2 ) + 36.2 Cal. All of the above facts illustrate the general principle that the larger amounts of intrinsic energy are associated with the more unstable forms. Thermoneutrality of Salt Solutions. In addition to the law of constant heat summation, Hess discovered two other important laws of thermochemistry, viz., the law of thermoneutrality of salt solutions, and the law governing the neutralization of acids by bases.* When two dilute salt solutions are mixed there is neither evolution nor absorption of heat. Thus when dilute solu- tions of sodium nitrate and potassium chloride are mixed, there is no thermal effect. The energy equation may be written as follows : NaN0 3 , aq. + KC1, aq. = NaCl, aq. + KNO 3 , aq. + Cal. According to this equation a double decomposition has taken place and we should naturally expect an evolution or an absorption of heat. While Hess could not account for the absence of any thermal effect, he recognized the fact as quite general and formu- lated the law of the thermoneutrality of salt solutions as fol- lows: The metathesis of neutral salts in dilute solutions takes place with neither evolution nor absorption of heat. The explanation of the phenomenon of thermoneutrality was furnished by the theory of electrolytic dissociation. When the above equation is written in the ionic form, it becomes Na + N0 8 ' + K* + Cl' = Na* + Cl' + K* + NO,'. From this it is apparent that the same ions exist on both sides of the equation, and in reality no reaction takes place. There are numerous exceptions to the law of thermoneu- trality. These can be satisfactorily accounted for by the theory of electrolytic dissociation. All of those salts, the behavior of which in dilute solution is contrary to the law, are found to be only partially * Pogg. Ann., 50, 385 (1840). THERMOCHEMISTRY 305 ionized, and, therefore, when their solutions are mixed, a chem- ical reaction actually occurs. The exceptions must be considered as furnishing additional evidence in favor of the theory of elec- trolytic dissociation. Heat of Neutralization. Hess also discovered * that when dilute solutions of equivalent quantities of strong acids and strong bases are mixed, practically the same amount of heat is evolved. The following energy equations may be considered as typical examples of such neutralizations : HC1, aq. + NaOH, aq. = NaCl, aq. + H 2 O + 13.75 Cal., HN0 3 , aq. + NaOH, aq. = NaN0 3 aq. + H 2 O + 13.68 Cal., HC1, aq. + KOH, aq. = KC1, aq. + H 2 O + 13.70 Cal., HN0 3 , aq. + KOH, aq. = KN0 3 , aq. + H 2 O + 13.77 Cal., HC1, aq. + LiOH, aq. = LiCl, aq. + H 2 O + 13.70 Cal. Here again it would be difficult to explain the phenomenon with- out the theory of electrolytic dissociation. In terms of this theory, however, the explanation is perfectly plausible. If MOH and HA represent any strong base and any strong acid respectively, then when equivalent amounts of these are dissolved in water, each solution being largely diluted to the same volume, the reac- tion may be written thus : M' + OH' + H- + A' = M- + A' + H 2 + 13.7 Cal. Disregarding the ions which occur on both sides of the equality sign, we have OH' + ET = H 2 + 13.7 Cal. It thus appears that the neutralization of a strong acid by a strong base in dilute solution consists solely in the combination of hydro- gen and hydroxyl ions to form undissociated water, the heat of this ionic reaction being 13.7 large calories. The heat of formation of water from its ions must not be con- fused with the heat of formation of water from its elements. When weak acids or weak bases are neutralized by strong bases or strong acids, or when weak acids are neutralized by weak bases, * Loc. cit. 306 THEORETICAL CHEMISTRY the heat of neutralization may differ widely from 13.7 Cal. This is shown by the following thermochemical equations : H-COOH, aq. + NaOH, aq. = H-COONa, aq. + H 2 + 13.40 Cal., HC1 2 -COOH, aq. + NaOH, aq. = HCl 2 -COONa, aq. + H 2 + 14.83 Cal., H-COOH, aq. + NH 4 OH, aq. = H-COONH 4 , aq. + H 2 + 11.90 Cal., HCN, aq. + NaOH, aq. = NaCN, aq. + H 2 O + 2.90 Cal. As will be seen, the heat of neutralization may be either greater or less than 13.7 Cal. The exceptions to the generalization of constant heat of neutralization are readily explained by the theory of electrolytic dissociation. Suppose a weak acid to be neutralized by a strong base. According to the dissociation theory, the acid is only slightly dissociated and, therefore, yields a comparatively small number of hydrogen ions to the solution. The base on the other hand is completely dissociated into hydroxyl and metallic ions. Therefore, as many hydroxyl ions disappear as there are free hydrogen ions with which they can combine to form water. When the equilibrium between the acid and the products of its dissociation has been thus disturbed, it undergoes further dissociation and the resulting hydrogen ions immediately combine with the free hydroxyl ions of the base. This process continues until all of the hydroxyl ions of the base have been neutralized. It is evident that the thermal effect in this case is the algebraic sum of the heat of dissociation of the weak acid, which may be positive or negative, and the heat of formation of water from its ions. A similar explanation holds for the neutrali- zation of a weak base by a strong acid, or for the neutralization of a weak acid by a weak base. This affords a method for estimat- ing the approximate value of the heat of dissociation of a weak acid or a weak base. For example, in the equation given above, HCN, aq. + NaOH, aq. = NaCN, aq. + H 2 O + 2.90 Cal., the difference between 2.9 and 13.7 or 10.8 Cal. represents approximately the heat of dissociation of hydrocyanic acid. THERMOCHEMISTRY 307 Since the acid is initially slightly dissociated in dilute solution, it is apparent that in order to obtain the true heat of dissociation, we must add to 10.8 Cal. the thermal value of the dissociation of that portion of the acid which has already become ionized. Heat of lonization. Since 13.7 Cal. is the heat of formation of water from its ions, this must also be the thermal equivalent of the energy required to dissociate one mol of water into its ions. It must be remembered that the dissociated molecule of water must be mixed with a very large volume of undissociated water, in order HEAT OF FORMATION OF IONS. Ion. Heat of Formation. Ion. Heat of Formation. Hydrogen 0.0 Copper (ic) -15.8 Potassium 61.9 Copper (ous) -16.0 Sodium . . 57.5 Mercury (ous) -19.8 Lithium 62 9 Silver 25 3 Ammonium 32 8 Lead 5 IVIagnesium 109 Tin (ous) 3.3 Calcium 109 Chlorine . ... 39.3 Aluminium Manganese 121.0 50.2 Bromine Iodine 28.2 13.1 Iron (ous). 22.2 Sulphate 214.4 Iron (ic) -9.3 Sulphite 151.3 Cobalt. 17.0 Nitrous 27.0 Nickel. . . . 16.0 Nitric 49.0 Zinc 35 1 Carbonate 161.1 Cadmium 18.4 Hyroxyl ... 54.7 that the dissociation may be permanent. Reference to the table of heats of formation (p. 296), will show that 68.4 Cal. are required to form one mol of water from its elements. Hence, it follows that 68.4 13.7 = 54.7 Cal., is the heat of formation of one equivalent of hydrogen and hydroxyl ions. It has been shown that an ex- tremely small amount of energy is necessary to ionize hydrogen when it is dissolved in water. It is evident, therefore, that 54.7 Cal. is a close approximation to the heat of formation of one equiva- lent of hydroxyl ions. On the assumption that the heat of ionization of gaseous hydro- gen in solution is zero, the values of the other ionic heats of forma- 308 THEORETICAL CHEMISTRY tion may be computed. For example, the heat of formation of KOH, aq. is 116.5 Cal. The ionic heat of formation of potassium ions must be 116.5 54.7 = 61.8 Cal. In like manner, the heat of formation of KC1, aq. is 101.2 Cal.; hence the ionic heat of formation of chlorine ions must be 101.2 61.8 = 39.4 Cal. The preceding table of ionic heats of formation has been calculated as in the above examples. The Principle of Maximum Work. A fundamental principle of the science of mechanics is, that a system is in stable equilib- rium when its potential energy is a minimum. In 1879, Ber- thelot * suggested that a similar principle applies to chemical systems. In terms of the kinetic theory, the temperature of a substance is to be regarded as a measure of the kinetic energy of its molecules. The development of heat by a chemical reaction would, therefore, be taken as an indication of a decrease in the potential energy of the system. Berthelot's theorem, known as the principle of maximum work, may be stated as follows: "Every chemical proc- ess accomplished without the intervention of any external energy tends to produce that substance or system of substances which evolves the maximum amount of heat." The table of heats of formation (p. 296), illustrates the general truth of this principle, but as will be seen, the theorem precludes the possibility of spontaneous endothermic reactions. Thus, for example, the formation of acetylene from its elements at the temperature of the electric arc is a well-known endothermic reaction, but according to the principle of maximum work, it could not take place spontaneously. Another serious objection to Berthelot's principle is, that accord- ing to it, all chemical reactions should proceed to completion, the reaction taking place in such a way as to evolve the greatest amount of heat. As is well known, many reactions, and theoretically all reactions, are never complete, but proceed until a condition of equilibrium is reached. The principle of maximum work, there- fore, denies the existence of equilibria in chemical reactions. Many attempts have been made to "explain away" these defects, but none of them have been successful. In referring to the generali- * Essai de Mecanique Chimique. THERMOCHEMISTRY 309 zation, Le Chatelier terms it "a very interesting approximation toward a strictly valid generalization. " The Theorem of Le Chatelier. As a result of his attempts to modify the principle of maximum work and render it generally applicable, Le Chatelier was led to the discovery of a rigorous law of wide-reaching usefulness. His generalization may be stated as follows: Any alteration in the factors which determine an equi- librium) causes the equilibrium to become displaced in such a way as to oppose, as far as possible, the effect of the alteration. If the temperature of a system which is in equilibrium be raised or lowered, the resulting displacement of the equilibrium is accom- panied by such absorption or evolution of heat as will tend to maintain the temperature constant. An interesting illustration of the behavior of a system when one of the factors controlling the equilibrium is varied, is afforded by the system 2N0 2 <=N 2 O 4 . The reaction proceeds in the direction indicated by the upper arrow with the evolution of 12.6 Cal. Increase of temperature favors the reaction which is accompanied by an absorption of heat, which in this case, is the reaction indicated by the lower arrow. Hence as the temperature rises, the percentage of NO 2 increases at the expense of N 2 O 4 . This fact can be demonstrated by the following experiment. Some liquefied N 2 4 is placed in each of three long glass tubes, which are sealed at one end. When enough N 2 O 4 has vaporized to displace the air, the open ends of the tubes are sealed. Changes in the equilibrium caused by varying the temperature can be followed by noting the changes in the color of the mixture. N 2 4 is an almost colorless substance, while N0 2 is reddish brown. At ordinary temperatures the contents of the tubes will .be brown in color. One tube is set aside as a standard of comparison, while the temperature of the second is lowered by surrounding it with a freezing mixture. As the temperature falls, the brown color of the contents of the tube becomes much lighter, showing an increased formation of NoOi- The third tube is heated by immersing it in a beaker of boiling water. As the temperature rises, the contents of the tube becomes 310 THEORETICAL CHEMISTRY much darker in color, indicating an increase in the amount of N(>2 in the mixture. Another example is afforded by the equilibrium between ozone and oxygen, represented by the equation 2O 3 <=3O 2 . The reaction indicated by the upper arrow is exothermic. In- crease of temperature causes a displacement of the equilibrium in the direction of the lower arrow, since under these conditions heat is absorbed. Thus, as the temperature rises ozone becomes increasingly stable. Nernst has calculated that at 6000 C., the temperature of the photosphere of the sun, 10 per cent of the above equilibrium mixture would be ozone. Other applications of the theorem of Le Chatelier will be given in subsequent chapters. PROBLEMS. 1. From the following data calculate the heat of formation of HN02 aq.- [NH 4 N0 2 ] = (N 2 ) + 2 H 2 + 71.77 Gal., 2 (H 2 ) + (0 2 ) = 2 H 2 + 136.72 Gal., (N 2 ) + 3 (H 2 ) + aq. = 2 NH 3 aq. + 40.64 Gal., NH 3 aq. + HN0 2 aq. = NH 4 N0 2 aq. + 9.110 Gal., [NH 4 N0 2 ] + aq. = NH 4 N0 2 aq. - 4.75 Cal. Am. (H) + (N) + (0 2 ) + aq. = HN0 2 aq. + 30.77 Cal. 2. By the combustion at constant pressure of 2 grams of hydrogen with oxygen to form liquid water at 17 C., 68.36 Cal. are evolved. What is the heat evolution at constant volume? Ans. 67.49 Cal. 3. The heats of solution of Na 2 S0 4 , Na 2 S0 4 .H 2 0, and Na 2 S0 4 .10H 2 O are 0.46, - 1.9 and - 18.76 Cal. respectively. What are the heats of hydration of Na 2 S0 4 ; (a) to monohydrate, (b) to decahydrate? Ans. (a) 2.36 Cal., (b) 19.22 Cal. 4. The heats of neutralization of NaOH and NH 4 OH by HC1 are 13.68 and 12.27 Cal. respectively. What is the heat of ionization of NH 4 OH, if it is assumed to be practically undissociated? Ans. 1.41 Cal. THERMOCHEMISTRY 311 5. From the following energy equations: [C] + (0 2 ) = (C0 2 ) + 96.96 Cal., 2 (H 2 ) + (0 2 ) = 2 H 2 + 136.72 Cal., 2 C 6 H 6 + 15 (0 2 ) = 12 (C0 2 ) + 6 H 2 + 1598.7 Cal., 2 (C 2 H 2 ) + 5 (0 2 ) = 4 (C0 2 ) + 2 H 2 + 620.1 Cal., all at 17 C. and constant pressure, calculate the heat evolved at 17 C. in the reaction 3 C 2 H 2 = Cell e, (a) at constant pressure, and (b) at constant volume. Ans. (a) 130.8 Cal., (b) 129.07 Cal. 6. Calculate the heat of formation of sulphur trioxide from the follow- ing energy equations: [PbO] + [S] + 3 (0) = [PbS0 4 ] + 1655 K. [PbO] + H 2 S0 4 .5 H 2 = [PbS0 4 ] + 6 H 2 + 233 K. [S] + 3 (0) + 6 H 2 = H 2 S0 4 .5 H 2 + 1422 K. [S0 3 ] + 6 H 2 = H 2 S0 4 .5 H 2 + 411 K. Ans. [S] + 3 (0) = [S0 3 ] + 1011 K. 7. What is the heat of formation of a very dilute solution of calcium chloride? (See table on p. 257.) Ans. 187.6 Cal. CHAPTER XV. HOMOGENEOUS EQUILIBRIUM. Historical Introduction. In this and the two succeeding chapters, the conditions which affect the rate and the extent of chemical reactions will be considered. When two substances react chemically, it is customary to refer the phenomenon to the existence of an attractive force known as chemical affinity. Ever since the metaphysical speculations of the Greeks, who endowed the atoms with the instincts of love and hate, the nature of chemical affinity has been under discussion. So little has been learned as to the cause of chemical reactions, that in recent years this question has been dismissed and attention has been directed to the more promising question as to how they take place. New- ton's discovery of the law of gravitation led him to consider the attraction between atoms and the attraction between large masses of matter as manifestations of the same force. Although Newton found that chemical attraction does not follow the law of the inverse square, yet his suggestion exerted a profound influence upon the minds of his contemporaries. Geoffrey and Bergmann arranged chemical substances in the order of their displacing power. Thus, if we have three sub- stances, A, J5, and C and the attraction between A and B is greater than that between A and C, then when B is added to AC it will completely displace C, as indicated by the following equa- tion:- AC + B = AB + C. These investigators overlooked a factor of fundamental importance in conditioning chemical reactivity, viz., the influence of mass. The importance of the relative amounts of the reacting substances in determining the course of a reaction was first clearly recognized by Wenzel * in 1777. It remained for Berthollet,t however, to * Lehre von der chemischen Verwandtschaft der Korper. t Essai de Statique Chimique. 312 HOMOGENEOUS EQUILIBRIUM 313 point out the significance of the views advanced by Wenzel. His first paper on this subject was published in 1799, while acting as a scientific adviser to Napoleon on his Egyptian expedition. Under ordinary conditions sodium carbonate and calcium chloride react according to the equation, NasCOs + CaCl 2 = 2 NaCl + CaC0 3 , the reaction proceeding nearly to completion. Berthollet observed the deposits of sodium carbonate on the shores of certain saline lakes in Egypt, and pointed out that this salt is produced by the reversal of the above reaction, the large excess of sodium chloride in solution in the water of the lakes conditioning the course of the reaction. The German chemist Rose* furnished much additional evidence in favor of the effect of mass on chemical reactions. He pointed out that in nature, the silicates, which are among the most stable compounds known, are undergoing a continual decomposition under the influence of such relatively weak agents as water and carbon dioxide. The relatively strong specific affinities of the atoms of the silicates are overcome by the preponderating masses of water and carbon dioxide in the atmosphere. In 1862 an important contribution to our knowledge of the effect of mass on the course of a chemical reaction was made by Berthelot and Pean de St. Gilles.f They investigated the formation of esters from alcohols and acids. The reaction between ethyl alcohol and acetic acid is represented by the equation C 2 H 5 OH + CH 3 COOH <= CH 3 COOC 2 H 5 + H 2 0. Starting with equivalent quantities of alcohol and acid, the reac- tion proceeds until about two-thirds of the reacting substances have been converted into ester and water. In like manner, if equivalent quantities of ethyl acetate and water are brought together, the reaction proceeds in the direction indicated by the lower arrow, until about one-third of the original substances have been converted into acid and alcohol. In other words the reac- tion is reversible, a condition of equilibrium resulting when the * Pogg. Ann., 94, 481 (1855); 95, 96, 284, 426 (1855). t Ann. Chim. Phys. [3], 65, 385; 66, 5; 68, 225 (1862-1863). 314 THEORETICAL CHEMISTRY speeds of the two reactions, indicated by the upper and lower arrows, become equal. If now a fixed amount of acid is taken, say 1 equivalent, and the quantity of alcohol is varied, a corre- sponding displacement of the equilibrium follows. The following table gives the results obtained by Berthelot and Pean de St. Gilles for ethyl alcohol and acetic acid. The first and third columns give the number of equivalents of alcohol to 1 equivalent of acetic acid, and the second and fourth columns give the percentage of ester formed. Equivalents of Alcohol. Ester Formed. Equivalents of Alcohol. Ester Formed. 0.2 19.3 2.0 82.8 0.5 42.0 4.0 88.2 1.0 66.5 12.0 93.2 1.5 77.9 50.0 100.0 The effect of increasing the mass of alcohol on the course of the reaction is very beautifully shown by the above results. The Law of Mass Action. While the influence of the relative , masses of the reacting substances in conditioning chemical reac- tions was thus fully established, it was not until 1867 that the law governing the action of mass was accurately formulated. In that year Guldberg and Waage,* two Scandinavian investiga- tors, enunciated the law of mass action as follows: The amount of chemical action at any stage of a reaction is proportional to the active masses of the reacting substances present at that time. Guld- berg and Waage defined the term "active mass" as the molecular concentration of the reacting substances. It is to be carefully noted that the amount of chemical action is not porportional to the actual masses of the substances present, but rather to the amounts present in unit volume. The law is generally applicable to homogeneous systems; that is, to those systems in which ordinary observation fails to reveal the presence of essentially different parts. The amount of chemical action exerted by a * Etudes sur les Affinite*s Chimiques, Jour, prakt. Chem. [2], 19, 69 (1879). HOMOGENEOUS EQUILIBRIUM 315 substance can be determined, either from its effect on the equili- brium, or from its influence on the speed of reaction. In order to apply the law of mass action practically, it must be formulated mathematically. Let a and b denote the molecular concentrations of the substances initially present in a reversible reaction. According to the law of mass action, the rate at which these substances combine is proportional to the active masses of each constituent, and therefore to their product, ah. The initial speed of the reaction at the time to is therefore, Speed< oo ab, or Speed^ = k ab, in which the proportionality factor fc, is known as the velocity constant. As the reaction proceeds, the molecular concentrations of the original substances steadily diminish, while the molecular concentrations of the products of the reaction steadily increase. Let us assume that after the interval of time t, x equivalents of the products of the reaction have been formed. The speed of the original reaction will now be Speedf = k (a x) (b x). As the reaction proceeds, the tendency of the products to combine and reform the original substances increases. At the time t, when the concentration of the products is x, the speed of the reverse reaction will be Speed* = ki x z , where ki is the velocity constant of the reverse reaction. We thus have two reactions proceeding in opposite directions: the speed of the direct reaction continuously diminishes while that of the reverse reaction continually increases. It is evident that a point must ultimately be reached at which the speeds of the direct and reverse reactions become equal, and a condition of equilibrium will be established. Let x represent the value of x when equilibrium is attained; we then have Speed direct = k (a Xi) (b Xi) = Speed reverse or (a - xi) (b - xi) /bi v _ __, = /< Xi 2 k 316 THEORETICAL CHEMISTRY in which K is known as the equilibrium constant. Since the veloc- ity constants k and k\ t are independent of the concentration, it follows that the above equation holds for all concentrations. Therefore, if the value of the equilibrium constant of a reaction is known, the equilibrium conditions can be calculated for any concentrations of the reacting substances. When more than one mol of a substance is involved in a reaction, each mol must be considered separately in the mass action equation. Thus, let niAi + nzAz-\- . . . +ni'Ai 4- nz'Az + . . . represent any reversible reaction, hi which HI mols of A\ and n^ mols of A 2 react to form n\ mols of A\ and n z ' mols of A 2 f . When equilibrium is attained, we shall have or c % c -h-K m C A l ' C A 1 ' ' ' in which the symbol c is used to denote the active mass, or molecular concentration of the substances involved in the reaction. This is a perfectly general form of the mass-action equation. Since at any one temperature, concentration and pressure are proportional, we may write equation (1) in the following form which, in the case of gaseous equilibria, is often a more convenient form of the equation. The relation between the two equilibrium constants, K c and K p , can be easily determined, as follows: Since c = - = -^m we have, on substituting this value of c in equation (1), Y* RT) T \RT HOMOGENEOUS EQUILIBRIUM 317 or indicating the sum of the initial number of mols by I>n and the sum of the final number of mols by Z??', we have K C = K P (RT) *'-*". It is evident, therefore, that in reactions where the same number of mols occur on both sides of the equality sign, K c = K P . Equa- tion (1) (or equation (2)) is sometimes known as the reaction isotherm. While the law of mass action may be proved thermody- namically, a much simpler kinetic derivation has been given by van't Hoff. If we assume that the rate of chemical change is pro- portional to the number of collisions per unit of time between the molecules of the reacting substances, then in the reaction niAi + n 2 A 2 + . . . = ni'Ai + nz'Az + ...:, the velocity of the direct change will be kc^c^ . . . and the velocity of the reverse reaction will be kic^,^,. . . . At equilibrium, the two velocities will be equal and, therefore, or C AfA, ' ' ' fcl ^ , , = -r~ = A. ft\ W2 L* c c As a consequence of the assumptions involved in both the thermo- dynamic and the kinetic proofs of the law of mass action, it fol- lows that the law is only strictly applicable to very dilute solutions. Notwithstanding this limitation, experimental results indicate that it frequently holds for moderately-concentrated solutions. Equilibrium in Homogeneous Gaseous Systems. (a) Decomposition of Hydriodic Acid. A typical example of equilibrium in a gaseous system is afforded by the decomposition of hydriodic acid, as represented by the equation This reaction has been thoroughly investigated by Hautefeuille, Lemoine and Bodenstein.* The reaction is well adapted for investigation since it proceeds very slowly at ordinary temper- * Zeit. Phys. Chem. 22, 1 (1897). 318 THEORETICAL CHEMISTRY atures, while at the temperature of boiling sulphur, 448 C., equilibrium is established quite rapidly. If the mixture of gases is maintained at 448 C. for some time and is then cooled quickly, the respective concentrations of the components of the mixture can be determined by the ordinary methods of chemical analysis. Various mixtures of the gases are sealed in glass tubes and heated for a definite time in the vapor of boiling sulphur. The tubes are then cooled rapidly to the temperature of the room and, after the iodine and hydriodic acid have been removed by absorption in potassium hydroxide, the amount of free hydrogen present in each tube is measured. Applying the law of mass action to the above equation, we have C HI Expressing the analytical results in mols, let a mols of iodine be mixed with b mols of hydrogen, and let 2 x mols of hydriodic acid be formed. Then when equilibrium is established, a x will be the amount of iodine vapor and b x will be the amount of hydro- gen present. The concentrations being directly proportional to the amounts present, we may substitute these values for c# 2 , c/ 2 , and CHI hi the mass-action equation. The following expression is thus obtained : (6 - z) (a - z) = A/.. Solving the equation for x, we obtain a + b - Va 2 + 6 2 - ab (2 - 16 K c ) Since, according to Avogadro's law, equal volumes of all gases contain the same number of molecules, volumes may be sub- stituted for a, 6, and x. Bodenstein expressed his results in terms of volumes reduced to standard conditions of temperature and pressure. On analyzing equilibrium mixtures, Bodenstein found that at 448 C., K c = 0.01984, and at 350 C., K c = 0.01494. Having determined the value of the equilibrium constant, he made use of this value in calculating the volume of hydriodic HOMOGENEOUS EQUILIBRIUM 319 acid which should t be obtained from known volumes of hydrogen and iodine. A comparison of the calculated and observed values showed excellent agreement. The following table contains a few of the results obtained by Bodenstein at 448 C. Hydrogen, b. Iodine, a. HI calculated, 2x. HI observed, 2z. 20.57 5.22 10.19 10.22 20.60 14.45 25.54 25.72 15.75 11.90 20.65 20.70 14.47 38.93 27.77 27.64 8.10 2.94 5.64 5.66 8.07 9.27 13.47 13.34 It is of interest to note that a change in pressure does not alter the equilibrium in this gaseous system. Making use of the partial pressures of the components of the gaseous system instead of the concentrations, we have Pff 2 * Ph _ j? rf ~ p ' PHI Now let the total pressure on the system be increased to n times its original value; then the partial pressures are all increased in the same proportion, and we have which is equivalent to the original expression, since n cancels out. The equilibrium is thus seen to be independent of the pres- sure. This is only true for those systems in which a change in volume does not occur. (b) Dissociation of Phosphorus Pentachloride. When phos- phorus pentachloride is vaporized it dissociates according to the following equation Applying the law of mass action, we have 320 THEORETICAL CHEMISTRY Starting with 1 mol of phosphorus pentachloride, which if undis- sociated would occupy the volume V, under atmospheric pressure, and letting a denote the degree of dissociation, the molecular con- centrations at equilibrium will be as follows: Cpa ' = Letting (1 + a) V = V, and substituting in the above equation, we have At 250 C. phosphorus pentachloride is dissociated to the extent of 80 per cent. Under atmospheric pressure 1 mol will be present 273 4- 250 in 22.4 "liters = V. The final volume will, therefore, be ZiO V = (1 + 0.8) (22.4 273 2 | 3 25 ) The value of the equilibrium constant, usually designated in cases of dissociation, the dissociation constant, is, therefore, (0.8) 2 (1-0.8) (1+0.8) ^22.4 273 + 250V 273 / Having obtained the value of K C) the direction and extent of the reaction at 250 C. can be determined, provided the initial molec- ular concentrations are known. The reaction is accompanied by a change in volume, and, therefore, the equilibrium is displaced by a change in pressure. Making use of the partial pressures of the components of the gaseous mixture, we have where pi and pz are the partial pressures of phosphorus penta- chloride and the products of the dissociation, phosphorus tri- HOMOGENEOUS EQUILIBRIUM 321 chloride and chlorine, respectively. Let the total pressure be increased n-times, then pi It is apparent from this equation, that the equilibrium is not independent of the pressure, an increase in pressure being accom- panied by a diminution of the dissociation. An important point in connection with dissociation, first observed by Deville,* is the effect on the equilibrium of the addition of an excess of one of the products of dissociation. For example, in the equilibrium an excess of chlorine or of phosphorus trichloride, drives back the dissociation. If pi denotes the partial pressure of phosphorus pentachloride, p 2 that of phosphorus trichloride, and ps that of chlorine, then we have Pz-ps _ K - -A-p. Pi Now let an excess of chlorine be added; this will cause the value of p 3 to increase. Since the value of K p is constant, the value of pz must diminish and that of p\ must increase. Hence, the addi- tion of an excess of either product of dissociation causes a diminu- tion of the amount of the dissociation. (c) Dissociation of Carbon Dioxide. Carbon dioxide dissociates according to the equation, This is a somewhat more complex gaseous system than either of the foregoing systems. When equilibrium is established, let pi be the partial pressure of the carbon dioxide, p 2 the partial pres- sure of carbon monoxide, and p z the partial pressure of oxygen, then we have At 3000 C. and under atmospheric pressure, carbon dioxide is * Leons sur la dissociation, Paris (1866). 322 THEORETICAL CHEMISTRY 40 per cent dissociated. The partial pressures of each of the com- ponents may be readily calculated as follows: 2(1-0.40) _ 2 (1 - 0.40) + 3 X 0.40 2 X 0.40 2 (1 - 0.40) + 3 X 0.40 _ 0.40 P * ~ 2 (1 - 0.40) + 3 X 0.40 ~ Substituting these values in the above equation, we obtain _ (0.33)' X 0.17 p (0.50) 2 The dissociation constant for carbon dioxide may have a different value if the equation is written in the form Applying the law of mass action, we have Substituting the above values of the partial pressures, we obtain K p = 0.272. Equilibrium in Liquid Systems. The reaction between an alcohol and an acid to form an ester and water may be taken as an example of equilibrium in a liquid system. In the reaction C 2 H 5 OH + CHsCOOH^ CH 3 COOC 2 H 5 + H 2 0, let a, 6, and c represent the number of mols of alcohol, acid and water respectively, which are present in V liters of the mixture, and let x denote the number of mols of ester and water which have been formed when the system has reached equilibrium. The active masses of the components will then be, n - a ~ x r - b ~ x r - x - AT _c + x ^alc. ~~ TT ) ^acid ~ -rr j ^ ester "TT > and U water ~ V ' Applying the law of mass action, we obtain (a - x) (b - x) __ x(c + x) - Ac ' HOMOGENEOUS EQUILIBRIUM 323 In this case the value of the equilibrium constant is independent of the volume. This reaction has been studied, as already men- tioned, by Berthelot and Pean de St. Gilles.* They found that when equivalent amounts of alcohol and acid are mixed, the reac- tion proceeds until two-thirds of the mixture is changed into ester and water. Hence, we find ixj "I xf K c 1. Having determined the value of K c , it may now be used to cal- culate the equilibrium conditions for any initial concentrations of the substances involved in the reaction. As an illustration, we will take 1 mol of acetic acid and treat it with varying amounts of alcohol, the initial mixture containing neither of the products of the reaction. The equation takes the form (a - x) (1 - x) _ 1 ~~tf~ ~4* Solving for x, we have x = f (1 + a - Va 2 -a + l). A comparison of the observed and calculated values given in the accompanying table shows that the agreement is excellent, even in the more concentrated solutions, where we might reasonably expect that the mass law would cease to hold. Alcohol, a. Ester (observed ) , z. Ester (calculated), z. 0.05 0.05 0.049 0.08 0.078 0.078 0.18 0.171 0.171 0.28 0.226 0.232 0.33 0.293 0.311 0.50 0.414 0.523 0.67 0.519 0.528 1.0 0.665 0.667 1.5 0.819 0.785 2.0 0.858 0.845 2.24 0.876 0.864 8.0 0.966 0.945 * Loc. cit. 324 THEORETICAL CHEMISTRY The Variation of the Equilibrium Constant with Temperature. Van't Hoff showed that the displacement of equilibrium due to change in temperature is connected with the heat evolved or ab- sorbed in a chemical reaction by the equations dT ' RT 2 ' and d(log e K p )_ Q p dT RT 2 ' where Q v and Q p are the heats of reaction at constant volume and constant pressure respectively, and where R and T have their usual significance. Equation (1) is known as the reaction isochore. Both equations show that the rate of change of the natural log- arithm of the equilibrium constant with temperature is equal to the total heat of reaction divided by the molecular gas constant times the square of the absolute temperature at which the reaction takes place. Equations (1) and (2) hold only for displace- ments of the equilibrium due to infinitely small changes in temper- ature. In order to render these equations applicable to concrete equilibria, it is necessary to integrate them. The integration of these expressions can only be performed if Q is constant. For small intervals of temperature, Q is practically independent of the temperature, and for larger intervals we may take the value of Q which corresponds to the mean of the two temperatures between which the integration is performed. Integrating equa- tions (1) and (2) on this assumption, we obtain log. K ct - log, K Cl = f (jr - f 9 ) ' (3) and log. Kp. - log. K = J (jr ~ j$ (4) Passing to Briggsian logarithms, and putting R = 1.99 calories, equations (3) and (4) become n /rr rr \ (5) HOMOGENEOUS EQUILIBRIUM 325 and We shall now proceed to show how these important equations may be applied to several typical equilibria. (a) Vaporization of Water. The equilibrium between a liquid and its vapor is conditioned by the pressure of the vapor, this in turn being dependent upon the temperature. In this case of physical equilibrium, we have K PI = pi, and K P2 = p 2 . The value of Q p for water can be calculated from the following data: Ti = 273, pi = 4.54 mm. of mercury, T 2 = 273 +11.54, p 2 = 10.02 mm. of mercury. Substituting in equation (6), we have 4.581 (log 10.02 - log 4.54) 273 X 284.5 ^ p ~ 273 - 284.5 or Q p = 10,670 calories. The value of Q p obtained by experiment is 10,854 calories. (b) Dissociation of Nitrogen Tetroxide. In the reaction the following values for the dissociation of N 2 04 have been ob- tained: 7 7 1 = 273+ 26.l, i = 0.1986, T 2 = 273 + 111.3, <* 2 = 0.9267. If the dissociation takes place under a pressure of 1 atmosphere, then the partial pressures of the component gases will be ~ The values of K PI and K^ are, then, according to the law of mass action as follows : 326 THEORETICAL CHEMISTRY and Substituting in equation (6) and solving for Q p , we obtain . 4 X (0.9267) 2 4XC0.1986) 2 4.581 log 2 - log 1 _ (Q 2 299.1 X 384.3 p 299.1 - 384.3 or Q p = - 12,260 calories per mol of N 2 4 . In a reaction which is accompanied by no thermal change, Q = 0, and the right-hand side of equations (1) and (2) becomes equal to zero. In other words, in such a reaction a change in temperature does not cause a displacement of the equilibrium. The reaction, C 2 H 5 OH + CH 3 COOH^ CH 3 COO.C 2 H 6 + H 2 0, is accompanied by such a small thermal change that it may be considered as zero, and according to the above reasoning there should be only a very slight displacement of the equilibrium when the temperature is varied. Berthelot found that at 10 C., 65.2 per cent of the alcohol and acid are changed into ester, and at 220 C., 66.5 per cent of the mixture is transformed into ester. As will be seen, an increase of 210 produces hardly any displace- ment of the equilibrium. PROBLEMS. 1. When 2.94 mols of iodine and 8.10 mols of hydrogen are heated at constant volume at 444 C. until equilibrium is established, 5.64 mols of hydriodic acid are formed. If we start with 5.30 mols of iodine and 7.94 mols of hydrogen, how much hydriodic acid is present at equilibrium at the same temperature? Ans. 9.49 mols. 2. At 2000 C., and under atmospheric pressure, carbon dioxide is 1.80 per cent dissociated according to the equation Calculate the equilibrium constant for the above reaction using partial pressures. Ans. 3 X 10~ 6 . HOMOGENEOUS EQUILIBRIUM 327 3. What is the equilibrium constant in the preceding problem, if the concentrations are expressed in mols per liter? Ans. 1.61 X 10~ 8 . 4. When 6.63 mols of amylene and 1 mol of acetic acid are mixed, 0.838 mol of ester is formed in the total volume of 894 liters. How much ester will be formed when we start with 4.48 mols of amylene and 1 mol of acetic acid in the volume of 683 liters? Ans. 0.8111 mol. 5. If 1 mol of acetic acid and 1 mol of ethyl alcohol are mixed, the reaction C 2 H 5 OH + CH 3 COOH <= CH 3 COOC 2 H 5 + H 2 0, proceeds until equilibrium is reached, when f mol of ethyl alcohol, mol of acetic acid, f mol of ethyl acetate, and f mol of water are present. If we start (a) with 1 mol of acid and 2 mols of alcohol; (b) with 1 mol of acid, 1 mol of alcohol, and 1 mol of water; (c) with 1 mol of ester and 3 mols of water, how much ester will be present in each case at equilibrium? Ans. (a) 0.845 mol, (b) 0.543 mol, (c) 0.465 mol. 6. In the reaction we find, since 3 = i 2 , for the values 3.02 at 386 C. and 2.35 at 419 C. Calculate the heat evolved by the reaction under constant pressure. Ans. 6827 cal. 7. Above 150 C. N0 2 begins to dissociate according to the equation At 390 C. the vapor density of N0 2 is 19.57 (H = 1), and at 490 C it is 18.04. Calculate the degree of dissociation according to the above equation at each of these temperatures; the equilibrium constants expressing the concentrations in mols per liter; and the heat of dissoci- ation of N0 2 . Ans. ai = 0.35, 2 = 0.55. K l = 2.884 X 10~ 2 . K 2 = 7.173 X 10- 2 . Q = -9407 cal. CHAPTER XVI. HETEROGENEOUS EQUILIBRIUM. Heterogeneous Systems. We have now to consider equilibria in systems made up of matter in different states of aggregation. Such systems are termed heterogeneous systems, as distinguished from those dealt with in the preceding chapter where the compo- sition is uniform throughout. The physically distinct portions of matter involved in a heterogeneous system are known as phases, each phase being homogeneous and separated from the other phases by definite bounding surfaces. Thus, ice, liquid water and vapor constitute a physically heterogeneous system. Another heterogeneous system is formed by calcium carbonate and its dissociation products, calcium oxide and carbon dioxide. The equilibrium between a solid, its saturated solution, and vapor affords an illustration of a still more complex heterogeneous system. Application of the Law of Mass Action to Heterogeneous Equilibria. It has been shown in the preceding chapter that the law of mass action may be applied to homogeneous equilibria provided the molecular condition of the reacting substances is known. When we attempt to apply the law of mass action to hetero- geneous equilibria, especially where solids are involved, the problem presents difficulties. In his investigation of the dis- sociation of calcium carbonate, according to the equation [CaCOJ <= [CaO] + (CO 2 ), Debray * showed that just as every liquid has a definite vapor pressure corresponding to a certain temperature, so there is a definite pressure of carbon dioxide over calcium carbonate at a definite temperature. Furthermore, the pressure was found to be independent of the amount of calcium carbonate present. * Compt. rend., 64, 603 (1867). 328 HETEROGENEOUS EQUILIBRIUM 329 Guldberg and Waage * showed that the law of mass action can be applied to such heterogeneous equilibria, provided that the active masses of the solids present are considered as constant. Nernst pointed out that this statement of Guldberg and Waage can be easily reconciled with experimental facts. In a hetero- geneous equilibrium involving solids, it is only necessary to con- sider the gaseous phase, the active mass of a solid being equivalent to its concentration in the gaseous phase. That is, every solid is to be looked upon as possessing, at a definite temperature, a defi- nite vapor pressure which is entirely independent of the amount of solid present. Such substances as arsenic, antimony, and cadmium are known to have appreciable vapor pressures at relatively low temperatures, and it is quite reasonable to suppose that every solid substance exerts a definite vapor pressure at a definite temper- ature, even though we have no method sufficiently refined to meas- ure such minute pressures. Since the active mass of a solid remains constant so long as any of it is present, the application of the law of mass action to certain heterogeneous equilibria is, in general, simpler than its application to homogeneous systems. The truth of this state- ment will be evident after a few typical heterogeneous systems have been considered. (a) Dissociation of Calcium Carbonate. In the reaction [CaC0 3 ]^[CaO] + (C0 2 ), let TTi and ir 2 represent the pressures due to the vapor of calcium carbonate and calcium oxide respectively, and let p denote the pressure of the carbon dioxide. Applying the law of mass action, we obtain But since TTI amd 7r 2 are constant at any one temperature, the equation becomes P = K,', or, the equilibrium constant at any one temperature is solely dependent upon the pressure of the carbon dioxide evolved. The * Loc. cit. 330 THEORETICAL CHEMISTRY accompanying table gives the values of the pressure of carbon dioxide corresponding to various temperatures. Temperature, Degrees. Pressure in Millimeters of Mercury. 547 27 610 46 625 56 740 255 745 289 810 678 812 753 865 1333 (b) Dissociation of Ammonium Hydrosulphide. When solid ammonium hydrosulphide is heated, it is almost completely dis- sociated into ammonia and hydrogen sulphide as shown by the following equation : This reaction was investigated by Isambert,* who found that the total gas pressure at 25. 1 C. is equal to 501 mm. of mercury. Since the partial pressures of the ammonia and hydrogen sulphide are necessarily the same, each must be approximately equal to 250.5 mm., the relatively small pressure due to the undissociated vapor of the ammonium hydrosulphide being neglected. Let IT be the partial pressure of the vapor of ammonium hydrosulphide, and let p\ and p 2 be the partial pressures of the ammonia and hydrogen sulphide. Applying the law of mass action, we have Pi -Pa _ -IT,-. (1) Since TT is constant at any one temperature, equation (1) becomes pi'pz = K p f . According to Dalton's law of partial pressures, we have P = Pi + PZ + 7T, * Compt. rend., 93, 595, 730 (1881). HETEROGENEOUS EQUILIBRIUM 331 where P is the total pressure. Neglecting the relatively small pressure w, we may write P = Pi + P2- Hence, since p\ = p*, P 2 = Pi = Pz Substituting these values in equation (1), we obtain (501)' P2 V T = 62,750 The value of the equilibrium constant may be checked by observ- ing the effect on the system of the addition of an excess of either one of the products of the dissociation. The accompanying table gives the results of a few of Isambert's experiments. Pressure of Ammonia. Pressure of Hydrogen Sulphide. PNH 3 'PH 2 S = KP- 208 138 417 453 294 458 146 143 61,152 63,204 60,882 64,779 Mean 62,504 As will be seen, the mean value of the equilibrium constant agrees well with the value found for equivalent amounts of the products of dissociation. (c) Dissociation of Ammonium Carbamate. The dissociation of ammonium carbamate takes place according to the equation NH 2 This dissociation has been investigated by Horstmann.