LIBRARY OF THE UNIVERSITY OF CALIFORNIA. Class WORKS OF PROF. J. L. R. MORGAN PUBLISHED BY JOHN WILEY & SONS. The Elements of Physical Chemistry. Third Edition, Revised and Enlarged, izmo, xii + sio pages. Cloth, $3 oo. Physical Chemistry for Electrical Engineers. i2mo, viii + 230 pages. Cloth, $1.50 net. TRANSLATION. The Principles of Mathematical Chemistry. The Energetics of Chemical Phenomena. By Dr. Georg Helm, Professor in the Royal Technical High School, Dresden. Authorized Translation from the German by J. Livingston R. Morgan, A.M., Ph.D., Adjunct Professor of Physical Chemistry, Columbia University. lamo, viii + 228 pages, cloth, $1.50. PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS BY J. LIVINGSTON R. MORGAN, PH.D. Professor of Physical Chtmistry in Columbia University FIRST EDITION FIRST THOUSAND NEW YORK JOHN WILEY & SONS LONDON: CHAPMAN & HALL, LTD. 1906 Copyright, 1906 BY J L. R. MORGAN ROBERT DRUMMOND, PRINTER, NEW YORK PREFACE. THIS book is intended not only for the professional electrical engineer, but also for the use of all those who have the same object in view, viz., the attainment of a knowledge of physical chemistry sufficient in its scope for the understanding of current work in electrochemistry. Although electrochemistry lies in that border-land between chemistry and electricity, it has been so con- sistently neglected by the majority of the workers in both fields, that its development has rested almost entirely in the hands of a comparatively small number of specialists. The time has now arrived, however, when the results obtained by these men are generally recognized as of exceeding importance, no less from the practical than the theoretical standpoint, and consequently a working knowledge of the subject has become a very essential part of the equipment of the student who is to devote himself to either branch of science. But even a glance on the part of such a student at one of the electrochemical contributions which at present fill so many pages of our journals is sufficient to make clear the fact that the entire work is based upon an utterly new and unfamiliar set of laws and concepts; in short, upon a knowledge of certain portions of physical iii IV PREFACE. chemistry, and that without this knowledge any at- tempt to understand the attaining of the result, or even its meaning when attained, is futile. The purpose of this book is to aid those who find them- selves in such a position, and to present, in as popular a form as is consistent with quantitative results, those laws and generalizations of physical chemistry which form the basis of the subject embracing the chemical application of electricity and the electrical applica- tion of chemistry. It is in nowise to be regarded as a text-book of electrochemistry, however, for although that subject is discussed in considerable detail, only those portions of it are considered which best illustrate the application, and methods of application, of the physical chemical principles already presented, and no com- plete account is attempted. But notwithstanding this the reader should gain a very clear and just idea of the elements of electrochemistry, and one which he can readily elaborate by further work. While a cursory glance might easily lead one to infer that some portions which are treated at length are utterly unnecessary for the electrochemist, such is not the case, for everything presented, if not vitally important in electrochemistry itself, is absolutely requisite for the understanding of something else, which is absolutely indispensable to it. The reader is cautioned, therefore, not to omit anything, thinking it unessential or trivial, for everything is given with a definite object in view. Since the subject of solution is also the most important portion of physical chemistry for those specializing in other branches of chemistry, the book is also adapted to the use of chemists who have but a limited time to devote to the subject, or whose need is restricted to a PREFACE. V knowledge of the behavior of substances in solution. And it may also serve as an introduction to a more com- plete course in physical chemistry, for the numerous references, both to originals and to a more elaborate work,* will enable the student to readily investigate further any point upon which his interest may happen to be aroused. In order that the reader may gain a working idea of the subject, a collection of problems, together with their answers, is given in the final chapter. The solution of these will not only make the principles of the subject more real and clear, but will serve to impress more deeply upon the reader's mind the essential and gener- ally useful portions of the various laws. But one other point need be mentioned here. Throughout the presentation / have avoided the use oj any hypothesis, feeling that, by placing the subject upon an absolutely experimental basis, giving a practical experi- mental definition of each concept as it is used, and draw- ing no inference not justified in all its parts by actual results, the reader's idea will be the more clear and scientific. J. L. R. M. COLUMBIA UNIVERSITY, September, 1905. * Morgan, The- Elements of Physical Chemistry, 30! edition, 1905. CONTENTS. CHAPTER I. PAGB SOME FUNDAMENTAL PRINCIPLES i Atomic and molecular weights, I. Energy, u. The factors of energy, 13. CHAPTER H. THE GENERAL PROPERTIES OF GASES 16 The gas laws, 16. Dissociation, 23. Partial pressures and concentrations, 30. CHAPTER III. HEAT AND ITS TRANSFORMATION INTO OTHER FORMS OF ENERGY 34 The first law of thermodynamics, 34. The second law of thermodynamics, 44. CHAPTER IV. SOLUTIONS 50 The formula (molecular) weight in the liquid and solid states, 50. Osmotic pressure, 54. Vapor pressure, 68. Boil- ing-point, 70. Freezing-point, 72. Coefficient of distribu- tion, 74. Electrolytic dissociation or ionization, 76. The thermal relations of electrolytes, 90. CHAPTER V. CHEMICAL MECHANICS. 