* Applying the law of mass action, we have Pi 2 ' P* = jr (2) * Lieb. Ann., 187, 48 (1877).| 332 THEORETICAL CHEMISTRY where pi and p 2 are the partial pressures of ammonia and carbon dioxide respectively, and where IT is the partial pressure of ammon- ium carbamate. Since IT is constant, equation (2) becomes If P denotes the total gaseous pressure, and TT is neglected as in the preceding example, we have, since three mols of gas are formed 4P 2 P Pi 2 : -g-, and p 2 = g- Substituting these values in equation (2), we have 4 27 = Kp'. This equation has also been tested by Isambert * by adding an excess of ammonia or carbon dioxide to the dissociating system. He found that the value of the equilibrium constant remains practically constant. The addition of a foreign gas was shown to be without effect on the dissociation. (d) Dissociation of the Hydrates of Copper Sulphate. Many inter- esting examples of heterogeneous equilibrium are furnished by hy- drated salts. Thus, if crystallized copper sulphate, CuSC^.S H 2 O, is placed in a desiccator, it gradually loses water of crystallization and ultimately only the anhydrous salt remains. If the desiccator be provided with a manometer and is so arranged that the tem- perature can be maintained constant, it is possible to observe the changes in vapor pressure accompanying the process of dehydra- tion. At the temperature of 50 C., the pressure over completely hydrated copper sulphate is found to remain constant at 47 mm. until the salt has been deprived of two molecules of water, when it drops abruptly to 30 mm. and remains constant until two more molecules of water have been lost. It then drops again to 4.4 mm. and remains constant until dehydration is complete. The successive stages of the dehydration are shown in the accom- panying diagram, Fig. 76. The constant pressures observed in the dehydration correspond to the successive equilibria involved. * Loc. cit. HETEROGENEOUS EQUILIBRIUM 333 At 50 C. the pentahydrate and the trihydrate are in equilibrium, a pressure of 47 mm. being maintained so long as any of the penta- hydrate is present. When all of the pentahydrate is used up, then the trihydrate begins to undergo dehydration into the monohydrate. This is a new equilibrium and the pressure of the 47mm 30 .mm 4.5, mm 3.H 2 O Composition Fig. 76. 1H 2 O OH 2 O aqueous vapor necessarily changes, and remains constant so long as any trihydrate remains. The last stage corresponds to the equilibrium between the monohydrate and the anhydrous salt. The following equations represent the three successive equilibria: (1) CuS0 4 5 H 2 ^ CuS0 4 3 H 2 + 2 H 2 0, (2) CuS0 4 3 H 2 + CuS0 4 H 2 O + 2 H 2 0, (3) CuS0 4 H 2 <=> CuS0 4 + H 2 O. Applying the law^of mass action to the first of the above equi- libria, we have in which TTI and ir 2 denote the partial pressures due to the hydrates CuSO 4 .5 H 2 O and CuSO 4 .3 H 2 respectively, and p denotes the 334 THEORETICAL CHEMISTRY pressure of aqueous vapor. Since TTI and 7r 2 are constant, the above expression simplifies to the following p 2 = K,'. In a similar manner it may be shown that the pressure of aqueous vapor in the other equilibria must be constant. It must be clearly understood that the observed pressure is only definite and fixed when two hydrates are present. If the dehydration CuS04 Temperature Fig. 77. were conducted at another temperature than 50 C. the equilibrium pressure would be different. The vapor pressure curves of the different hydrates are shown in the temperature-pressure diagram of Fig. 77. Heat of Dissociation of Solids. When the products of the dissociation" of a solid are gaseous, it has been pointed out by De Forcrand '* that the ratio of the heat of dissociation of 1 mol of * Ann. Chim. Phys. [7], 28, 545. HETEROGENEOUS EQUILIBRIUM 335 solid to the absolute temperature at which the dissociation pres- sure is equal to 1 atmosphere, is constant. Or, denoting the heat of dissociation by Q and the absolute temperature by T, De For- crand's relation may be expressed thus, ~ = constant 33. Nernst has shown that the value of the constant in this relation is not independent of the temperature. Thus, the value of the ratio at 100 C. is 29.7, while at 1000 C. it is 37.7. Up to the present time no expression has been derived in which the variation of the ratio with the temperature is included. Distribution of a Solute between Two Immiscible Solvents. When an aqueous solution of succinic acid is shaken with ether, the acid distributes itself between the ether and the water in such a way that the ratio between the two concentrations is always constant. It will be seen that the distribution of the succinic acid between the two solvents is analogous to that of a substance between the liquid and gaseous phases (see page 170), and there- fore the laws governing the latter equilibrium should apply equally to the former. Nernst * has shown that (a) // the molecular weight of the solute is the same in both solvents, the ratio in which it distributes itself between them is constant at constant temperature, or in other words, Henry's law is applicable; and (b) // there are several solutes in solution the distribution of each solute is the same as if it were present alone. This is clearly Dalton's law of partial pressures. The ratio in which the solute distributes itself between the two solvents is termed the coefficient of distribution or partition. The following table gives the results of three experiments on the distribution of succinic acid between ether and water. Concentration in Water. Concentration in Ether. Distribution Coefficient. 43.4 7.1 6.1 43.8 7.4 5.9 47.4 7.9 6.0 Zeit. phys. Chem., 8, 110 (1891), 336 THEORETICAL CHEMISTRY As will be seen the distribution coefficient is constant, show- ing that Henry's law applies. When the molecular weight of a solute is not the same in both solvents the distribution coefficient is not constant, and conversely, if the distribution coefficient is not constant, we infer that the molecular weights of the solute in the two solvents are not identical. Let us assume that a solute whose normal molecular weight is A, when shaken with two immiscible solvents undergoes polymeri- zation in one of them, its molecular weight being A n . We then have the equilibrium A n ^nA: applying the law of mass action, we have If the molecular weight in one solvent is twice the molecular weight in the other, then n = 2, and or 7^ = constant, since the two ratios are not equal. C2 VCa Thus Nernst found the following concentrations of benzoic acid when it was shaken with benzene and water. Cl (Water). c 2 (Benzene). C 2 V = ' 0.0150 0.0195 0.0289 0.242 0.412 0.970 0.062 0.048 0.030 0.0305 0.0304 0.0293 As will be seen, the values of the ratio Ci/c 2 steadily Decrease, while on the other hand, the values of the ratio Ci/Vc^ remain constant. This shows, therefore, that benzoic acid has twice the normal molecular weight in benzene. The Solution of a Solid in a Non-dissociating Solvent. When a solid is brought in contact with a non-dissociating solvent, it continues to dissolve until the solution becomes saturated. A condition of equilibrium then obtains, the rates of solution and HETEROGENEOUS EQUILIBRIUM 337 precipitation being the same. This is plainly a case of hetero- geneous equilibrium. If c is the concentration of the dissolved substance, and IT is the concentration of the undissolved solid, then according to the law of mass action c K -= K *> or since IT is constant, c = K c r . Variation of the Constant of Heterogeneous Equilibrium with Temperature. The reaction isochore equation of van't Hoff d (logg) _Q_ dT RT 2 ' which has been shown to connect the displacement of a homo- geneous equilibrium with change in temperature, applies equally well to heterogeneous equilibria. The following examples will serve to illustrate its application in such cases. (a) Dissociation of Ammonium Hydrosulphide. In the reaction representing the dissociation of ammonium hydrosulphide, [NH 4 HS]^(NH 3 ) + (H 2 S), let pi and p 2 be the partial pressures of ammonia and hydrogen sulphide, and let TT be the partial pressure of ammonium hydro- sulphide. Then as has been shown (see page 331), where P is the total gaseous pressure. From the following data: Ti = 273 + 9.5, Pi = 175 mm. of mercury, and r 2 = 273 + 25.l, P 2 = 501 mm. of mercury, we have, on applying the reaction isochore equation, and solving forQ p , 4.581 [log (^i) 2 - log (lpj] 282.5 X 298.1 p = 282.5 - 298.1 or Q p = 22,740 calories. 338 THEORETICAL CHEMISTRY This result agrees well with the value, 22,800 calories, found by direct experiment. (b) Solution of Succinic Acid. The concentration of succinic acid (in a saturated solution) and the temperature, are the factors which determine the equilibrium in this case. In the equation d (\og e K c ) Q c dT RT 2 ' K c = c, where c is the concentration of succinic acid in a saturated solution. The following experimental data, due to van't Hoff, enables us to calculate the heat of solution of the acid. Ti = 273 c = 2.88 grams per 100 grams of water, and T 2 = 273 + 8.5, c = 4.22 grams per 100 grams of water. Substituting in the reaction isochore equation and solving for Q c , we have Q 4.581 (log 4.22 - log 2.88) 273 X 281.5 ^ c = 273 - 281.5 or Q = 6871 calories. The value of the heat of solution for 1 mol of succinic acid as found by direct experiment is 6700 calories. The Phase Rule. While it is possible to apply the law of mass action to certain heterogeneous equilibria there are numerous cases where its application is either difficult or impossible. To deal with such heterogeneous systems we make use of a general- ization discovered by J. Willard Gibbs,* late professor of mathe- matical physics in Yale University. This generalization was first stated by Gibbs in 1874, and is commonly known as the phase rule. Before entering upon a discussion of the phase rule, it will be necessary to define a few of the terms employed. The composition of a system is determined by the number of independent variables or components involved. Thus in the system ice, water, and vapor there is but a single com- ponent. In the system CaC0 3 <=CaO + CO 2 , * Trans. Connecticut Academy, Vols. II and III, 1875-8. HETEROGENEOUS EQUILIBRIUM 339 while there are three constituents of the equilibrium, only two of these need be considered as components, for the amount of any one constituent is not independent of the amounts of the other two, as the following equations show : CaO + CO 2 = CaC0 3 , CaC0 3 - CaO = CO 2 , CaC0 3 - C0 2 = CaO. In general, the components are chosen from the smallest number of independently-variable constituents required to express the composition of each phase entering into the equilibrium, even negative quantities of the components being permissible. The number of variable factors, temperature, pressure, and concentration, of the components which must be arbitrarily fixed in order to define the condition of the system, is known as the degree of freedom of the system. For example, a gas has two degrees of freedom since two of the variables, temperature, pres- sure or volume, must be fixed in order to define it; a liquid and its vapor has only one degree of freedom, since for equilibrium at a certain temperature, there can be but a single pressure; and in a system consisting of a substance in the three states of aggregation, equilibrium can only exist at a single temperature and pressure. Derivation of the Phase Rule. The following derivation of the phase rule is due to Nernst. Let us assume a complete hetero- geneous equilibrium made up of y phases of n components, and let us fix our attention upon one single phase. This phase will con- tain a certain amount of each one of the n components, the con- centrations of which may be designated by Ci, c 2 , c 3 , . . . c n . Since we have assumed complete equilibrium to exist, the slightest change in concentration, temperature or pressure will alter the composition of this phase. This may be expressed by the equation / (ci, C2, c 3 , . . . c n , p, T) = 0, where / is any function of the variables. Since any change in one phase implies a corresponding change in the remaining y 1 phases, it follows that the composition of all the phases is a certain determined function of the same variables. 340 THEORETICAL CHEMISTRY The above equation is, then, of the form ascribed to each sepa- rate phase, and since there are y phases we have y separate equa- tions. There are, however, n + 2 variables in each equation, so that if y = n + 2, that is if we have two more phases than com- ponents, each unknown quantity has a definite known value. In this case there is only one value for ci, c*, c 3 , c 4 , . . . c n , p and T at which the system can be in equilibrium. Hence when n components are present in n + 2 phases, we have equilibrium only for a certain temperature, a certain pressure, and a certain ratio of concentrations of the single phases. That is, n + 2 phases of n substances can only exist at a certain point in a co-ordinate system. This point is termed the transition point. If one value be altered then one phase vanishes, and there remain n + 1 phases of n components, and the problem becomes indeterminate. Thus it is proved that n components are necessary in order that a system containing n -j- 1 phases may exist in complete equilibrium. The- phase rule may be stated as follows: A system made up of n components in n + 2 phases can only exist when pressure, temperature and concentration have definite fixed values; a system of n components in n + 1 phases can exist only so long as one of the factors varies; and a system of n components in n phases can exist only so long as two of the factors vary. If P denotes the number of phases, C the number of components, and F the number of degrees of freedom, then the phase rule may be conveniently summarized by the expression, C - P + 2 = F. Equilibrium in the System, Water, Ice, and Vapor. In this system we may have one, two, or three phases present, according to the conditions. Under ordinary circumstances of temper- ature and pressure, water and water vapor are in equilibrium. The vapor pressure curve of water is represented by the line OA in the pressure-temperature diagram (Fig. 78). It is only at points on this curve that water and its vapor are in equilibrium. Thus, if the pressure be reduced below that corresponding to any point on OA, all of the water will be vaporized; if on the other hand, the pressure be raised above the curve, all of the vapor will HETEROGENEOUS EQUILIBRIUM 341 ultimately condense to the liquid state. When the temperature is reduced below C., only ice and vapor are present, the curve OC representing the equilibrium between these two phases. It is to be observed that the curve OC is not continuous with OA. At Solid Vapor 0.0075 Temperature Fig. 78. the point 0, where the two curves intersect, ice, water, and water vapor are in equilibrium. At this point ice and water must have the same vapor pressure, otherwise distillation of vapor from the phase having the higher vapor pressure to that with the lower vapor pressure would occur, and eventually the phase having the higher vapor pressure would disappear. This result would be in contradiction to the experimentally-determined fact that both solid and liquid phases are in equilibrium at the point 0. The temperature at which ice and water are in equilibrium with their vapor under atmospheric pressure is C. Since increase of pressure lowers the freezing-point of water, the point 0, repre- senting the equilibrium between ice and water under the pressure of their own vapor, viz., 4.57 mm., must be a little above C. The exact temperature corresponding to the point has been found to be O.0075 C. 342 THEORETICAL CHEMISTRY The change in the melting-point of ice due to increasing pressure is represented by the line OB. This line is inclined toward the vertical axis because the melting point of ice is lowered by in- creased pressure. The point is called a triple point because there, and there only, three phases are in equilibrium. As is well known, water does not always freeze exactly at C. If the containing vessel is perfectly clean, and care is taken to exclude dust, it is possible to supercool water several degrees below its freezing-point and measure its vapor pressure. The dotted curve OA', which is a continuation of OA, represents the vapor pressure of supercooled water. It will be noticed that (1) there is no break in the vapor-pressure curve so long as the solid phase does not separate, and (2) the vapor pressure of super- cooled water, which is an unstable phase, is greater than that of ice, the stable phase, at that temperature. We now proceed to apply the phase rule to this system. In the formula, C - P + 2 = F, C = 1. It is evident that if P = 3, then F = 0; or the system has no degree of freedom. We have seen that the triple point 0, represents such a condition. At this point ice, water, and water vapor are co-existent, and if either one of the variables, temperature or pressure, is altered, one of the phases disappears; in other words, the system has no degree of freedom. Such a system is said to be non-variant. If in the above formula, P = 2, then F = 1, and the system has one degree of freedom, or is univariant. Any point on any one of the curves OA, OB, or OC represents a univariant system. Take, for exam- ple, a point on the curve OA. In this case the temperature may be altered without altering the number of phases in equilibrium. If the temperature is raised, a corresponding increase in vapor pressure follows and the system will adjust itself to some other point on the curve OA. In like manner, the pressure may be altered without causing the disappearance of one of the phases. If, however, the temperature is maintained constant, then a change in the pressure will cause either condensation of water vapor or vaporization of liquid water. Under these conditions the system has only one degree of freedom. Again, if P = 1, then F = 2, and the system is bivariant, or has two degrees of freedom. The HETEROGENEOUS EQUILIBRIUM 343 areas included between the curves in the diagram are examples of bivariant systems. Consider the vapor phase; the temperature may be fixed at any desired value within the vapor area AOC, and the pressure may be altered along a line parallel to the vertical axis without causing a change in the number of phases, provided the curves OA and OC are not intersected. The System, Sulphur (Rhombic, Monoclinic), Liquid and Vapor. This system is more complicated than the preceding one-component system, since there are two solid phases in addition to the liquid and vapor phases. At ordinary temperatures, rhom- bic sulphur is the stable modification. When this is heated rapidly it melts at 115 C., but if it is maintained in the neighbor- hood of 100 C. it gradually changes into monoclinic sulphur which melts at 120 C. Monoclinic sulphur can be kept indefin- itely at 100 C. without undergoing change into the rhombic modification, or in other words it is the stable phase at this temper- ature. It is evident, therefore, that there must be a temperature above which monoclinic sulphur is the stable form and below which rhombic sulphur is the stable modification. This temperature at which both rhombic and monoclinic modifications are in equi- librium with each other and with their vapor, is termed the transition point. Its value has been determined to be 95.6 C. The change from one form into the other is relatively slow, so that it is possible to measure the vapor pressure of rhombic sul- phur up to its melting-point, and that of monoclinic sulphur below its transition point. The vapor pressure of solid sulphur, although very small, has been measured as low as 50 C. The complete pressure-temperature diagram for sulphur is shown in Fig. 79. At the point 0, rhombic and monoclinic sul- phur are in equilibrium with sulphur vapor, this being a triple point analogous to the point in Fig. 78. The vapor pressure curves of rhombic and monoclinic sulphur are represented by OB and OA respectively. The dotted curve OA' which is a continu- ation of OA is the vapor-pressure curve of monoclinic sulphur in a metastable region. In like manner OB' represents the vapor- pressure curve of rhombic sulphur in the metastable condition, B* 344 THEORETICAL CHEMISTRY being a metastable melting point. As in the pressure-temper- ature diagram for water, the metastable phases have the higher vapor pressures. The effect of increasing pressure on the transi- tion point 0, is represented by the line OC. This is termed a Vapor Temperature Fig. 79. transition curve, and, since increase in pressure raises the transi- tion point, the line slopes away from the vertical axis. The effect of increased pressure on the melting-point of monoclinic sulphur is shown by the curve AC. This also slopes away from the vertical axis, but the change in the melting-point of monoclinic sulphur produced by a given change in pressure being less than the corresponding change in the HETEROGENEOUS EQUILIBRIUM 345 transition point, the two curves, OC and AC, intersect at the point C. The point C corresponds to a temperature of 131 C. and a pressure of 400 atmospheres. The vapor-pressure curve of stable liquid sulphur is represented by the curve AD. The vapor-pressure curve of the metastable liquid phase is represented by the curve AB r which is continuous with AD. The diagram is completed by the curve B'C which represents the effect of pressure on the metastable melting-point of rhombic sulphur. Mono- clinic sulphur does not exist above the point C; hence when liquid sulphur is allowed to solidify at pressures exceeding 400 at- mospheres, the rhombic modification is formed, whereas under ordinary pressures the monoclinic modification appears first. The phase rule enables us to state the exact conditions required for equilibrium in this system and to check the results of exper- iment. Thus, according to the formula, C P + 2 = F, since C = 1, the system will be non-variant when P = 3. Since there are four phases involved, theoretically any three of these may be co-existent and four triple points are possible. The theoreti- cally-possible triple points are as follows: (1) Rhombic sulphur, monoclinic sulphur, and vapor (0); (2) Rhombic sulphur, monoclinic sulphur, and liquid (C); (3) Rhombic sulphur, liquid, and vapor (B f ); (4) Monoclinic sulphur, liquid and vapor (A). In this particular system all of the four possible triple points .can be realized experimentally. That this is the case is due to the comparative slowness of the change from rhombic to monoclinic sulphur above the triple point. If this change were rapid it is evident that all of the theoretically-possible non-variant systems could not be realized experimentally. As in the case of water, the curves in the diagram represent univariant systems and the areas bivariant systems. The student is advised to tabulate the univariant and bivariant systems repre- sented in the pressure-temperature diagram for sulphur. Two-component Systems. Turning now to two-component systems we are confronted with a more difficult problem, and one which includes many special cases. Thus, we may have cases of 346 THEORETICAL CHEMISTRY anhydrous salts and water, hydrated salts and water, volatile solutes, two liquid phases, consolute liquids, and solid solutions. To enter upon a discussion of these would not be profitable, since they only serve to give greater emphasis to the general truth of the phase rule. We shall select a few typical two-component systems for consideration here. (a) Anhydrous Salt and Water. In the equilibrium diagram of water (here represented by dotted lines, Fig. 80), we desig- Temperature Fig. 80. nate the triple point by 0. At this point ice and water have the same vapor pressure. Similarly, a solution at its freezing-point has the same vapor pressure as the ice which separates. The intersection of the vapor-pressure curve for ice, OB, and the vapor- pressure curve of the solution of the anhydrous salt, 0"A", deter- mines a new triple point 0" '. Since the presence of the dissolved salt tends to diminish the vapor pressure of water, the curve 0" A" is situated below the curve OA, and for the same reason the triple point 0" is found to the left of 0. If now we keep an excess of dissolved substance continually present, all of the liquid phases HETEROGENEOUS EQUILIBRIUM 347 which are formed will of necessity be saturated solutions. When these solutions finally freeze they will furnish, not pure ice, but a mixture of ice and solid salt, known as a cryohydrate. By a partial freezing we can therefore obtain the system: Solid salt, ice, saturated solution and vapor, or in other words, a system of n -f- 2 phases of which the existence is only possible at the freez- ing temperature T' of the saturated solution, and under the pres- sure p' corresponding to the vapor pressure of ice and the saturated solution. These conditions are represented in the diagram by the quadruple point O 1 '. If now we pass from the point 0', increas- ing the temperature and pressure as prescribed by the curve O'A', the ice disappears, while the salt, the saturated solution, and the vapor furnish a series of 3-phase systems. Again start- ing from the point 0' and lowering the temperature and the pres- sure as indicated by the curve O'B, the liquid phase disappears, while the solid salt, ice, and vapor constitute another series of 3-phase systems. This, of course, is on the supposition that the vapor pressure of the solid salt is negligible. Finally, a consider- able increase in pressure causes a slight lowering of the temper- ature corresponding to the quadruple point, the conditions being represented by the curve O'C". All possible non-saturated solutions of the salt will be repre- sented by points within the area, AOO'A'. Thus, let 0"A" repre- sent the vapor-pressure curve of a dilute solution of the salt in water. The freezing-point of this solution is represented by the point 0", while 0"C" represents the variation of the freezing-point of the solution with pressure. The following table summarizes the possibilities indicated by the phase rule : , 4 phases; salt, ice, saturated solution, vapor (point, 0')', 3 phases; salt, saturated solution, vapor (curve, O'A'); 3 phases; salt, ice, vapor (curve, O'B)', 3 phases; salt, ice, saturated solution (curve,, O'C'); 2 phases; salt, saturated solution (area, C'O'A')', 2 phases; salt, water vapor (area, BO' A')', 2 phases; salt, ice (area, BO'C')] 348 THEORETICAL CHEMISTRY 3 phases; ice, non-saturated solution, vapor (curve, 00'); 2 phases; non-saturated solution, vapor 1 phase; non-saturated solution 2 phases; non-saturated solution, ice 1 phase; non-saturated solution (area, (area, COO'C'). As will be seen, there is only one non-variant point in the entire diagram, viz., the point 0'. In this system there are three degrees of freedom, since in addition to temperature and pressure the con- centration of the solution may also be varied. The pressure-temperature diagram (Fig. 80) having been dis- cussed, we now turn to the concentration-temperature diagram for the same system, Fig. 81. In this diagram the abscissae Temperature Fig. 81. represent temperatures and the ordinates, concentrations. For convenience, corresponding points in Figs. 80 and 81 will be desig- nated by the same letters. The equilibrium between ice, water and water vapor is represented by the point 0. If now a small HETEROGENEOUS EQUILIBRIUM 349 amount of anhydrous salt be added to the water, the freezing-point will be lowered to 0". As the proportion of salt is increased the temperature of equilibrium is lowered along the curve 00" 0'. A point is ultimately reached at which the solution becomes saturated, and on further addition of salt it is not dissolved, but remains in contact with the ice and saturated solution. This is the cryo- hydric point, and represents the lowest temperature which can be obtained in this particular system. The diagram is completed by the solubility curve of the salt, O'A'. Each point on this curve represents the concentration of the saturated solution at all temperatures, from the critical temperature of the solution to the cryohydric temperature. The meaning of the concentration- temperature diagram may be made clearer by a consideration of the behavior of a solution when gradually cooled. Let a repre- sent a dilute solution of the anhydrous salt. On lowering the temperature along ab, no change will occur until the curve 00' is reached; then ice will begin to separate and as the cooling is continued, the composition of the solution will change along 00' until it reaches the cryohydric point 0'. Here both salt and ice will separate, and the solution will solidify completely at the temperature corresponding to the point 0'. In like manner, if we start with a concentrated solution represented by the point c and cool along cd no change will take place until the curve O'A' is reached; then solid salt will separate and the composition of the solution will alter along O'A' until the temperature is reduced to that corresponding to the cryohydric point, when the whole solution will solidify as in the previous case. This phenomenon was first systematically investigated by Guthrie * who concluded that such mixtures of constant composition and definite melting- point are chemical compounds, and, therefore, he proposed to call them cryohydrates. It has since been shown that cryohydrates are not definite chemical compounds. Among the various reasons which have been advanced to prove the incorrectness of Guthrie's views, the following are the most cogent: (1) the physical properties of a cryohydrate are the mean of the corresponding properties of the constituents, this being rarely true of chemical * Phil. Mag, [4], 49, 1 (1875); [5], i, 49 and 2, 211 (1876). 350 THEORETICAL CHEMISTRY compounds; (2) the lack of homogeneity of a cryohydrate can be detected under the microscope; and (3) the constituents are seldom present in simple molecular proportions. Applying the phase rule to the above two-component system, it is evident that there is but one non-variant system: this is represented by the point 0' '. When three phases are co-existent the system is univa riant, when only two phases are present the system is bivariant, and finally, when only one phase is present the system acquires three degrees of freedom or is trivariant. It is evident that a system having three degrees of freedom cannot be completely represented by a diagram in a single plane. It is possible, however, to construct a three-dimensional model which will represent the equilibrium very satisfactorily. Such a model is Temperature Fig. 82. I. Unsaturated Solution. II. Salt and Saturated Solution. III. Ice and Unsaturated Solution. IV. Ice and Cryohydrate. V. Salt and Cryohydrate. shown in Fig. 82, the lettering being made to correspond with that of the two diagrams, Figs. 81 and 82, from which it is derived. HETEROGENEOUS EQUILIBRIUM 351 (b) Hydrated Salt and Water. An interesting example is fur- nished by the system ferric chloride and water. This system has been very carefully investigated by Roozeboom.* The con- centration-temperature diagram, plotted from Roozeboom's data, Temperature - Fig. 83. is given in Fig. 83. The freezing-point of pure water is repre- sented by A, and the lowering of the freezing-point produced by the addition of ferric chloride is indicated by the curve AB. At the cryohydric temperature, - 55 C., ice, Fe 2 Cle 12 H 2 0, sat- urated solution, and vapor are in equilibrium, and the system is non-variant. On adding more ferric chloride, the ice phase dis- appears, and the univariant system, Fe 2 Cl6 12 H 2 0, saturated solution, and vapor results. The equilibrium is represented by the curve BC which may be regarded as the solubility curve of the dodecahydrate. On continuing the addition of ferric chloride, the temperature continues to rise until the point C is reached. Here the composition of the solution is identical with that of the dodecahydrate, and, therefore, the temperature corresponding to this point, 37 C., may be looked upon as the melting-point of Fe 2 Cl 6 12 H 2 0. Further addition of ferric chloride will naturally * Zeit. phys. Chem., 4, 31 (1889); 10, 477 (1892). 352 THEORETICAL CHEMISTRY lower the melting point and the equilibrium will alter along the curve CD. It is thus possible to have two saturated solutions, one of which contains more water and the other less, than the hydrate which is in equilibrium with the solution. These solu- tions are both stable throughout and are nowhere supersaturated. Roozeboom was the first investigator to discover a saturated solution containing less water than the solid hydrate with which it is in equilibrium. This discovery led him to define supersatu- ration as follows: "A solution is supersaturated with respect to a solid phase at a given temperature if its composition is between that of the solid phase and the saturated solution. " At the point D the curve reaches another minimum which is analogous to the point B, except that the heptahydrate, Fe 2 Cl 6 7 H 2 0, takes the place of ice. Here we have equilibrium between the dodecahydrate, the heptahydrate, saturated solution, and vapor, and the system is non-variant. On further addition of ferric chloride another maximum is reached at E, corresponding to the melting-point of the heptahydrate. In a similar manner, two other maxima at greater concentrations of ferric chloride reveal the existence of the hydrates, Fe 2 Cl 6 5 H 2 O, and Fe 2 Cl 6 4 H 2 0. At the three remaining quadruple points the following phases are in equilibrium : At F, Fe 2 Cl 6 '7 H 2 0, Fe 2 Cl 6 -5 H 2 0, saturated solu- tion and vapor; at H, Fe 2 Cle 5 H 2 0, Fe 2 Cl 6 4 H 2 O, saturated solution and vapor; and at K, Fe 2 Cle 4 H 2 0, Fe 2 Cle, saturated solution and vapor. The solubility of the anhydrous salt is represented by the curve KL. Metastable solubility and melting- point curves are represented by dotted lines. The student should apply the phase rule to this system. If a fairly dilute solution of ferric chloride is evaporated at 31 C., the water gradually disappears and a residue of the dodecahydrate remains. This residue then liquefies and again dries down, the composition of the residue corresponding to the heptahydrate: on further standing the phenomenon is repeated, the final and permanent residue having a composition corresponding to the pentahydrate. The dotted line ab shows the isothermal along which the composition varies. It would have been a difficult matter to explain the alternations of moisture and dryness ob- HETEROGENEOUS EQUILIBRIUM 353 served in this experiment without the concentration-temperature diagram. Alloys. Among the most interesting two-component systems known are those involving mixtures of metals, or alloys. These have been made the subject of systematic investigations by num- erous experimenters among whom may be mentioned Roberts- Austen, Charpy, Roozeboom, and Heycock and Neville. We have space to consider only two comparatively-simple cases. (a) Alloys of Silver and Copper. The conditions of equilibrium in this binary system have been studied by Heycock and Neville.* The two components, silver and copper, are not miscible in the solid state and do not combine chemically. To determine the curves of equilibrium, mixtures of the two metals in varying pro- portions were fused and then allowed to cool slowly, the rate of cooling being observed with a thermocouple, one junction of which was maintained at constant temperature, while the other junction was placed in the mixture of molten metals. The terminals of the thermocouple were connected to a sensitive galvanometer graduated to read directly in degrees, and the rate of cooling was followed by the movement of the needle of the galvanometer. As the mixture cooled, two " breaks" were observed; the first of these varied with the composition of the mixture, while the second remained practically constant at 777 C. When the temperatures corresponding to the first break are plotted as ordinates against the composition of the mixture as abscissae, the diagram shown in Fig. 84 is obtained. The point A represents the freezing-point of pure silver, B that of pure copper, the curve AO represents the effect of the gradual addition of copper upon the freezing-point of silver, and BO the effect of silver on the freezing-point of copper. The intersection of the two curves at corresponds to an alloy containing 40 atomic per cent of copper. This lowest melting mixture is known as the eutectic (ev = well, and TTTJKUV = melt) mixture. At the system is non-variant, silver, copper, solution and vapor being in equilibrium. The solid which separates at 0, having a more uniform texture than that of all other mixtures of the two com- * Phil. Trans., 189, 25 (1897). 354 THEORETICAL CHEMISTRY ponents, is known as the eutectic alloy. When the composition of a mixture of two metals corresponds to that of the eutectic alloy, the two metals crystallize simultaneously in minute separate Solution Aff+Solution Cu+Solution AK+ Eutectic Cu+Eutectlc 1081.5 Ag 40 at. per cent Concentration Fig. 84. Cu crystals. When examined under the microscope the solid eutectic alloy will be seen to be a conglomerate of very small crystals, whereas all of the other alloys of the same metals will be found to contain large crystals of either one or the other component em- bedded in the conglomerate. While the composition of the eutec- tic alloy in the above system is found to correspond very closely to the formula Ag 3 Cu 2 , yet the nature of the equilibrium curves proves it to be nothing more than a mechanical mixture of the two metals. The meaning of the diagram will be clearer from a careful consideration of the phenomena accompanying the cooling of a mixture of the molten metals. Take for example, a fused mixture relatively rich in silver. As the temperature falls, a point will ultimately be reached at which HETEROGENEOUS EQUILIBRIUM 355 pure silver begins to separate, and since the temperature remains constant during the solidification, a break occurs in the cooling curve. This first break corresponds to a point on the curve AO. As silver continues to separate, the composition of the mixtures changes along AO, until when is reached, the mixture is satu- rated with respect to copper, and both metals separate as a con- glomerate having the same composition as the fused mixture. The separation of the eutectic alloy causes the second break in the cooling curve, the temperature remaining constant until the en- tire mass has solidified. It will be noticed that this system is the exact analogue of the system anhydrous salt and water; the eutec- tic point and the cryohydric point representing identical conditions. (b) Alloys of Gold and Aluminium. This system has been studied by Roberts- Austen.* The equilibrium curves in the concentration temperature diagram, Fig. 85, reveal the existence I Composition Fig. 85. * Phil. Trans. A., 194, 201 (1900). 356 THEORETICAL CHEMISTRY of definite compounds, AusAl 2 , Au 2 Al, and AuAl 2 , corresponding to the points D, E, and H respectively. The discontinuities at B and G suggest the possibility of two other compounds, viz., Au4Al and AuAl. The diagram shows that the following substances will crystallize in succession from the molten alloy, these being the different solids with which the liquid mixture is saturated in its successive stages of equilibrium : - Curve AD, pure gold at A ; Curve BC, Au4Al, nearly pure at B; Curve CD, Au 5 Al 2 or Au 8 Al 3 , nearly pure at D; Curve DEF, Au 2 Al, pure at E\ Curve FG, AuAl, maximum undetermined ; Curve GHI, AuAl 2 , pure at H', Curve 7J, Al, pure at J. The points C, F, and / represent non-variant systems, the melt- ing points of the respective eutectic alloys being 527, 569, and 647. This system in many respects resembles the system ferric chloride and water. Three-component Systems. When three components are pres- ent, the equilibria become much more complicated. Applying the formula, C P + 2 = F, we find that it is necessary to have five phases co-existent for a non-variant system, four for a univariant, three for a bivariant, and two for a trivariant. The most satisfactory method of representing equilibria in three-com- ponent systems is that in which use is made of the triangular diagram. The three corners of an equilateral triangle are taken to represent the pure components, and the composition of any mixture, expressed in atomic percentages, is represented by the position of the center of mass of the three components within the triangle. IJor example, in the system, potassium nitrate, sodium nitrate, and lead nitrate, carefully investigated by Guthrie,* the three components are placed at the corners of the triangle shown in * Phil. Mag., 5, 17, 472 (1884). HETEROGENEOUS EQUILIBRIUM 357 Fig. 86. The melting-point of pure potassium nitrate is 340 and that of pure sodium nitrate is 305. The melting-point of Fig. 86. pure lead nitrate cannot be determined since the salt decomposes before its melting-point is reached. The eutectic mixtures of the three pairs of salts are represented by the points D, E, and F respectively. In like manner represents the melting point of the non-variant system, potassium nitrate, sodium nitrate, lead nitrate, fused mixture of the three salts, and vapor. In order to represent temperature, use is frequently made of a triangular prism in which the altitude is taken as the temperature axis, the resulting surface within the prism representing the variation of the equilibrium with temperature.* PROBLEMS. 1. The vapor pressure of solid NH 4 HS at 25. 1 is 50.1 cm. Assuming that the vapor is practically completely dissociated into NH 3 and H 2 S, calculate the total pressure at equilibrium when solid NH 4 HS is allowed * For a complete treatment of three-component systems as well as for a clear presentation of the phase rule, the student should consult " The Phase RuJe and Its Applications," by Alexander Findlay. 358 THEORETICAL CHEMISTRY to dissociate at 25. 1 in a vessel containing ammonia at a pressure of 32 cm. Ans. 59.5 cm. 2. In the partition of acetic acid between CCU and water, the con- centration of the acetic acid in the CC1 4 layer was c gram-molecules per liter and in the corresponding water layer w gram-molecules per liter. c 0.292 0.363 0.725 1.07 1.41 w 4.87 5.42 7.98 9.69 1.07 Acetic acid has its normal molecular weight in aqueous solutions. From these figures show that, at these concentrations, the acetic acid in the carbon tetrachloride solution exists as double molecules. 3. Acetic acid distributes itself between water and benzene in such a manner that in a definite volume of water there are 0.245 and 0.314 gram of the acid, while in an equal volume of benzene there are 0.043 and 0.071 gram. What is the molecular weight of acetic acid in benzene, assuming it to be normal in water? Ans. 121.3. 4. The salt Na 2 HP0 4 .12 H 2 has a vapor pressure at 15 of 8.84 mm., and at 17. 3 of 10.53 mm. Calculate the heat of vaporization, i.e., the thermal change during the loss of 1 mol of water of crystallization by evaporation. Ans. 12,651 cal. 5. The solubility of boric acid in water is 38.45 grams per liter at 13, and 49.09 grams per liter at 20. Calculate the heat of solution of boric acid per mol. Ans. 5,822 cal. 6. Plot the pressure-temperature diagram for calcium carbonate from the table given on p. 280, and apply the phase rule. 7. Is it possible to decide by the phase rule whether the eutectic alloy is a mixture or a compound? CHAPTER XVII. CHEMICAL KINETICS. Velocity of Reaction. In the two preceding chapters we have considered the equilibrium which is established when the speeds of the direct and reverse reactions have become equal. We now proceed to consider the velocity of individual reactions. By far the greater number of the reactions between inorganic substances proceed with such rapidity that it is impossible to measure their velocities. Thus, when an acid is neutralized by a base, the indi- cator changes color almost instantly. There are a few well- known reactions which are exceptions to this rule; among these may be mentioned the oxidation of sulphur dioxide and the de- composition of hydrogen peroxide. Both of these reactions are well adapted to kinetic experiments. In organic chemistry, on the other hand, slow reactions are the rule rather than the excep- tion. Thus the reaction between an alcohol and an acid forming an ester and water, proceeds very slowly under ordinary condi- tions and the progress of the reaction may be easily followed. By means of the law of mass action it is possible to derive equations expressing the velocity of a reaction at any moment in terms of the concentrations of the reacting substances present at that time. Let the equation represent a reversible reaction and let a, 6, c, and d be the respec- tive initial concentrations of the reacting substances A\, A 2 , A\, and A 2 '. The velocity of the direct reaction will then be AT g = k (a - x) (b - x), (D where k is the velocity constant, and dx is the infinitely small increase in the amount of x during the infinitely small interval 359 360 THEORETICAL CHEMISTRY of time dt. Similarly the velocity of the reverse reaction will be dx' ^ = *i (c + x) (d + x). (2) It is evident that the substances on the right-hand side of the equa- tion will exert an ever-increasing influence upon the velocity of the direct reaction, which must accordingly decrease. .When, however, the velocities of the direct and reverse reactions become equal, equilibrium will be established, and the ratio of the amounts of the reacting substances on the two sides of the equation will remain constant. The total velocity due to these opposing reac- tions will be ^j = ^-~ = k(a-x)(b-x)-k t (c + x)(d + x) (3) j~y and at equilibrium, when -jr = 0, k(a-x)(b-x) = k, (c + x) (d + x), or (c + ap (d + x) _kjf m (a - x) (b - x) " /d " This equation has been thoroughly tested in the two preceding chapters. Thus, in the reaction C 2 H 5 OH + CH 3 COOH<=> CH 3 COOC 2 H 5 + H 2 O, K c has been shown to have the value, 2.84, at ordinary temper- atures. The velocity constants of the direct and reverse reactions have also been determined, the values being, k = 0.000238 and ki = 0.000815. When these values are substituted in the equa- k tion, T- = K c , we obtain K c = 2.92, a value which agrees well KI with that found by direct experiment. The application of equa- tion (3) is much simplified by the fact that most reactions proceed nearly to completion in one direction, so that the term k\ (c + x) (d + #) will be so small that it may be neglected. We then have 2 = k (a - *) (6 - x), (5) CHEMICAL KINETICS 361 an equation expressing the velocity of the direct reaction in terms of the concentrations of the reacting substances. Unimolecular Reactions. The simplest type of chemical reaction is that in which only one substance undergoes change and in which the velocity of the reverse reaction is negligible. The decomposition of hydrogen peroxide is an example of such a reac- tion. In the presence of a catalyst, such as certain unorganized ferments or collodial platinum, hydrogen peroxide decomposes as represented by the equation, This reaction is usually allowed to take place in dilute aqueous solution so that there is no appreciable alteration in the amount of solvent throughout the entire course of the reaction. Further- more, the activity of the catalyst remains constant so that the course of the reaction is wholly determined by the concentration of the hydrogen peroxide. A very satisfactory catalyst is haemase, an enzyme derived from blood. The concentration of hydrogen peroxide present at any time during the reaction can be deter- mined very simply by removing a definite portion of the reaction mixture, adding an excess of sulphuric acid to destroy the activity of the hsemase, and then titrating with a standard solution of potassium permanganate. The following table gives the results of such an experiment: t (minutes). a'x, cc. KMnO 4 . z. cc. KMnO 4 . k o 46 1 o 5 37.1 9.0 0.0435 10 29.8 16.3 0.0438 20 19.6 26.5 0.0429 30 12.3 33.8 0.0440 50 5.0 41.1 0.0444 Mean 0.0437 The second column of the table gives the number of cubic centimeters of the potassium permanganate solution required to oxidize 25 cc. of the reaction mixture when the time intervals 362 THEORETICAL CHEMISTRY recorded in the first column have elapsed after the introduction of the catalyst. Since the numbers in the second column repre- sent the actual concentration of hydrogen peroxide present at the end of the successive intervals of time, it is evident that the difference between these numbers and 46.1 cc. the initial concentration of hydrogen peroxide will give the amounts of peroxide decomposed in those intervals. These numbers are recorded in the third column of the table. It will be seen that as the concentration of the hydrogen peroxide decreases the rate of the reaction diminishes. Thus, hi the first interval of 10 min- utes, an amount of hydrogen peroxide corresponding to 46.1 29.8 = 16.3 cc. of potassium permanganate is decomposed, while in the second interval of 10 minutes, the amount of hydrogen peroxide decomposed is equivalent to 29.8 19.6 = 10.2 cc. of potassium permanganate. Since only a single substance is under- going change, equation (5) simplifies to the following form:- dX 7 / N = k (a - x). * It is impossible to apply the equation in this form, since in order to obtain accurate titrations, dt must be taken fairly large and during this interval of time a x would have diminished. Approx- imate values of k may be obtained by taking the average value of a x during the interval dt within which an amount dx of hydro- gen peroxide is being decomposed. For example, let us take the interval between 5 and 10 minutes; dx = 16.3 9.0 = 7.3 cc., dt = 5 min., and the average value of a x is 37.1 + 29.8 33.45 cc. Substituting in the equation we have and = k X 33.45, k = 0.0436. CHEMICAL KINETICS 363 Similarly taking the next interval between 10 and 20 minutes; dx = 26.5 16.3 = 10.2 cc., dt = 10 minutes, and the average 29 8 - 1 19 6 value of a x is - - = 24.7 cc. Substituting in the 2i equation as before, we obtain and k = 0.0413. As will be seen these two values of k are not in good agreement, although the first value of k agrees closely with the mean value of k given in the fourth column of the table. In order to apply the equation dx ^ t . 37 = k (a x) at it must be integrated.* The integration of this equation may be performed as follows: dx j f , = *(-*), therefore dx = kdt a x Integrating, we have /j / -- / k dt = constant = C, a x J therefore loge (a x) kt = C. In order to determine C, the constant of integration, we make use of the experimental fact that when t = 0, x = 0. Substitut- ing these values, we have log e a = C. , Consequently log e a loge (a x) = kt, or 1 a * The student who is unfamiliar with the Calculus must take the result of this calculation for granted. 364 THEORETICAL CHEMISTRY Passing to Briggsian logarithms, we obtain _ I g a = 0.4343 k. t a x By substituting in this equation the corresponding values of a, a x, and t from the preceding table, the values of k given in the fourth column of the table are obtained. The equation dx may also be thrown into an exponential form, as follows: - Since 1, a we may write, or a x = ae~ kt , and x = a (1 - e~ kt ). In this equation k may be regarded as the fraction of the total amount of substance decomposing in the unit of time, provided this unit is so small that the quantity at the end of the time unit is only slightly different from that at the beginning. The time required for one-half of the substance to change, is known as the period of half-change, T, and may be calculated from k by means of the equation log 2 = 0.4343 kT, therefore T = 0.6943 i, k or | = 1.443 T. Reactions in which only one mol of a single substance undergoes change are known as unimolecular reactions, or reactions of the CHEMICAL KINETICS 365 first order. In a unimolecular reaction, the velocity constant fc, is independent of the units in which concentration is expressed. If, in the integrated equation t becomes infinite, then x = a. In other words, for finite values of t, x must always remain less than a and the reaction will never proceed to completion. Another unimolecular reaction which has been thoroughly investigated, is the hydrolysis of cane sugar. When cane sugar is dissolved in water containing a small amount of free acid it is slowly transformed into d-glucose and d-fructose. The velocity of the reaction is very small and is dependent upon the strength of the acid added. The progress of the reaction may be very easily followed by means of the polarimeter. Cane sugar itself is dex- tro-rotatory, while d-fructose rotates the plane of polarization more strongly to the left than d-glucose rotates it to the right. Therefore, as the hydrolysis proceeds, the angle of rotation to the right steadily diminishes until, when the reaction is complete, the plane of polarization will be found to be rotated to the left. On this account the hydrolysis of cane sugar is commonly termed inversion and the molecular mixture of d-fructose and d-glucose constituting the product of the reaction is called invert sugar. Let ao denote the initial angle of rotation, at the time t = 0, due to a mols of cane sugar, let a Q ' denote the angle of rotation when inversion is complete and let a be the angle of rotation at any time t; then since rotation of the plane of polarization is pro- portional to the concentration x, the amount of cane sugar in- verted, will be OQ a x = a 7- CiQ-\- OQ In the equation C 12 H 22 On + H 2 <= C 6 H 12 6 +C6HJ206, representing the inversion of cane sugar, the velocity of the reaction will be, according to the law of mass action, proportional to the molecular concentrations of the cane sugar and the water. 366 THEORETICAL CHEMISTRY Since the reaction takes place in the presence of such a large excess of water, its effect may be considered to be constant. The velocity of the reaction is then proportional to the active mass of the sugar alone, or in other words the reaction is unimolecular. In the differential equation expressing the velocity of a unimolec- ular reaction, dx we have , 1 K = - t a - x and since a and x are measured in terms of angles of rotation of the plane of polarization, .we have 7 , fc = 7 log e - , t a + a The following table gives the results obtained with a 20 per cent solution of cane sugar in the presence of 0.5 molar solution of lactic acid at 25 C. t (minutes). a /; 34. 5 1,435 31. 1 '0^2348' 4,315 25. 0.2359 7,070 20. 16 0.2343 11,360 13. 98 0.2310 14,170 10. 01 0.2301 16,935 7. 57 0.2316 19,815 5. 08 0.2991 29,925 - 1.65 0.2330 Inf. -10. 77 Bimolecular Reactions. When two substances react and the concentration of each changes, the reaction is bimolecular or of the second order. Let a and b represent the initial molar con- centrations of the two reacting substances and let x denote the amount transformed in the interval of time t; then the velocity of the reaction will be expressed by the equation dx dt = k (a x) (b x). CHEMICAL KINETICS 367 The simplest case is that in which the two substances are present in equivalent amounts. Under these conditions the velocity equation becomes This equation may be integrated as follows: * Mi = ^W' ' k r\dt= r* dx J i, J*, (a - x)* therefore or fe - ti) (a - xi) (a - Xz) ' If time be reckoned from the beginning of the reaction, then Xi = and t = 0, and we have . iv ' , t \ t a (a x) If the reacting substances are not present in equivalent amounts then the velocity equation becomes ~ = k (a - x) (b - x). Assuming that time is measured from the beginning of the reaction, the integration of this equation may be performed as follows: _ x) (b x) Decomposing into partial fractions, _ a b Jo b x * The student who is unfamiliar with the Calculus must take the results of these calculations for granted. 368 THEORETICAL CHEMISTRY therefore 14 1 fi a-xf kt = r log c r i a - b \_ 3 6 - x Jo or u = 1 6 (o - re) Or passing to Briggsian logarithms, 0.4343 k t (a 6) *a(b x) The value of fc in a bimolecular reaction is not independent of the units in which the concentration is expressed, as is the case with a unimolecular reaction. Suppose that a unit I/nth of that originally selected is used to express concentration, then the value of k in the equation t a(a x) ' becomes , , _ 1 nx 1 t na n (a x) t na (a x) Thus, the value of k varies inversely as the numbers expressing the concentrations. As an illustration of a bimolecular reaction we may take the hydrolysis of an ester by an alkali. The reaction CH 3 COOC 2 H 5 + NaOH <= CH 3 COONa + C 2 H 5 OH, has been studied by Warder,* Reicher,f Arrhenius, t Ostwald and others. Arrhenius employed in his experiments 0.02 molar solutions of ester and alkali. These solutions were placed in separate flasks and warmed to 25 C. in a thermostat maintained at that temperature; equal volumes were then mixed, and at frequent intervals a portion of the reaction mixture was removed * Berichte, 14, 1361 (1881). t Lieb. Ann., 228, 257 (1885). J Zeit. phys. Chem., i, 110 (1887). Jour, prakt. Chem., 35, 112 (1887). CHEMICAL KINETICS 369 and titrated rapidly with standard acid. The accompanying table contains some of the results obtained : t (minutes). ax 1 8.04 4 5.30 0.0160 6 4.58 0.0156 8 3.91 0.0164 10 3.51 0.0160 12 3.12 0.0162 Mean 0.0160 The numbers in the second column of the table represent concentrations of sodium hydroxide and of ethyl acetate, expressed in terms of the number of cubic centimeters of standard acid required to neutralize 10 cc. of the reaction mixture. Owing to the high velocity of the reaction it is difficult to avoid large experi- mental errors, nevertheless the values of k given in the third column of the table will be observed to differ very slightly from the mean value. Reicher investigated the same reaction when the reacting sub- stances were not present in equivalent proportions. In this case, the progress of the reaction was followed by titrating definite portions of the reaction mixture from time to time, the excess of sodium hydroxide being determined by titrating a portion of the mixture at the expiration of twenty-four hours, when the ester was completely hydrolyzed. His results are given in the following table : t (minutes). a x (alkali). b-x (ester). k o 61 95 47 03 4.89 11.36 29.18 Inf 50.59 42.40 29.35 14 92 35.67 27.48 14.43 0.00093 0.00094 0.00092 370 THEORETICAL CHEMISTRY Reicher also studied the effect of different bases upon the ve- locity of the reaction. He found for strong bases approximately equal values of k, but for weak bases the values were irregular and smaller than those obtained with the more completely ionized bases. Arrhenius pointed out that the hydrolyzing power of a base is proportional to the number of hydroxyl ions which it yields. Writing the equation for the above hydrolysis in terms of ions, we have CH 3 COOC 2 H 5 + Na* + OH' <= CH 3 COO' + Na* + C 2 H 5 OH. It is evident from this equation that all bases furnishing the same number of hydroxyl ions should give identical values of k. We may, therefore, modify the fundamental differential equation as rfollows : Af ^ = k'a (a -x)(b- x), where a is the degree of ionization of the base. Trimolecular Reactions. When equivalent quantities of three substances react, the reaction is trimolecular or of the third order. If the initial molar concentrations of the reacting substances are denoted by a, b, and c, and if x denotes the proportion of each which is transformed in the interval of time t, the velocity of the reaction will be represented by the differential equation dx , , -^ = k (a - x) (b - x}(c - x). If the substances are present in equivalent amounts, the equation becomes -^ = k (a z) 3 , an expression which is much less difficult to integrate. The integration of this equation may be performed as follows.* S* A /"_ 1 * The student who is unfamiliar with the Calculus must take the results of these calculations for granted. CHEMICAL KINETICS 371 therefore *_., hence *[-J- T 2 |_(a- *)*_] ' = if 1 -II 2l(a-x)* a"J , 1 x (2 a - x) ~ * t 2a 2 (a-a0 2 When the reacting substances are not taken in equivalent amounts, the integration of the velocity equation may be performed as follows : ~ = k (a - x) (b - x) (c - x), therefore , ,. dx (a x) (b x) (c x) Decomposing into partial fractions, B C (a x) (b x) (c x) a x b x c x Multiplying through by (a z), we obtain 1 , ( B C (a x) < 7 ' \ \ (b - x) (c - x) ' (b - x c - x Let x = a, then A ___ i (a - 6) (c - a) Similarly, multiplying by (b x) and (c x), and then placing x = b, and x = c, we have B= ~ (a-b) (b-c)' and C= 1 (b-c)(c- a)' Then we ottain by substitution r* _ dx_ _ = __ i r* dx Jo (a - x) (b - x) (c- x) i (a - b) (c- a)J a- x' __ 1 r* dx ___ 1 C x dx (a - b) (b - c) Jo b - x (b-c) (c-a)J c - x 372 THEORETICAL CHEMISTRY Therefore, or /c = 4 - > ~ c > - [ & - (a - 6)6 - c) ~ a > (a - 6) (b - c) (c - a) In a trimolecular reaction, k is inversely proportional to the square of the original concentration. A typical trimolecular reaction is that between ferric and stannous chlorides. This reaction, represented by the following equation 2 FeCl 3 + SnCl 2 ?=* 2 FeCl 2 + SnCl 4 , has been investigated by A. A. Noyes.* Dilute solutions of the reacting substances were mixed at constant temperature, and definite portions of the reaction mixture were removed at meas- ured intervals of time and titrated for ferrous iron. Before titrating with a standard solution of potassium permanganate it was necessary to decompose the stannous chloride present with mercuric chloride. The following table gives the results obtained with 0.025 molar solutions of ferric chloride and stannous chloride. t (minutes). a x X it 2.5 0.02149 0.00351 113 3 0.02112 0.00388 107 6 0.01837 0.00663 114 11 0.01554 0.00946 116 15 0.01394 0.01106 118 18 0.01313 0.01187 117 30 0.01060 0.01440 122 60 0.00784 0.01716 122 Mean 116 Noyes also found that the velocity of the reaction is accelerated more by an excess of ferric chloride than by an equal excess of stannous chloride. * Zeit. phys. Chem., 16, 546 (1895). CHEMICAL KINETICS 373 Reactions of Higher Orders. Reactions of the fourth, fifth and eighth orders have recently been investigated, but examples of reactions of orders higher than the third are extremely rare. This fact is at first sight surprising since the equations of many chemical reactions involve a large number of molecules, and we would naturally expect the order of such reactions to be corre- spondingly high. For example, the reaction represented by the equation, involves six molecules of the substances initially present and, therefore, we should infer it to be a reaction of the sixth order. Kinetic experiments by van der Stadt have shown it to be a bimolecular reaction, the velocity of reaction being proportional to the concentrations of the phosphine and the oxygen. On allowing the gases to mix slowly by diffusion, it was discovered that the reaction actually takes place in several successive stages, the first stage being represented by the equation of the bimolec- ular reaction P H 3 + 2 = HPO 2 + H 2 . The subsequent changes, involving the oxidation of the products of this reaction, take place with great rapidity. It is highly probable that the equations which are ordinarily employed to represent chemical reactions really represent only the initial and final stages of a series of relatively simple reactions. Larmor* has shown that when chemical reactions are considered from the molecular standpoint, the bimolecular reaction is the most prob- able. He says, " Imagine a substance, say gaseous for simplicity, formed by the immediate spontaneous combination of three gas- eous components A, B, and C. When these gases are mixed, the chances are very remote of the occurrence of the simultaneous triple encounter of an A, a B, and a C, which would be necessary to the immediate formation of an ABC', whereas if ever formed, it would be liable to the normal chance of dissociating by collisions; it would thus be practically non-existent in the statistical sense. But if an intermediate combination AB could exist, very tran- siently, though long enough to cover a considerable fraction of the * Proc. Manchester Phil. Soc., 1908. 374 THEORETICAL CHEMISTRY mean free path of the molecules, this will readily be formed by ordinary binary encounters of A and B, and another binary encounter of AB with C will now form the triple compound ABC in quantity. " Determination of the Order of a Reaction. It has been shown in the foregoing pages that the time required to complete a certain fraction of a reaction is dependent upon the order of the reaction in the following manner : (1) In a unimolecular reaction the value of k is independent of the initial concentration; (2) In a bimolecular reaction the value of k is inversely pro- portional to the initial concentration; (3) In a trimolecular reaction the value of k is inversely pro- portional to the square of the initial concentration. Hence, in general, in a reaction of the nth. order, the value of k is inversely proportional to the (n 1) power of the initial con- centration. If the value of k is determined with definite concen- trations of the reacting substances, and then with multiples of those concentrations, the order of the reaction can be determined according to the above rules by observing the manner in which k varies with the concentration. The order of a reaction may also be readily determined by means of a graphic method. Thus, to determine the order of a reaction we ascertain by actual trial which one of the following expressions, in which C denotes concentration, will give a straight line when plotted against times as abscissae : (1) log C reaction unimolecular; (2) 1/C -reaction bimolecular; (3) 1/C 2 reaction trimolecular; (4) l/C n reaction n -j- 1 molecular. Complex Reaction Velocities. Thus far we have considered the velocity of reactions which are practically complete. There are numerous cases, however, in which the course of the reaction is complicated by such disturbing factors as (1) counter reactions, (2) side reactions, and (3) consecutive reactions. These disturb- ing causes will now be considered. CHEMICAL KINETICS 375 (1) Counter Reactions. In the chemical change represented by the equation CH 3 COOH + C 2 H 5 OH <= CH 3 COOC 2 H 5 + H 2 O, the speed of the direct reaction steadily diminishes owing to the ever-increasing effect of the reverse or counter reaction. Ulti- mately, when two-thirds of the acid and alcohol are decomposed, the velocities of the two reactions become equal and a condition of equilibrium results. Starting with 1 mol of acid and 1 mol of alcohol, and letting x represent the amount of ester formed, we have When equilibrium is attained, K = = 4- By observing the change for any time t, we have 7/ 3 . 2 - x k ~ k = T7 lo goTf-;- Having the values of k/k' and k k', the velocity constant k of the direct reaction can be determined. The value of k so obtained has been shown by Knoblauch * to vary in those reac- tions where the concentration of the hydrogen ion changes. (2) Side Reactions. When the same substances are capable of reacting in more than one way with the formation of different products, the several reactions proceed side by side. Thus, benzene and chlorine may react in two ways as shown by the equations, (1) C 6 H 6 + C1 2 = C 6 H 5 C1 + HC1, and (2) C 6 H 6 + 3 C1 8 = C 6 H 6 C1 6 . It is generally possible to regulate the conditions under which the substances react so as to promote one reaction and retard the other. * Zeit. phys. Chem., 22, 268 (1897). 376 THEORETICAL CHEMISTRY (3) Consecutive Reactions. By consecutive reactions we under- stand those reactions in which the products of a certain initial chemical change react, either with each other or with the original substances to form new substances. Attention has already been called to the fact that many of our common chemical equations really represent the summation of a number of consecutive reac- tions. If the system A is transformed into the system C through an intermediate system B, then we shall have the two reactions (1) A-+B, and (2) 5->C. If reaction (1) should have a very much greater velocity than reaction (2), then the measured velocity of the change from A to C will be practically the same as that of the slower reaction. This fact has been illustrated by means of the following analogy, due to James Walker: * "The time occupied by the transmission of a telegraphic message depends both on the rate of transmission along the conducting wire, and on the rate of progress of the messenger who delivers the telegram; but it is obviously this last, slower rate that is of really practical importance in determin- ing the time of transmission." The saponification of ethyl succinate may be taken as an illustration of consecutive reactions. This reaction proceeds in two stages as follows: (1) C 2 H 4 + NaOH<=C 2 H 4 +C 2 H 5 OH, .COONa (2) C 2 H 4 + NaOH vc/ V3/ vz/ w i> i V^/ \i/ M/ 00 0|0 !0 1 i i i ii I 1 ! I 000 |0 k 1 1 i 1 IV i I 000 i jo Fig. 87. during electrolysis. He showed that different ions migrate with different speeds, and that the faster moving ions carry a greater proportion of the current than the slower moving ions. The effect of unequal ionic velocities on the concentrations of the solutions around the electrodes is clearly shown by the ac- companying diagram (Fig. 87) due to Ostwald. The anode and * Pogg. Ann., 89, 177; 98, 1; 103, 1; 106, 337, 513 (1853-1859). 394 THEORETICAL CHEMISTRY cathode in an electrolytic cell are represented by the vertical lines A and C respectively. The cell is divided into three compart- ments by means of porous diaphragms, represented by the ver- tical dotted lines. The cations are represented by dots (") and the anions by dashes (') Before the current passes through the cell, the concentration of the solution is uniform throughout, the conditions being represented by I. Now let us imagine that only the anions move when the current is established. The conditions when the chain of anions has moved two steps toward the anode are shown in II. Each ion which has been deprived of a partner is supposed to be discharged. It will be observed that although the cations have not migrated toward the cathode, yet an equal number of positive and negative ions are discharged, and that while the concentration in the anode compartment has not changed, the concentration in the cathode compartment has diminished to one-half its original value. Let us now suppose that both anions and cations move with the same speed, and as before, let each chain of ions move two steps toward their respective electrodes, as indicated in III. It will be seen that four positive and four negative ions have been dis- charged, and that the concentration of the electrolyte in the anode and cathode compartments has diminished to the same extent. Finally, let us assume that the ratio of the speeds of the cations to that of the anions is as 3 : 2. When the cations have moved three steps toward the cathode and the anions have moved two steps toward the anode, the conditions will be as shown in IV. It is evident that five positive and five negative ions have been discharged, and that the concentration in the cathode compart- ment has diminished by two molecules while the concentration in the anode compartment has diminished by three molecules. It will be observed that the change in concentration in either of the electrode compartments is proportional to the speed of the ion leaving it. Thus, in II, the concentration in the cathode compartment diminishes while that in the anode compartment remains unchanged, since only the anion moves. In like manner, the change in concentration about the electrodes in III corre- sponds with the fact that both ions migrate at the same rate. ELECTRICAL CONDUCTANCE 395 In IV the ratio of the change in concentration in the cathode compartment to that in the anode compartment is as 2 : 3. It will be apparent from these examples, that the relation between the speeds of the ions and the corresponding changes in concen- tration at the electrodes may be expressed by the following pro- portion : Change in concentration at anode _ speed of cation Change in concentration at cathode speed of anion If the relative speed of the cations is represented by u, and that of the anions by v, then the total quantity of electricity trans- ported will be proportional to u -f- v: of this total, the fractions carried by the anion and cation respectively, will be n = - , u + v and 1 n = - The values of these ratios, n and I n. u + v are called the transport numbers of the anion and cation respec- tively. It is apparent from the diagram, that if the electrolysis is not carried too far, the concentration of the solution in the inter- mediate compartment will undergo no change. In order to deter- mine transport numbers, therefore, it is simply necessary to remove portions of the solutions in the immediate vicinity of the two electrodes and determine the concentration of the electrolyte analytically. The success of the experiment depends upon keep- ing the concentration of the intermediate compartment unaltered. Experimental Determination of Transport Numbers. Various forms of apparatus have been constructed for the determination of transport numbers, among which one of the most satisfactory is that devised by Jones and Bassett,* and shown in Fig. 88. It consists of two vertical tubes of wide bore connected by a U-tube fitted with a stop-cock. Into each of two electrodes, made of a suitable metal, is riveted a short piece of stout platinum wire, which is then sealed into heavy-walled glass tubes. The exposed end of the platinum wire on the under side of each electrode is covered with a drop of fusion glass. The tubes carrying the electrodes are fitted into holes bored through the ground glass * Am. Chem. Jour., 32, 409 (1904). 396 THEORETICAL CHEMISTRY stoppers which close the right and left arms of the apparatus. Two small graduated tubes are sealed to the two vertical tubes just below the stoppers. These tubes allow for any slight dis- placement of the solution due to expansion or the formation of Fig. 88. gas, and at the same time make it possible to level the apparatus accurately. When electrolysis has proceeded far enough, the circuit is broken and the stop-cock closed, thus preventing the mixing of the solutions in the anode and cathode compartments. The solutions in the two halves of the apparatus are then rinsed out into separate beakers and the concentration of each is determined analytically. Knowing the initial concentration of the solution and the final concentrations at the two electrodes, together with the total quantity of electricity which has passed through the apparatus during the experiment, we have all of the data neces- sary for the calculation of the transport numbers of the two ions. ELECTRICAL CONDUCTANCE 397 The following example will serve to make the method of calcu- lation clear : In an experiment to determine the transport numbers of the ions of silver nitrate, a solution containing 0.00739 gram of that salt per gram of water was prepared. The solution was introduced into the migration apparatus and, after inserting silver electrodes, a small current was passed through the appa- ratus for two hours. A silver coulometer was included in the cir- cuit, and 0.0780 gram of silver was deposited by the current. This mass of silver is equivalent to 0.000723 gram-equivalent. After the circuit was broken, the anode solution was rinsed out and its concentration determined analytically. It was found to con- tain 0.2361 gram of silver nitrate to 23.14 grams of water. This amount of solution contained originally 23.14 X 0.00739 = 0.1710 gram of silver nitrate. Thus, the amount of silver nitrate in the anode compartment had increased by 0.2361 0.1710 = 0.0651 gram of silver nitrate, or 0.000383 gram-equivalent of silver. Obviously the increase in the concentration of the nitrate ion must have been the same. The amount of silver dissolved from the anode must have been equal to that deposited in the coulometer, or since 0.000723 gram-equivalent of silver was deposited and the actual increase found was 0.000383 gram- equivalent, the difference, 0.000723 - 0.000383 = 0.000340 gram- equivalent, is the amount of silver which migrated away from the anode. At the same time 0.000383 gram-equivalent of nitrate ions migrated into the anode compartment. The ratio of the speed of migration of the silver ions to that of the nitrate ions is as 0.000340 : 0.000383. Since 0.000723 gram-equivalent of silver ions measures the total quantity of electricity transported, the transport numbers of the two ions will be as follows : Transport number of Ag = n = = 0.470, Transport number of NO 3 ' = 1 - n = ^ f = 0.530. 0.000 1 Z*j These numbers can be checked by a similar calculation based on the change in concentration in the cathode compartment. 398 THEORETICAL CHEMISTRY The following table gives the transport numbers of the an ions of various electrolytes at different dilutions, V being the number of liters of solution containing one gram-equivalent of solute. The transport numbers of the corresponding cations can be found by subtracting the transport numbers of the anions from unity. TRANSPORT NUMBERS OF ANIONS. 7 = 100 50 20 10 5 2 i 0.5 KC1 } KBr KI NH 4 C1 J NaCl 0.506 0.507 0.507 614 0.508 617 0.509 620 0.513 626 0.514 637 0.515 KNO 3 497 496 492 487 479 AgNO 3 0.528 6.528 528 528 527 0.519 501 476 KOH 0.735 0.736 HC1 0.172 0.172 0.172 0.173 0.176 \ BaCl 2 . 640 657 1 K 2 CO 3 0.435 434 413 |CuSO 4 0.620 0.626 0.632 6.643 0.668 0.696 720 H 2 i3O 4 0.182 0.174 It is apparent from the table that the transport numbers are not entirely independent of the concentration. They also vary slightly with the temperature and approach the limiting value, 0.5, at high temperatures. Specific, Molar and Equivalent Conductance. As is well known, the resistance of a metallic conductor is directly propor- tional to its length and inversely proportional to its area of cross- section. Similarly, the resistance of an electrolyte is proportional to the length and inversely proportional to the cross-section of the column of solution between the two electrodes. The specific resistance of an electrolyte may be defined as the resistance in ohms of a column of solution one centimeter long and one square centimeter in cross-section. Specific conductance is the reciprocal of specific resistance. Since the conductance of a solution is almost wholly dependent upon the amount of solute present, it is more convenient to express conductance in terms of the molar or equivalent concentration. The molar conductance p, is the ELECTRICAL CONDUCTANCE 399 conductance in reciprocal ohms, of a solution containing one mol of solute when placed between electrodes which are exactly one centimeter apart. The equivalent conductance A is the conduc- tance in reciprocal ohms of a solution containing one gram- equivalent of solute when placed between electrodes which are one centimeter apart. If K denotes the specific conductance of a solution and V mj the volume in cubic centimeters which contains one mol of solute, then and in like manner A = where V e is the volume of solution in cubic centimeters which contains one gram-equivalent of solute. The following table gives the specific and molar conductance of solutions of sodium chloride at 18 C.: Concentration. Dilution. Sp. Cond. Molar Cond. 1 1,000 0.0744 74.4 0.1 10,000 0.00925 92.5 0.01 100,000 0.001028 102.8 0.001 1,000,000 0.0001078 107.8 0.0001 10,000,000 0.00001097 109.7 It will be observed that the molar conductance increases with the dilution up to a certain point beyond which it remains nearly constant. That the molar conductance should change but little will become apparent from the following considerations: - Imagine a rectangular cell of indefinite height and having a cross- sectional area of one square centimeter, and further assume that two opposite walls can function as electrodes. Let 1000 cc. of a solution containing one mol of solute be introduced into the cell, and let its conductance be determined. Now let the solution be diluted to 2000 cc. and the conductance of the diluted solution be measured. While the specific conductance of the diluted solution is reduced to one-half of its original value, yet since the electrode 400 THEORETICAL CHEMISTRY surface in contact with the solution is doubled, owing to the fact that the solution stands at twice the original height in the cell, the total conductance due to one mol of solute remains un- changed. This, of course, is only the case with completely ionized solutes. Determination of Electrical Conductance. The determination of the electrical conductance of a solution resolves itself into the determination of its resistance by a simple modification of the familiar Wheatstone-bridge method. The arrangement of the apparatus for this method devised by Kohlrausch * is represented diagrammatically in Fig. 89, where ab is the bridge wire, B is a Fig. 89. resistance box, and C is a cell containing the solution whose resistance is to be measured. The points d and c are connected to a small induction coil / which gives an alternating current. This is necessary in order to prevent polarization which would occur if a direct current were used. The use of the alternating current necessitates the substitution of a telephone, T, for the galvanometer usually employed in measuring resistance. The positions of the induction coil, and telephone are sometimes inter- changed, but the arrangement shown in the diagram is to be pre- ferred, since it insures a high electromotive force where the sliding * Wied. Ann., 6, 145 (1879); u, 653 (1880); 26, 161 (1885). ELECTRICAL CONDUCTANCE 401 contact c touches the wire, this being the most uncertain connec- tion in the entire arrangement. A small accumulator A, serves to operate the induction coil. In making a measurement, the coil is connected with the accumulator and the vibrator adjusted so that a high mosquito-like tone is emitted; then the sliding contact c is moved along the wire ab until the sound in the tele- phone reaches a minimum, the position of the point of contact with the "bridge-wire being read on the millimeter scale placed below. According to the principle of the Wheatstone bridge, it follows that C = bc B ac Since the resistance B and the lengths be and ac are known, the resistance C can be calculated. Various types of conductance cells are in use, depending upon whether the solution has a high or a low resistance. The form shown in Fig. 90 is widely used. The two electrodes are made of platinum foil, connection with the mercury in the two glass tubes tt being established by means of two pieces of stout platinum wire sealed through the ends of these tubes. The tubes tt are fastened into a tight-fitting vulcanite cover so that the electrodes may be re- moved, rinsed and dried without altering their relative positions. Before the cell is used, the electrodes must be coated electro- lytically with platinum black. It is not necessary to know the area of the elec- trodes or the distance between them, since it is possible to determine a factor, termed the resistance capacity, by means of which the results obtained with the cell can be transformed into reciprocal ohms. To this Kit:. HO. end the specific conductances of a number of standard solu- tions have been carefully determined by Kohlrauscb; thus, for 402 THEORETICAL CHEMISTRY a 0.02 molar solution of potassium chloride he found the following values : Ki 8 = 0.002397 and K 25 = 0.002768, or 119.85 and A2 5 = 138.4. Let the resistance of the cell when filled with 0.02 molar potassium chloride be C, then according to the principle of the Wheatstone bridge we have r K bc L = > j ac or denoting the conductance of the solution by L, we obtain T 1 - ac ~C~~B^bc Since the specific conductance K must be proportional to the observed conductance, we have where K is the resistance capacity of the cell. If the measure- ment is made at 18 C., then we have 0.002397 B-bc K ac Having determined the resistance capacity of the cell we may then proceed to determine the conductance of any solution. For example, suppose that when the resistance in the box is B f , the point of balance on the bridge-wire is at c' } then the specific con- ductance of the solution will be K > = K ^L. B'bc' , If K' is multiplied by the volume of the solution, we obtain the equivalent conductance, or A = K f V. Relative Conductances of Different Substances. The study of the electrical conductance of various solutes in aqueous solu- tion, reveals the fact that electrolytes differ greatly in their con- ducting power. They may be roughly divided into two classes: - ELECTRICAL CONDUCTANCE 403 those with high conducting power, such as strong acids, strong bases, and salts; and those with low conducting power, such as ammonia and most of the organic acids and bases. Further- more, the equivalent or molar conductance increases with the dilu- tion until a dilution of about 10,000 liters is reached, beyond which it remains constant. The following table gives the equiv- alent conductances of three typical electrolytes, V representing the volume of the solution in liters, and A the equivalent con- ductance : HYDROCHLORIC ACID. V A (18). 0.333 201.0 1.0 278.0 10.0 324.4 100.0 341.6 1000.0 345.5 SODIUM HYDROXIDE. V A (18). 0.333 100.7 1.0 149.0 10.0 170.0 100.0 187.0 500.0 186.0 POTASSIUM CHLORIDE. V A (18). 0.333 82.7 1.0 91.9 10.0 104.7 100.0 114.7 1,000.0 119.3 10,000.0 120.9 404 THEORETICAL CHEMISTRY The curves shown in Fig. 91 are plotted from the data of the foregoing table, and bring out very clearly the differences in con- ducting power possessed by the three electrolytes. In general the conductance of pure liquids is small. Thus, the specific conductance of pure water at 18 is approximately 1 X 10~ 6 Dilution, V Fig. 91. reciprocal ohms and, as Walden * has shown, the specific conduct- ance of a number of other solvents is of the same order as that for water. Mixtures of two liquids, each of which is practically non-conducting, may have a conductance differing but little from that of the two components; or the mixture may have a very high conductance. For example, the conductance of a mixture * Zeit. phys. Chem., 46, 103 (1903). ELECTRICAL CONDUCTANCE 405 of water and ethyl alcohol is of the same order of magnitude as that of the two components, while on the other hand, a mixture of water and sulphuric acid, each of which in the pure state is practically a non-conductor, has great conducting power. The variation of the specific conductance of mixtures of water and sulphuric acid is represented in Fig. 92, the concentrations of sul- phuric acid being plotted on the axis of abscissae and the specific conductances on the axis of ordinates. It appears that as the 30 40 60 60 70 80 90 100 110 P&r Cent Sulphuric Add Fig. 92. concentration of the sulphuric acid increases, the specific conduct- ance of the mixture increases until 30 per cent of acid is present, beyond which point it gradually diminishes. When pure sul- phuric acid is present the value of the specific conductance is practically zero. On dissolving sulphur trioxide in the pure acid, the specific conductance increases slightly to a maximum and then falls rapidly to zero. There is a minimum in the curve corre- sponding to about 85 per cent of acid, a concentration which 406 THEORETICAL CHEMISTRY corresponds almost exactly with the hydrate H 2 SO4.H 2 O. Why some liquid mixtures should have marked conducting power and others hardly any, it is difficult to explain. Many fused salts, such as silver nitrate and lithium chloride, are excellent conductors and are thus exceptions to the general rule, that pure substances belonging to the second class of conductors possess little conduct- ing power. The Law of Kohlrausch. The electrical conductance of solu- tions was systematically investigated by Kohlrausch who showed that the limiting value of the equivalent conductance, which may be represented by A, is different for different electrolytes and may be considered as the sum of two independent factors, one of which refers to the cation and the other to the anion. This experi- mental result is commonly known as the law of Kohlrausch. The limiting value of the equivalent conductance is reached when the molecules are completely . broken down into ions, and under these conditions the whole of the electrolyte participates in conducting the current. The accompanying table, giving the equivalent conductances at infinite dilution of several binary electrolytes, illustrates the truth of the law of Kohlrausch. EQUIVALENT CONDUCTANCES AT INFINITE DILUTION. K Na Li NH 4 H Ag Cl 123 103 95 122 353 NO 3 118 98 350 109 OH 228 201 C1O 3 115 103 C 2 H 3 2 94 73 83 The differences between two corresponding sets of numbers in the same vertical column, and of any two corresponding sets of numbers in the same horizontal row will be found to be nearly equal. This could only occur when the limiting conductance is the sum of two entirely independent quantities. Each ion invariably carries the same charge of electricity and moves with ELECTRICAL CONDUCTANCE 407 its own velocity quite independent of the nature of its compan- ion ion. Therefore, at infinite dilution, we have in which l c and l a are the equivalent conductances of the ions of the electrolyte at infinite dilution. From this it follows that and -^k t i^^kl^ij or l c = nAoo, and l a = (1 n)A<. Thus, the equivalent conductance of silver nitrate at infinite dilu- tion at 18 is 115.5, while n = 0.518 and 1 - n = 0.482; there- fore l c = 0.518 X 115.5 = 60.8, and l c = 0.482 XI 15.5 = 55.7; or one gram-equivalent of silver ions possesses a conductance of 60.8 when placed between electrodes one centimeter apart and large enough to contain between them the entire volume of solu- tion in which the Ag* ions exist; and one gram-equivalent of NO 3 ' ions under the same conditions have a conductance equal to 55.7. The values of the ionic conductances at infinite dilution remain constant in all solutions in the same solvent at the same temper- ature, so that it is possible to calculate the equivalent conductance for any substance at infinite dilution. In the subjoined table are given the ionic conductances of various ions at 18 and infinite dilution, together with their temper- ature coefficients. 