97 The law of mass action, 97. Equilibrium in gaseous sys- tems, IQI. Equilibrium in liquid systems, 114. The effect vii viu CONTENTS. PAGE of temperature upon the equilibrium-constant, 118. Velocity of a chemical reaction, 124. Reactions of the first order. Catalysis, 128. Reactions of the second order, 130. CHAPTER VI. EQUILIBRIUM IN ELECTROLYTES 133 Organic acids and bases. The Ostwald dilution law, 133. Acids, bases, and salts which are ionized to a considerable extent. Empirical dilution laws, 140. The heat of ionization, 145. Solubility or ionic product, 148. Hydroly tic dissociation, or hydrolysis, 157. Ionic equilibria, 168. The color of solu- tions, 170. CHAPTER VII. ELECTROCHEMISTRY. 172 The migration of ionized matter. Faraway's law. Electrical units, 172. The migration of ionized matter, 174. The conductivity of electrolytes. The specific, molar, and equivalent conductivities, 180. Ionic conductivities, 181. Empirical relations, 187. The ionization of water, 188. The solubility of difficultly soluble salts, 189. Electromotive force. The chemical or thermodynamical theory of the cell, 191. The osmotic theory of the cell, 193. Differences of potential. Calculation of the electrolytic solution pressure, 202. The heat of ionization, 204. Concentration cells, 205. Determina- tion of ionization from electromotive-force measurements, 209. The processes taking place in the cells in common use, 211. Electrolysis and polarization. Decomposition values, 214. Primary and secondary decom- position of water, 218. CHAPTER VIII. PROBLEMS > 220 OF THE ( UN1VER- PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. CHAPTER I. SOME FUNDAMENTAL PRINCIPLES. Atomic and molecular weights. From the very defi- nition of physical chemistry that portion of science which has for its object the study and investigation of the laws- governing chemical phenomena it is apparent that, we have to do with the most general and inclusive of all the branches of chemistry. Certain of its funda- mental concepts, therefore, are those which are also fundamental to all these branches, and it is essential that we at least recall them to mind before proceeding to the consideration of the things based upon them. Two of the concepts common to all branches of chem- istry, the atomic weight and the molecular weight, play an especially important r61e in physical chemistry, as they do, indeed, in all the other branches of chemistry. It is unfortunate, however, that the student's usual idea of these is such an utter confusion of fact and hypothesis that no very clear conception of their meaning is pos- sible. This is particularly evident at the present time, 2 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. when it is by no means an uncommon thing to find those who actually fear for the security of the quantitative, experimental results of chemistry, because of the pos- sibility of the so-called atom proving divisible; in other words, making the definition of the hypothetical atom, and hence also that of atomic weight, a meaningless col- lection of words. Without in any way condemning these hypotheses more than the others in common use, it must be ad- mitted as far better to recognize that our science is based upon no such insecure and uncertain foundation, but upon facts which will continue to exist under any hypothesis, or lack of hypothesis, and which at any time can be confirmed by experiment. That such is the foundation of chemistry is undeniably true, but, un- fortunately, the lay mind, and, still more unfortunately, the average professional mind, refuses utterly to recog- nize and acknowledge it. As in our present work we are to depend solely upon fact, avoiding all hypothesis, it is quite essential that we secure from the very beginning a hypothesis-free idea of atomic and molecular weight. The first step toward this end, however, is to define exactly what we mean by the word hypothesis, for common usage indicates that its significance is not uniform. A hypothesis, in the sense we are to use it, is a theory into which is incor- porated something that is foreign to the facts observed; the word theory, as we shall continue to employ it, signifying a generalization of observed facts, containing nothing beyond what is expressed by these facts. A theory, then, is a law of nature holding between certain well-defined, if narrow, limits, which becomes a general law when the limits have been sufficiently extended. A SOME FUNDAMENTAL PRINCIPLES. 3 mathematical formula, thus, provided each term is de- terminable by experiment, is no hypothesis, but a theory or a general law, according as its limits are restricted or wide. A hypothesis, on the other hand, when in the. form of an equation, contains a term or terms which cannot be determined by direct experiment; and in any other form contains assumptions foreign to the facts observed, assumptions which sooner or later must be found to be in disagreement with fact.