408 THEORETICAL CHEMISTRY IONIC CONDUCTANCES AT INFINITE DILUTION. Ion. i e Temp. Coeff . Li'. . 33.44 0.0265 Na" 43.55 0.0244 KV. 64.67 0.0217 Rb'.. 67.6 0.0214 CsV. 68.2 0.0212 NH 4 '.. 64.4 0.0222 IT 66.0 0.0215 A.g' 54 02 0229 F' 46 64 0.0238 or.. 65 44 0.0216 Br'. 67 63- 0.0215 ['.. 66.40 0.0213 3CN'.. 56.63 0.0211 D1O 3 '.. 55.03 0.0215 tO 3 '.. 33.87 0.0234 NO 3 ' 61 78 0205 H' 318 3H' 174 j Zn". . . . 45 6 0251 & Me". 46 0256 [Ba".. 56 3 0238 fc Pb" . . 61.5 0243 tSO 4 ".. 68.7 0.0227 fcCO," 70.0 0.0270 In the case of weak electrolytes the value of A cannot be determined directly from conductance measurements, since before the limiting value is reached, the solution has become so dilute as to render accurate measurements of the specific conductance impossible. The law of Kohlrausch enables us to get around this difficulty. Thus, the value of A> for acetic acid must be equal to the sum of the conductances of the H* and CH 3 COO' ions. The conductance of the H* ion at 18 is, according to the preceding table, 318. The value of the conductance of the CH 3 COO' ion can be determined from the conductance of sodium acetate at infinite dilution, A* for this salt being 78.1 at 18. Since the ionic conductance of the Na* ion is 43.55 at 18, it follows that the conductance of the CH 3 COO' ion must be 78.1 43.55 = 34.55. Therefore, for acetic acid we have A*. = l c + la = 318 + 34.55 = 352.55 at 18. ELECTRICAL CONDUCTANCE 409 Bredig * has shown that the ionic conductance of elementary ions is a periodic function of the atomic weight. When the ionic 80-- 60- .2 Ba Cd 80 ,120 Atomic Wj3igh.t> TFig.93. 200 conductances are plotted as ordinates against the atomic weights as abscissae, the curve shown in Fig. 93 is obtained. A glance at the curve shows the periodic nature of the relation. Absolute Velocity of the Ions. Thus far we have considered only the relative velocities of the ions and their conductances; we now proceed to the consideration of their absolute velocities in centimeters per second. Let a current of electricity pass through a centimeter cube of a solution of a binary electrolyte. If the solution contains m mols of solute per liter, then ra/1000 will be the number of mols in the centimeter cube. The charge on either the cation or the anion s , where F = 96,540 coulombs. If C represents the total current, we have m 1000 (Ic + U Zeit. phys. Chem., 13, 242 (1894). 410 THEORETICAL CHEMISTRY since the current is the charge which passes through one face of the cube in one second. In a centimeter cube, the current is equal to the product of the specific conductance and the difference of potential E, the latter being numerically equal to the potential gradient, the distance between the electrodes being one centi- meter. Hence, we have 1000 KE = Fm (l c + l a ). HE is expressed in volts and K in reciprocal ohms, l c and l a will be expressed in centimeters per second, for on passing to absolute electromagnetic units, we have 1000 (K X 10~ 9 ) (E X 10 8 ) (F X 10- 1 ) ". + , or where l c and l a are the ionic velocities for unit potential gradient 1 volt per centimeter. From this it follows that The equivalent conductance of a 0.0001 molar solution of potas- sium chloride at 18 is 128.9; the total velocity of the two ions is then, IOQ q = 0.001345 cm. per sec. This total velocity is made up of the two individual ionic velocities. The transport numbers of the two ions, K" and Cl', are respectively 0.493 and 0.507. Hence the absolute velocities of the ions, ex- pressed in centimeters per second, in a 0.0001 molar solution of potassium chloride at 18 are as follows : u = 0.001345 X 0.493 = 0.00066 cm. per sec., and v = 0.001345 X 0.507 = 0.00068 cm. per seCc ELECTRICAL CONDUCTANCE 411 The absolute velocities of some of the more common ions at 18 are given in the following table : ABSOLUTE IONIC VELOCITIES. Ion. Velocity. Ion. Velocity. K* cm. per sec. 00066 H* cm. per sec. 00320 NH 4 ' 00066 Cl' 00069 Na" ... . 00045 NO 3 ' 00064 Li* 00036 C1O 3 ' 00057 Ag* 00057 OH' 00181 Cr 2 O 7 " 0.000473 Cu" 00031 The velocities of certain ions have been determined directly. Thus, the velocity of the hydrogen ion was. measured by Lodge * in the following manner: The tube B, Fig. 94, 40 cm. long and 8 cm. in diameter, was graduated and bent at right angles at the Fig. 94. ends. This was filled with an aqueous solution of sodium chloride in gelatine, colored red by the addition of an alkaline solution of phenolphthalein. When the contents of the tube had gelatinized, the latter was placed horizontally, connecting two beakers filled with dilute sulphuric acid as shown in the diagram. A current of electricity was passed from one electrode A to the other elec- trode C. The hydrogen ions from the anode vessel were thus carried along the tube, and discharged the red color of the phenolphthalein as they migrated toward the cathode. In this manner the velocity * Brit. Assoc. Report, p. 393 (1886). 412 THEORETICAL CHEMISTRY of the hydrogen could be observed under a known potential gra- dient. The observed and calculated values agree excellently. It was shown that the velocity of the hydrogen ions suffered almost no retardation from the high viscosity of the gelatine solution. Whetham,* in his experiments on ionic velocity, employed two solutions one of which possessed a colored ion, the progress of the latter being observed and its velocity determined under unit potential gradient. For example, consider the boundary line between two equally dense solutions of the electrolytes AC and BC, C being a colorless and A a colored ion. When a current passes through the boundary between the two electrolytes, the anion C will migrate toward the positive electrode while the two cations, A and B, will migrate toward the negative electrode and the color boundary will move with the current, its speed being equal to that of the colored ion A . In this way Whetham measured the absolute velocities of the ions, Cu", Cr 2 O 7 ", and Cl'. Ionic velocities have also been determined by Steele f who observed the change in the index of refraction of the solution as the ions migrated. The accompanying table gives a comparison of the calculated and observed velocities of some of the ions. Ion. Velocity (obs.). Velocity (calc.). H* . cm. per sec. 0.0026 cm. per sec. 0032 Cu" 0.00029 0.00031 cr 0.00058 . 00069 Cr 2 O 7 " 00047 000473 Conductance and lonization. We have already seen that solutions of strong acids, strong bases and salts exert abnormally- great osmotic pressures. According to the molecular theory, this abnormal osmotic activity has been ascribed to the presence in the solutions of a greater number of dissolved particles than would be anticipated from the simple molecular formulas of the solutes. The ratio of the observed to the theoretical osmotic pressure was represented, according to van't Hoff, by the factor "i." * Phil. Trans. A., 184, 337 (1893); 196, 507 (1895). t Phil. Trans. A., 198, 105 (1902). ELECTRICAL CONDUCTANCE 413 In 1887, Arrhenius showed that there is an intimate connection between electrical conductance and abnormal osmotic activity, only those solutions conducting the electric current which exert abnormally-high osmotic pressures. It had already been pointed out by Kohlrausch, that the equivalent conductance of a solution increases at first with the dilution and then ultimately becomes constant. Arrhenius explained this behavior by assuming that the molecules of the solute are dissociated into ions, the con- ductance of the solution being solely dependent upon the number of ions present. The dissociation increases with the dilution until finally, when the equivalent conductance has reached its maximum value, it is complete, the molecules of solute being entirely broken down into ions. This theory of Arrhenius, known as the theory of electrolytic dissociation, is based, as has been pointed out, upon the views advanced by Clausius. Arrhenius showed how the degree of dissociation of an electrolyte can be calculated from the electrical conductance of its solutions. According to the theory of electrolytic dissociation, the conductance of a solution is dependent upon the number of ions present in the solution, upon their charges, and upon their velocities. Since the electric charges carried by equivalent amounts of the ions of different electrolytes are equal, and since the velocities of the ions for the same electrolyte are practically independent of the dilution of the solution, it follows that the increase in equivalent conductance with dilution must depend almost wholly upon the increase in the number of ions present. The equivalent conductance at infinite dilution has been shown by the law of Kohlrausch to be Aoo =l c + l a , and, therefore, the equivalent conductance at any dilution v, must be A v = a (l c + l a ), where a is the degree of dissociation of the electrolyte. Dividing the second equation by the first, we obtain 414 THEORETICAL CHEMISTRY This equation enables us to calculate the degree of ionization of an electrolyte at any dilution, provided the conductance of the solution at the particular dilution is known, together with its conductance, at infinite dilution. For example, A v at 18 for a molar solution of sodium chloride is 74.3, and Aoo is 110.3; there- fore, a. = 74.3 -r- 110.3 = 0.673, or in a molar solution, the mole- cules of sodium chloride are dissociated to the extent of 67.3 per cent. A comparison of the values of i based upon conductance and osmotic data has already been given in the table on page 230. Since A v = a. (l c + l a ), we may also write The Dissociation of Water. Water behaves as a very weak binary electrolyte, dissociating according to the equation, The specific conductance of water, purified with the utmost care, has been determined by Kohlrausch and Heydweiller.* Their results are given in the following table : Temperature, degrees. Specific Conduct- anceX10-. 0.014 18 0.040 25 0.055 34 0.084 50 0.170 The conductance of pure water at is so small that one milli- meter of it has a resistance equal to that of a copper wire of the same cross-section and 40,000,000 kilometers in length, or in other words, long enough to encircle the earth one thousand times. Knowing the specific conductance of water, its degree of dissoci- ation can be easily calculated. The ionic conductances of the two ions of water at 18 are as follows: H" = 318, and OH' = * Zeit. phys. Chem., 14, 317 (1894). ELECTRICAL CONDUCTANCE 415 174. Therefore, the maximum equivalent conductance of water should be Aoo = 318 + 174 = 492. The equivalent conductance at 18, of a liter of water between electrodes 1 cm. apart is, according to the data of Kohlrausch, 0.04 X 10~ 6 X 10 3 = 0.04 X 10~ 3 ; therefore 0.04 X 10~ 3 492 = 0.8 X 10~ 7 = c, the concentration of the ions, H* and OH', in mols per liter at 18. Conductance of Difficultly-Soluble Salts. In a saturated solution of a difficultly-soluble salt, the solution is so dilute that in general we may assume complete ionization, or A v = Aoo . When this is the case, we have ^solution *H 2 O = K > and A v = Aoo = 1000 K.V. Hence = HXXU' or if m denotes the concentration in gram-equivalents per liter, we have 1 1000 K Thus, Bottger found for a saturated solution of silver chloride at 20, K' = 1.374 X 10~ 6 . Deducting the specific conductance of the water at this temperature, we have K = 1.374 X 10- 6 - 0.044 X 10~ 6 = 1.33 X lO" 6 . Since the value of A, at 20, for silver chloride, determined from the table of ionic conductances, is 125.5, we have TO = 1000XL33Xlfr- = j Q6 x ]0 _ 5 gr _ equiv AgC1 per nter . 416 THEORETICAL CHEMISTRY Temperature Coefficient of Conductance. When the temper- ature of a solution of an electrolyte is raised, the equivalent con- ductance usually increases. The increase in conductance is due, not to an increase in the ionization, but to the greater velocity of the ions caused by the diminution of the viscosity of the solution. According to Kohlrausch, the relation between conductance and temperature may be approximately expressed by the following equation, A|=A 18 o}l +0($ where /3 is the temperature coefficient, or change in conductance for 1 C. Solving the equation for /3, we have A, - A 180 A a iz\ i *-18(l io) The temperature coefficients of several of the more common elec- trolytes are given in the accompanying table. TEMPERATURE COEFFICIENTS OF CONDUCTANCE. Electrolyte. Temperature Coefficient. Nitric acid. . 0163 Sulphuric acid 0164 Hydrochloric acid Potassium hydroxide Potassium nitrate 0.0165 0.0190 0211 Potassium iodide . 0212 Potassium bromide .... 0216 Potassium chlorate. . . 0.0216 Silver nitrate 0216 Potassium chloride . . . 0.0217 Ammonium chloride Potassium sulphate Copper sulphate 0.0219 0.0223 0225 Sodium chloride Sodium sulphate. . 0.0226 0234 Zinc sulphate . . 0250 The temperature coefficient of conductance is not, however, a simple linear function of the temperature. The following empiri- ELECTRICAL CONDUCTANCE 417 cal equations, expressing equivalent conductance at infinite dilu- tion at any temperature t in terms of the conductance at 18, have been derived by Kohlrausch: A** = Aooiso { 1 + a (t - 18) + ft (t - 18) 2 }, and ft = 0.0163 (a - 0.0174). When the values of Aooi 8 o, a, and ft, as determined for a large number of electrolytes, are substituted in the above equation, he showed that Aoo< becomes equal to zero at a temperature approx- imating to 40. Kohlrausch suggested that each ion moving through the solution carries with it an " atmosphere" of solvent, and that the resistance offered to the motion of the ion is simply the frictional resistance between masses of pure water. This view is in harmony with the solvate theory discussed in an earlier chapter. Washburn * has calculated the degree of ionic hydration for several ions. He finds, for example, that the hydrogen ion carries with it 0.3 molecule of water, while the lithium ion is hydrated to the extent of 4.7 molecules of water. Conductance at High Temperatures and Pressures. The conductance of several typical electrolytes, at temperatures rang- ing from that of the room up to 306, have been measured by A. A. Noyes and his co-workers, f These determinations were made in a conductance cell especially constructed to withstand high pressures. The results show that the values of A for binary electrolytes become more nearly equal with rise of temperature. This may be taken as an indication of the fact that the ionic velocities tend to become more nearly equal as the temperature rises. The conductance of ternary electrolytes increases uniformly with the temperature, and attains values which are considerably greater than those reached by binary electrolytes. This is what might be expected, since if an ion is bivalent, as in a ternary electrolyte, the driving force is greater, and the ion must move faster, and, consequently, the conductance must be greater. * Jour. Am. Chem. Soc., 30, 322 (1909). t Publication of Carnegie Institution, No. 63. 418 THEORETICAL CHEMISTRY The temperature coefficient of conductance for binary elec- trolytes is greater between 100 and 156, than below or above these temperatures. The temperature coefficients of ternary electrolytes increases uniformly with rising temperature. In the case of acids and bases, the rate of increase in conductance steadily diminishes as the temperature rises. The ionization decreases regularly with rise in temperature, the temperature coefficient of ionization being small between 18 and 100. The effect of pres- sure on conductance was studied by Fanjung.* He found that the conductance increases slightly with increasing pressure. This result he interprets as being due to increased ionic velocity rather than to an increase in the number of ions present in the solution. Conductance of Non-aqueous Solutions. A large amount of interesting and important work has been done in recent years upon the electrical conductance of solutions in non-aqueous sol- vents. It is impossible to give even a brief survey of the results of these investigations, and we must limit ourselves to the statement of the following general conclusions : f (1) The conditions in non-aqueous solutions are much more complex than in aqueous solutions. (2) In general, the laws which have been found to apply to aque- ous solutions also apply to non-aqueous solutions. (3) Different solvents appear to have different dissociating powers. (4) The dissociating power appears to run parallel with the dielectric constant of the solvent. Many interesting phenomena present themselves in connec- tion with the conductance of electrolytes in mixed solvents, but for an account of this work the student must consult the original papers of Jones and his students, f * Zeit. phys. Chem., 14, 673 (1894). t " Elektrochemie der nichtwassrigen Losungen," by G. Carrara, Ahren'a "Sammlung Chemischer und chemisch-technischer Vortraege," Vol. XII. J Publication of the Carnegie Institution, No. 80. ELECTRICAL CONDUCTANCE 419 Ionizing Power of Solvents. Thomson * and Nernst f pointed out that if the forces which hold the atoms in the molecule are of electrical origin, then those liquids which possess large dielectric constants should have correspondingly great ionizing power. This is a direct consequence of Coulomb's law of electrostatic attraction, which may be expressed by the equation, / = Kd? in which 51 and # 2 denote two electric charges, d the distance between them, / the force of attraction and K the dielectric con- stant. Obviously the larger K becomes, the smaller will be the value of/; i. e., the more likely the molecule will be to break down into ions. That the above relation is approximately true may be seen from the following table : DIELECTRIC CONSTANTS. Solvent. K Ionizing Power. Benzene 2.3 Extremely weak Ethyl ether 4.1 Weak Ethyl alcohol 25 Fairly strong Formic acid 62 Strong Water 80 Very strong Hydrocyanic acid 96 Very strong Dutoit and Aston J have suggested that there is a connection between the ionizing power of a solvent and its degree of associa- tion, and Dutoit and Friderich conclude that the values of A, for a given electrolyte dissolved in different solvents, are a direct function of the degree of association and an inverse function of the viscosity of the solvents. Water and the alcohols furnish good illustrations of the truth of this generalization. * Phil. Mag., 36, 320 (1893). t Zeit. phys. Chem., 13, 531 (1894). t Compt. rend., 125, 240 (1897). Bull. Soc. Chim. [3], 19, 321 (1898). 420 THEORETICAL CHEMISTRY Conductance of Fused Salts. While solid salts are exceed- ingly poor conductors of electricity, yet as the temperature is raised their conductance increases until at their melting point they may be grouped with good conductors. There is no sudden increase in conductance at the melting point. The specific conductance of a fused salt may exceed the specific conductance of the most concentrated aqueous solutions, but owing to the high concentration the equivalent conductance is much less. The following table gives the specific and equivalent conductance of fused silver nitrate : Temperature, degrees. Sp. Cond. Equiv. Cond. 218 (melt, pt.) 0.681 29.2 250 0.834 36.1 300 1.049 46.2 350 1.245 55.4 The specific conductance of a 60 per cent aqueous solution of silver nitrate at 18 is 0.208 reciprocal ohms. If the salts are impure the conductance is raised, the effect of impurities being apparent even before the salts have reached their melting points. This is analogous to the behavior of solutions, and suggests that the impurity functions in the salt mixture as a dissolved solute.* PROBLEMS. 1. An aqueous solution of copper sulphate is electrolyzed between copper electrodes until 0.2294 gram of copper is deposited. Before elec- trolysis the solution at the anode contained 1.1950 grams of copper, after electrolysis 1.3600 grams. Calculate the transport numbers of the two ions, Cu" and S0 4 ". Ans. n = 0.28, 1 - n = 0.72. 2. A solution containing 0.1605 per cent of NaOH was electrolyzed between platinum electrodes. After electrolysis 55.25 grams of the cathode solution contained 0.09473 gram of NaOH, whilst the concen- tration of the middle portion of the electrolyte was unchanged. In a * For a complete treatment of fused electrolytes the student is advised to consult, "Die Elektrolyse geschmolzener Salze, " by Richard Lorenz. ELECTRICAL CONDUCTANCE 421 silver coulometer the equivalent of 0.0290 gram of NaOH was deposited during electrolysis. Calculate the transport numbers of the Na" and OH' ions. Am. n = 0.791, 1 - n = 0.209. 3. In a 0.01 molar solution of potassium nitrate, the transport num- bers of the cation and anion are, respectively, 0.503 and 0.497. Find the equivalent conductances of the two ions in this solution having given that its specific conductance is 0.001044. Am. l c = 52.5, l a = 51.9. 4. The absolute velocity of the Ag* ion is 0.00057 cm. per sec., and that of the Cl' ion is 0.00059 cm. per sec. Calculate the equivalent con- ductance of an infinitely dilute solution of silver chloride. 5. The equivalent conductance of an infinitely dilute solution of am- monium chloride is 130; the ionic conductances of the ions OH' and Cl' are 174 and 65.44 respectively. Calculate the equivalent conductance of ammonium hydroxide at infinite dilution. Am. A = 238.56. 6. The equivalent conductance of a molar solution of sodium nitrate at 18 is 66; its conductance at infinite dilution is 105.3. What is the degree of ionization in the molar solution? Ans. a = 62.6 per cent. 7. The specific conductance of a saturated solution of AgCN at 20 is 1.79 X 10~ 6 and the specific conductance of water at the same temper- ature is 0.044 X 10~ 6 reciprocal ohms. The equivalent conductance at infinite dilution is 115.5. Calculate the solubility of AgCN in grams per liter. Ans. 2.02 X 10~ 3 gram/liter. 8. The equivalent conductance at 18 of a solution of sodium sulphate containing 0.1 gram-equivalent of salt per liter is 78.4, the conductance at infinite dilution is 113 reciprocal ohms. What is the value of i for the solution? What is its osmotic pressure? Ans. i = 2.388; osmotic pressure = 2.85 atmos. 9. The freezing-point of a 0.1 molar solution of CaCl 2 is 0.482. (a) Calculate the degree of ionization (freezing point constant = 1.89 for one mol per liter), (b) Calculate the degree of ionization from the equivalent conductance at 18, which is 82.79 reciprocal ohms, whilst the equivalent conductance of CaCU at infinite dilution is 115.8 reciprocal ohms. Ans. (a) a = 0.774; (6) a = 0.715. CHAPTER XIX. ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS. Ostwald's Dilution Law. It has been shown in preceding chapters that the law of mass action is applicable to chemical equilibria in both gaseous and liquid systems. We now proceed to show that it applies equally to electrolytic equilibria. When acetic acid is dissolved in water it dissociates according to the equation CHsCOOH <= CH 3 COO' + H*. Let one mol of acetic acid be dissolved in water and the solution diluted to v liters, and let a denote the degree of dissociation. Then, the concentration of the undissociated acid is - , and the concentration of the ions is-- Applying the law of mass action, we have or where K is the equilibrium or ionization constant. This equation expressing the relation between the degree of ionization and dilution, was derived by Ostwald* and is known as the Ostwald dilution law. Since a = - - , we may substitute Aoo this value of a in equation (1) and obtain the expression oo (A w A r ) v = K. (2) * Zeit. phys. Chem., 2, 36 (1888); 3, 170 (1889). 422 ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 423 The dilution law may be tested by substituting the value of a, corresponding to any dilution v, in the equation and calculating the value of the ionization constant, K\ the value of a at any other dilution may then be calculated and compared with the value determined_by direct experiment. The following table gives the results obtained with acetic acid at 14.l, K being equal' to 0.0000178: (in liters). aX102 (calc.). aX102 (obs.). 0.994 0.42 0.40 2.02 0.60 0.614 15.9 1.67 1.66 18.1 1.78 1.78 1,500.0 15.0 14.7 3,010.0 20.2 20.5 7,480.0 30.5 30.1 15,000.0 40.1 40.8 As will be seen, the agreement between the observed and cal- culated values is very close. The table also shows to how small an extent the molecules of acetic acid are broken down into ions, a molar solution being dissociated less than 0.5 per cent. The dilution law holds for nearly all organic acids and bases, but fails to apply to salts, strong acids, and strong bases. When a is small, the term (1 a) does not differ appreciably from unity, and equation (1) becomes or (3) On the other hand, when a cannot be neglected, we have, on solving equation (2) for a, V J\. , A I T ^ . C/ J.*. The method of derivation indicates that the dilution law is only strictly applicable to binary electrolytes and, therefore, 424 THEORETICAL CHEMISTRY it is improbable that it will hold for electrolytes yielding more than two ions. It has been found, however, that organic acids whether they are mono-, di-, or polybasic always ionize as a monobasic acid up to the dilution at which a = 50 per cent. This means that the dilution law is applicable to poly- basic acids up to that dilution at which the acid is 50 per cent ionized. Strength of Acids and Bases. There are several methods by which the relative strengths of acids can be estimated. A method which has proved of great value is that in which two different acids are allowed to compete for a certain base, the amount of which is insufficient to saturate both of them. Suppose equiva- lent weights of nitric and dichloracetic acids together with sufficient potassium hydroxide to saturate one acid completely are taken: we then determine the position of the equilibrium represented by the equation HN0 3 + CHC1 2 COOK <= CHC1 2 COOH + KNO 3 . In order to determine the conditions of equilibrium we may make use of any method which does not disturb this equilibrium. Since ordinary chemical methods are excluded on this account, we employ any physical property which is capable of exact measure- ment and differs sufficiently in the two systems, as for example, the change in volume, or the thermal change, accompanying neutralization. Thus, Ostwald * found that when one mol of potassium hydroxide is neutralized by nitric acid in dilute solu- tion, the volume increases approximately 20 cc. When one mol of potassium hydroxide is neutralized by dichloracetic acid, how- ever, the increase in volume is 13 cc. It is evident, therefore, that if nitric acid completely displaces dichloracetic acid as represented by the above equation, the increase in volume will be 20 13 = 7 cc.; if no displacement occurs, then the volume will remain constant. He found that the volume actually increased 5.67 cc. Therefore, the reaction represented by the upper arrow has pro- ceeded to the extent of 5.67 -^ 7 = 80 per cent. That is to say, * Jour, prakt. Chem. [2], 18, 328 (1878). ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 425 in the competition of the two acids for the base, the nitric acid has taken 80 per cent and the dichloracetic acid has taken 20 per cent, or the relative strengths of the two acids are in the ratio of 80 : 20, or 4 : 1. The relative strengths of acids can also be determined from their catalytic effect on the rates of certain reactions, such as the hydrolysis of esters or the inversion of cane sugar. The order of the activity of acids is the same whether measured by equilibrium or kinetic methods. Arrhenius pointed out that the relative strengths of acids can be readily determined from their electrical conductance. The order of the strengths of acids as determined by equilibrium and kinetic methods is the same as that of their electrical conductances in equivalent solutions. This is well illustrated by the following table in which the three methods are compared, hydrochloric acid being taken as the standard of comparison: Acid. Method Employed. Equilibrium. Kinetic. Conductance. HC1.. 100 100 49 9 100 100 53.6 4.8 0.4 100 99.6 65.1 4.8 1.4 HNO 3 H 2 SO 4 ... CH 2 C1COOH CHsCOOH. The results of these and other experiments warrant the con- clusion that the strength of an acid is determined by the number of hydrogen ions which it yields. It is important to note that the electrical conductance of an acid is not directly proportional to its hydrogen ion concentration; the relatively high velocity of the H ion is the cause of the approximate proportionality between these two variables. In the case of a weak acid, the value of the ionization constant may be taken as a measure of the strength of the acid. The following table gives the values of the ionization constants at 25 for several different acids. 426 THEORETICAL CHEMISTRY IONIZATION CONSTANTS OF ACIDS. Acid. lonization Constant. Acetic acid 0000180 Monochloracetic acid .... 00155 Trichloracetic acid . . 1 21 Cyanacetic acid Formic acid 0.0037 0.000214 Carbonic acid 3040 XlO- 10 Hydrocyanic acid 570 XlO- 10 Hydrogen sulphide 13 XlO- 10 Phenol 1.3 XlO- 10 Since for a weak acid, a = \^vK, it follows that for two weak acids at the same dilution, we may write 01 0:2 or the ratio of the degrees of ionization of the two acids is equal to the square root of the ratio of their ionization constants. Thus, from the data given in the foregoing table for acetic and mono- chloracetic acids, we have '0.000018 1 ai = /0.0 a, V 0.( a 2 * u.00155 9.3 or the effect of replacing one atom of hydrogen in the methyl group of acetic acid increases the strength of the acid about nine times. Just as the hydrogen ion concentration of acids determines their strength, so the strength of bases is determined by the concen- tration of hydroxyl ions. The strength of bases may be estimated by methods similar to those employed in determining the strength of acids. Thus, two different bases may be allowed to compete for an amount of acid sufficient to saturate only one of them; or a catalytic method developed by Koelichen * may be used. This method is based upon the effect of hydroxyl ions on the rate of condensation of acetone to diacetonyl alcohol, as represented by the equation 2CH 3 COCH 3 = CH 3 COCH 2 C (CH 3 ) 2 OH. * Zeit. phys. Chem., 33, 129 (1900). ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 427 In addition to these two methods, the method of electrical con- ductance is also applicable. The agreement between the results obtained by the three methods is quite satisfactory. The alkali and alkaline earth hydroxides are very strong bases and are dis- sociated to about the same extent as equivalent solutions of hydrochloric and nitric acids, while on the other hand, ammonia and many of the organic bases are very weak. The following table gives the ionization constants of several typical bases: IONIZATION CONSTANTS OF BASES. Base. Ionization Constant. Ammonia. . ... 000023 Methylamine ... 00050 Trimethylamine 000074 Pyridine 2.5X10- 10 Aniline 1 . 1 X 10- 10 Mixtures of Two Electrolytes with a Common Ion. Just as the dissociation of a gaseous substance is diminished by the addi- tion of an excess of one of the products of dissociation, so the ionization of weak acids and bases is depressed by the addition of a salt with an ion common to the acid or the base. If the degree of ionization of a salt with an ion in common with an acid, or a base is represented by a', and n denotes the number of molecules of salt present, then the equation of equilibrium of the acid or base will be (not + a)a = Kv(l- a), where a is the degree of ionization of the acid or base. For very weak acids and bases, a is so small that 1 a does not differ appreciably from unity, and since a is practically independent of the dilution, we obtain no. = Kv or Kv a = n 428 THEORETICAL CHEMISTRY That is, the ionization of a weak acid or base, in the presence of one of its salts, is approximately inversely proportional to the amount of salt present. In many of the processes of analytical chemistry, advantage is taken of the action of neutral salts on the ionization of weak acids and bases. Thus, while the concentration of hydroxyl ions in ammonium hydroxide is sufficient to precipitate magnesium hy- droxide from solutions of magnesium salts, the presence of a small amount of ammonium chloride depresses the ionization of the ammonium hydroxide to such an extent that precipitation no longer takes place. Isohydric Solutions. Arrhenius * was the first to point out what relation must exist between solutions of two electrolytes with a common ion, in order that, when mixed in any proportions, they may not exert any mutual influence. He showed that when the concentration of the common ion in each of the two solutions is the same before mixing, no alteration in the degree of ionization will occur after mixing. Such solutions are said to be isohydric. Thus, an aqueous solution containing one mol of acetic acid in 8 liters, is isohydric with an aqueous solution containing one mol of hydrochloric acid in 667 liters. On mixing these two solutions the hydrogen ion concentration remains unchanged, and if the mixture is treated with a small amount of sodium hydroxide, equal amounts of sodium acetate and sodium chloride will be formed. That isohydric solutions may be mixed without altering their respective ionizations may be shown in the following manner : Let C and c denote the concentrations of the undissociated por- tions, and CA, C z , CA, and c 2 denote the concentrations of the dis- sociated portions of two electrolytes, and let C% and Cz correspond to two different ions. Then, we have kc = c A Cv, (1) and KC - C A C*. (2) * Wied. Ann., 30, 51 (188,7). ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 429 If v liters of the first solution be mixed with V liters of the second solution, the concentrations of the undissociated portions, and of the dissimilar ions, will be Cv cv C 2 V w while the concentration of the common ion A, becomes C A V + c A v Applying the law of mass action, we have j CA.V + CA.V KC = pr C2, (3) and "W /~i ^- / A. ' T~ ^ 'Ay s~i t j\ KC v + v C 2 (4) But equations (3) and (4) only become identical with equations (1) and (2) when C A = C A , or in other words, no change in the degree of dissociation takes place after the two solutions are mixed. lonization of Strong Electrolytes. It has already been men- tioned that the Ostwald dilution law, which is a direct conse- quence of the law of mass action, applies only to weak electrolytes. Just why the law of mass action should fail to apply to strong electrolytes is not known, but several possible causes have been suggested to account for its failure. One of the most plausible explanations is that advanced by Biltz,* who attributes the failure of the law of mass action when applied to strong electrolytes, to hydration of the solute. If the ions become associated with a large proportion of the solvent, the effective ionic concentration would then be the ratio of the amount of the ion present to that of the free solvent, instead of to the total solvent, as ordinarily calculated. This view is in harmony with certain facts which have been adduced in favor of the theory of solvation. While the Ostwald dilution law does not apply to strongly ionized elec- trolytes, certain empirical expressions have been derived which * Zeit. phys. Chem., 40, 218 (1902). 430 THEORETICAL CHEMISTRY hold fairly well over a wide range of dilution. Thus, Rudolph! * showed that the equation (1 - a) Vv gives approximately constant values for K' for strong electrolytes. The following table gives the results obtained with solutions of silver nitrate at 25; the numbers in the third column being cal- culated by means of the Ostwald dilution law, while those in the fourth column are calculated by means of Rudolphi's dilution law. a K K' 16 0.8283 0.253 .11 32 0.8748 0.191 .16 64 0.8993 0.127 .06 128 0.9262 0.122 .07 256 0.9467 0.124 .08 512 0.9619 0.125 .09 The Rudolphi equation was modified by van't Hoff f to the form /I \2 This equation holds even more closely than that of Rudolphi. Of the more recent empirical equations which have been derivec to express the change of conductance of an electrolyte with dilu- tion, the equations of Kraus * and Bates f deserve special men- tion. The equation of Kraus has the following form: /A \ 9 /~^ I A?? V C , , , \A ?7o/ (i - ^) In this equation A is the conductance of the electrolyte whose con- centration C is expressed in mols per liter, 77/770 is the ratio of the viscosity of the solution to that of the solvent, and k, k', A, and A * Jour. Am. Chem. Soc., 35, 1412 (1913). t Ibid., 37, 1421 (1915). ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 431 are empirical constants the values of which are so chosen as to insure close agreement between the observed and calculated values of the conductance. The equation of Bates is similar to that of the preceding equa- tion, except that the logarithm of the left-hand side of the equation is substituted for the original expression of Kraus. The equation of Bates takes the form : = k + k >(cf\ h . Ar?\ V i / lvr i lOglO (-T The constants k, k', and h are purely empirical as in the equation of Kraus, but A denotes the equivalent conductance at infinite dilution. A comparison between the two equations is afforded by the fol- lowing table in which is recorded the observed and calculated values of the "corrected " equivalent conductance, A^/r/o, for solutions of potassium chloride at 18. COMPARISON OF THE EQUATIONS OF KRAUS AND BATES. c n/no A (obs.) AIJ/IJO A,/ % (K.) A.,/,,, (B.) 3 0.9954 88.3 87.89 87.4 89.3 2 0.9805 92.53 90.73 90.9 91.9 1 0.982 98.22 96.5 96.4 96.53 0.5 0.9898 102.36 101.32 101.1 101.29 0.2 0.9959 107.90 107.46 107.6 107.43 0.1 0.9982 111.97 111.77 1U. 9 111.73 0.05 0.9991 115.69 115.59 115.5 115.58 0.02 0.9996 119.90 119.85 119.8 119.83 0.01 0.9998 122.37 122.35 122.4 122.32 0.005 0.9999 124.34 124.33 124.4 124.38 0.002 1.0000 126.24 126.24 126.3 126.31 0.001 127.27 127.27 127.2 127.32 0.0005 128.04 128.04 127.6 128.05 0.0002 128.70 128.70 127.9 .128.68 0.0001 129.00 129.00 128.1 128.96 0.0 129 50 129.50 128.3 129.50 It will be observed that for dilute solutions, the ratio 77/170 is practically unity and, furthermore, that the value of (7A/A is so small that the second term on the right-hand side of both equations is negligible in comparison with the value of k. 432 THEORETICAL CHEMISTRY Therefore, under these conditions, both equations reduce to the form A 2 C TT TT = constant, Ao (A A) which will be recognized as identical with Ostwald's dilution law as given on page 422. Heat of lonization. The heat of ionization of an electrolyte can be calculated by means of the reaction isochore equation of van't Hoff (see p. 324), provided the degree of ionization at two different temperatures is known. Since and l (1 - K a * Z (1 - 2 ) v' it follows that the heat of ionization may be calculated by means of the equation 2.306 R log ,, (1 ~ - log ,, V (1 Arrhenius * has shown that this equation also applies to those electrolytes which do not obey the Ostwald dilution law. Some of the results obtained by Arrhenius are given in the accompany- ing table : Electrolyte. Temperature. Calories. Acetic acid \ 35 386 Propionic acid \ 21.5 35 -28 557 Butyric acid ] 21.5 35 183 935 Phosphoric acid < 21. 5 35 427 2458 Hydrochloric acid 21. 5 35 2103 1080 Potassium chloride . . 35 362 Potassium bromide 35 425 Potassium iodide 35 916 Sodium chloride 35 454 Sodium hydroxide 35 1292 Sodium acetate 35 391 Zeit. phys. Chem., 4, 96 (1889). ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 433 It will be found that the values of the heats of ionization given in this table do not agree with the values calculated for these same substances from the data given in the table on page 307. The reason for this lack of agreement is, that the data of the earlier table refer to the heat of formation of the ions from the dissolved substance, whereas the data of the table just given represent the combined thermal effects of solution and ionization. The Solubility Product. While the law of mass action does not in general apply to the equilibrium between the dissociated and undissociated portions of an electrolyte, except in the case of organic acids and bases, it does apply with a fair degree of accuracy to saturated solutions of electrolytes. A saturated solution of silver chloride affords an example of such an equilibrium. This salt is practically completely ionized in a saturated solution, as represented by the equation Applying the law of mass action to this equilibrium, we obtain CAg* X CC1 _ jr CAgCl Since the solution is saturated, the value of CA g ci must remain constant at constant temperature, and therefore CAg- X cci' = constant = s, where the product of the ionic concentrations s is called the solu- bility or ionic product. The equilibria in the above heterogeneous system may be repre- sented thus : (in solution) (solid) The solubility product for silver chloride at 25 is 1.56 X 10~ 10 , the ionic concentrations being expressed in mols per liter. Hence, since the two ions are present in equivalent amounts, a saturated solution of silver chloride at 25 must contain Vl.56 X 10~ 1( = 1.25 X 10~ 5 mols per liter of Ag* and Cl' ions. In general, if 71 A < 434 THEORETICAL CHEMISTRY represents the equilibrium between an electrolyte and its products of dissociation in saturated solution, we have rii n^ Cm / O A ' L . o. A! A 2 The solubility product may be defined as the maximum product of the ionic concentrations of an electrolyte which can exist at any one temperature. Just as the dissociation of a gaseous substance or of an organic acid is depressed by the addition of one of the products of dis- sociation, so when a substance with a common ion is added to the saturated solution of an electrolyte, the dissociation is depressed and the undissociated substance is precipitated. The following example will serve to illustrate how the solu- bility product of a substance can be determined, and how the change in solubility due to the addition of a substance containing a common ion may be calculated. The solubility of silver bromate at 25 is 0.0081 mol per liter. If we assume complete ionization, the concentration of the ions, Ag" and BrCY will be the same and equal to 0.0081 mol per liter, or (0.0081) (0.0081) = s. The solubility in a solution of silver nitrate containing 0.1 mol of Ag' ions can be calculated from the equation (0.0081) 2 = (0.0081 + 0.1 - x) (0.0081 - x), where x represents the amount of silver bromate thrown out of so- lution by the addition of 0.1 mol of Ag* ion. Since (0.0081 x) represents the concentration of Ag* and BrO 3 ' ions after the addition of the silver nitrate, it also represents the solubility of silver bromate under similar conditions. The effect of adding a solution of a soluble bromate containing 0.1 mol of Br(V ion will be the same as that produced by 0.1 mol of Ag* ion. The Basicity of Organic Acids. The Ostwald dilution law holds strictly for all monobasic organic acids, and also for poly- basic organic acids which are less than 50 per cent ionized. The neutral salts of these acids, however, are much more highly ionized, and the difference in conductance between two dilutions of a neu- tral salt of a polybasic acid is greater than the difference in ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 435 conductance between the same dilutions of a neutral salt of a monobasic acid. Ostwald * has shown that it is possible to estimate the basicity of an organic acid from the difference in the equivalent conductance of its sodium salt at two different dilutions. As the result of a long series of experiments, he found that the difference between the equivalent conductance of the sodium salt of a monobasic organic acid at v = 32 liters and at v = 1024 liters is approximately 10 units. Similarly, the difference for a dibasic acid between the same dilutions is 20 units, and for an n-basic acid the difference is 10 n. Hence, to estimate the basicity of an organic acid, the equivalent conductance of its sodium salt at v = 32 liters and at v = 1024 liters is determined; then, if A is the dif- ference between the values of the conductance at the two dilutions, the basicity will be n = -^. The following table gives the values of A and n for the sodium salts of several typical organic acids : Acid. A n Formic 10 3 1 Acetic 9 5 1 Propionic 10 2 1 Benzoic 8.3 1 Quininic . . 19.8 2 Pyridine-tricarboxylic (1, 2, 3) 31.0 3 Pyridine-tricarboxylic (1, 2, 4) f Pyridine-tetracarboxylic ... 29.4 41.8 3 4 Pyridine-pentacarboxylic 50.1 5 Influence of Substitution on lonization. Attention has already been called to the marked difference in the strength of acetic acid produced by the replacement of the hydrogen atoms of the methyl group by chlorine. In the accompanying table the ioniza- tion constants for various substitution products of acetic acid are given : * Zeit. phys. Chem., i, 105 (1887); 2, 902 (1888). 436 THEORETICAL CHEMISTRY Acid. Acetic, CHsCOOH Propionic, CH 3 CH 2 COOH. . . . Chloracetic, CH 2 C1COOH. . . . Bromacetic, CH 2 BrCOOH. . . . Cyanacetic, CH 2 CNCOOH. . . Glycollic, CH 2 OHCOOH Phenylacetic, C 6 H 5 CH 2 COOH Amidoacetic, CH 2 NH 2 COOH. lonization Constant (25). 0.000018 0.000013 0.00155 0.00138 0.00370 0.000152 0.000056 3.4X 10- 1C This table affords an interesting illustration of the influence of different substituents on the strength of acetic acid. Thus, the activity of the acid is increased by the replacement of alkyl hydro- gen atoms by Cl, Br, CN, OH, or C 6 H 5 , while the substitution of the CH 3 or NH 2 groups diminishes its activity. If we assume that the substituents retain their ion-forming capacity on enter- ing into the molecule of acetic acid, these differences in activity can be readily explained. Thus, Cl, Br, CN, and OH tend to form negative ions, and hence increase the negative character of the group into which they enter. On the other hand, basic groups, such as NH 2 , diminish the tendency of the group into which they .enter to yield negative ions. The influence of an alkyl residue on the strength of an organic acid is conditioned by its distance from the carboxyl group. This is well illustrated by the ionization constants of propionic acid and some of its derivatives. Acid. Ionization Constant (25). Propionic acid, CH 3 CH 2 COOH . . 0000134 Lactic acid, CH 3 CHOHCOOH 000138 /8-oxypropionic acid, CH 2 OHCH 2 COOH . 0000311 The effect of the OH group in the a-position is seen to be much more marked than when it occupies the /3-position. The position of a substituent in the benzene nucleus exerts a marked influence on the strength of the derivatives of benzoic ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 437 acid. The ionization constants of benzoic acid and the three chlorbenzoic acids are given in the following table: Acid. Ionization Constant (25). Benzoic acid, C 6 H 5 COOH 000073 o-Chlorbenzoic acid, C 6 H 4 C1COOH 00132 m-Chlorbenzoic acid CeEUClCOOH 000155 p-Chlorbenzoic acid, C 6 H 4 C1COOH . 000093 When the halogen enters the ortho-position, the strength of the acid is greatly augmented, while in the meta- and para- positions the effect is much smaller, meta-chlorbenzoic acid being stronger than para-chlorbenzoic acid. It is a general rule that the influence of substituents is always greatest in the ortho-position, and least in the meta- and para- positions, the order in the two latter being uncertain. Hydrolysis. When a salt formed by a weak acid and a strong base, such as sodium carbonate, is dissolved in water, the solution shows an alkaline reaction, while on the other hand, when a salt formed by a strong acid and a weak base, such as ferric chloride, is dissolved in water, the solution shows an acid reaction. The process which takes place in the aqueous solution of a salt, causing it to react alkaline or acid, is termed hydrolysis or hydro- lytic dissociation. If MA represents a salt, in which M is the basic and A is the acidic portion, then the hydrolytic equilibrium may be represented by the equation MA + H 2 <= MOH + HA. If the base formed is insoluble or undissociated and the acid is dissociated, the solution will react acid. If the acid formed is insoluble or undissociated and the base is dissociated, the solution will react alkaline. Finally, if both base and acid are insoluble or undissociated, the salt will be completely transformed into base and acid, and, as there will be no excess of either H' or OH 7 ions, the solution will remain neutral. It is evident, then, that hydrolysis is due to the remove! of either one or both of the ions of water by the ions of the salt to 438 THEORETICAL CHEMISTRY form undissociated or insoluble substances. As fast as the ions of water are removed, the loss is made good by the dissociation of more water, until eventually a condition of equilibrium is estab- lished. The conditions governing hydrolytic equilibrium may be determined from a knowledge of the solubility or ionic constant of the substances involved. Thus, if the product of the concen- trations of the ions M" and OH' exceeds that which can exist in pure water, then some undissociated or insoluble substance will be formed. This will disturb the equilibrium of H* and OH' ions, and a further dissociation of water must occur until the ionic product of water is just reached. If now the ions H* and A' do not unite to form undissociated acid, the presence of an excess of H" ions will disturb the equi- librium between pure water and its products of dissociation; or, since CH* X COH' = Sn 2 o, the concentration of OH' ions present, when CH- represents the total concentration of H* ions, will be -3fi . CH A similar readjustment will take place when an undissociated or insoluble acid and a dissociated base are formed. We may now proceed to consider three different cases of hydroly- sis, viz., when the reaction is caused (1) by the base, (2) by the acid, and (3) by both base and acid. CASE I. The formation of an undissociated or insoluble base is pri- marily the cause of the hydrolysis, the acid formed being dissociated. Let the hydrolytic equilibrium be represented by the equation MA + H 2 O <=* MOH + HA. The reaction will proceed in the direction of the upper arrow until the product, CM- X COH', exceeds that which can exist in the ab- sence of an undissociated base. When equilibrium is established, we have final CM- X final COH- = #MOH X CMOH formed, (1) or if the base formed is practically insoluble, the equilibrium equa- tion simplifies to the form final CM' X final COH' = SMOH, (2) ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 439 where SMOH is the solubility product of the base. The condition of equilibrium represented by the equation CH- X COH' = SH Z O must be fulfilled. It follows that the final concentration of the OH' ions will be the quotient obtained by dividing the ionic product for water, at the temperature of the experiment, by the final con- centration of the H* ion, this latter being wholly dependent upon the extent of the reaction and the degree of ionization of the acid formed. If the degree of hydrolysis of the salt be represented by x, and the degree of dissociation of the unhydrolyzed portion of the salt be denoted by a s) then, if one mol of salt be dissolved in V liters of solution, the final concentration of M" ions will be - y and the final concentration of the undssiociated base will be y> The total acid formed will be y, and if a a denotes the degree of dissociation of the acid, the concentration of the H* 1* ions will be a a y Substituting these values in equations (1) and (2), we obtain ds (1 - X) SR 2 ~ X ( . ~V~ ~ = A - MOHX y' d a y and (4) Simplifying equations (3) and (4), we have & d a _ SH Z O . a M \ T7 * - ~ IF - ~~ - x) V a s AMOH and . . (1 x) a s SMOH From equations (5) and (6) it appears that the constant of hydrol- ysis can be found either from the ionic product for water and the 440 THEORETICAL CHEMISTRY ionization constant of the base, or from the ionic product for water and the solubility product of the base. Furthermore, if the base formed is insoluble, equation (6) shows that the degree of hydrolysis, x, is independent of the dilution of the salt, V. CASE II. The formation of an undissociated or insoluble add is primarily the cause of the hydrolysis, the base formed being dis-^ sociated. In this case hydrolysis takes place until the product CH* X CA' exceeds that which can exist in the absence of undis- sociated acid. When equilibrium is established, we have final CH- X final CA' = KHA X CHA formed, (7) or if the acid formed is practically insoluble, the equilibrium equa- tion simplifies to the form final CH- X final CA' = SHA. (8) Since the final CH' = SH Z O -*- final COH', we have, final CA' = " -, yj final COH' = ctb y , where ah is the degree of dissociation of the base formed, and the final CHA = tions (7) and (8), we obtain formed, and the final CHA = Substituting these values in equa- (9) T and Oity Simplifying equations (9) and (10), we have and SH.O _ jf ~ (1 x) a s It is evident from equations (11) and (12), that the constant of hydrolysis can be found either from the ionic product for water and ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 441 the ionization constant of the acid, or from the ionic product for water and the solubility product of the acid. CASE III. The formation of an acid and a base, both being slightly dissociated, is the cause of the hydrolysis. In this case let us assume that KHA is smaller than Since the final COH' = XMQH x CMOH and since both HA and MOH CM* are slightly dissociated, we may write CHA = CMOH = y, and a s (1 - x) CA' = CM- = -- y Substituting these values in equation (7), we obtain x Simplifying equation (13), we obtain (1-s) c? KHAX^MOH From equation (14) we see that the constant of hydrolysis can be found from the ionic product for water and the ionization constants of the acid and the base. If both acid and base are practically insoluble, the reaction will be complete at all dilutions. As an illustration of the application of the foregoing equations, we may take the calculation of the degree of hydrolytic dissociation of potassium cyanide in 0,1 molar solution at 25. Potassium cyanide being a salt of a weak acid, the degree of hydrolysis can be calculated by means of the equation X 2 Cib _ SH 2 _ K ' ~ Since at 25, K HA = 13 X 10" 10 and s H2 o = (0.91 X 10~ 7 ), we have SH,O (0.91 XlO- 7 ) 13 X 10- 10 ' 442 THEORETICAL CHEMISTRY and since in dilute solution a a = <*& = 1, we have x 2 (0.91 X 1Q- 7 ) 2 (1 - x) - 10 13 X ID" 10 or x = 0.00798. Experimental Determination of Hydrolysis. The degree of hydrolysis can be determined experimentally in several different ways. A very convenient method is that based upon measure- ments of electrical conductance. When a salt reacts hydrolyti- cally with one mol of water, the limiting value of its equivalent conductance will be A A + A^, where A A and AS denote the equiv- alent conductances of the acid and base formed. If A is the equiv- alent conductance of the unhydrolyzed salt, and A/ t is the actual conductance of the salt at the same dilution, then the increase in conductance corresponding to a degree of hydrolysis x will be Ah A. The value of A may be found by determining the conductance of the salt in the presence of an excess of one of the products of hydrolysis and deducting from it the conductance of the substance added. Since if the hydrolysis were complete, the equivalent conductance would be AA + AB A, we have A A - A " A A + AB- A' all conductances being measured at the same dilution and the same temperature. The following example will illustrate the use of this equation : At 25, the equivalent conductance of an aqueous solution of aniline hydrochloride is 118.6, the dilution being 99.2 liters. The equivalent conductance in the presence of an excess of aniline is 103.6, while the equivalent conductance of hydrochloric acid at the same dilution is 411. The conductance of pure aniline is so small as to be negligible. Substituting these values in the equation, we find 118.6 - 103.6 nnA . '- 411 - 103.6 = ' 488 - Lunden * has shown how this method may be extended to cases where both acid and base are slightly dissociated. * Jour. chim. phys., 5, 145, 574 (1907). ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 443 The lonization Constant of Water. One of the most accurate methods known for the determination of the ionization constant of water is based upon measurements of the degree of hydrolytic dissociation of different salts. Thus Shields * found that a 0.1 molar solution of sodium acetate is 0.008 per cent hydrolyzed at 25. We may consider the salt, as well as the sodium hydroxide formed from its hydrolysis, to be completely dissociated at this dilution. The ionization constant of the acetic acid formed is 0.000018 at 25. Solving equation (11) (on page 440) for SH.O, and remembering that a s = a b = 1, we have Substituting the above values in this expression, we obtain SHl o = 0.000018 - _ 10 = U6 X 10-, and since the ions, H* and OH', are present in equivalent amounts, we have CH- = COH' = Vl.16 X 10- 14 = 1.1 X 10~ 7 mol per liter. Kohlrausch obtained from his measurements of the conductance of pure water at 25, CH- = COH' = 1.05 X 10~ 7 mol per liter (see p. 415). PROBLEMS. 1. At 25 the specific conductance of butyric acid at a dilution of 64 liters is 1.812 X 10" 4 reciprocal ohms. The equivalent conductance at infinite dilution is 380 reciprocal ohms. What is the degree of ioniza- tion and the concentration of H* ions in the solution? What is the ioni- zation constant of the acid? Ans. a = 0.0305, CH- = 4.765 X 10~ 4 mol per liter, K = 1.5 X 1Q- 5 . 2. The heat of neutralization of nitric acid by sodium hydroxide is 13,680 calories, and of dichloracetic acid, 14,830 calories. When one equivalent of sodium hydroxide is added to a dilute solution containing one equivalent of nitric acid and one equivalent of dichloracetic acid, 13,960 calories are liberated. What is the ratio of the strengths of the two acids? Ans. HN0 3 : CHC1 2 COOH :: 3.1 : 1. * Zeit. phys. Chem., 12, 167 (1893). 444 THEORETICAL CHEMISTRY 3. For potassium acetate we have the following data: r A(18) 2 67.1 10 78.4 100 87.9 1000 91.9 and ZOQ = 64.67, and 1 M = 35. Compare the constants obtained by the K' CH 3 COO' Ostwald, Rudolphi, and van't Hoff dilution laws. 4. The ionization constant of a 0.05 molar solution of acetic acid is 0.0000175 at 18, and 0.00001624 at 52. Calculate the heat of ionization of the acid. To what temperature does this value correspond ? Ans. 416 calories at 35. 5. At 20 the specific conductance of a saturated solution of silver bromide was 1.576 X 10~ 6 reciprocal ohms, and that of the water used was 1.519 X 10~ 6 reciprocal ohms. Assuming that silver bromide is completely ionized, calculate the solubility and the solubility product of silver bromide, having given that the equivalent conductances of potas- sium bromide, potassium nitrate, and silver nitrate at infinite dilution are 137.4, 131.3, and 121 reciprocal ohms respectively. Ans. CAgBr = 4.49 X 10~ 7 mol per liter, SAgBr = 2.02 X 10~ 13 . 6. The solubility of silver cyanate at 100 is 0.008 mol per liter. Cal- culate the solubility in solution of potassium cyanate containing 0.1 mol of K' ions. Ans. 6.4 X 10~ 4 mol per liter. 7. Calculate the degree of hydrolytic dissociation of a 0.1 molar solu- tion of ammonium chloride, having given the following data: a a = 0.86, a = 0.87, tf NH 4 OH = 0.000023, and SH 2 o = (0.91 X 10~ 7 ) 2 at 25. Ans. x = 0.006 per cent. 8. In the reaction represented by the equation MA 3 + 3 H 2 = M (OH), + 3 HA, the base formed is insoluble. Derive an expression for the constant of hydrolysis. V S 3 H 2 O (3z) 3 a a 3 A VI O fi __ _ i ^ -n/t'O. -ivp , \ o SM(OH) 3 (1 - X)V Z a, 9. The equivalent conductance of aniline hydrochloride at a dilution of 197.6 liters is 126.7 reciprocal ohms, at 25. The equivalent con- ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 445 ductance of aniline hydrochloride in the presence of an excess of aniline is 106.6; and the equivalent conductance of hydrochloric acid at the same dilution is 415. If the conductance of pure aniline is negligible, calculate the degree of hydrolytic dissociation and the constant of hydrol- ysis, assuming a s = a a = 1. Am. x = 6.52 per cent, K h = 2.30 X 10~ 5 . 10. The hydrolysis constant of aniline is 2.25 X 10~ 5 , and the ioniza- tion constant is 5.3 X 10~ 10 . Calculate the concentration of the H" and OH' ions in water, Ans. CCH- = CCH' = 1.09 X 10~ 7 . CHAPTER XX. ELECTROMOTIVE FORCE. Galvanic Cells. Since the year 1800, when Volta invented his electric pile, many different forms of galvanic cell have been introduced. It is not our purpose to give a detailed account of these cells, but rather to give a brief outline of the theories which have been advanced in explanation of the electromotive force developed in such cells. When two metallic electrodes are immersed in a solu- tion of an electrolyte, a current will flow through a wire connect- ing the electrodes, provided the two metals are dissimilar, or that a difference exists between the solutions surrounding the electrodes. An electric current can be obtained from a combination of two different metals in the same electrolyte, from two different metals in two different electrolytes, from the same metal in different elec- trolytes, or from the same metal in two different concentrations of the same electrolyte. In order that the electromotive force of the combination shall remain constant, it is necessary that the chemical changes involved in the production of the current shall neither destroy the difference between the electrodes, nor deposit upon either of them a non- conducting substance. A galvanic combination which fulfils these conditions very satisfactorily is the Daniell cell. This cell consists of zinc and copper electrodes immersed in solutions of their salts, as represented by the scheme Zn - Sol. of ZnS0 4 || Sol. of CuS0 4 - Cu, in which the two vertical lines indicate a porous partition separat- ing the two solutions. When the zinc and copper electrodes are connected by a wire, a current of positive electricity passes from the copper to the zinc along the wire. Zinc dissolves from the zinc electrode, an equivalent amount of copper being displaced from 446 ELECTROMOTIVE FORCE "447 the solution and deposited simultaneously on the copper electrode. As long as only a moderate current flows through the cell, the original nature of the electrodes is not modified, the only change which occurs being the gradual dilution of the copper sulphate, owing to the separation of copper and its replacement by zinc. If the loss of copper sulphate is replaced, the electromotive force of the cell will remain constant. If, after the cell is assembled no current be allowed to flow, the copper sulphate will slowly diffuse into the solution of zinc sulphate, and metallic copper will ulti- mately be deposited on the zinc electrode. In this way miniature, local galvanic cells will be formed on the surface of the zinc, and the metal dissolves as though the main circuit were closed. Until this deposition takes place, the cell may be left on open circuit without danger of deterioration. Unless chemically pure zinc is used, local action is likely to occur, owing to the formation of local galvanic couples between the impurities in the electrode, chiefly iron, and the zinc. This action may be prevented by amalga- mating the zinc electrode. In this process the mercury dissolves the zinc and not the iron, a uniform surface of the former metal being produced. An interesting experiment due to Ostwald * illustrates the con- ditions essential to the continuous production of an electric current. Two electrodes, one of amalgamated zinc and the other of platinum, are each immersed in a solution of potassium sulphate, the two solutions being separated by a porous cup. When the two elec- trodes are connected by means of a wire, no permanent current passes. An inappreciable quantity of zinc goes into solution, since any current must necessarily first liberate potassium at the platinum electrode, the potassium thus set free reacting with the water. This process requires the expenditure of more energy than the solution of the zinc supplies. If sulphuric acid is added to the compartment containing the zinc, the condition of the system will be unchanged, the zinc remaining undissolved. If, on the other hand, a few drops of sulphuric acid are added to the compartment containing the platinum electrode, bubbles of hydrogen will appear and the zinc will dissolve with the simulta- * Phil. Mag. [5], 32, 145 (1891). 448 THEORETICAL CHEMISTRY neous development of an electric current. This experiment shows that in order that positively charged ions may enter a solution, an equivalent amount of negatively charged ions must be intro- duced, or an equivalent amount of positively charged ions must be removed. Reversible Cells. Galvanic cells are either reversible or non- reversible, according as the processes taking place within them can be reversed or not. If we disregard the slow processes of diffusion, the Daniell cell may be taken as an example of an almost perfect reversible element. If an electromotive force slightly less than that of the cell be applied to it in the reverse direction, the current within the cell will flow from the zinc to the copper electrode as usual. On the other hand, if the external electromotive force slightly exceeds that of the cell, the current within the cell will flow in the reverse direction, zinc being deposited and copper dissolved. Any cell from which gas is evolved is non-reversible, since the passage of a current in the reverse direction cannot restore the cell to its original condition. Relation between Chemical Energy and Electrical Energy. Helmholtz and Thomson were the first to propose a theory of the action of the reversible cell. According to this theory the energy of the chemical process taking place within the cell was considered to be completely transformed into electrical energy. It was soon shown that this theory is inadequate, since, with the exception of the Daniell cell, the chemical energy is not equivalent to the elec- trical energy produced. Subsequently, Gibbs* and Helmholtz f showed independently that only in those cells in which the elec- tromotive force does not vary with the temperature, is the chem- ical energy completely transformed into electrical energy. They also derived an equation expressing the relation between the chemical and electrical energies in any reversible cell. Let us imagine a reversible element in which an amount of heat q, is either liberated or absorbed, when one faraday of electricity has passed through the cell. Let the cell be immersed in a bath, which is so * Proc. Conn. Acad., 3, 501 (1878). t Sitzungsbericht., Ber. Akad., 22, 825 (1882). ELECTROMOTIVE FORCE 449 arranged that the temperature of the cell can be maintained con- stant under any working conditions. If the chemical process within the cell is accompanied by an evolution or an absorption of heat, then of necessity, heat energy must be removed or supplied in order to maintain the temperature of the system constant. It is evident that this will involve a corresponding decrease or increase in the electrical energy produced by the t cell. The effect of the evolution or absorption of heat upon the electrical energy of the cell may be derived in the following manner: Let the cell be heated from its initial temperature T to the temperature (T + dT), and let the corresponding change in the electromotive force of the cell be dE. If now the circuit be closed and one faraday of electricity be allowed to pass through the cell, F (E + dE) units of electrical work will be done. In order that the temperature of the cell may not change, (q + dq) units of heat must be absorbed. The cell is now cooled to the temperature T, at which the electromotive force of the cell is E, and F units of electricity are sent through the cell in the reverse direction, thus increasing the energy of the cell by FE. In order to maintain the temperature of the cell unchanged, q units of heat must be removed. If the cell is completely reversible, when this cycle of operations, is completed, it will be restored to its original condition. The total work done during the cycle is F (E + dE) FE, and the amount of heat transformed into work is (q + dq) q- } therefore, applying the second law of thermo- dynamics, we have dq _ FdE _ dT ~q = q = T ' or Since the electrical energy is equal to FE, the relation between this and Q, the chemical energy of the cell, expressed in calories, becomes FE = Q + q. (2) 450 THEORETICAL CHEMISTRY Substituting in equation (2) the value of q given in equation (1), we obtain or Q dE When -y = 0, E becomes equal to ^ , or, when the temperature coefficient of the cell is zero, the electrical energy is equal to the chemical energy. This is true of the Daniell cell, which has an extremely small temperature coefficient. For cells in which the electromotive force varies appreciably with the temperature, it is possible to calculate the value of the electromotive force at any temperature by means of the Gibbs- Helmholtz equation, provided the temperature coefficient is known. In. the Grove gas cell, E = 1.062 and Q = 34,200 calories, hence T dE - 1 062 - 34 ' 200 -0 418 1 df 96,540 X 0.2394 " The value determined by direct experiment is 0.416 volt. The Gibbs-Helmholtz equation shows that the amount of heat accom- panying a chemical process does not alone furnish a measure of the electrical energy which may be obtained from it, since the heat which is absorbed from the surrounding medium may also be transformed into electrical energy, or the output of electrical energy may be less than the heat evolved by the chemical reaction within the cell. Solution Pressure. It is a familiar fact that water has a tendency to assume the form of vapor, and if the vapor be contin- ually removed from its surface, a definite mass of water will grad- ually be completely transformed into the state of vapor. The pressure of the vapor at any one temperature is a measure of the tendency of water to undergo this transformation. This tendency of water to assume another form than that in which it actually exists, is typical of all substances. Attention has already been directed to this fact in connection with the application of the law ELECTROMOTIVE FORCE 451 of mass action to heterogeneous equilibria. It was then pointed out that all solids have a definite vapor pressure at a definite temperature, which is independent of the amount of solid present. When a solid, such as cane sugar, is brought in contact with water, it tends to pass into solution. This tendency is constant at constant temperature, since the active mass of the solid is constant. From the close analogy between the vapor state and the dissolved state, the tendency of a solid to pass into solution is termed the solution pressure. A dissolved solid, on the other hand, also shows a tendency to separate from the solution as the concentration is increased. When the solution becomes supersaturated, the tendency of the solute to separate in the solid form is greater than the tendency of the solid to dissolve. It is evident from these considerations that the pressure exerted by the dissolved solid is its osmotic pressure, and whether the solid will dissolve or separate from the solution depends upon whether the solution pressure is greater or less than the osmotic pressure. This conception of solution pressure was introduced by Nernst,* and in conjunction with the theory of electrolytic dissociation it has proved of great value in affording a much deeper insight into the mechanism of the development of differences in potential within a galvanic cell. Thus, when a metal is dipped into water it tends to dissolve owing to its solution pressure P and, in con- sequence of this tendency, it sends a certain number of positive ions into solution. The solution thus becomes positively charged, and the metal, which was initially neutral, acquires a negative charge due to the loss of a certain amount of positive electricity. This process will cease when the solution becomes so strongly charged with positive electricity that it prevents the separation of any more positive ions from the metal. Relatively few ions leave the metal before equilibrium is established, since the charge on each ion is so great; in fact, the concentration of metal ions in the solution is much too small to be detected analytically. When a metal is dipped into a solution of one of its salts, the conditions are altered. In this case, the positive ions of the metal already present in the solution oppose the entrance of more positive ions, * Zeit. phys. Chem., 4, 150 (1889). 452 THEORETICAL CHEMISTRY and the equilibrium between these two opposing tendencies will be conditioned by the relative values of the solution pressure P, of the metal, and the osmotic pressure p, of the ions of the dissolved salt. It is evident that the three following conditions are possible: (1) If P > p, the metal will continue to send ions into the solution until the accumulated charges in the solution oppose further action. The solution acquires a positive charge and the metal a negative charge. (2) If P < p, the positive ions of the dissolved salt will sepa- rate on the metal until the accumulated charges oppose further action. The metal acquires a positive charge and the solution a negative charge. (3) If P = p, no action will take place and no difference of potential will be established between the metal and the solution. These three cases are represented diagrammatically in Fig. 95. ?=P -h Fig. 96. When equilibrium is established and the metal is negative against the solution, the metal is surrounded by a layer of positively charged ions. This constitutes what is known as a Helmholtz electrical double layer. If positive electricity be communicated to the metal, the double layer will be broken and more ions will pass from the metal into the solution, but as soon as the supply ELECTROMOTIVE FORCE 453 of positive electricity is cut off, the double layer will again be formed. Similarly, when the metal is positive against the solu- tion, an electrical double layer will be formed, the metal being surrounded by a layer of negatively charged ions. The actual existence of a Helmholtz double layer has been demonstrated by Palmaer.* In his experiments, Palmaer allowed exceedingly minute globules of mercury to fall into a dilute solu- tion of mercurous nitrate contained in a tall vessel, the bottom of which was covered with a layer of pure mercury, as shown in Fig. 96. Since the solution pressure of mercury is less than the Fig. 96. osmotic pressure of the Hg* ions, each drop of mercury as it enters the solution will acquire a positive charge, and if the theory of the electrical double layer is correct, this positively charged globule should attract negatively charged ions and drag them down through the solution. When the globule reaches the mercury at * Zeit. phys. Chem., 25, 265 (1898); 28, 257 (1899); 36, 664 (1901). 454 THEORETICAL CHEMISTRY the bottom of the vessel, it will give up its positive charge and as many Hg" ions will pass into solution as there are NCV ions in the double layer. The solution will thus become more concentrated just above the layer of mercury on the bottom of the vessel. Pal- maer's experiments showed that this difference in concentration is actually produced, in some cases the concentration in the upper part of the solution being reduced as much as 50 per cent. The metals sodium, potassium, . . . zinc, cadmium, cobalt, nickel, and iron are negative against solutions of their salts, or P > p. The noble metals are generally positive against solutions of their salts, or P < p. The anions are, so far as is known, posi- tive to solutions of their salts. Electrolytic solution pressure varies with the temperature, with the nature of the solvent, and also with the concentration of the active substance in the elec- trode. The Difference of Potential between a Metal and a Solution. From the foregoing considerations, it is possible to derive an equation expressing the difference of potential between a metallic electrode and a solution of one of its salts. Let us imagine one gram-ion of a metal to be transferred from the electrolytic solution pressure P, to the osmotic pressure p. The osmotic work done will be p Integrating this expression, we have p Osmotic work = RT log The corresponding electrical energy gained is nFir, where TT is the difference of potential between the metal and the solution, F 1 faraday = 96,540 coulombs, and n is the valence of the metal. Since the osmotic work done is equivalent to the electrical energy gained, we may equate these two expressions, as follows: nFir = RT\og e -> or RT ELECTROMOTIVE FORCE 455 96,540 X n X 0.4343 X 0.2394 Tlog p ' Expressing both sides of equation (1) in electrical units, and trans- forming to Briggsian logarithms, we obtain p p or 0.0002 P * = -^-T\og-. (2) For univalent ions at 17, we have TT = 0.0575 log-- (3) In a galvanic cell composed of two metals, each immersed in a solution of one of its salts, a difference of potential may be estab- lished (1) at the junction of the two metals, (2) at the junction of the two solutions, and (3) at the points of contact of the metals with their respective solutions. If the temperature remains con- stant, (1) is negligible, and in general, (2) is exceedingly small; therefore, the electromotive force of the cell may be considered as due to the differences of potential arising at the two electrodes. Assuming the temperature to be 17, the electromotive force of the cell will be ,, 0.0575, Pi 0.0575, P 2 E = TTi - 7T 2 = - log ---- log n & pi n & pz The Measurement of Electromotive Force. The value of the electromotive force of a cell may vary with the conditions of meas- urement. Since, according to Ohm's law, E = C (R + r), where R is the resistance of the external circuit and r is the internal resistance of the cell, it follows that the fall of potential CR, in the external circuit, will only be equal to E when r is negligible in comparison with R. Furthermore, when the circuit is closed, the electrodes of the cell frequently become polarized, owing to the deposition of the products of electrolysis, and an opposing electromotive force is set up. To avoid these difficulties, the electromotive force is usually measured on open circuit by the Poggendorff compensation method. In this method the electromotive force to be measured is just 456 THEORETICAL CHEMISTRY balanced by an equal and opposite electromotive force, so that no current passes. The arrangement of the apparatus for such measurements is shown in Fig. 97. If the two ends of the wire AB Fig. 97. of a Wheatstone bridge are connected to a lead accumulator C, there will be a uniform fall of potential along its length. The amount of fall along any portion AD will be proportional to the AD length AD, and equal to the fraction - of the total fall of poten- tial along the entire length of the wire. Now let one terminal of a cell whose electromotive force is less than that of C be con- nected to Aj and the other terminal be connected through a galvanometer G, with a sliding contact D, the two cells E and C working in opposition. A current will flow through the cir- cuit AEGD, and will be indicated by the galvanometer at all positions, except that at which the fall of potential along the wire from A to D is equal to the electromotive force of E. Hence we have e.m.f. of C : e.m.f. of E :: AB : AD, from which the value of the electromotive force of the cell E, can be calculated. Since the electromotive force of a lead accumu- lator is not quite constant, it is customary, after having deter- mined the point D, to substitute a standard cell for E, and balance ELECTROMOTIVE FORCE 457 this against the accumulator, finding a new point of balance D'. We now have the proportion e.m.f. of C : e.m.f. of standard :: AB : AD'. Combining these two proportions, we obtain e.m.f. of E : e.m.f. of standard : : AD : AD'. Instead of using a galvanometer as a "null " instrument for indi- cating when the point of balance has been reached, a capillary electrometer may be employed. Standard Cells. It is apparent that the accuracy of all measurements of electromotive force is dependent upon the cell employed as a standard. Much time has been devoted to the study of various reversible elements with a view to establishing a standard of electromotive force. As a result, we have the com- plete specifications for two standard cells, either of which may be readily reproduced. (a) The Weston, or Cadmium Standard Cell. The most widely used standard of electromotive force is the so-called Weston cell, made up according to the scheme Hg - Solution Hg 2 SO 4 1| Solution CdS0 4 - Cd. A diagram of the usual form of the Weston cell is given in Fig. 98. Fig. 98. 458 THEORETICAL CHEMISTRY A short platinum wire is sealed through the bottom of each limb of the H-shaped vessel. In one limb is placed a small amount of a 10 to 15 per cent cadmium amalgam, A; B is a layer of small g crystals of CdSO4~H 2 0. In the other limb is placed a small o amount of pure mercury, over which is a layer, D, of a paste composed of solid mercurous sulphate and a saturated solution of cadmium sulphate. The cell is then filled with crystals of cadmium sulphate and a saturated solution of cadmium sulphate. The two limbs of the cell are closed with a thin layer of paraffin E, cork F, and sealing wax G. If carefully prepared, this cell will remain unaltered for years and will have an electromotive force at 20 of 1.0183 volts. In addition to the fact that it can be so easily reproduced, the temperature coefficient of the cell is almost negligible. The electromotive force of a Weston standard cell at any temper- ature t, is given by the formula e.m.f. at t = 1.0183 - 0.000038 (t - 20). (b) The Clark, or Zinc Standard Cell. Until about ten years ago, the Clark cell was considered to be the most trustworthy standard of electromotive force. This cell is made up according to the scheme Hg - Solution Hg 2 S0 4 1| Solution ZnS0 4 - Zn. The construction of the cell is similar to that of the Weston cell. It may be reproduced with great accuracy and with no more trouble than the Weston cell, but its relatively large temperature coefficient renders it less satisfactory. The electromotive force of the Clark standard cell at any temperature t, may be calculated by means of the formula e.m.f. at t = 1.4328 - 0.00119 (t - 15) - 0.00007 (t - 15) 2 . The Capillary Electrometer. When pure mercury is covered with sulphuric acid, its surface tension is diminished. This may be shown by the following experiment: In a small evaporating dish place about 5 cc. of pure mercury, and cover it with a 10 per ELECTROMOTIVE FORCE 459 cent solution of sulphuric acid to which has been added enough potassium dichromate to impart a light yellow color to the so- lution. The globule of mercury will immediately flatten out, indicating that its surface tension has diminished. If now the mer- cury be touched with a piece of iron wire, it will instantly contract until the contact with the wire is broken; it will then flatten out, until it again comes in contact with the wire, when the globule of mercury will once more contract. In this way a regular pul- sation of the mercury may be obtained. This interesting phenom- enon was observed early in the nineteenth century by Henry, but was first satisfactorily explained by Lippmann * in 1873. Lipp- mann showed that when the globule of mercury is negatively electrified, its surface tension increases and the drop shrinks. If sufficient negative electricity is imparted to the mercury it is possible to restore the globule to its original form. On ap- plying more negative electricity, the globule of mercury again expands. When the iron wire touches the globule it charges it negatively, because when the iron dis- solves, it furnishes positively charged ions to the solution and thus acquires a negative charge which it imparts to the mercury. At the same tune, the chromic acid in the so- lution undergoes reduction to chromium sulphate. Lippmann concluded from his experiments that the difference of poten- tial arises at the surface of contact between the mercury and the solution of the electro- lyte, and that the surface tension of the mercury is a function of the difference of J I potential. Making use of this principle he f A constructed the capillary electrometer, a con- VHvA venient form of which is shown in Fig. 99. The bulb A, through the bottom of which is sealed a platinum wire, contains pure mercury and dilute sulphuric acid (1 : 6). Pure mercury is poured into the other limb of the electrometer * Pogg. Ann., 149, 546 (1873). Fig. 99. 460 THEORETICAL CHEMISTRY until it stands at B in that tube, and at C in the capillary tube. Owing to the capillary depression of the mercury, C lies below B. Electrical connection with the mercury at B is established by means of a platinum wire. The position of the mercury in the capillary is determined by its surface tension; if the surface tension is increased, the mercury will descend; if it is diminished, the mercury will ascend. If a, negative charge is communicated to the mercury at B, the surface tension will be increased and the meniscus will descend ; if a posi- tive charge is imparted to the mercury at B, the surf ace tension will be diminished and the meniscus will ascend. The amplitude of the movement of the meniscus is an inverse function of the diameter of the capillary tube. If the meniscus be observed through a microscope provided with an eye-piece microm- eter, Fig. 100, it is possible to detect very slight movements, and to meas- ure differences of potential less than 0.0001 volt. The capillary electrometer is an ex- cellent "null" instrument. In using the electrometer no large electromo- tive force should be applied, since the meniscus surface becomes polar- ized very easily. If this should occur, a new surface may be secured by blowing gently at B and forcing a drop of mercury out of the capillary into the bulb. Lippmann studied the effect of steadily increasing potentials on the movement of the meniscus. Plotting movements of the meniscus on the axis of ordinates, and potentials on the axis of abscissae, he found that there is a maximum in the curve corresponding to about 0.8 volt. This is the electromotive force which must be applied in order to counterbalance the differ- ence of potential produced by the contact of dilute sulphuric acid Fig. 100. ELECTROMOTIVE FORCE 461 Sulphuric Acid- Mer.curj/5. Fig. 101. with the surface of the mercury. At the meniscus surface an electrical double layer is formed. The mercury is positively charged, and above it there must be a layer of negatively charged ions, as shown in Fig. 101. Just how this double layer is formed is not known with cer- tainty, but it has been suggested that the slight film of oxide which is prob- ably present on the surface of the pur- est mercury, dissolves in the sulphuric acid forming a solution of mercurous sulphate, and from this solution the positively charged Hg" ions deposit on the mercury, giving it a positive charge. Whether this explanation is correct or not, the fact remains that the mercury is positive against the solution. Normal Electrodes. The method commonly employed for the measurement of the difference of potential between a metal and a solution, is based upon the use of an electrode in which the difference of potential between the electrode and a certain solution of one of its salts is known. Such an electrode is called a normal electrode. If a cell is made up by combining the normal electrode with the electrode whose potential is to be determined, it is possible, from measurements of the resulting electromotive force, to cal- culate the value of the unknown difference of potential. The most convenient electrode to prepare is the normal calomel electrode, a satisfactory form of which is shown in Fig. 102. The bottom of the electrode vessel is covered with a layer of pure mercury, upon which is poured a paste, prepared by rubbing together in a mortar mercury and calomel, moistened with a molar solution of potassium chloride. The vessel is ihen filled with a molar solution of the same salt which has been saturated with calomel by prolonged shaking with the latter. Connection with the mercury is estab- lished by means of a platinum wire sealed into a glass tube A , the latter being passed through the rubber stopper which closes the vessel. In using the calomel electrode, the bent side tube C is 462 THEORETICAL CHEMISTRY filled with molar potassium chloride by applying suction at the side tube B, which is then closed by means of a pinch-cock. The difference of potential, at any temperature t, of the calo- mel electrode prepared as described, and represented by the scheme Hg - Solution HgCl in molar KC1, is TT = -0.560 1 1 + 0.0006 (t - 18) j volt. The negative sign indicates that the solution is negative to the electrode. In order to measure the potential of another electrode Fig. 102. Fig. 103. by means of the calomel electrode, the arrangement shown in Fig. 103 is commonly used. Here A represents the " half -element" of which the potential is to be determined, B represents the side tube of the calomel electrode, and C represents an intermediate, connecting vessel containing a molar solution of potassium chloride. In cases where potassium chloride forms a precipitate with the electrolyte in A, the solution in C may be replaced by a molar ELECTROMOTIVE FORCE 463 solution of potassium nitrate without altering the value of the electromotive force of the cell. The original measurement of the potential of the calomel electrode was made by forming a cell with this and another electrode whose potential against its solution is zero. Such an electrode is known as a null electrode. Thus, if a copper electrode is immersed in a solution of copper sulphate, the Cu" ions will leave the solution and charge the electrode positively. If now a solution of potassium cyanide is added, the nearly undissociated salt, K 2 Cu 2 (CN) 4 , will be formed, and by adding a sufficient amount of the solution, the concentration of the Cu" ions may be reduced until the metal and the solution have the same potential. The addition of more potassium cyanide will still further dimmish the osmotic pressure of the Cu" ions, and the electrode will acquire a negative charge. Similarly, mer- cury in a solution of a double cyanide may be used as a null elec- trode. Another form of null electrode is the so-called dropping elec- trode of Helmholtz.