* In our work, therefore, we must test each concept as it arises in order to see that it is free from hypothesis, and to be assured that we have to do solely with facts. Later, naturally, these facts, when supplemented by others, may be included in more general laws, but facts they will always remain, notwithstanding, and independ- ent of any hypothesis which at that time may happen to be in vogue. Our progress will thus be continuous, and it will never become necessary to halt for the purpose of inquiring what is fact and what hypothesis. At worst, in this way (retaining hypotheses), one may be certain that the greater emphasis is laid upon the facts themselves, and that the hypotheses can lead to no serious confusion. At best, on the other hand (relin- quishing all hypotheses, as we shall do), we can devote all our energy to the facts and their generalization, and thus avoid wasting the time and effort requisite for the making of unwarranted assumptions, which the progress of science must inevitably prove to be utterly untenable. That any one who has read of the determination of * For a masterly discussion of these things see Ostwald, Vorlesungen uber Naturphilosophie, pp. 202-227, r 9 O2 > a book to which I am in- debted for this point of view. 4 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. an atomic weight can retain the idea that it depends in any way other than name upon hypothesis, seems impossible, and yet the proof of the non-existence of the hypothetical atom would seriously disturb the ideas of many such, notwithstanding. That an event of the sort could affect the actual results of the science, and certainly the actual results constitute the science, is not to be imagined for an instant, however. In the words of Ostwald the stoichiometrical laws will continue to be a part of chemistry, even after the time when the atom is only to be found in the dust of the libraries. The atomic weight of an element, as we determine it, is simply the weight which will combine with 16 units of weight of oxygen, or some small multiple or sub- multiple of 16; or which will combine with the atomic weight so determined, or some small multiple or sub- multiple of it, of any other element. In each case, there- fore, the atomic weight is an experimentally determined combining weight. Which combining ratio is to be selected in case two are discovered (CO and CO 2, for example), and which weight of oxygen is to be employed as the standard for any one element, are matters that are de- pendent only upon other facts, i.e., those relating to com- pound substances. An atomic weight, therefore, has no vital connection with the atomic hypothesis, but is based solely upon the experimental law that elements combine with one another only in the proportions of their combining weights, or of small multiples of them. Naturally, if atoms exist (in accordance with the usual definition) they must be individually related in weight as the experi- mentally determined combining weights, called atomic weights; but, whether atoms exist or not, these com- SOME FUNDAMENTAL PRINCIPLES. 5 bining weights must continue to be true, for they are facts, independent of time, and can always be tested by experiment. A moment's thought, indeed, will show that the connection between the hypothesis and the actual results is always as slight as this. Whenever we make use of the word atom, in practical work (for no other use is warranted), it is the atomic weight, the gram- atom, the experimentally determined combining weight that is intended, and never the hypothetical atom. It is only the name atomic weight, then, which leads to the inference that the atomic hypothesis is fundamental to chemistry. To avoid all possibility of such a mis- conception here, we shall employ throughout the book the word combining weight, meaning by it that combining weight which is usually designated as the atomic weight. Much that has been said concerning the atomic weight is also true of the molecular weight. For the molecular weight, as we determine it in the gaseous state (we shall consider the other states of aggregation later), is the actually observed weight of gas which occupies the same volume as 32 units of weight of oxygen under like con- ditions of temperature and pressure. And again here, if we assume the gas to be composed of ultimate particles (molecules) of uniform size, the weight of the molecule of one substance will be related to that of another as are the molecular weights. But, whether molecules exist or not, the so-called molecular weights are experiment- ally observed facts, and as such are independent of hypothesis. According to hypothesis the molecular weight of a substance is equal to the sum of the atomic weights of the constituents. Speaking solely from the experi- mental standpoint, the molecular weight is that weight, 6 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. found by a summation of the respective combining weights (p. 5), which will occupy the same volume as 32 units of weight of oxygen. As the various symbols of the elements represent the combining weights of these elements, and the number of combining weights (atoms) is always given by the subnumerals, the molecular weight thus being indicated by the formula, we shall designate throughout, in our later work, the so-called molecular weight as the formula weight of the substance. Any method, then, for the determination of the formula (excluding the analytical methods, for they enable us only to ascertain the ratios of the weights of the elements combining) will offer a means of finding the molecular, i.e. the formula weight. And in place of the expression, the molecular (formula) weight in grams, we shall employ the abbreviation suggested by Ostwald, and call it the mole.