* The principle involved "in this electrode has already been discussed in connection with Palmaer's experi- ment (p. 453). An extremely fine stream of mercury is allowed to flow from a funnel having a minute capillary orifice: the stem of the funnel dips below the surface of a molar solution of potas- sium chloride containing mercurous ions. As each little globule enters the solution, it acquires a positive charge and attracts the negatively charged ions of the electrolyte, dragging them down with itself. When the globule reaches the layer of mercury at the ^bottom of the vessel, its surface and capacity are diminished, and as many Hg* ions leave the layer of mercury and enter the solution as there were negatively charged ions carried down by the globule. This process continues until the osmotic pressure of the remaining ions is equal to the solution pressure of the metal: the mercury, both in the stream and at the bottom of the vessel, has the same potential as the solution. If now the difference of potential between the mercury in the funnel and the mercury in the vessel be measured, we shall obtain the potential of mercury against a molar solution of potassium chloride. The dropping * Ann. der Phys., 44, 42 (1890). 464 THEORETICAL CHEMISTRY electrode was for a long time regarded as an ideal standard of potential, but Nernst has quite recently pointed out a number of serious objections to it. Until a wholly satisfactory standard of potential is obtained, he proposes that the potential of the hydrogen electrode be adopted as the standard. This consists of a strip of platinized platinum, half in pure hydrogen gas and half in a solution of sulphuric acid of such concentration that it shall contain 1 gram of hydrogen ions per liter. The use of the hydrogen electrode as a standard is, of course, purely arbi- trary, but there are many advantages in referring differences of potential to this standard. Owing to certain experimental diffi- culties attending the use of this electrode, it is customary to make the actual measurements with the calomel electrode, and then refer them to the hydrogen standard, taking the potential of the calomel electrode to be 0.283 volt when referred to the hydrogen electrode as zero. The negative sign indicates that the solution is negative against the electrode. Measurement of the Difference of Potential between a Metal and a Solution. The difference of potential between a metal and a solution of one of its salts is easily determined by means of the calomel electrode. For example, in order to determine the potential of zinc against a molar solution of zinc sulphate, the electromotive force E, of the combination Zn - m ZnSO 4 1| m KC1, HgCl - Hg, (cal. electrode) is measured and found to be 1.08 volts, the mercury being the positive terminal of the cell. Applying the Nernst equation, ,-, RT , .Pi RT , PZ ._ = tog.-' ..log.-, in which PI and p\ denote the solution pressure and the osmotic pressure of the zinc ions, and P 2 and p 2 denote the solution pressure and the osmotic pressure of the mercury ions, we have prrr p 1.08 =log.-(- 0.56), ELECTROMOTIVE FORCE 465 or D/77 T> nUog e = 1.08 - 0.56 = 0.52 volt. & r pi That is, the zinc electrode is negative against a molar solution of zinc sulphate, the difference of potential being 0.52 volt. As an example of a cell in which the mercury of the calomel electrode is the negative terminal of the cell, we may take the following combination : Cu - m CuS0 4 1| m KC1, HgCl - Hg. The electromotive force of this cell is 0.025 volt. Since the current flows from the copper to the mercury, we have E = 0.025 = -0.56 - or ^log,- 1 = -0.025 - 0.56 = -0.585 volt. Z r pi That is, the copper electrode is positive against a molar solution of copper sulphate. From the above results it is possible to cal- culate the electromotive force of the combination Zn - m ZnSO 4 [| m CuS0 4 - Cu. Since, according to Nernst's equation RT, Pi RT^P* where PI and p L refer to the zinc, and P 2 and pz refer to the copper, we have E = 0.52 - (- 0.585) = 1.105 volts. Concentration Elements. We now proceed to consider cells in which the electromotive force depends primarily on differences in concentration, the so-called "concentration elements." Concentration elements may be conveniently divided into two classes: (a) elements in which the electrodes are of different concen- trations, and (b) 'elements in which the solutions are of different concentrations. 466 THEORETICAL CHEMISTRY (a) Elements in which the Electrodes are of Different Concentra- tions. (Amalgams and Alloys.) If in the equation RT P! RT, P 2 PI = Pz, as is the case when the ionic concentrations of the two solutions are identical, then we have where PI and PZ are the respective solution pressures of the metal dissolved in the electrodes. If the amalgams are dilute, the osmotic pressure of the dissolved metal will be proportional to the solution pressure of the electrode, and since osmotic pressure is proportional to concentration, we may replace PI and P 2 in the above formula by the proportional terms, Ci and (%, the respec- tive concentrations of the metal in the two electrodes. Hence, we have E = lo - The accuracy of this equation has been fully established by the experiments of Meyer,* and Richards and Forbes. f Meyer's results for zinc amalgams in solutions of zinc sulphate are given in the accompanying table. A T ' degrees. Cl C 2 E (obs.). E (calc.). 284.6 0.003366 0.00011305 0.0419 0.0416 291.0 0.003366 0.00011305 0.0433 0.0425 285.4 0.002280 0.0000608 0.0474 0.0445 333.0 0.002280 0.0000608 0.0520 0.0519 The agreement between the observed and calculated values of E is all that can be desired. That the above formula holds for zinc amalgams may be considered as a proof of the fact that the zinc dissolves in the mercury as monatomic molecules. Thus, suppose the zinc to be present in the mercury in the form of dia- * Zeit. phys. Chem., 7, 477 (1891). t Publication of the Carnegie Institution, No. 56. ELECTROMOTIVE FORCE 467 tomic molecules; then while the electrical energy would be equal to 2 FE, the osmotic work required to develop this energy would be ^= RT log e , hence we should have 2 c or the calculated value of the electromotive force would be just one-half of the observed value. The mercury in the amalgam has been shown to exert no effect upon the electromotive force of the cell so long as the dissolved metal has the greater potential. (b) Elements in which the Solutions are of Different Concentra- tions. In this type of cell we have two electrodes of the same metal immersed in solutions of different ionic concentrations of the metal. Hence, we may put PI = P 2 in the equation RT, P 1 RT, P 2 E = F, loge p, log e , nF 3 pi nF & i p 2 which then takes the form Since osmotic pressure is proportional to concentration, pi and pz may be replaced by the proportional terms Ci and 02, and the foregoing equation becomes ,, RT, d E = w l0 ^' or RT where mi and m% are the molar concentrations of the two solutions and cti and az are the corresponding degrees of ionization. As an example of a concentration element of this class we may take the following : Ag - 0.01 m AgNO 3 1| 0.1 m AgN0 3 - Ag. 468 THEORETICAL CHEMISTRY The degrees of ionization of the two solutions at 18 are as follows: -for 0.01 m AgN0 3 , a = 0.93, and for 0.1 m AgNO 2 , a = 0.81. Substituting in the equation E = 0.058 log we have E = 0.058 log* 0.0545 volt. The value of E found by direct experiment is 0.055 volt. In the example just given, the electrodes are reversible with respect to the positive ion of the electrolyte. Such electrodes are known as electrodes of the first type. It is also possible to construct cells with electrodes which are reversible with respect to the nega- tive ion of the electrolyte. These are termed electrodes of the second type. The calomel electrode is an example of an electrode of this latter type. If positive electricity passes from the metal to the solution, the mercury combines with the Cl' ions forming mercu- rous chloride, and if positive electricity passes in the reverse direction, chlorine dissolves and mercurous chloride is formed. In other words the electrode behaves like a chlorine electrode, giving up or absorbing the element according to the direction of the current. A typical combination involving an electrode of the second type is the following : Ag - 0.1 m AgNO 3 - KNO 3 - 0.1 m KC1, AgCl - Ag. This particular combination was studied by Goodwin * with a view to determining the solubility of silver chloride. If we assume a saturated solution of silver chloride to be completely ionized, then the solubility product will be CAg- X CCV = S. Since the concentrations of the two ions, Ag" and Cl', are equal, it follows that Vs will be equal to the solubility of the silver chloride. The electromotive force of a concentration cell at 25 is given by the equation ,-, 0.0595 . log n * Zeit. phys. Chem., 13, 577 (1894). ELECTROMOTIVE FORCE 469 or En 00595 The value of E at 25 for the above cell was found to be 0.450 volt. The degrees of ionization of the two electrolytes are as follows : for 0.1 molar AgNO 3 , a = 0.82, and 0.1 molar KC1, a = 0.85. Substituting in the preceding equation, we obtain 0.1 X 0.82 0.450 ~^ = 0^595' therefore, cs = 2.24 x 10~ 9 . Or, 1.94 X 10~ 9 is the concentration of the Ag* ion in mols per liter in a 0.1 molar potassium chloride solution of silver chloride. Hence the solubility product s will be s = 2.24 X 10- 9 X 0.085 = 1.91 X lO" 10 , and Vs = 1.38 X 10~ 5 ; that is, the solubility of silver chloride in a saturated aqueous solution is 1.28 X 10~ 5 mol per liter at 25. The Difference of Potential at the Junction of the Solutions of Two Electrolytes. Thus far we have not taken into consider- ation the potential differences which may be established at the junction of two solutions. Nernst * has shown that in many cases it is possible to calculate these differences of potential by means of his osmotic theory of the origin of electromotive force, and the values obtained are in close agreement with the results of experi- ment. Let us imagine that two solutions of hydrochloric acid of different concentrations are brought together so as to avoid mix- ing, the acid in each solution being highly ionized. The hydro- gen and chlorine ions will diffuse independently, and since the former move with the greater velocity, the more dilute solution will soon contain an excess of H' ions and the more concentrated solution an excess of Cl' ions. The more dilute solution will be- come positively charged owing to the presence of an excess of H" * Zeit. phys. Chem., 4, 129 (1889). 470 THEORETICAL CHEMISTRY ions, while the more concentrated solution will acquire a negative charge due to the presence of an excess of CI' ions. The accumu- lation of positive electricity, however, will retard the velocity of the H* ions and accelerate the velocity of the CY ions, so that ultimately the two ions will move with the same velocity. The difference of potential produced in this way will cease to exist when the two solutions have acquired the same concentration. In general, it may be stated that the difference of potential set up at the junction of the two solutions is caused by the differ- ence in the rates of migration of the two ions, the more dilute solution acquiring a charge corresponding to that of the faster moving ion. Electromotive Force of Concentration Cells with Transference. Let u and v be the migration velocities of the cation and the anion respectively, and let p\ be the osmotic pressure of the ions in the concentrated solution and pz the osmotic pressure of the ions in the dilute solution. Now let one faraday of electricity pass through the two solutions, the current entering on the concen- trated side; then - - gram-equivalents of positive ions will migrate from the concentrated to the dilute solution, while u + v gram-equivalents of negative ions will migrate from the dilute to the concentrated solution. The maximum work done by the process will be RT log, 2-' - -L_ RT log.** = RT log, & & 6 corres have , - - -_ . = u + v & p 2 u + v & p 2 u + v The corresponding electrical energy developed is irF, hence we u-v RT or, since osmotic pressure is proportional to concentration, we may substitute Ci and c^ for pi and pz in the preceding equation, and obtain the following expression for the electromotive force at the junction of the two solutions: - u-v RT ELECTROMOTIVE FORCE 471 As an example of the use of the above formula, we may take the calculation of the electromotive force of the following combina- tion : Ag - 0.1 m AgNO s - 0.01 m AgN0 3 - Ag. (a) (b) (c) Taking the junctions (a), (b), and (c) in order, we obtain E = R T l C_j. u - v RT * c i RT, C 2v RT u + v * F 2 v But = n a , the transport number of the anion, and therefore we may write the preceding equation in the form (1) V 01 Assuming the temperature to be 17 and passing to Briggsian logarithms, we have E = 0.116 n a log-- The transport number for the anion of silver nitrate is 0.522, while ci = mien = 0.1 X 0.82 = 0.082, and ca = m** = 0.01 X 0.94 = 0.0094. Hence, E = -2 X 0.522 X 0.058 X ' or E = -0.057 volt. The value found by direct experiment is 0.055 volt. , It is to be noted that if the electromotive force at the junction of the two solutions is negative, equation (1) takes the form where n c is the transport number of the cation. Electromotive Force of Concentration Cells without Trans- ference. A less familiar type of concentration cell is that which 472 THEORETICAL CHEMISTRY does not involve transference. 'The following combination may be taken as an example of such a cell : Ag|AgCl,KCl|K(Hg), | K(Hg), KC1, AgCl | Ag. Ci C 2 It will be observed that this cell is in reality made up of two independent cells and involves no liquid junction. While the elimination of diffusion appreciably simplifies the theoretical treatment of cells of this type, practically it is difficult to find electrodes which are reversible toward both ions of the electrolyte. The passage of one faraday of electricity through the cell results in the formation on the dilute side of one equivalent of potassium chloride from the silver chloride and the amalgam, while on the concentrated side a corresponding amount of potassium chloride is decomposed. If we assume the solutions to be so dilute as to be completely ionized, one equivalent of potassium chloride will function as two mols of gas, and the electromotive force of the cell will be Formulas for the Difference of Potential at Liquid Junctions. As has been pointed out, the total electromotive force of a con- centration cell with transference is made up of the algebraic sum of the potentials at the two electrodes and the potential at the junction of the two solutions. Since the value of the potential at the electrodes alone is often desired, numerous formulas have been derived for the calculation of the potential at the liquid junction. In addition to the formula of Nernst, already mentioned, for- mulas have been proposed by Planck,* Henderson, f Gumming, j: Lewis and Sargent, and Maclnnes. || Of these different formulas that of Maclnnes possesses distinct advantages. * Wied. Ann., 40, 561 (1890). f Zeit. phys. Chem., 59, 118 (1906); 63, 325 (1908). j Trans. Faraday Soc., 8, 86 (1912); 9, 174 (1913). Jour. Am. Chem. Soc., 31, 363 (1909). || Ibid., 37, 2301 (1915). ELECTROMOTIVE FORCE 473 If one faraday of electricity be passed through the cell Ag|AgCl, KC1|KC1, AgCl|Ag, Cl Cz one equivalent of chloride ions will enter the dilute solution while a corresponding amount will be electrolyzed out of the more con- centrated solution. The current will be carried across the liquid junction by the migration of n c equivalents of potassium ions in the direction of the current and by the migration of (1 n c ) equivalents of chloride ions in the opposite direction. The total effect of the passage of one faraday of electricity will be the trans- ference of one equivalent of salt from the more concentrated to the more dilute solution. The osmotic work at the junction of the two solutions will correspond to the algebraic sum of the number of ion equivalents which have migrated from the concentrated to the dilute solution, or n c (1 n c ) = 2 n c 1. ( In order to obtain the electrical energy necessary to perform this amount of osmotic work, let us consider the following cell involving no transference : Ag | AgCl, KC1 1 K (Hg) z || K (Hg) x | KC1, AgCl | Ag. Ci C 2 The passage of one faraday of electricity through this cell in- volves, as we have seen, the formation of one equivalent each of potassium and chloride ions in the dilute solution, and the re- moval of one equivalent of each ion from the more concentrated solution. The electrical energy accompanying the transfer of two ion equivalents from one solution to the other will be equal to the product of the electromotive force of the cell and one faraday, or EF. The electromotive force IT, at the liquid junction, can now be obtained by the simple proportion EF : irF = 2 : 2 n c - 1, . = g ^- 1) -. CD Since the ratio of E () the electromotive force of the cell with trans- ference, to E, the electromotive force of the cell without trans- 474 THEORETICAL CHEMISTRY ference, is equal to n c , it follows that equation (1) may be written in the form (2) It is to be observed that equations (1) and (2) involve no assump- tions concerning the concentrations of the ions in the solutions, one of the characteristics which distinguishes these formulas from any of the others hitherto proposed. Lewis and Sargent * have proposed a modification of a formula derived by Planck for the difference of potential at the junction of two equally concentrated electrolytes having a common ion. If AI and A 2 are the equivalent conductances of the two electrolytes, the value of the liquid junction potential TT is given by the formula The junction between two different concentrations of two elec- trolytes with a common ion may be replaced by two junctions, one of which connects two different concentrations of the same electrolyte, and the other connects two different electrolytes of the same concentration. For example, the junction 0.1NNaCl|0.05NHCl may be replaced by 0.1 N NaCl | 0.05 N NaCl | 0.05 N HC1 (A) (B) in which the potential of the junction (A) may be calculated by equations (1) or (2) and the potential of junction (B) by equa- tion (3). The electromotive force at the junction of solutions of differently concentrated uni-univalent ionogens which do not contain a com- mon ion may, in like manner, be calculated by employing three junctions. Thus, the junction 0.1 N KN0 3 1 0.05 N NaCl * Loc. cit. ELECTROMOTIVE FORCE 475 may be replaced by 0.05 N NaCl | 0.1 N NaCl | 0.1 N KC1 1 0.1 N KNO 3 . (A) (B) (BO The potential at (A) may be calculated by means of equations (1) or (2) and the potentials at (B) and (B') by means of equation (3). By the use of equations (1), (2), and (3) in the manner indi- cated, it is possible to calculate the junction-potential between the solutions of any two uni-univalent ionogens to within a few tenths of a millivolt. Normal Electrode Potential. The difference of potential at a reversible electrode, when the concentration of the ions of the electrolyte is normal, is known as the normal electrode potential. According to Nernst, the expression for the difference of potential at a single electrode is RT RT where P is the solution pressure of the electrode, p is the osmotic pressure of the ions of the electrolyte, and where C and c are con- centrations proportional to P and p respectively. If the concen- tration of the ions is unity, c = 1, and the expression for the normal electrode potential E becomes (2) On subtracting (1) from (2), we obtain (3) where c is the concentration of the ions in a solution whose elec- trode potential is TT. In the actual determination of normal elec- trode potentials, the value of c in equation (3) is usually very much less than unity. The following table gives the most important electrode poten- tials, the data in the second column being referred to the normal 476 THEORETICAL CHEMISTRY calomel electrode as zero, and the data of the third column being referred to the normal hydrogen electrode as zero. NORMAL ELECTRODE POTENTIALS. Electrode. N. E. Potential, N. Calomel Electrode . =- \. H. Potential, N. Hydrogen Electrode = 0. Cations. Copper +0.046 +0.329 Hydrogen 283 +0 000 Lead 412 -0 129 Mercury . . +0 467 +0 750 Platinum +0.580 oa. +0 863 ca. Potassium -3.48 -3 20 Silver Sodium +0.515 +2.998 +0.718 +3.281 Tin -0.475 -0.192 Zinc -1.053 -0.770 Anions. Bromine +0.797 + 1.08 Chlorine + 1.067 +1 35 Iodine +0.345 +0 628 Oxygen +0.110 +0 393 It should be observed that the value of the normal electrode potential cannot be calculated directly by means of equation (2) owing to the uncertainty attaching to the quantity C. In fact the best interpretation that we have of the physical significance of C is, that it is a quantity whose logarithm is proportional to the normal electrode potential. Electrometric Determination of Valence. The equation of Nernst for the electromotive force of a concentration cell may be employed to determine the valence of the ions. For example, the chemical behavior of mercurous salts is such as to justify the use of single or double formulas involving either Hg* or Hg2*. To determine which of these two formulas is correct, Ogg * set up the following cell : Hg | 0.5 N Mercurous Nitrate || 0.05 N Mercurous Nitrate | Hg 0.1 N HN0 3 0.1 N HN0 3 * Zeit. phys. Chem., 27, 285 (1898). ELECTROMOTIVE FORCE 477 and found its electromotive force at 17 to be 0.029 volt. If we neglect the difference of potential at the junction of the two solu- tions, the electromotive force of this cell may be calculated by the familiar formula Assuming that c^/Ci is equal to 10, and passing to Briggsian loga- rithms, we have 0.029 = 0.058/n, or n = 2. Hence the valence of the mercurous ion is 2 and it must, in con- sequence, be represented by Hg2*. It follows that the correct for- mula of mercurous nitrate is Hg 2 (NO 3 ) 2 . Electronic trie Determination of Transport Numbers. The formula for the electromotive force of a concentration cell with transference has been shown to be > If the electromotive force of such a combination is measured and the values of Ci and & are accurately known, obviously the value of the transport number of the cation n c , can be calculated. If, after having measured E t) a similar cell without transference be set up and its electromotive force measured, it is possible to obtain an expression for n c which does not involve either Ci or c%. As has been shown, the electromotive force of a cell without transference, E, is given by the formula Dividing equation (1) by equation (2), we obtain which expression gives the transport number in terms of the two measured electromotive forces. This method for the determina- tion of transport numbers was first suggested by Helmholtz.* The transport number of the lithium ion has recently been de- * Ges. Abhl. I, 840: II, 979. 478 THEORETICAL CHEMISTRY termined in this manner by Pearce and Mortimer * using cells made up according to the following schemes : Ag | AgCl, LiCl || LiCl, AgCl | Ag (with transference), Ci Cz Ag | AgCl, LiCl | Li (Hg) x 1 1 Li (Hg) x | LiCl, AgCl | Ag / without \ c i ^2 Vtransference/ A comparison between their results and the values given by Kohlrausch and Holborn f is afforded by the accompanying table. Concentration Ratio ...... 1.0-0.1 0.5-0.05 0.1-0.01 0.05-0.005 Mean n c (K. and H.) ..... 0.285 0.300 0.340 0.360 Mean n c (P. and M.) ..... 0.279 0.322 0.343 0.365 It will be observed that the agreement between the two series of results is satisfactory. Concentration and Activity. If in the equation for the elec- tromotive force of a concentration cell without transference, viz., 2RT we substitute the observed value of E and solve the equation for the ratio, GZ/CI, we find that the value thus obtained does not agree with the value derived from conductance measurements. In other words, if mi and mz are the actual molar concentrations of the two solutions forming the cell, and c*i and <* 2 are the corresponding de- grees of ionization, we find that the ratio mzaz/miai is not equal to the ratio GZ/CI as calculated from the electromotive force of the cell. This is shown by the following table which gives the ratios of the ionic concentrations of solutions of potassium chloride corresponding to various ten-to-one salt concentrations. RATIOS OF ION CONCENTRATIONS IN SOLUTIONS OF POTASSIUM CHLORIDE. Concentration Ratio. C 2 /C, (Conductance). C 2 /Ci (Electromotive Force). 0.5 :0.05 0.1 :0.01 0.05 : 0.005 0.01 : 0.001 8.85 9.16 9.30 9.62 8.09 8.33 8.64 9.04 * Jour. Am. Chem. Soc., 40, 518 (1918). t Leitvermogen der Elektrolyte, p. 201. ELECTROMOTIVE FORCE 479 If the values of c^/Ci in the table are plotted as ordinates against the higher concentration of each pair of solutions as abscissae, we obtain the curves shown in Fig. 104. It will be seen that notwith- standing the fact that the ratios differ widely when calculated by the two methods, both sets of figures approach the limiting value, 10, at infinite dilution. From this it is evident that the equation 10.0 0.4 0.5 0.2 ' 0.3 Concentration Fig. 104. of Nernst is only applicable to cells involving solutions which are practically infinitely dilute. The figures in the last column of the table are known as activity ratios, and the individual values of c\ and GZ are termed activi- ties. If it be assumed that in 0.001 KC1 the activity of the ions is identical with their concentration, as obtained by multiplying the total concentration of the salt by the ionization as determined from the conductance ratio, then the activities in 0.01 N KC1 and 0.1 N KC1 can be calculated. The values so obtained are given in the following table together with the corresponding values of the ionization calculated, (1) from the conductance ratio, and (2) from the so-called " thermodynamically effective " ionization. this latter being the quotient obtained by dividing the activity by the corre- sponding total concentration. That the degree of ionization determined from conductance measurements does not agree with the value obtained from meas- urements of electromotive force has been known for some time. 480 THEORETICAL CHEMISTRY ACTIVITY AND IONIZATION OF POTASSIUM CHLORIDE SOLUTIONS. Salt Concentration. Activity. A/A O . " Thermodynamically Effective " lonization. 0.001 N 0.01 N 0.1 N 0.000979 0.00885 0.0738 97.9 94.1 86.1 (97.9) 88.5 73.8 It is thought that the decrease in activity of the ions with increas- ing concentration is due to a change in the nature of the solvent brought about by the electric charges on the ions. Electrometric Determination of Hydrolysis. One of the most satisfactory methods which we possess for the determination of the degree of hydrolysis of salts depends upon the measurement of the electromotive force of cells made up as follows : Pt H2 1 salt solution | sat. NH 4 NO 3 * || N KC1, H&Cl, | Hg. From the measured electromotive force of the cell, the concentra- tion of the hydrogen ion in the solution can be determined, and from this the degree of hydrolytic dissociation of the salt can be calculated. The method is especially valuable in cases where the concentration of the hydrogen ion is very small. Unfortunately the application of the method is restricted to the salts of metals less noble than hydrogen. In other words, it cannot be used to deter- mine the hydrolysis of the salts of metals which would be precipi- tated upon the platinum electrode. The applicability . of the method is further limited by the fact that certain ions, such as Fe"", N(V, and C1CV, are either wholly or partially reduced by the hydrogen of the hydrogen electrode. The method has been successfully applied by Denham f to the determination of the hydrolysis of various salts, among which may be mentioned, aluminium chloride, aluminium sulphate, nickel chloride, nickel sulphate, cobalt sulphate, and ammonium chloride. As an illustration of the method we may take the case of am- * A saturated solution of ammonium nitrate is interposed between the solutions to eliminate any difference of potential at the junction of the solu- tions. t Jour. Chem. Soc., 93, 41 (1908); Zeit. anorg. Chem., 57, 361 (1908). ELECTROMOTIVE FORCE 481 monium chloride which dissociates hydrolytically according to the equation NH 4 C1 + H.OH <= NH 4 OH + HC1. Here we have a weak base and a very strong acid as the products of hydrolytic dissociation. While the ammonium hydroxide may be regarded as practically non-ionized, the hydrochloric acid is to be considered as having undergone complete ionization. If one mol of ammonium chloride is dissolved in v liters of water and the degree of hydrolytic dissociation is x, then the concentrations of the products of the reaction, ammonium hydroxide and hydro- chloric acid, will be x/v. Since the acid is completely ionized, x/v will also represent the concentrations of the hydrogen and chloride ions. Assuming that the active mass of the water remains con- stant, we have, according to the law of mass action, where KH is the hydrolytic constant. Since v is known, it only re- mains to determine x/v, or the concentration of the hydrogen ions, in order to be able to calculate K h . The potential at the hydrogen electrode is given by the familiar formula RT, CH- or denoting the normal electrode potential R T log e C by ir Q , we may write . RT, x According to Denham, the electromotive force of the cell, Pt H2 1 N/32 NH 4 C1 1 sat. NH 4 NO 3 1| N KC1, Hg 2 Cl 2 1 Hg, at 25 is 0.6056 volt, the cu rent flowing outside the cell in the direction of the arrow. The potential of the normal calomel electrode being + 0.56 volt, it follows that the potential of the hydrogen electrode, TT = 0.56 - 0.6056 = - 0.0456 volt. There- fore, PT x - 0.0456 = *o +^p log, - 482 THEORETICAL CHEMISTRY The absolute value of the potential of the normal hydrogen elec- trode 7r , referred to the normal calomel electrode as + 0.56 volt and not as zero, is + 0.277 volt. Substituting this value in the foregoing equation and passing to Briggsian logarithms, we have 0.059 log- = 0.0456 - 0.277 = - 0.2314. Solving this equation we find x/v = 0.340 X 10 6 gram-ions of hydrogen per liter. Since the value of x/v would be ^ if the salt were completely hydrolyzed, the percentage of hydrolysis under the conditions of the experiment, i.e., when v = 32, is 0.340 x W X 100 The value of the hydrolytic constant is given by the equation . (0.340 X Kh - 32(1- 0.340 X10-)- The degree of hydrolytic dissociation of ammonium chloride has been determined by Noyes from measurements of electrical con- ductance. The value of x for 0.01 N NH 4 C1 was found by him to be 0.02 at 18, while Denham found x = 0.018 at 25 by extrapola- tion of his electrometric data. Oxidation and Reduction Elements. When a dissolved sub- stance passes from a lower to a higher state of oxidation, the change in the positive ion may be considered as due to an increase in the number of electrical charges on the ion; thus, when a ferrous salt is oxidized to the ferric state, the change may be represented by the equation Fe"-K(+)-Fe~ Similarly, the reduction of a ferric salt to the corresponding ferrous salt may be represented by the reverse equation, or The formation of zinc ions from metallic zinc may be considered as an oxidation, and may be represented by the equation Zn + 2 (+) -* Zn~. ELECTROMOTIVE FORCE 483 The formation of negative ions from a non-metallic element may be considered as a reduction, as for example, the change of potas- sium ferricyanide to potassium ferrocyanide, which may be rep- resented by the equation Fe(CN) 6 '" + (-) -Fe(CN) 6 "". The foregoing considerations lead to the following definition of the terms oxidation and reduction: Oxidation is the process in which a substance takes up positive or parts with negative charges t and reduction is the process in which a substance takes up negative or parts with positive charges. The potential difference between a metal and a solution of one of its salts is a measure of the tendency of the metal to form ions, or in other words is the criterion of its tendency towards oxidation. The tendency of an ion in a lower state of oxidation to pass over into a higher state of oxidation may be determined by means of electromotive force measurements. A typical oxidation cell is the following: Pt - Fe" || Fe*" - Pt. If the platinum electrodes are connected to a lead accumulator, and a current is passed in the direction of the arrow Pt - Fe" || Fe"" - Pt, the ferrous ions on one side of the element will all be oxidized to the ferric state, while the ferric ions on the other side will all be reduced to the ferrous state, thus: Pt - Fe'" || Fe" - Pt. If now the connection with the accumulator is broken and the two platinum electrodes are connected with a wire, the following proc- ess will take place: ferric ions will give up positive charges to one electrode, thereby being reduced to ferrous ions, while the charges given up pass along the wire to the other electrode, there converting some ferrous ions to ferric ions. This process will continue until an equilibrium is established, in which the ratio 484 THEORETICAL CHEMISTRY Fe" : Fe*" will be the same in both solutions. The electromotive force corresponding to this change gives a measure of the tendency of the ions to undergo oxidation, and is directly proportional to the ionic concentrations. The following modification of the Nernst equation .gives the relationship between ionic concentrations and the resulting elec- tromotive force: where E (CO -> C .) is the electromotive force produced by the passage from the lower, or "ous " state of oxidation to the higher, or "ic " state of oxidation; P is the potential when the concentrations of the "ous " and "ic " ions are equal, c and d are the ionic con- centrations of the lower and higher states of oxidation respectively, and n is the difference in valence of the two kinds of ions. A number of oxidation and reduction elements have been studied by Bancroft.* Heat of lonization. If the difference of potential between a metal and the solution of one of its salts is known, together with its temperature coefficient, it is possible to calculate the heat of ionization of the metal by means of the Gibbs-Helmholtz equation, * = nF +T dT' Solving the equation for Q, the heat evolved when one mol of ions is formed at the electrode, we have dir For example, the potential of zinc against a molar solution of zinc chloride is 0.497 volt at 25, and the temperature coefficient of electromotive force is 0.000664 volt per degree. Substituting in the equation, we have Q = [- 0.497 - (273 + 25) X 0.000664] (2 X 96,540 X 0.2394) or Q = 32,120 calories. * Zeit. phys. Chem., 10, 387 (1892). ELECTROMOTIVE FORCE 485 That is, the heat of the reaction Zn + 2 (+) -> ZiT is 32,120 calories per mol of zinc. Gas Cells. It is interesting to note that gases may function as electrodes in much the same way as metals or amalgams. Gas electrodes are usually prepared by partially immersing strips of platinized platinum in a solution of a suitable electrolyte, and bubbling the gas through the solution until a constant difference of potential is estab- lished between it and the electrode. A very satisfactory form of gas electrode is shown in Fig. 105. Reference has already been made to the hydrogen electrode in connection with the measurement of single electrode potentials. This electrode is completely reversible and behaves like a plate of metallic hydro- gen, the reaction at the electrode being represented by the equation The amount of energy developed by the passage of a certain quantity of gas into the ionic state is precisely the quantity nec- essary and sufficient to produce the reverse Fi 8- 105 - action. This being true, the metal of the electrode can exert no influence upon the electromotive force. A hydrogen concentration cell can be formed by connecting two hydrogen electrodes, containing the gas at different pressures, through an intermediate electrolyte. The direction of the current is such that the pressures on the two sides of the cell tend to become equal, molecular hydrogen being ionized on the high pressure side, and ionized hydrogen being dis- charged on the low pressure side. The electromotive force of such a cell can be calculated by means 486 THEORETICAL CHEMISTRY of the Nernst equation. Let us consider a cell composed of two hydrogen electrodes, each at atmospheric pressure, the H* ion concentration in each being ci, then f. til , G! fll where C\ is the molecular concentration of the hydrogen dissolved in the platinum at atmospheric pressure p\. Since the hydrogen is present in the form of diatomic molecules, n = 2. If now the pressure of the gas at one electrode be increased to p z , and the corresponding molecular concentration of the hydro- gen in the electrode be C 2 , then we shall have T-. 1,1 i GI til. i L>2 or snce Ci : C 2 :: p\ : pz, * -??"* Equation (1) applies equally well to cells in which two different gases are employed. If solution pressures be used in the calcu- lation of the electromotive force of a gas cell, equation (1) becomes g e g, (2) where PI and P 2 are the solution pressures of the two gases. Since the values of E obtained by equations (1) and (2) must be equal, we may write RT, Pl RT, Pi and therefore That is, the ratio of the actual gas pressures is equal to the ratio of the squares of the corresponding solution pressures. ELECTROMOTIVE FORCE 487 lonization of Water. An important application of the gas cell is its use in determining the degree of ionization of water. If we measure the electromotive force of the cell Pt H2 - 0.1 m NaOH - 0.1 m HC1 - Ptn 2 , and determine the concentration of the H" ions on one side and the concentration of the OH' ions on the other side, we can calculate the concentration of the H* ions in the sodium hydroxide solution. The reaction which produces the current is represented by the equation NaOH + HC1 = NaCl + H 2 O, or more correctly H- + OH' = H 2 O. At 25 the electromotive force of the above cell is 0.646 volt. At the junction of the two solutions an electromotive force of 0.0468 volt is set up; hence the true electromotive force of the cell is 0.646 + 0.0468 = 0.6928 volt. The degree of ionization of a 0.1 molar solution of hydrochloric acid is i = 0.924, and the degree of ionization of a 0.1 molar solution of sodium hydroxide is a z = 0.847. Introducing these values into the Nernst equation we have 0.6928 = 0.0595 log ' 1 X0 - 924 . C2 Solving this equation for C2, the concentration of the H* ions in the sodium hydroxide solution, we find Cz to be equal to 1.66 X 10~ 13 . Therefore s = CH- X COR' = 1.66 X 10- 13 X 0.1 X 0.847 = 1.406 X lO" 14 , and CH- = COH' = V7 = Vl.406 X 10~ 14 = 1.187 X 10~ 7 . This value is in excellent agreement with the values obtained from measurements of the electrical conductance of water and the speed 488 THEORETICAL CHEMISTRY of hydrolysis of esters. The values of the degree of ionization of water at 25, as determined by these three methods, are as fol- lows : Electrical conductance of pure water 1.05 X 10~ 7 Velocity of hydrolysis of methyl acetate 1.2 X 10~ 7 E.M.F. of hydrogen-oxygen cell 1.18 X 10" 7 When we consider the exceedingly small extent to which water is ionized, the close agreement between these results is most satis- factory. The correctness of these figures can be further checked by taking the values of the degree of ionization of water at two temperatures, and calculating the heat of the reaction H' + OH' = H 2 0, by means of van't HofPs isochore equation. Thus according to Kohlrausch, the degree of ionization of water is 0.35 X 10~ 7 at 0, and 2.48 X 10~ 7 at 50. Introducing these values into the equation RX 2.3026 (log K 2 - and solving for q, we obtain 13,810 calories, a value agreeing well with that found by the direct measurement of the heat of neu- tralization of completely ionized acids and bases, viz., 13,700 calories. Storage Cells or Accumulators. Storage cells or accumulators, as the name implies, are devices for the storage of electrical energy in the form of chemical energy. Any reversible cell may be employed as an accumulator. Thus, the oxygen-hydrogen cell Pto 2 Solution of Sulphuric Acid Ptn 2 may be used as an accumulator, if the gases resulting from the electrolysis of water are collected at the electrodes and then used to produce a current. Practically, the lead accumulator is used almost exclusively. If two lead plates are immersed in a 20 per cent solution of sulphuric acid, a minute amount of lead sulphate will be formed on the surface of each plate. If now a current of electricity is passed through the solution, the lead sulphate on the ELECTROMOTIVE FORCE 489 cathode will undergo reduction to metallic lead, and the lead sul- phate on tho anode will be oxidized to lead peroxide. In this way we form the cell Pb - 20 per cent Sol. H 2 SO 4 - Pb0 2 , the electromotive force of which is about 2 volts. The amounts of lead and lead peroxide produced in this way are so small that the cell can only furnish a very small amount of electrical energy. In order to increase its capacity, the electrodes should be given as large an amount of surface as possible. This may be brought about by the method of Plante, in which the solution is elec- trolyzed first in one direction and then in the other, thus causing the plates to become spongy; or by the method of Faure, in which a lead "grid " is charged with a paste of lead oxide and red lead, and is then introduced into the solution and the current passed until we obtain spongy lead at the cathode and lead peroxide at the anode. If now the charging circuit is broken and the two elec- trodes are connected by a wire, a current will flow from the peroxide plate to the lead plate, lead sulphate being slowly formed at each. In charging the accumulator, the lead sulphate on the negative electrode is reduced to metallic lead, the reaction being represented by the following equation : PbS0 4 + 2 (-) = Pb" + SO/'. At the positive electrode SO/' ions are liberated and they react with the lead sulphate and the water of the electrolyte in the following manner : PbS0 4 + 2 H 2 + SO/' + 2 (+) = Pb0 2 + 4 H* + 2 SO/'. When the cell is discharged SO/' ions are liberated at the lead plate forming lead sulphate, as shown by the equation Pb + SO/' + 2 (+) = PbS0 4 . At the peroxide plate H* ions are discharged and, in the presence of the electrolyte, convert the lead peroxide into lead sulphate, according to the equation PbO 2 + 2 H* + H 2 SO 4 + 2 (-) = PbSO 4 + 2 H 2 O. 490 THEORETICAL CHEMISTRY Combining the foregoing equations, we obtain the following single equation summarizing the chemical changes involved in the pro- duction of the current: - Pb0 2 + Pb + 2 H 2 S0 4 ^ 2 PbS0 4 + 2 H 2 0. The upper arrow represents the reaction on discharging, while the lower arrow represents the reaction on charging. The electromotive force of the storage cell is approximately 2 volts. It is not completely reversible, but under favorable conditions its efficiency is about 90 per cent; that is, 90 per cent of the electrical energy supplied to it in charging can be recov- ered on discharging.* PROBLEMS. 1. Calculate the heat of amalgamation of cadmium at from the following data: the electromotive force of a cell made up of a 1-per cent cadmium amalgam in a solution of cadmium sulphate is 0.06836 volt at 0, and 0.0735 volt at 24.45. Ans. 510 calories per mol of Cd. 2. The electromotive force of the cell Pb - 0.01 m Pb(N0 3 ) 2 - sat. NH 4 N0 3 - m KC1, HgCl - Hg is - 0.469 volt at 25. The lead nitrate is 62 per cent ionized. What is the potential of lead against a solution containing 1 mol of Pb ions per liter, referred to the calomel electrodes as zero? Ans. - 0.405 volt. 3. Calculate the electromotive force of the cell Cu amalgam (a) Solution CuS0 4 Cu amalgam (b), at 20.8, having given that the concentrations of the amalgams (a) and (b) are 0.0004472 and 0.00016645 respectively. Ans. 0.0125 volt. 4. Calculate the electromotive force of the cell Cu - m CuS0 4 - 0.01 m CuS0 4 - Cu, at 25, having given the following values for the degree of ionization of the two solutions: for m copper sulphate, a = 0.21, and for 0.01 m * For a thorough treatment of the theory of the lead accumulator the student is recommended to consult "Die Theorie des Bleiaccumulators " by F. Dolezalek. For a detailed account of the primary cells in common use the reader will find Carhart's "Primary Batteries" most satisfactory. ELECTROMOTIVE FORCE 491 copper sulphate, a = 0.61. The electromotive force at the junction of the two solutions may be neglected. Am. 0.0458 volt. 5. At 25 the electromotive force of the cell Zn - 0.5 m ZnS0 4 - 0.05 m ZnS0 4 - Zn is 0.018 volt. Neglecting the potential developed at the junction of the solutions, and assuming the dilute solution of zinc sulphate to be ionized to the extent of 35 per cent, find the degree of ionization of the concen- trated solution. Ans. 0.142. 6. What is the electromotive force of the cell Zn - 0.1 m ZnS0 4 - 0.01 m ZnS0 4 - Zn, at 18? For ZnS0 4 , - - = 0.601, for 0.1 molar ZnS0 4 , = 0.39, and for 0.01 molar ZnS0 4 , a = 0.63. Ans. 0.078 volt. 7. The electromotive force of the cell Ag - 0.001 m AgN0 3 - m KN0 3 - m KI, Agl - Ag is 0.22 volt at 18. A molar solution of KI is 78 per cent ionized, and a 0.001 molar solution of AgN0 3 is 98 per cent ionized. Calculate the solu- bility of Agl. Ans. 3.53 X 10~ 3 mols per liter. 8. The potential of zinc against a solution containing one mol of Zn ions is 0.493 volt, at 18. Assuming complete dissociation, calculate the solution pressure of zinc in atmospheres. 9. The electromotive force of the Daniell cell Cu - CuS0 4 - ZnS0 4 - Zn is 1.0960 volt at 0, and 1.0961 volt at 3. Calculate the heat of the reac- tion taking place in the cell. Ans. 51,070 calories. 10. Calculate the electromotive force of the cell Pt H2 - 0.1 m KOH - m HC1 - Pt Hl) at 25, having given that 0.1 molar KOH is 85 per cent ionized and molar HC1 is 70 per cent ionized. Ans. 0.757 volt. CHAPTER XXL ELECTROLYSIS AND POLARIZATION. Polarization. If a difference of potential of about 1 volt is applied to two platinum electrodes immersed in a concentrated solution of hydrochloric acid, it will be found that the current which passes at first, steadily diminishes and ultimately becomes zero. The cessation of the current has been shown to be due to the accumulation of hydrogen on the cathode and chlorine on the anode, these two gases setting up an opposing electromotive force called the electromotive force of polarization. If in the above case the applied electromotive force is increased to 1.5 volts, the counter electromotive force is no longer sufficient to reduce the current to zero. In fact, at any voltage above 1.35 volts a continuous current passes; this is termed the decomposition potential of hydrochloric acid. At all voltages above the decomposition potential, the current C may be calculated by means of the formula E - e = CR, where E is the applied electromotive force, e the counter electro- motive force, and R the resistance of the electrolyte. As the applied electromotive force and the current increase, the polarization increases, since the gases are liberated under a pres- sure greater than that of the atmosphere; but since the gases escape from the solution the value of e can never become equal to E. The decomposition potential of an electrolyte can be determined in two different ways, viz.: (1) by gradually raising the applied electromotive force E until it exceeds e, when the current will suddenly increase; or (2) by charging the electrodes up to atmospheric pressure by means of an electromotive force greater than e, and then breaking the external circuit and measur- ing the counter electromotive force. 492 ELECTROLYSIS AND POLARIZATION 493 The arrangement of apparatus for the measurement of the electromotive force of polarization, as suggested by Le Blanc,* is indicated in Fig. 106. A is the cell in which polarization occurs, B is the source of external electromotive force, C is a capil- lary electrometer, D is a source of variable potential, and E is one prong of an electrically-driven tuning fork which serves to make and break contact with the points F and G in rapid alternation. When the tuning fork makes contact at F the polarizing current B Fig. 106. Fig. 107. flows through A, polarizing the electrodes; when contact is made at G, the counter electromotive force due to this polarization causes a current to flow through D and C. The counter electro- motive force is balanced by varying D, until C indicates zero current: the potential of D is then equal to the electromotive force of polarization. Just as the electromotive force of a galvanic cell is due to the combined action of several differences of potential, so also the electromotive force of polarization is due to the individ- ual differences of potential located at the electrodes. The method employed for the measurement of polarization at a single elec- trode was devised by Fuchs, and is illustrated diagrammatically in Fig. 107. Into the vessel containing the electrolyte, dip the * Zeit. phys. Chem., 5, 469 (1890). 494 THEORETICAL CHEMISTRY two electrodes A and B, and the side tube of the calomel normal electrode C. D is the source of external electromotive force, E is a capillary electrometer, and F is a source of variable potential. Before closing the external circuit DAB, the potential of the electrode B against the solution is first measured. Then the cir- cuit DAB is closed, thereby polarizing the electrode B. The external circuit is again broken and the potential of B against the solution remeasured. The difference between the final and initial values of the electrode potential gives the polarization at B. In like manner, the polarization at the electrode A can be measured. The small amount of electricity which is necessary to polarize an electrode is termed the polarization capacity of the electrode. This factor is dependent upon the extent of surface of the electrode, and also upon the nature of the metal of which the electrode is made. For electrodes of equal surface, the polarization capacity of palladium is greater than that of platinum, when hydrogen is liberated on each. The solubility of hydrogen is greater in palladium than in platinum, and consequently, because a larger amount of hydrogen is dissolved, a greater quantity of electricity will be required to bring the pressure of the hydrogen up to that of the hydrogen dissolved in the platinum. If, through the processes of solution or diffusion, or through chemical action, the substance which causes the polarization is removed, the electrode is said to be depolarized. Thus, when a reducing agent, such as ferrous chloride, is electrolyzed, the oxygen liberated at the anode immediately combines with the electrolyte, forming ferric chloride and preventing polarization of the electrode. If water be electrolyzed between platinum electrodes, the cathode becomes saturated with hydrogen and the anode with oxygen, until, when tne electromotive force of polarization becomes equal to that of the external circuit, the current ceases. The two gases, hydrogen and oxygen, are soluble, however, and conse- quently diffuse away from the electrodes, either escaping from the solution or recombining to form water. In order to compen- sate for this continuous loss of gas at the electrodes, a small current continues to flow, thus maintaining the initial electromotive force constant. This small current is termed the residual current. ELECTROLYSIS AND POLARIZATION 495 If oxygen is bubbled over the surface of the cathode during elec- trolysis, the hydrogen is removed as rapidly as it is liberated. Such an electrode on which no new substance is formed during electrolysis is called an unpolarizable electrode. Decomposition Potentials. The decomposition potential of an electrolyte can be determined, as has already been pointed out, by immersing two platinum electrodes in the solution and connect- ing with a source of electricity, the electromotive force of which can be varied at will. The voltage is gradually increased and the corresponding current is observed. The current increases at first Electromotive Eonce Fig. 108. and then drops almost to zero every time the voltage is raised, until the decomposition potential is reached. Beyond this point the current is directly proportional to the electromotive force. If the applied electromotive forces are plotted as abscissae and the corresponding currents as ordinates, we obtain curves of the form shown in Fig. 108. Some of the decomposition potentials of molar solutions determined by Le Blanc * are given in the accom- panying tables. * Zeit. phys. Chem., 8, 299 (1891). 496 THEORETICAL CHEMISTRY SALTS. Salt. Decomp. Potential. Salt. Decomp. Potential. ZnSO 4 Volts. 2.35 Cd(NO 3 ) 2 .. Volts. 1.98 ZnBr 2 1 80 CdSO 4 2 03 NiSO 4 2 09 CdCl 2 1 88 NiCl 2 1 85 CoSO 4 1 92 Pb(NO 3 ) 2 1 52 CoCl 2 1 78 AgN0 3 0.70 ACIDS. Acid. Decomp. Potential. H 2 SO 4 . Volts. 67 HNO 3 . . 69 H 3 PO 4 .70 CH 2 C1.COOH .72 CHC1 2 .COOH CH 2 (COOH) 2 HC1O 4 HC1 (COOH) 2 HBr HI .66 .69 .65 .31 0.95 0.94 0.52 BASES. Base. Decomp. Potential. NaOH Volts. 1 69 KOH NH 4 OH 1.67 1.74 It will be observed that while there is considerable variation in the decomposition potentials of salts, there is very little variation in the decomposition potentials of acids and bases. There is a maximum value of about 1.70 volts to which many acids and bases closely approximate. It is found that all acids and bases which decompose at 1.70 volts give off hydrogen and oxygen at ELECTROLYSIS AND POLARIZATION 497 the electrodes. Those acids and bases which decompose at potentials less than the maximum, do not liberate hydrogen and oxygen. When their solutions are sufficiently diluted, however, hydrogen and oxygen are evolved and the decomposition potential rises to the maximum value. Thus, Le Blanc found the following values for the decomposition potential of different dilutions of hydrochloric acid. Concentration. Decomp. Potential. 2mHCl Volts. 1.26 imHCl 1.34 imHCl 1.41 AmHCl 1.62 AmHCl 1.69 / When 2 m hydrochloric acid is electrolyzed, hydrogen and chlorine are given off at the electrodes, whereas when the concen- tration of the acid is reduced to 1/32 m, hydrogen and oxygen are the products of electrolysis, and the decomposition potential increases to 1.70 volts. It is found that the values of the decom- position potentials vary slightly with the nature of the electrodes used. The above values were determined with platinum elec- trodes. The Theory of Polarization. Our knowledge of the process taking place at the electrodes during electrolysis is largely due to the investigations of Le Blanc. He determined the electro- motive force of polarization at each electrode, varying the external electromotive force from zero up to the decomposition potential of the solution. When the decomposition value was reached, he found the potential of the electrode against the solution to be the same as the difference of potential between the solution and the element liberated at the electrode. Thus, the decomposition potential of a molar solution of zinc sulphate is 2.35 volts; the corresponding difference of potential between the electrode and the solution is found to be 0.493 volt. If a piece of pure zinc is immersed in a molar solution of zinc sulphate, the difference 498 THEORETICAL CHEMISTRY of potential is found to be 0.493 volt, the metal being negative to the solution. It frequently happens that the electrode exhibits the potential due to the deposited metal before the decomposition point of the solution is reached. For example, in a molar solution of silver nitrate the electrode acquires the potential of pure silver in molar silver nitrate before" the decomposition value, 0.70 volt, is reached. This is due to the negative solution pressure of the silver which causes the deposition of the ions of the metal without the application of any external electromotive force. When an indifferent electrode, such as platinum, is immersed in a solution of a salt, a very small amount of ionic deposition must occur, otherwise, according to the Nernst equation, an infinite electro- motive force must be established. Thus, in the equation RT P if the solution^ pressure P = 0, it is evident that TT = oo and a perpetual motion must result. We are thus forced to the con- clusion that when an indifferent electrode is immersed in a salt solution, ions will continue to separate upon it until the tendency for the deposited metal to go back into solution in the ionic state exactly counterbalances the tendency to separation. Hence, the electrode will become positive toward the solution. The magnitude of this difference of potential will be dependent upon the amount of -metal deposited. It is to be noted that this difference of potential need not be equal to that between the massive metal and the solution. If the electrodes be connected with an external source of electromotive force, the value of which can be varied at will, and a small electromotive force be applied, more metal will separate on the cathode. This will cause an increase in the solu- tion pressure P, tending to offset further deposition. A still further increase in the external . electromotive force will cause the deposition of more metal, and as a result of the corresponding increase in P, further deposition at that voltage will be pre- vented. Ultimately, when the applied electromotive force is such that P acquires its maximum value, equivalent to that of the massive metal, continuous deposition will occur. An exactly ELECTROLYSIS AND POLARIZATION 499 analogous process takes place at the anode. If a gas is liberated, its concentration steadily increases until the maximum pressure is reached, when it will escape from the solution. When strong currents are employed, P does not remain constant, as has been assumed above, but gradually diminishes causing the difference of potential at the electrode to increase. From the above considerations it becomes clear why a definite electromotive force is necessary to bring about a continuous decomposition of an electrolyte: this will only take place when the concentrations of the substances separating at the electrodes have attained their maximum values. When the decomposition point is reached, the electrode exhibits the potential characteristic of the massive metal. It is evident from the behavior of silver nitrate and the salts of other metals having negative solution pressures, that the maximum values of concentration at the the electrodes need not necessarily be attained simultaneously. When the products of electrolysis are gaseous, the value of the decomposition potential depends upon the nature of the electrodes. Thus, the cell Pt Hj - m H 2 S0 4 - Pto 2 gives an electromotive force of 1.07 volts if platinized platinum electrodes are used. If an external electromotive force slightly greater than 1.07 volts be applied to this cell in the reverse direc- tion, water will be steadily decomposed, hydrogen and oxygen being evolved at the electrodes. If on the other hand, the plati- nized electrodes are replaced by electrodes of polished platinum, the decomposition potential rises to 1.68 volts. The reverse electromotive force of polarization, however, is only 1.07 volts. That is, the liberation of gas at a polished platinum electrode is an irreversible process. The difference in the behavior of the two electrodes is explicable when it is remembered that platinum is capable of occluding large amounts of gas. A platinized elec- trode absorbs the liberated gas very slowly, and when thoroughly saturated, if it is not entirely immersed in the solution, it gradually gives up the gas by diffusion, no bubbles being formed. Thus, if the external electromotive force be raised to 1.07 volts, the system 500 THEORETICAL CHEMISTRY will be in equilibrium, while if the applied electromotive force be greater or less than the equilibrium value, a current will flow in one direction or the other, gas being either liberated or dissolved. In other words, the cell is completely reversible. Where polished platinum or gold electrodes are used, however, the decomposition potential is, as has been stated, 1.68 volts. Polished electrodes have relatively small absorbing power. Hence, if an electromotive force between 1.07 and 1.68 volts be applied, the gases cannot diffuse away from the electrode rapidly enough, and, when the solution in the vicinity of the electrodes becomes saturated with gas, the current ceases to flow. A very slow process of diffusion from the solution into the air is constantly taking place however, and this permits the continu- ous evolution of an exceedingly small amount of gas, while a corre- spondingly small current traverses the solution. In order to produce a steady electrolysis, it is necessary to raise the external electromotive force to such an intensity that it is able to bring about the formation of bubbles at the surface of the electrodes. This calls for the expenditure of an amount of work depending upon the condition of the electrode surfaces, the sur- face tension of the solution, and various other factors. In cases where bubbles are formed, a portion of the available energy of the chemical process is not expended in effecting electrical separa- tion; consequently the reverse electromotive force is less than the applied, and the system is irreversible. The reactions at the electrodes are catalytically accelerated by the metal of which the electrodes are made. Thus, platinized platinum is the most effective catalyst for the reaction represented by the equation H 2 + 2(+)<=2H'. Hydrogen is liberated on platinized platinum at the potential volt, on polished platinum at 0.09 volt, and on zinc at 0.70 volt. The electromotive force necessary to overcome the resistance of the chemical reaction at an electrode is termed the overvoltage. Thus, we say that hydrogen is liberated on polished platinum with an overvoltage of 0.09 volt, and on zinc with an overvoltage of ELECTROLYSIS AND POLARIZATION 501 0.70 volt. The following table gives the overvoltage necessary for the liberation of hydrogen and oxygen on electrodes of differ- ent metals. ELECTRODE OVERVOLTAGES. Hydrogen Liberation. Oxygen Liberation. Metal. Overvoltage. Metal. Overvoltage. Pt (platinized) Au 0.00 0.01 0.08 0.09 0.15 0.21 0.23 0.46 0.53 0.64 0.70 0.78 Au .75 .67 .65 .65 .63 .53 .48 .47 .47 .36 .35 .28 Pt (polished) Pd Cd Fe (in NaOH) Pt (polished) Ag Ni Cu Ag Pb. Cu. Pd Fe Sn Pb Zn Hg Pt (platinized) . . Co... Ni (polished) Ni (spongy) Primary Decomposition of Water in Electrolysis. The decom- position potential of an electrolyte giving off hydrogen and oxygen at the electrodes, is dependent upon the concentrations of the two ions, H* and OH', and is independent of the nature of the electrolyte. As has already been stated, the decomposition potential of all acids and bases giving off hydrogen and oxygen approximates to 1.70 volts. According to the law of mass action, the product of the concentrations of the H* and OH' ions is constant and inde- pendent of the other substances which may be present; hence, although the potentials of the individual electrodes may differ considerably, their sum remains practically constant. Excluding solutions of salts which undergo reduction by hydro- gen, and solutions of chlorides, bromides, and iodides reducible by oxygen, the ions H* and OH', according to Le Blanc, are to be regarded as the sole factors in the electrolysis of solutions, and not the ions of the dissolved electrolyte. In other words, elec- trolysis involves a primary decomposition of water. The electrical conductance of the solution is due to the ions of the electrolyte together with the ions of water, but at the electrode 502 THEORETICAL CHEMISTRY that process takes place which involves the expenditure of the minimum amount of energy, and this is, under ordinary conditions the separation of the H" and OH' ions. Thus, when a solution of potassium sulphate is electrolyzed, only a moderately strong current being used, it is not rational to assume the discharge of the K" and SO 4 " ions at the electrodes, and then subsequent reaction between these discharged ions and water. This may be made clear by considering the process taking place at the cathode. According to the explanation based upon so-called ''secondary action," the K" ions give up their positive charges to the electrode and then react with water as indicated by the equation K + H- + OH' -K' + OH' + H. This explanation involves the transfer to the potassium atom of the positive charge of the H* ion of water; this can only take place if the H* ion holds its charge less tenaciously than the K" ion. Hence, if the H" ion parts with its charge more readily than the K' ion, the former will be discharged primarily at the cathode. Similar reasoning may be employed to explain the action at the anode. Therefore, in electrolysis all of the ions participate in conducting the current and collect around the electrodes, but since the H* and OH' ions separate more easily, these are dis- charged. With stronger currents it is possible to cause the separ- ation of the K* and S0 4 " ions also, since the number of H* and OH' ions present is too small to carry all of the current, and the energy required to discharge the ions of the electrolyte is less than that necessary to remove the small number of residual H* and OH' ions. The formation and decomposition of water are reversible processes, so that no loss of energy is involved, as would be the case if secondary actions occurred. Electrolytic Separation of the Metals. Freudenberg * was the first to recognize the possibility of effecting the quantitative separation of different metals by means of graded electromotive forces. He showed that it was only necessary to select a salt of each metal the decomposition potentials of which differ as widely as possible, and electrolyze at an electromotive force intermediate * Zeit. phys. Chem., 12, 97 (1893). ELECTROLYSIS AND POLARIZATION 503 between these potentials. The salt having the lower decomposi- tion potential will decompose first, and when the deposition of the metal is complete, the current will practically cease; then if the applied electromotive force be raised above the decomposition potential of the second salt, the second metal will be deposited. In practice it is foiind necessary to increase the applied electro- motive force slightly because of the gradual decrease in the num- ber of ions of the salt having the lower decomposition potential. The amount of this increase may be readily calculated from the familiar equation RT . P "K FT lOffe nF * p Suppose a mixture of the nitrates of cadmium, lead and silver is subjected to electrolysis, the decomposition potentials of the salts being as follows: Cd(N0 3 ) 2 = 1.98 volts, Pb(N0 3 ) 2 = 1.52 volts, and AgNO 3 = 0.70 volt. The applied electromotive force is made a little less than 1 volt and all of the silver is deposited ; then the electromotive force is raised to about 1.6 volts, thus depositing all of the lead; and finally, with an electromotive force of about 2 volts the cadmium is deposited. In the subjoined table are given the separation potentials of some of the ions, the separation potential of the H* ion being assumed to be equal to zero. SEPARATION VALUES OF IONS FOR MOLAR CONCENTRA- TION. Ion. Separation Potential. Ion. Separation Potential. Ae* -0 78 I' 52 or" -0 34 Br' 94 H* 0" 1 08 (in acid) Pb" 17 Cl' 1 31 Cd" 38 OH' 1 68 (in acid) Zn". 74 OH' 0.88 (in base) SO 4 " 1.9 According to this table the decomposition potential of water is equal to the sum of the separation potentials of its ions, or 1.68 volts. CHAPTER XXII. PHOTOCHEMISTRY. Radiant Energy. The visible portion of the spectrum is com- prised between the extreme red at one end and the extreme violet at the other; the wave-length corresponding to the former is approximately 0.7 micron, while that corresponding to the latter is about 0.4 micron. The visible portion of the spectrum, how- ever, is but a small fraction of the entire spectrum. Beyond the red of the visible spectrum lies the region of the so-called infra- red, comprising all wave-lengths from 0.76 micron up to 300 mi- crons. Beyond the infra-red, between 300 and 2000 microns, is an unmeasured region, which is succeeded by the region of elec- trical waves, extending from 2000 microns to an undetermined maximum. On the other hand, extending beyond the violet of the visible spectrum is the so-called ultra-violet or actinic region, comprising all wave-lengths, between 0.4 micron and 0.1 micron. It thus appears that heat, light, and electricity are all forms of radiant energy, the only distinction between them being a differ- ence in wave-length. Very little is known concerning radiant energy, and up to the present time all attempts to resolve it into a capacity and an intensity factor have failed. Whatever may be the nature of this form of energy, we know that the effects produced by it are dependent upon the wave-length of the radiation. We have already devoted several chapters to the consideration of thermochemistry and electrochemistry, and it now remains to study very briefly the connection between chemical energy and that subdivision of radiant energy called light. This branch of theoretical chemistry is termed photochemistry. The ultra-violet or actinic rays are the most active chemically, although light of every wave-length, including the invisible infra-red, is capable of producing chemical action. When light falls upon a substance, a portion of the incident radiation is reflected, a portion is absorbed, and a portion is transmitted. It has been shown that only that 504 PHOTOCHEMISTRY 505 portion of the incident radiation which is absorbed is effective in producing chemical change. Radiant energy has been shown by Lebedew,* and also by Nichols and Hull,t to exert a definite, though extremely small pressure. Thus, the pressure of solar radiations on the earth is equivalent to that of a column of mercury 1.4 X 10~ 9 mm. high. Source of Radiant Energy. According to the electromagnetic theory of light, the emission of waves of light from a material source is due to the vibrations of minute charged particles called radiators. These radiators, which may be either atoms or elec- trons, give rise to electromagnetic waves of the same period as their own, that is, to light waves of definite length. The energy required to produce these electromagnetic waves is derived from the vibrating system itself, and unless an equivalent amount of energy is constantly supplied to the system, the amplitude of the vibrations will steadily diminish and ultimately cease. There are two different ways in which this supply of energy can be maintained. First, the temperature of the vibrating system as a whole may be kept high. This type of radiation, which is maintained by purely physical means, is called pure temperature radiation. Every substance whose temperature is above the abso- lute zero ( 273) gives rise to pure temperature radiation. The higher the temperature, the more rapid and the more energetic the atomic and electronic vibrations become. With increase in rapid- ity of vibration, there results a corresponding diminution in wave- length, so that, as the temperature is raised, the longer heat waves are succeeded by the shorter waves of the visible region of the spectrum. When a sufficiently high temperature is reached, the period of vibration becomes so rapid as to cause the radiation of waves corresponding to the entire range of the visible spectrum. At higher temperatures, the rate of vibration is such that the vibrating particles must of necessity possess extremely small mass: it is commonly believed that under these conditions the vibration is wholly electronic. The second way in which energy may be supplied to the vibrating * Rapp. pres au Congres de Physique, 2, 133 (1900). t Phys. Rev., 13, 293 (1901). 506 THEORETICAL CHEMISTRY atoms or electrons is by chemical or electrical means. The general term luminescence has been proposed by Wiedemann for all cases where luminous energy is derived from other sources than high temperature. It is to be observed that luminescence is frequently exhibited by systems whose temperatures are comparatively low. For example, notwithstanding the fact that the flame resulting from the combustion of carbon disulphide has a temperature of only 150, it has been found to be capable of affecting the photo- graphic plate. Pure temperature radiation alone at 150 would correspond to long waves in the infra-red region of the spectrum and, as is well known, such waves are incapable of exerting appre- ciable photographic action. Emission and Absorption. The relation between the emissive and absorptive powers of different bodies was first clearly enun- ciated by Kirchhoff * in 1859. This law may be stated as follows: Light of any given wave-length emitted by a body can also be absorbed by the same body at a lower temperature. This law, it will be seen, offers a satisfactory explanation of the Fraunhofer lines in the solar spectrum. The sun is surrounded by a gaseous atmos- phere resulting from the vaporization of the elements present in the body of the sun. Each element in the cooler gaseous envelope, according to KirchhofFs law, absorbs those wave-lengths which it emits at the higher temperature of the solar nucleus. The result- ing dark lines of the solar spectrum have enabled the astronomer to determine the elementary composition of the sun. If the emissive and absorptive powers of a body be denoted by E and A respectively, then according to Kirchhoff's law, E/A = S, where S is a constant. When absorption is complete, A is unity and S = E. Under these conditions the constant, S, may be de- fined as the emissivity of a body which absorbs all of the incident radiation and reflects none. Such a body was called by Kirch- hoff a perfectly black body. The emissivity of a perfectly black body is equal to the ratio of the emissive to the absorptive power of any body at the same temperature. A familiar qualitative illus- * Ostwald's Klassiker, No, 100 (1898). PHOTOCHEMISTRY 507 tration of KirchhofFs law is that afforded by the appearance of a fragment of white chinaware possessing a dark pattern when heated to a high temperature. The dark parts of the design absorb light, while the white parts reflect it. On heating the fragment to redness, the pattern will be reversed, the dark portions of the design appearing bright and the white portions dark. It has been shown that the law of Kirchhoff is a necessary con- sequence of the application of the second law of thermodynamics to the thermal equilibrium within an enclosure whose walls are impervious to heat. The Stefan-Boltzmann Law. From a study of the experi- ments of Dulong and Petit on the rate of cooling of different bodies, Stefan * discovered an empirical relation between the total radia- tion of a body and its temperature. Later, Boltzmann f derived the same relation thermodynamically and showed that instead of being general, as Stefan supposed, it is only strictly applicable to a perfectly black body. The Stefan-Boltzmann law may be stated as follows : The total radiation from a perfectly black body is directly proportional to the fourth power of the absolute tempera- ture. If the total radiation be denoted by S, we may write S = CT 4 , where C is a constant. If the radiation from the sun be con- sidered solely as a temperature effect, its temperature may be calculated by the above equation expressing the Stefan-Boltzmann law. Employing available bolometric data, the temperature of the sun may thus be shown to be 6200 absolute. The Displacement Law of Wien. Having considered the total energy radiated by a given source, we now come to the considera- tion of the distribution of energy throughout the entire spectrum in its relation to temperature. The results of the experiments of Lummer and Pringsheim J on the distribution of energy in the normal spectrum of a black body are shown by the curves of Fig. 109. The values of the energy radiated by the source are plotted * Sitz. Ber. Wiener Akad., 79 (II), 391 (1879). t Wied. Ann., 22, 291 (1884). * Verb, deutsch. phys. Ges., i, 230 (1889); 3, 36 (1901). 508 THEORETICAL CHEMISTRY as ordinates against the corresponding values of the wave-length as abscissae. It will be observed that the energy corresponding to a definite wave-length increases with the temperature, and that each curve, or isothermal, exhibits a distinct maximum. The 1646 Wave-length f Fig. 109. position of this maximum is displaced in the direction of decreasing wave-length as the temperature is raised. In 1893, Wien * discovered the law governing this displacement of the energy maximum with temperature. If X ma x. denotes the * Wied. Ann., 58, 662 (1896). PHOTOCHEMISTRY 509 wave-length corresponding to the energy maximum, and T is the absolute temperature of the radiating black body, Wien showed that \ max x T = constant. Furthermore, Wien found that S max . = BT*, where ma x. is the maximum emissivity corresponding to X max .. In general, the emissivity S is the amount of energy radiated per second from a narrow strip of the spectrum corresponding to the mean wave-length X. It should be mentioned that while the experimental realization of a perfectly black body is impossible, a very close approximation can be obtained by the employment of a hollow blackened sphere perforated by a small hole to permit the passage of the radiation. When this sphere is heated, practically all of the radiation is absorbed by multiple reflections at the inner surface. It has been found that where such a black body is not available, or where extreme accuracy is not required, a thin strip of platinum foil coated with ferric oxide and heated electrically proves a satisfactory substitute. It is hardly necessary to call attention to the fact that the term black body, as here used, does not imply a total absence of color. At high temperatures, a " black " source of radiation may be red or even white. To avoid confusion, it has been proposed to substitute the term full radia- tor for the older term, black body. Distribution of Energy throughout the Spectrum. The experi- mental determination of the radiant energy corresponding to a given wave-length, really resolves itself into the measurement of the energy emitted between two contiguous wave-lengths, X and X + d\. In other words, we actually measure the energy of a very small portion of the spectrum included between two wave- lengths which lie very close together. Several different formulae have been proposed for the calcula- tion of the distribution of energy throughout the spectrum. Of these, the formulae of Rayleigh and Wien have been found to reproduce experimental values with considerable accuracy. The formula of Rayleigh has the following form: 510 THEORETICAL CHEMISTRY while that of Wien may be written thus: In these two formulas, C and c' are constants, while the other symbols have their usual significance. The formula of Rayleigh has been found to hold better in the region of the longer wave- lengths while the reverse is true of the formula of Wien. It should be remembered that both of these formulas apply only to bodies emitting continuous spectra. High Temperature Thermometry.* Various optical methods for the measurement of high temperatures have been developed, but a detailed treatment of these methods is obviously out of place in a book of this character. Mention should be made, however, of two instruments which have proven of great value in high tem- perature measurements. The optical pyrometer of Fe*ry is based upon Stefan's law of total radiation. It consists of a telescope fitted with an objective of fluorite, at the focus of which is placed a sensitive thermo- couple. In order to determine the temperature of a source of radiant energy, such as a crucible of molten metal, the telescope is directed toward the contents of the crucible and the image is focussed on the thermo-j unction by means of an adjustable eye- piece. The resulting electric current is then measured by means of a galvanometer. In the optical pyrometer of Holborn and Kurlbaum, use is made of the luminous radiations only. In this instrument the current through a small incandescent lamp is varied until its light is just eclipsed by that from the hot body. When this point of balance has been reached, the incandescent filament and the hot body have the same temperature. The pyrometer is calibrated by de- termining the current necessary to raise the filament of the lamp to the temperature of the standard black body, the temperature of the latter being determined by means of a thermocouple. Of * For a detailed account of optical pyrometers, the student is referred to "High Temperature Measurements" by Le Chatelier and Boudouard, trans- lated by Burgess (John Wiley and Sons, Inc.). PHOTOCHEMISTRY 511 course, when the instrument is used to determine the temperature of sources of radiant energy which differ widely in character from that of a perfectly black body, the accuracy of the measure- ments is lessened, but even in an extreme case, such as that pre- sented by polished platinum at 950, the error does not exceed 74. The importance of optical pyrometers in photochemical investigations lies chiefly in the determination of energy curves of light sources and in the absolute measurement of radiant energy. Luminescence. As has already been pointed out, it is cus- tomary to distinguish between pure temperature radiation and luminescence. The latter term is applied to all cases where chemi- cal or electrical energy is transformed directly into radiant energy. The various types of luminescence may be conveniently classified in the following manner: Type of Luminescence. Origin of Radiation. (1) Photo-luminescence Preliminary exposure of the lumi- (a) Fluorescence, nous substance to some external (6) Phosphorescence. source of radiant energy. (2) Thermo-luminescence. Stimulation by heat, but at a tem- perature considerably lower than that required for pure temperature radia- tion. (3) Chemi-Iuminescence. Chemical reaction. (4) Tribo-luminescence. Fracture or cleavage of crystals. (5) Cathodo-luminescence Electric discharge. (6) Radio-luminescence. Radioactivity. By the term fluorescence is meant the phenomenon of the emission, by an illuminated medium, of light of a different wave-length from that of the incident radiation. In general, the wave-length of the transformed radiation is greater than that of the incident radiation. This law, to which several exceptions have been discovered, was first enunciated by Stokes. When the incident radiation is cut off, fluorescence ceases. On the other hand, there are many substances which continue to emit light for some time after the external light-stimulus is removed. This phenomenon is termed phosphorescence and appears to be governed by Stokes law for fluorescence. The 512 THEORETICAL CHEMISTRY property of phosphorescence appears to be limited to anhydrous substances. Among the numerous substances which are known to exhibit the phenomenon of fluorescence may be mentioned fluorite (from which the phenomenon derived its name), uranium glass, petro- leum, solutions of organic dyestuffs, and quinine sulphate. The vapors of sodium, mercury, and iodine have recently been found by Wood to fluoresce brilliantly. The sulphides of the alkaline earths may be mentioned as ex- amples of phosphorescent substances. The investigations of Lenard and Urbain have revealed the interesting fact that the presence of a trace of one of the heavy metals greatly intensifies the light emitted by a phosphorescent substance. The phenomenon of thermo-luminescence calls for little com- ment. There seems to be an intimate connection between thermo- luminescence and phosphorescence, since the substances exhibit- ing the former phenomenon must be exposed initially to light, otherwise they do not emit any visible radiation on gentle heating. Chemi-luminescence is a phenomenon accompanying many chemical reactions. Thus, the precipitation of sodium chloride from its saturated solution by hydrochloric acid gas is accompanied by an emission of light which may readily be seen if the reaction is carried out in a dark room. When certain crystals, such as those of cane sugar, are either crushed or simply rubbed together, flashes of light are emitted. This phenomenon is known as tribo-luminescence. Cathodo- and radio-luminescence may be considered as sub- divisions of the more inclusive term, electro-luminescence. At- tention has already been called to the fact that the residual gas in a vacuum tube is rendered luminous by the passage of the electric discharge, and also that certain minerals become phosphorescent when placed in the path of the cathode rays. These may be taken as examples of cathodo-luminescence. The luminosity of a screen coated with crystals of zinc sulphide, when subjected to the action of the a-particles shot out from a radio-active substance, has also been mentioned in a previous chapter. This is clearly an instance of radio-luminescence. PHOTOCHEMISTRY 513 Having briefly reviewed the different processes involved in the production of light we now turn to a consideration of the chemical phenomena resulting from exposure to light. Photochemical Action. The development of the green color of plants under the influence of the rays of the sun, and the reverse process of bleaching in darkness, were probably the first photo- chemical reactions to be observed. To-day it is known that light has the power of initiating or accelerating every variety of chemical change. This statement may be illustrated by the following typi- cal photochemical reactions : The polymerization of anthracene, the depolymerization of ozone, the transformation of maleinoid into fumaroid forms, the hydrolysis of acetone, the oxidation of lead sulphide, and the reduction of silver salts. That such a variety of photochemical reactions should result from exposure to mixed, or heterogeneous, light is due to the selective absorption of each par- ticular chemical system. Photochemical action is not limited to the waves of the visible spectrum alone, but extends from the red end of the spectrum (wave-length 800 MM) * into the ultra-violet region (wave-length 300 MM)- In fact, the shorter wave-lengths of the ultra-violet region of the spectrum have been found to be the most active photochemically . Laws of Grotthuss. The two fundamental generalizations of photochemistry were first enunciated by Grotthuss in 1818. These generalizations may be stated as follows : (1) Only those rays of light which are absorbed produce chemical action. (2) The action of a ray of light is analogous to that of a voltaic cell. It has recently been shown by Bancroft | that the second of these two laws is inadequate to account for all of the known facts and, therefore, he proposes the following modification: All of the radiations which are absorbed by a substance tend to eliminate that substance. It is merely a question of chemistry whether any re- action occurs and what the products of the reaction will be. * 1 MM~ 7 = 10~ 7 cm. t Jour. Phys. Chem., 12, 209, 318, 417 (1908); 13, 1, 181, 269, 449, 538 (1909); 14, 292 (1910). 