* A glance at the methods of using the conceptions of atomic and molecular weight is amply sufficient to show that it is upon the practical results that everything is based, and not the hypothesis. According to hypothesis, the equation H 2 + C1 2 =2HC1 means that i molecule (2 atoms) of hydrogen and i mole- cule (2 atoms) of chlorine unite to form 2 molecules of hydrochloric acid gas. This, of course, may be true, * Since the normal solution of analytical chemistry contains one equiva- lent mole per liter, we shall use the word molar when speaking of the number of moles per liter. In certain cases, then, hydrochloric acid for example, the two expressions will be identical, while in others, sulphuric acid, for instance, the molar solution will contain twice as much as the normal one, etc. SOME FUNDAMENTAL PRINCIPLES. 7 and it may not, for we can neither prove it nor yet dis- prove it, but, whether it be true or false, the fact still remains that 2 units of weight (i.e. 2 combining weights, or i formula weight) of hydrogen and 71 units of weight (i.e. 2 combining weights, or i formula weight) of chlor- ine unite to form 73 units of weight (i.e. 2 formula or combining weights) of hydrochloric acid gas. And this was true before the hypothesis was formulated, and can always be proven to be true at any time in the future, independent of the hypothesis. In the latter case we have simply expressed facts which have been observed, and can be observed at any time; in the former, we have added to the observed facts the assumption of an atomic structure of matter, which is foreign to the facts themselves, and which cannot, like them, be proven by experiment to be true. That is all very simple and true, the reader may say, now that we have formed the conceptions of atomic weight and molecular weight, but how could such results have been attained without the aid of hypothesis? The answer to this question is that everything which has been attained is only the result of experiment, and that such conceptions, similar in all but name, can be obtained without difficulty directly from the actually observed relations. Starting, for example, with the law of com- bining weights that elements combine with one another only in the proportions of their combining * weights or of small multiples of them and the law of combining volumes that gaseous elements combine in simple relations as to volume, or in small multiples of them, the volume of the gaseous product formed standing in simple relation to the volume occupied by the constitu- * Used here in its general sense, not as on pp. 5 and 6. V 8 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. ents originally it is quite evident that the weights oj equal volumes oj gases are related as their combining weights, or as small multiples of them. We might infer from this, then, that the combining (atomic) weights could be so selected that the density of the various gases would be proportional to their combining weights.* Experiment, however, shows this assumption to be incorrect. For example, it has been found that 2 volumes of hydrogen unite with i volume of oxygen to form 2 volumes of gaseous water; that i volume of chlorine with i volume of hydrogen forms 2 volumes of hydro- chloric acid gas, and that i volume of gaseous phos- phorus with 6 volumes of hydrogen produces 4 volumes of phosphine. And the assumption that the unit of volume is that volume which is occupied by the com- bining (atomic) weight of an element leads to the result that the density (i.e. the weight of unit volume) of water vapor is but one-half the sum of the combining (atomic) weights of its constituents, which is also true for the gaseous hydrochloric acid, while the density of phos- phine is but one-fourth the sum of the combining (atomic) weights of the constituents. If we retain the combining weights as they have been determined, however, and assume the unit of volume to be that occupied by a small number of combining (atomic) weights, where the number is dependent upon the element, it is at once evident that a general result is obtained. And this assumption is perfectly justi- fiable and unconnected with hypothesis, for, although * The combining weight of a compound being identical, naturally, with its formula weight, i.e., equal to the sum of the combining (atomic) weights of its constituents. Thus for water the value is 18, i e., 0+H+H. SOME FUNDAMENTAL PRINCIPLES. 9 the combining ratios (both for volumes and weights) are experimental facts, the choice of the units for their expression is perfectly arbitrary, as is the choice of any other unit, and purely a matter of convenience. In the case of hydrochloric acid gas, then, for example, assuming the unit of volume to be that occupied by 2 units of weight of hydrogen, we obtain the following result: i volume of hydrogen (2 units of weight) will combine with i volume of chlorine (2X35.5 units of weight) to form 2 volumes (73 units of weight) of hydro- chloric acid gas, and the density (i.e. the weight of unit volume) of this will be equal to the sum of the com- bining (atomic) weights of the constituents (i.e. 35.5 + 1 = 36.5 units of weight). And this will also be true for water vapor and phosphine when we use the factor 2 for oxygen and 4 for phosphorus. Representing, then the combining weight of an ele- ment by its symbol, and designating by a