514 THEORETICAL CHEMISTRY Quantitative Relations Concerning the Absorption of Light. When a ray of light enters an absorbing medium only a certain proportion of the incident radiation is absorbed. The intensity of the light entering an absorbing medium is not equal to that which is incident on the surface of the medium, owing to the fact that a portion of the incident beam is reflected. It has been found that if the thickness of the medium be increased in arithmetical pro- gression, the intensity of the transmitted light decreases in geomet- rical progression. If the intensity of the light traversing a layer dl be denoted by 7, then - dl/dl = kl, (1) where k is a constant depending upon the nature of the absorbing medium and the wave-length of the light. This constant k is known as the absorption index. If the initial intensity of the light is /o and the total thickness of the medium is d, equation (1) be- comes, on integrating, / = / . e - kd . (2) Or equation (2) may be written in the form, If we replace e~ k by a, then equation (2) becomes ///o = of. (4) The constant a is called the transparency or the transmission co- efficient. Bunsen and Roscoe introduced the term extinction co- efficient. This quantity may be defined as the reciprocal of that thickness of the medium which reduces the intensity of the trans- mitted light to one-tenth of its initial value. If the extinction coefficient be denoted by e, its value may be calculated from equa- tion (2) in the following manner : r= i . io-*, (5) or 6= 3 loge JT (6) It is evident that er k is identical with 10~ or e = mk, where m represents the modulus of the Naperian system of logarithms. PHOTOCHEMISTRY 515 In 1852 Beer * enunciated an important law concerning the in- fluence of concentration on absorption. Beer's law may be stated as follows : The absorption of light by different concentrations of ike same solute dissolved in the same solvent is an exponential function of the concentration, provided the thickness of the absorbing medium be maintained constant. It follows from this law that (7) where c is the concentration of the solution. If Beer's law is valid, the ratio, c/e = A, known as the absorption ratio, should be constant. Beer's law has been tested with a large number of solutions and has been found to hold quite generally where no change in the solute occurs when the concentration of the solution is altered. Photochemical Extinction. According to the first law of Grotthuss, only the absorbed light is chemically active. The con- verse of this law, viz., that every substance which absorbs light undergoes chemical change, apparently does not hold. Further- more, only a portion of the rays absorbed by a light-sensitive sub- stance are directly involved in effecting chemical change. For example, while an alkaline copper tartrate solution shows marked absorption in the infra-red, red, yellow, and ultra-violet, it has been proven that the photochemical reduction of the copper salt to cuprous oxide is due to the action of the ultra-violet rays alone. The first quantitative measurements of the absorption of light by a reacting system were made by Bunsen and Roscoe.f These investigators found that the absorption of light by hydrogen and chlorine, taken separately, was less than that of the reacting mixture of the two gases. From this result they concluded that in a photochemical reaction, the absorption of light by the re- acting system is greater than the sum of the individual absorptions of the reacting substances. The absorption of light by the re- acting system, over and above the ordinary or "optical " absorp- tion of the reacting substances, they called photochemical extinc- tion. While the phenomenon of photochemical extinction is re- * Pogg. Ann., 86, 78 (1852). t Ostwald's Klassiker, No. 38. 516 THEORETICAL CHEMISTRY garded by many physical chemists as having doubtful significance, nevertheless the distinction between purely optical absorption on the one hand, and chemical absorption on the other, is quite generally accepted.* If the distinction drawn by Bunsen and Roscoe between optical and chemical absorption be accepted, then all photochemical re- actions may be regarded as belonging to one or the other of two classes, as follows: (1) reactions in which light does work against chemical affinity, the work being equivalent to the photo- chemical extinction; or (2) reactions in which the light functions merely as a catalyst. In reactions belonging to the first class, the light is considered to be the agent which actually initiates chemical change, whereas in reactions of the second class, the light is assumed to accelerate reactions which would otherwise proceed at a slower rate. Kinetics of Photochemical Reactions. Let A and B repre- sent two chemically distinct substances, and let us assume that the reaction takes place under the influence of light. The course of the re- action can be followed in the usual manner by determining the amount of either constituent which is present at any definite time. There are certain factors, however, which render such kin- etic measurements more difficult in the case of a photochemical reaction than in that of an ordi- nary chemical reaction. Let us assume that the above reaction Fig. 110. is homogeneous and of the first order and also that it takes place in homogeneous solution. It is apparent that if the system is illuminated from one side, as shown in Fig. 110, the rate at which A undergoes transformation into B will depend upon the thick- * For a condensed summary of the different theories of light-absorption, the student should consult Sheppard's " Photochemistry " (Longmans, Green & Co.). PHOTOCHEMISTRY 517 ness, d, of the absorbing layer. Hence, if the velocity of the transformation be denoted by dB/dt, we may write dB/dt = k[A], in which the value of the constant k will not only be a function of the intensity and wave-length of the light but also of position. Furthermore, the variation in the velocity of the reaction in suc- cessive layers will cause differences in concentration which will tend to become equalized by the process of diffusion. From this ex- ample it is obvious that the extent to which light is absorbed is dependent upon the dimensions of the absorbing system. If light be regarded as a material substance, then we may con- veniently consider its absorption as analogous to the diffusion of a gas into a liquid and the light intensity at any point as the analogue of concentration or active mass. According to the law of mass action, the velocity of a chemical reaction is expressed by the equation, dx/dt = /cci n i >ef*' .-. . -- k'ci n *' c^' . . . where Ci, Cz, etc., are the concentrations of the substances entering into the reaction, and n\, n 2 , etc., are the coefficients derived from the chemical equation and indicating the order of the reaction. In applying this equation to photochemical reactions, the following possibilities must be borne in mind : (1) The reaction may take place in successive stages. Under these conditions the experimentally measured velocity will corre- spond to that of the slowest reaction. (2) Side reactions may occur with the formation of products quite different from those resulting from the main reaction. (3) There may be catalysis. In fact, this phenomenon is fre- quently met with in the study of photochemical reactions. Nernst has pointed out that the velocity constants, k and k', in the foregoing equation, may be conveniently considered as being directly proportional to the intensity of the light, for light of the same kind. Owing to absorption, this intensity will be a function of position in the absorbing medium. In applying the law of mass action to photochemical reactions, it is important to note that, as a general rule, the exponents n\, nt, 518 THEORETICAL CHEMISTRY etc., in the equation of the so-called dark-reaction are not identical with the exponents v\, i>2, etc., of the light-reaction. In other words, the order of a chemical reaction is usually different in the light from what it is in the dark. The photochemical exponents v\, vz, etc., are never greater than, and are seldom equal to, the corresponding exponents n\, n?, etc., of the dark-reaction. Thus, Bodenstein * showed that the dissociation of hydriodic acid, in the dark, is a reaction of the second order and may be rep- resented by the equation 2 HI <=> H 2 + I 2 , whereas when hydriodic acid dissociates in the light, the reaction is of the first order, as shown by the equation HI < H + I. In this case, the light acts as a catalyst, merely accelerating the velocity of the dark-reaction. The action of light on the reverse reaction may be such as to oppose its usual course in the dark. In this case the resulting photochemical equilibrium, or so-called photo-stationary state, will not be identical with the corresponding chemical equilibrium. It should be observed that the photo-stationary state differs from ordinary chemical equilibrium, in that its permanency is wholly dependent upon the constancy of the source of illumination; i.e., when the light is cut off, the photo-stationary state shifts to the ordinary chemical equilibrium, provided the reaction is reversible. . It has also been found that the temperature coefficients of most photochemical reactions are negligible. Becquerel f discovered that silver chloride, which had been precipitated in the dark, was only sensitive to short wave-lengths of light, whereas silver chloride, which had been exposed for a few moments to sunlight, became sensitive to all wave-lengths r in the visible spectrum and to the shorter wave-lengths of the infra-red. This phenomenon, which has been observed with various sub- stances, was first studied systematically by Bunsen and Roscoe | who termed it photochemical induction. * Zeit. phys. Chem., 22, 23 (1897). t Ann. Chim. Phys. [3], 9, 257 (1843). j Pogg. Ann., 100, 481 (1857). PHOTOCHEMISTRY 519 Employing a mixture of hydrogen and chlorine gases they found that under constant illumination, the velocity of formation of hydrochloric acid, which was hardly appreciable at first, increased rapidly to a maximum and then remained constant. The interval of time required for the reaction to attain its maximum velocity is known as the period of induction. It has been found that in almost every photochemical reaction there is a similar period of initial perturbation. The phenomenon has been thoroughly investigated by Burgess and Chapman * who arrived at the conclusion that induction effects are to be ascribed to the presence of minute traces of various impurities, such as gases or water vapor, adsorbed by the walls of the reaction-vessel. We may therefore conclude that induction effects are not char- acteristic of photochemical reactions. Classification of Photochemical Reactions. According to Sheppard ("Photochemistry") photochemical reactions may be conveniently classified in the following manner : (1) Reversible reactions, i.e., reactions in which the products formed under the influence of light react to reproduce the original system when the light is removed. (2) Irreversible reactions, i.e., reactions in which the light pro- motes transformation to a more stable system. Irreversible re- actions are subdivided into (a) Complete reactions; and (b) Pseudo-reversible reactions. The polymerization of anthracene may be taken as an example of a reversible photochemical reaction. This reaction may be represented as taking place according to the equation light 2 Ci 4 Hi <= C 2 sH 20 , dark the upper arrow indicating the direction of the light-reaction (poly- merization) and the lower arrow that of the dark-reaction (de- polymerization) . I An illustration of a completely irreversible photochemical re- action is afforded by a mixture of hydrogen and chlorine gases * Jour. Chem. Soc., 89, 1402 (1906). 520 THEORETICAL CHEMISTRY which combine, on exposure to light, to form hydrochloric acid gas, according to the equation light H 2 + C1 2 - 2 HC1. If the hydrochloric acid is removed by solution in water as fast as it is formed, the velocity of the reaction will be directly proportional to the intensity of the light. Bunsen and Roscoe * made use of the foregoing facts in the construction of their actinometer. The reduction of ferric oxalate may be taken as an example of a pseudo-reversible photochemical reaction. This substance is reduced to ferrous oxalate on exposure to light as shown by the equation light Fe 2 (C 2 4 ) 3 - 2 Fe (C 2 O 4 ) + CO 2 . In the dark, ferrous oxalate is re-oxidized to ferric oxalate by the oxygen of the air. While the initial substance is reproduced, it is evident that the reaction is not strictly reversible. Actinometers. A number of different forms of apparatus have been devised for measuring the chemical action of light: such in- struments are known as actinometer s. The hydrogen-chlorine actinometer is based upon the well- known fact that the speed of the reaction between hydrogen and chlorine varies greatly with the intensity of illumination. Bun- sen and Roscoe, f guided by the experiments of Draper, con- structed an actinometer in which the rate of combination of hydro- gen and chlorine could be measured by allowing the hydrochloric acid formed to dissolve in water, and noting the resulting diminu- tion of volume. A diagram of this apparatus is given in Fig. 111. The apparatus is filled with a mixture of equal parts of hydrogen and chlorine, obtained by the electrolysis of a solution of hydro- chloric acid. The bulb A, containing water, is connected at one end with a tube fitted with a stop-cock B, and at the other end with a horizontal tube terminating in a reservoir D, which also contains water. When the water has become saturated with the constitu- * " Photochemische Untersuchungen," Ostwald's Klassiker, No. 34. t Fogg. Ann., 100, 43 (1857); 101, 235 (1857). PHOTOCHEMISTRY 521 ents of the gaseous mixture, B is closed and the entire apparatus is protected from light. When it is desired to measure the photo- chemical action of a source of light, the bulb A is uncovered and the light is allowed to fall upon it. Some of the hydrogen and chlorine will combine, and the hydrochloric acid formed will be absorbed by the water in A ; the column of water in the horizontal Fig. 111. tube will move to the right, the magnitude of the movement being measured on the scale C. In this way the amount of photochemi- cal action can be determined. An objection to the use of hydrogen and chlorine in the actinometer is the danger of violent explosions when the illumination is too intense. To remove this objection, Burnett replaced the hydrogen of the mixture by carbon monoxide. Action of Light on the Silver Halides. Probably the most familiar, and undoubtedly one of the most important, photo- chemical reactions is that which takes place on the exposure of a photographic plate.* Luther f has shown that when a pure silver halide is exposed to light, it undergoes reduction according to the equation light 2 AgX -> Ag 2 X + X, where X may be chlorine, bromine, or iodine. On removing the light, the sub-halide recombines with the free halogen as shown by the equation dark Ag 2 X + X -> 2 AgX. In other words, the reaction is strictly reversible and a well- defined photo-stationary state results from a given intensity of * For an excellent treatment of the chemistry of photography the student is recommended to consult "Photography for Students of Physics and Chem- istry," by Louis Derr (Macmillan); or "Photochemie und Beschreibung der photographischen Chemikalien," by H. W. Vogel. t Zeit. phys. Chem., 30, 628 (1899). 522 THEORETICAL CHEMISTRY illumination. The investigations of Baker * make it appear quite probable that the photochemical reduction of the silver halides is dependent upon the presence of a minute trace of water vapor as a catalyst. In the photographic plate, the silver halide is embedded in gela- tine, which not only accelerates the rate of reduction of the silver halide but also causes the reaction to become irreversible. The alteration in the behavior of the silver halide, brought about by the presence of gelatine, is due to the fact that the latter reacts with the free halogen according to the equation light 2 AgBr + gelatine Ag 2 Br + brominated gelatine. The continuous removal of the liberated bromine by the gelatine is an example of what is known as photochemical sensitization. Another example of photochemical sensitization is afforded by a mixture of benzene and silver chloride. The normal darkening of the silver salt is markedly increased by the presence of the benzene, which combines with the chlorine as rapidly as it is set free by the action of the light on the silver halide. While the silver bromide of the photographic plate is extremely sensitive to the shorter wave-lengths corresponding to the violet and ultra-violet regions of the spectrum, it is only reduced by the longer wave-lengths after prolonged exposure. It has been found, how- ever, that the addition of certain dyestuffs, such as eosine and Congo-red, renders the silver halide sensitive to the longer wave- lengths of the spectrum. This phenomenon, which is known as optical sensitization, must be carefully distinguished from chemical sensitization to which reference has already been made. It is to be noted that an optical sensitizer does not absorb chemically any of the products of the reaction. Photographic plates which have been sensitized in this way are commonly known as ortho-chro- matic plates. All of the dyestuffs which can function as optical sensitizers have been found to exhibit anomalous refraction; i.e., for wave- lengths slightly longer than those absorbed, these substances possess an abnormally large refractive index, in consequence of * Jour. Chem. Soc., 61, 782 (1892). PHOTOCHEMISTRY 523 which the refracted waves exert the same effect upon the silver halide as the shorter wave-lengths of the spectrum. Although it is not an essential property, it is generally found that optical sen- si tizers are fluorescent. As an example of optical sensitization, where the sensitizer is non-fluorescent, we may take the photochemical reduction of mercuric chloride in the presence of ammonium oxalate. This re- action takes place according to the equation light 2 HgCl 2 + (NH 4 ) 2 C 2 4 - Hg 2 Cl 2 + NH 4 C1 + 2 CO 2 . The presence of the non-fluorescent ferric ion, Fe"", has been shown by Winther * to be an effective optical sensitizer in the re- action. The ferric ion is reduced to the ferrous state while the oxalic acid undergoes oxidation by the mercuric chloride. It was pointed out by Eder f that this reaction is well adapted for actinometric measurements, since the amount of mercurous chlo- ride precipitated is directly proportional to the intensity of the light. Photochemical After-Effect. Certain photochemical reac- tions have been discovered in which the reaction proceeds even after the light stimulus is removed. This phenomenon is known as the photochemical after-effect. The velocity of a reaction of this kind is different from that of either the light- or the dark- reaction. It has also been found that if a portion of the reac- tion-mixture, which has already been exposed to the light, be added to a fresh unexposed portion, the latter immediately com- mences to decompose. Thus, a solution of iodoform in chloro- form becomes brown, on exposure to light, due to liberation of iodine. If the solution is removed from the light, the decomposition will continue for several days, and if a small portion of the partially de- composed solution be added to a freshly prepared solution, the latter will commence to decompose. The photochemical after- effect can be readily explained if we assume that the action of light gives rise to heterogeneous nuclei which persist for a sufficient time * Zeit. wiss. Phot., 7, 409 (1909). t Sitz. her. Wien. Akad. (1879). 524 THEORETICAL CHEMISTRY after the light is removed to act as centers around which the reaction can proceed throughout the unexposed portion of the mixture. Assimilation of Carbon Dioxide. A photochemical reaction of special interest to the biologist is that in which atmospheric carbon dioxide is taken up by plants under the influence of solar radiation. While the reaction is exceedingly complex, it may be regarded as taking place according to the hypothetical equation light 6 C0 2 + 5 H 2 O -> C 6 H 10 5 + 6 2 . (starch) This reaction represents a gain in energy amounting to approxi- mately 685 calories per formula- weight of starch. It is quite probable, however, that the initial product of the reaction is for- maldehyde and that subsequently the latter substance undergoes polymerization with the formation of starch and other carbo- hydrates. The green coloring matter of the leaves of plants, known as chlorophyll, also plays an important part in the assimi- lation of carbon dioxide, but beyond the fact that its action is not catalytic, little can be stated as to the manner in which it functions in the reaction. The velocity of the reaction has been found to attain its maximum value in yellow and green light, a result which is in complete agreement with the fundamental law of Grott- huss that only those rays which are absorbed are active chemi- cally. As has already been stated, the temperature coefficients of photochemical reactions are generally very small. This is not true, however, of the reaction under consideration. It has been found that the rate at which carbon dioxide is assimilated by a plant is nearly doubled for a rise in temperature of 10 (see p. 377). Photochemical Synthesis. While it has long been known that light is capable of effecting the synthesis of complex organic compounds in the living plant from carbon dioxide and water, it is only recently that its efficiency in bringing about a great variety of organic reactions has been fully recognized. Owing to the re- searches of Ciamician and others in this field, numerous photo- chemical syntheses of considerable practical value to the organic PHOTOCHEMISTRY 525 chemist have been discovered. It must suffice here to mention a few typical photochemical syntheses. Alcohols may be oxidized in successive steps, the action of the light closely resembling the action of ferments. Methyl alcohol and acetone react to form iso- butylene glycol according to the equation CH 3 ^ ht CH 3 CH 3 - OH + CO COH - CH 2 OH CH 3 CH 3 The action of light has been found to be especially favorable to such processes as the foregoing, involving reciprocal oxidation and re- duction. Ortho-nitro-benzaldehyde in the presence of ethyl alcohol reacts to form the ethyl ester of o-nitroso-benzoic acid as shown by the equation N0 2 ,0 *t NO ,0 '\ H ^ + CH 2 -OH~ Photoelectric Cells. The absorption of light by any one of the three states of matter is invariably accompanied by a change in electrical condition. The different electric effects accompany- ing the absorption of radiant energy have been classified by Shep- pard in the following manner: - (1) lonization with a corresponding increase of conductance in gases, liquids, and solids caused by transmission of light. (2) Indirect ionization of a gas due to reflection or emission of electrons from the surface of a contiguous denser phase. (3) Development of electromotive forces in cells of the following type: Electrode A (light) Conductor, Di-electric Electrode A (dark) where the two electrodes are separated by a medium which is par- tially a conductor and partially a di-electric. 526 THEORETICAL CHEMISTRY The treatment of the first and second of these three classes of photoelectric phenomena properly lies within the domain of pure physics. The third class, however, includes a number of photo- galvanic combinations of considerable interest and importance to the physical chemist. The first investigation of photoelectric combinations was under- taken by E. Becquerel * in 1839. He prepared a cell consisting of identical plates of pure silver, coated with a silver halide, and immersed in dilute sulphuric acid as an electrolyte. On exposing one electrode to the light, while the other was kept in the dark, an appreciable electromotive force was developed, the current flowing in the solution from the darkened to the illuminated electrode. If a galvanometer be included in the circuit, the deflection of the needle may be taken as a measure of the intensity of the light. The action of the Becquerel actinometer has been explained by Ostwald as follows : The incident light lessens the stability of the silver iodide, which undergoes ionization according to the equation AgI->Ag+I': the Ag* ions give up their charges to the electrode while the I' ions enter the solution. For every Ag' ion which is discharged at the illuminated electrode, an equal number of Ag* ions enter the solution at the darkened electrode, thus charging the latter nega- tively. Hence the current flows in the solution from the dark- ened to the illuminated electrode. Rigollot f has constructed an electrical actinometer in which the silver plates used by Becque- rel are replaced by two oxidized copper electrodes, while instead of dilute sulphuric acid a dilute solution of sodium chloride is used. A number of photoelectric combinations have been investigated by Wildermann, J among which he recommends the following as being especially suitable for actinometers : * Ann. Chim. Phys. [3], 9, 257 (1843). t Jour, de Phys. [3], 6, 520 (1897). j For a thorough treatment of theoretical photochemistry, the student is referred to Sheppard's "Photochemistry," while Plotnikow's " Photochemische Versuchstechnik " is recommended as an excellent guide to practical laboratory methods. PHOTOCHEMISTRY 527 (a) Ag | AgBr, 0.1 N KBr | AgBr | Ag (light) (dark) Reaction, 2 AgBr > Ag2Br, (b) Cu | CuO | NaOH | CuO | Cu (light) (dark) Reaction, 2 CuO -> Cu 2 O. Of these two cells, the latter is perhaps the better, since it gives an appreciable electromotive force for light of the same intensity and varying wave-length. While all of the theoretical interpretations of the action of photo- electric cells are more or less inadequate, the investigations of Scholl and others make it appear probable that negative electrons enter the electrolyte from the electrode, while positive ions are discharged on the electrode, thereby causing anodic polarization. In general the direction of the photoelectric current inside the cell is from the non-exposed to the exposed electrode. INDEX OF NAMES Abegg, 218. Alexieeff, 172, 174. Amagat, 74, 75. Andrews, 105. Angstrom, 141. Arrhenius, 211, 227, 368, 377, 392, 413, 425, 428, 432. Aston, 63, 419. Avogadro, 9, 76. Babo, 208. Baker, 521. Baly, 140, 141, 142. Barlow, 159. Barnes, 262. Bassett, 395. Bates, 430. Beccaria, 385. Bechhold, 259. Beckmann, 220. Becquerel, 42, 518, 526. Beer, 515. Bergmann, 312. Berkeley, Earl of, 197, 200, 201. Bernoulli, 76. Berthelot, 289, 308, 313, 323, 326. Berthollet, 4, 312. Berzelius, 16, 386. Bigelow, 381. Biltz, 260, 429. Bingham, 119. Biot, 132. Blagden, 217. Blake, 253, 254. Bodenstein, 317, 518. Boettger, 415. Boguski, 377. Boltwood, 44, 56. Boltzmann, 76, 507. Boyle, 2, 49, 72, 76, 170, 192, 197, 286. Bragg, W. H., 160. Bragg, W. L., 160. Bredig, 276, 381, 409. Brown, 279, Bruhl, 127. Bruni, 250. Budde, 81, Buff, 121. Bunsen, 169, 170, 514, 515, 518, 520. Burgess, 519. Burton, 253. Cailletet, 113. Callendar, 201. Carlisle, 385. Chapman, 519. Charpy, 353. Ciamician, 524. Claude, 115. Clausius, 76, 392, 413. Clement, 96, 382, 383. Cotton, 252. Crookes, 33, 47, 48. Gumming, 472. Curie, 43, 44, 49, 56. Dale, 126, 152. Dalton, 1, 7, 16, 168. Davy, 385, 386. Debierne, 44. Debray, 328. 529 530 INDEX OF NAMES Debye, 161. De Chancourtois, 21. De Forcrand, 334. Denham, 480. Desch, 142. Desormes, 96, 382, 383. Deville, 88, 321. De Vries, 201, 202, 203, 204, 230. Dewar, 114. Dobereiner, 21. Draper, 520. Drude, 151. Duclaux, 248. Dulong, 11, 162. Dumas, 20. Dutoit, 419. Eder, 523. Einstein, 283, 285. Eotvos, 146. Fanjung, 418. Faraday, 7, 138, 150, 268, 389. Faure, 499. F&y, 510. Fick, 206, 208. Forbes, 466. Fletcher, 285. Freudenberg, 502. Freundlich, 254, 264, 269, 270. Friderich, 419. Fuchs, 493. Gay-Lussac, 8, 76, 192, 196, 197, 208. Geiger, 40, 52, 54, 55, 57. Geoffrey, 312. Gibbs, 273, 274, 338, 448. Gladstone, 126, 152. Goodwin, 468. Gouy, 279. Graham, 206, 237, 274, 276. Grotthuss, 391, 513, 524. Grove, 392. Guldberg, 314, 329. Guthrie, 349. Hall, 66. Hamburger, 204, 205. Hampson, 114. Hardy, 253, 256. Harkins, 63, 66. Hartley, 140, 142, 144, 197. Hautefeuille, 317. Hatty, 159. Hedin, 205. Helmholtz, 448, 477. Henry, 170, 184. Henderson, 472. Hess, 290, 304, 305. Heycock, 353. Heydweiller, 414. Hittorf, 393. Holborn, 478, 510. Horstmann, 331. Hulett, 181. Hull, 505. Isambert, 330. Jones, 219, 233, 395, 418. Jurin, 144. Kirchhoff, 506. Knoblauch, 375. Koelichen, 426. Kohlrausch, 400, 413, 414, 415, 416, 417, 478, 488. Konowalow, 175, 176. Kopp, 14, 88, 120, 121, 127. Kraus, 430. Kroenig, 76. Kundt, 98. Kurlbaum, 510. Laborde, 56. Landolt, 4. Langbein, 302. Laplace, 98, 290. Larmor, 373. Lavoisier, 2, 286, 290. Lebedew, 505. INDEX OF NAMES 531 Le Bel, 134, 135, 137. Le Blanc, 493, 495, 497, 501. Le Chatelier, 95, 309. Lehmann, 165. Lemoine, 317. Lenard, 36. Lewis, 162, 472, 474. Liebig, 382. Lillie, 249. Linde, 114. Lindemann, 164. Linder, 243, 251, 254. Lippmann, 459. Lodge, 411. Loomis, 218. Lorentz, 126, 152. Lorenz, 126, 152. Lessen, 121. Ludeking, 267. Lummer, 507. Lund&i, 442. Luther, 521. Mac Innes, 472. Marignac, 5, 16. Maxwell, 76. Mayer, 94, 290. Mendeleeff, 22. Menschutkin, 378. Meyer, 466. Meyer, Lothar, 22, 28. Meyer, Victor, 83. Millikan, 40, 285. Mitscherlich, 14. Morgan, 149, 150. Morley, 16. Morse, 194, 196, 197, 198, 200. Moseley, 60. Mouton, 252. Natanson, 91. Nernst, 119, 141, 151, 164, 297, 335, 336, 419, 451, 472, 475, 476, 486, 517. Neumann, 13. Neville, 353. Newlands, 22. Newton, 312. Nicholson, 385. Nollet, Abb6, 188. Nordlund, 285. Noyes, 372, 377. Nuttall, 52. Ohm, 388. Olszewski, 113. Onnes, 116. Ostwald, W., 8, 147, 169, 191, 231, 273, 287, 368, 379, 383, 393, 422, 424, 435, 447, 526. Ostwald, Wo., 245. Palmaer, 453. Pappada, 250. Pasteur, 132, 133, 134. Pauli, 254. Pebal, 90. Perkin, 138. Perrin, 35, 250, 279, 280, 281, 282, 283, 285. Petit, 11, 162. Pfeffer, 188, 189, 190, 191, 192, 193, 194, 198. Philip, 170. Pictet, 113. Picton, 243, 251, 254. Planck, 40, 207, 472, 474. Plant6, 499. Pope, 159. Pringsheim, 507. Proust, 5. Prout, 20. Pulfrich, 124. Quincke, 250. Ramsay, 49, 146, 147, 148, 149, 150, 168. Raoult, 209, 210, 212, 215, 217, 218, 223. 532 INDEX OF NAMES Rayleigh, 509. Regener, 40, 54. Regnault, 83. Reicher, 368. Reinitzer, 165. Reuss, 250. Richards, 16, 159, 466. Rigollot, 526. Roberts- Austen, 185, 35$, 355. Rodewald, 267. Roentgen, 42. Roozeboom, 351, 353. Roscoe, 177, 514, 515, 518, 520. Rose, 313. Royds, 55. Rudolphi, 430. Rutherford, 40, 45, 46, 48, 49, 50, 54, 55, 57. Sargent, 472, 474. Schiff, 121. Scholl, 527. Schroeder, 265. Shields, 146, 147, 148, 149, 150. Siedentopf, 241. Snell, 124. Soddy, 48, 49, 50, 51, 52, 60. Soret, 208. St. Gilles, Pean de, 313, 323. Stas, 16, 20. Stefan, 507. Steno, 155. Stokes, 39, 283. Svedberg, 277. Tammann, 206. Thilorier, 113. Thomsen, 302. Thomson, J. J., 35, 45, 68, 282, 387, 419. Thorpe, 121. Traube, 122, 129, 188. Trouton, 118. Van der Stadt, 373. Van der Waals, 31, 81, 82, 107, 121, 171. Van't Hoff, 135, 137, 184, 185, 192, 193, 196, 197, 213, 217, 218, 223, 226, 324, 337, 378, 412, 430. Volta, 385, 446. Waage, 314, 329. Walden, 404. Walker, 117, 211, 376. Warder, 368. Washburn, 417. Weber, 12, 13. Weimarn, 245. Wenzel, 312. Whetham, 412. Whitney, 253, 254, 377. Wiedemann, 250, 267. Wien, 508, 509. Wiener, 279. Wildermann, 526. Williamson, 392. Wilson, C. T. R., 45. Wilson, E. D., 63. Winther, 523. Wladimiroff, 205. Wood, 512. Wroblewski, 113. Wullner, 208. Young, 183. Zsigmondy, 241, 259. INDEX OF SUBJECTS Abnormal solutes, 225. Absolute index of refraction, 125. Absorption and emission, 506. Absorption index, 514. Absorption of light, quantitative re- lation concerning, 514. Accumulators, 488. Actinometers, 520. Active deposits, 49. Activity and concentration, 478. Adsorption, 267. of gases, 268. in solution, 269. isotherm, 269, 270. Alloys, 353. Amicrons, 239. Association in solution, 225. Atomic theory, 6, 7. number, 58. structure, 57. structure, hydrogen-helium system of, 63. weights, 16, 18. weights and atomic numbers, re- lation between, 64. Autocatalysis, 381. Avogadro constant, 40. hypothesis of, 9, 79. Basicity of organic acids, 434. Bimolecular reactions, 366. Boiling point constant, 213. and critical temperature, 119. elevation of, 213. elevation and osmotic pressure, 216. Blood corpuscle method, 204. Brownian movement, 279. recent investigations of, 285. Calorimeter, combustion, 289. Capillary electrometer, 458. Carbon dioxide, assimilation of, 523. Catalysis, 379. mechanism of, 352. negative, 382. Cataphoresis, 251. Cathode particle, charge carried by, 38. ratio of charge to mass of, 38. velocity of, 36. Cathode rays, 33. properties of, 33. Cathodo-luminescence, 512. Cells, electromotive force of con- centration, 411. galvanic, 446. gas, 485. photo-electric, 525. reversible, 448. standard, 457. storage, 488. Chemical constitution, 132, 142. Chemi-luminescence, 512. Colloidal solutions, 238. density of, 245. electrical conductance of, 253. osmotic pressure of, 247. preparation of, 274. surface tension of, 247. viscosity of, 246. Colloids, 237. 533 534 INDEX OF SUBJECTS Colloids, molecular weight of, 249. precipitation by electrolytes, 254. surface energy of, 271. Components, 338. Compounds, 2, 7. Compressibilities of solid elements, 159. Concentration and activity, 478. elements, 465. Conductance, and ionization, 412. at high temperatures and pressures, 417. determination of, 400. equivalent, 398. molar, 398. specific, 398. Conductance of difficultly soluble salts, 415. of different substances, 402. of fused salts, 420. of non-aqueous solutions, 418. temperature coefficient of, 416. Conduction of electricity through gases, 31. Connection between gaseous and liquid states, 104. Consecutive reactions, 376. Corresponding conditions, 110. Co- volume, 121. Critical pressure, 105. temperature, 105. volume, 105. Crookes' dark space, 32. Cryohydrate, 347. Crystal form and chemical composi- tion, 158. Crystallography, 154. Crystalloids, 237. Crystals, liquid, 165. mixed, 14. properties of, 156. Crystallization methods, 275. Decomposition potential, 495. Degree of dissociation, calculation of, 91. Degree of dissociation, freedom, 339. Density of colloidal solutions, 245. Dialysis, 237. Dielectric constants, 150. Diffusion, coefficient of, 206. Dilution law, 422. Disintegration hypothesis, 50. series, 52. Dispersity, 238. Dispersoids, 239, 245. classification of, 244. Dissociation constant, 320. hypothesis, 437. in solution, 226. Distillation fractional, 178. steam, 179. Distribution of a solute between two immiscible solvents, 335. Drop weight, 149. Electrical conductance of colloidal solutions, 253. double layer, 452. Electrode, dropping, 463. hydrogen, 464. normal, 461. null, 463. Electrodes, of first type, 468. of second type, 468. Electrpendosmosis, 250. Electro-luminescence, 512. Electrolytes, ionization of strong, 429. mixtures with a common ion, 427. Electrolytic dissociation theory, 227, 229. separation of metals, 502. Electrometer capillary, 458. Electromotive force, measurement of, 455. Electrons, sources of, 41. Electron theory, 31. Elements, 2. classification of, 20. oxidation and reduction, 482. INDEX OF SUBJECTS 535 Elements, periodic series of, 24. Elevation of boiling point, 213. Emanations, 48. Emission and absorption, 506. Emulsions, 239. Emulsoids, 239. action of heat on, 257. precipitation of, 257. Energy evolved by radium, 56. distribution in spectrum, 509. Equation of condition, reduced, 111. Equation of Nernst-Lindemann, 164. van der Waals, 81, 107. Equilibrium constant, 316. variation with temperature, 324. Equilibrium, homogeneous, 312. in homogeneous gaseous systems, 317. in liquid systems, 322. Equivalent, chemical, 6. electrochemical, 390. Etch figures, 158. Eutectic, 353. Faraday dark space, 32. Ferments, inorganic, 381. Fluorescence, 511. Formation, heat of, 294. Freezing point, lowering of, 217. of concentrated solutions, 233. Fusion, heat of, 153. Galvanic cells, 446. Gas, ideal, 75. perfect, 75. Gases, adsorption of, 268. specific heat of, 100. Gas constant, molecular, 73. specific, 73. Gas laws, deviations from, 74. Gaseous and liquid states, connection between, 104. Gaseous molecule, mean velocity of translation of, 80. Gelation, 239. Gels, elastic, 265. hydration and dehydration of, 263. non-elastic, 264. physical properties of, 262. Gold-number, 259. Hsematocrit method, 205. Heat, of combustion, 302. of dilution, 297. of dissociation of solids, 334. of formation, 294. of fusion, 153. ol imbibition, 266. of ionization, 397, 432. of neutralization, 305. of reaction, variation with tempera- ture, 300. of solution, 295. of vaporization, 11$. Helium atoms and a-particles iden- tical, 55. Helium, rate of production of, 56. Helix of de Chancourtois, 21. Heterogeneous systems, 328. Homogeneous gaseous systems, equi- librium in, 317. Hydrogel, 328. Hydrolysis, 437. determination of, 442, 480. Hydrosol, 238. Imbibition, 266. in solutions, 267. velocity of, 266. Immiscibility, 178. Inorganic ferments, 381. Ionic product, 433. Ionization constant, 422. of water, 443. Ionization of gases, 45. Ionization, heat of, 432, 484. influence of substitution on, 435. of water, 487. lonogen, 229. Ions, 45. 536 INDEX OF SUBJECTS Ions, absolute velocity of, 409. existence of free, 391 . migration of, 393. Isohydric solutions, 428. Isomorphism, 14. Isothermals, 105. Isotopes, 63. Isotonic coefficient, 203. Kinetic equation, deductions from, 78. derivation of, 77. theory of gases, 100. Kinetics, chemical, 359. Labile solutions, 183. Law of Beer, 141, 515. of Boyle, 73, 78. of combining .proportions, 5. of conservation of mass, 3. of constant heat summation, 290. of definite proportions, 4. of Dulong and Petit, 11. of Faraday, 389. of Gay-Lussac, 73, 79. of Graham, 80. of Grotthuss, 513. of Guldberg and Waage, 314. of Henry, 170, 171. of Hess, 290, 305. of Jurin, 144. of Kirchhoff, 506. of Kohlrausch, 406. of Lavoisier and Laplace, 290. of mass action, 314. of Mitscherlich, 14, 15. of molecular displacement, 283. of multiple proportions, 5. of Neumann, 14. of octaves, 22. of Ohm, 388. of Raoult, 209. of Richter, 4. of Stefan-Boltzmann, 507. of volumes, 8. of Wien, 507. Light, action on silver halides, 521. Liquefaction of gases, 112. Liquids, characteristics of, 104. refractive power of, 123. vapor pressure of, 117. Liquid systems, equilibrium in, 322. Luminescence, 511. Lyotrope series, 239. Magnetic rotation, 138. Mass, conservation of, 3. Mass action, law of, 314. Maximum work, principle of, 308. Mechanism of catalysis, 382. Membranes, semi-permeable, 191. Metastable solutions, 183. Microns, 238. Miscibility, complete, 174. partial, 172. Mixtures, specific refraction of, 128. Molar volume, 10. Molecular gas constant, evaluation of, 74. magnetic rotation, 139. refraction, 126. rotation, 131. vibration, 142. volume, 120. weight, 83, 146. weight by boiling point method, 215. weight by freezing point method, 220. weight in solution, 222. weight of colloids, 249. weight in solution, 222. vibration, 142. volume, 120. Neutralization, heat of, 305. Nitrogen peroxide, dissociation of, 93. Non-dissociating solvent, solution of solid in, 336. Normal electrodes, 461. INDEX OF SUBJECTS 537 Octaves, law of, 22. Optical activity, 132. Osmotic pressure, 187, 191. and boiling point elevation, 216. and diffusion, 206. and freezing point depression, 221. Osmotic pressure and lowering of vapor pressure, 211. comparative values of, 201. measurement of, 189. of colloidal solutions, 247. recent work on direct measure- ment of, 194. theoretical value of, 192. Oxidation and reduction elements, 482. Particles, a-, 55. Peptization, 276. Periodic law, 22, 64. applications of, 26. defects in, 29. Periodicity of physical properties, 25. of radio-elements, 61. Phase rule, 338. derivation of, 339. Phosphorescence, 511. Photochemical action, 513. after effect, 523. extinction, 515. induction, 518. reaction, kinetics of, 516. sensitization, 522. synthesis, 524. Photo-electric cells, 525. Photo-stationary state, 518. Physical properties of ionized solu- tions, 231. Plasmolytic methods, 201. Polarization, 492. capacity of electrodes, 494. electromotive force of, 492. theory of, 497. Polarized light, rotation of plane of, 129. Polymorphism, 158. Potential, between metal and solu- tion, 464. decomposition, 495. difference at junction of two solu- tions, 469. difference at liquid junctions, 472. normal electrode, 475. Precipitation and valence, 256. Precipitation of colloids by electro- lytes, 254. of emulsoids, 257. of suspensoids, 254. Preparation of colloidal solutions, 274. Principle of maximum work, 308. of Soret, 208. Properties of cathode rays, 33. of gels, 262. Proportions, law of combining, 5. law of definite, 4. law of multiple, 5. Protective colloids, 258. Prout's hypothesis, 20. Quantum Theory, 164. Radiant energy, 504. source of, 505. Radiations, nature of, 46. phosphorescence induced by, 46. photographic action of, 45. Radioactive constant, 50. equilibrium, 52. substances, 44. Radioactivity, 42. discovery of, 42. Radio-elements, periodicity of, 61. Radio-luminescence, 512. Radium, 43. discovery of, 43. energy evolved by, 56. Rays, a-, /3-, and 7-, 47. Ratio of charge to mass, 38. of specific heats, 96. 538 INDEX OF SUBJECTS Ratio of charge to mass, of specific heats, determination of, 97, 98. Reaction, determination of order of, 374. influence of solvent on velocity of, 378. velocity, 359, 374, 377. Reactions, at constant pressure, 298. at constant volume, 298. bimolecular, 366. counter, 374. heterogeneous, velocity of, 376. of first order, 364. of higher orders, 373. of second order, 366. of third order, 370. photochemical, classification of, 519. side, 374. trimolecular, 370. Reciprocal precipitation, 260. Refraction, index of, 124. molecular, 126. specific, 126. Residual current, 495. Resistance capacity, 401. Reversible cells, 448. Rotation, magnetic, 139. molecular, 131. of plane polarized light, 129. specific, 131. Salt solutions, thermoneutrality of, 305. Saturated solution, 181. Semi-permeable membranes, 187. Side reactions, 374. Solation, 239. Solids, general properties of, 153. heat capacity of, 161. Solubility coefficient, 171. i product, 433. Solutes, abnormal, 225. Solution, association in, 225. dissociation in, 226. Solution, heat of, 295. methods, 276. pressure, 450. Solutions, 167. absorption in, 269. classification of, 167. colloidal, 238. conductance of non-aqueous, 418. imbibition in, 267. of gases in gases, 167. of gases in liquids, 167. of liquids in liquids, 172. of solids in liquids, 181. isohydric, 428. labile, 183. metastable, 183. properties of ionized, 231. saturated, 169. solid, 184. supersaturated, 169. volume-normal, 196. weight-normal, 196. Solvate theory, 234. Solvent, ionizing power of, 419. Specific heat, 92, 93. and atomic weight, 11. at constant pressure and volume, 93. of gases, 100. of solid elements, 11, 161. ratio of, 96. Specific refraction, 126. rotation, 131. Spectra, absorption, 139. Standard cells, 457. Steam distillation, 179. Strength of acids and bases, 424. Sublimation, 153. Submicrons, 239. Sulphur, equilibrium between the phases of, 343. Surface concentration, 272. Surface energy of colloids, 271. Surface tension, 144, 146, 149. of colloidal solutions, 247. INDEX OF SUBJECTS 539 Suspensions, 239. Suspensoids, 239. precipitation of, 254. Systems, three-component, 356. two-component, 345. Theorem of Le Chatelier, 309. Theory, electron, 31. kinetic, 100. of electrolytic dissociation, 227, 229. of solvation, 234. Thermal units, 286. Thermochemical equations, 287. measurements, 288. Thermo-luminescence, 512. Thermoneutrality of salt solutions, 304. Transition point, 340. Transmission coefficient, 514. Transport numbers, 395. determination of, 395. electrometric determination of, 477. Triads, of Dobereiner, 21. Tribo-luminescence, 512. Tri molecular reactions, 370. Triple point, 342. Tyndall phenomenon, 241. Ultrafiltration, 242. Ultramicroscope, 241. Unimolecular reactions, 361, 364. Units, electrical, 388. Valence, 15. electrometric determination of, 476. Vapor density, 83. abnormal, 88. determinations of, 86. Vaporization, heat of, 118. Vapor pressure, 117. and osmotic pressure, connection between, 211. lowering of, 209. of liquids, 117. Variation of equilibrium constant with temperature, 337. of heat of reaction with tempera- ture, 300. Velocity constant, 315. of gaseous molecule, 80. of imbibition, 266. Vibration, curves of molecular, 142. Viscosity of colloidal solutions, 246. Voltaic pile, 385. Volume, molar, 10. Volumes, Gay-Lussac's law of, 8. Water, dissociation of, 414. equilibrium between phases of, 340. ionization constant of, 443. primary decomposition of, 501. Weight, atomic, 8, 10, 17, 18. combining, 5, 8. gram-molecular, 10, 17. molar, 10, 17. X-rays and atomic structure, 58. and crystal structure, 159. spectra, 60. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. 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