sub-numeral the number of these combining weights which will occupy, under standard conditions, the unit of volume, we shall obtain the formula of -the element, and the weight thus represented by this formula will occupy the same volume, under like conditions, as the formula weight of any other element or compound. Applying this to the three cases just considered, we obtain where although the weights represented by the terms H 2 , C1 2 , O 2 , P 4 , HC1, H 2 O, and PH 3 , i.e. the formula weights, are all different, the volumes occupied by them 10 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. are identical, provided the conditions of pressure and temperature are the same. But these are the formula weights, i.e. are what, accord- ing to hypothesis, have been and are designated as molec- ular weights, although here they have been arrived at wholly without the aid o) hypothesis. Since our customary unit of weight is the gram, and the combining weight is usually based upon oxygen ( = 16), we can define a formula (molecular) weight in the gaseous state as that weight which will occupy the volume oj 2X16 grams 0} oxygen under like conditions. From this definition, however, it is at once obvious that equal volumes o) gas under like conditions must contain the same number 0} formula weights. But this, in its practical meaning, is identical with what has long been known, from its hypothetical origin, as Avagadro's hypothesis, which now, having been derived from experi- mental results, without the aid of any hypothesis, becomes Avagadro's law. As has been mentioned, the factor necessary to trans- form the combining (atomic) weight of an element into the formula weight according to this definition (i.e. the number of combining weights which will occupy the normal volume under standard conditions) varies for the different elements. These factors, however, are usually small, ranging from i, for most of the gaseous metallic elements, to 8 for gaseous sulphur under cer- tain conditions.* We have arrived thus at a purely experimental hypoth- esis-free conception of the atomic and molecular weight * For a list of these factors, and a similar derivation of these con- cepts, see Ostwald, Grundriss der allgemeinen Chemie, 1899, pp. 65-73. SOME FUNDAMENTAL PRINCIPLES. H of a substance in the gaseous state. It is not to be assumed, though, that the formula (molecular) weight necessarily remains the same in the other states of aggre- gation, although this is true for the combining weight, so far as we know. Experiment shows, indeed, that the formula weight depends, even in the gaseous state, according to this definition, upon the temperature (sul- phur for example) ; while in the state of solution (accord- ing to an experimental definition derived later) it often depends upon the nature of the solvent (acetic acid in benzene and in water) ; and in the liquid state upon the presence of another liquid (alcohol alone and with water.) Energy. Energy is work or anything which can be transformed into work or produced from work. Although energy may appear in many different forms, it is to be remembered that all these forms can be transformed, the one into the other. The principal forms under which the common manifestations of energy may be grouped are as follows : Kinetic energy, i.e. the energy of motion, distance or potential energy, i.e. the energy of position, electrical energy, magnetic energy, heat, chemical energy, surface energy, volume energy, and radiant energy. But since these forms are only phases, as we may say, of the fundamental concept of energy, and each can be transformed into the other, one form, viz. that which can be most readily defined, has been chosen as the standard of reference. In other words, all kinds of energy are measured and expressed in terms of a standard form, into which they could all be transformed. This standard form is that manifested in the ordinary mechan- ical relations. The unit of work is the erg, which is the work done 12 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. when unit force is overcome through unit distance, the unit of force being the dyne, i.e. that force which, acting for i second, will impart to i gram the velocity of i centimeter per second. Naturally, then, n dynes will impart a velocity of n centimeters per second to one gram. Since at Washington a body falling freely will acquire the velocity of 980.1 centimeters per second, the force of gravitation there is equivalent to 980.1 dynes, and the work of raising i gram through i centimeter is 980.1 ergs. Or, in general, the force of gravitation is equal to g dynes, and the work of raising i gram through i centimeter is g ergs, where g is the gravita- tional constant, i.e. the velocity per second acquired by i gram in falling freely. As this unit is exceedingly small, and results expressed in it are cumbersome to handle, a larger unit, i.e. one that is equal to ten mil- lion ergs, is often employed. This larger unit is called the joule and is equal to io 7 ergs, while a still larger one, without a name and designated by J (the joule being abbreviated to j) is equal to io 10 ergs. Although this method of expressing the amount of energy involved, independent of the special form in which it is at the time, is ideal, it is not in general use in all cases, for many of the forms of energy were studied long before the system was commonly accepted. For this reason we still find, for certain forms of energy, that the older, arbitary units are still in vogue (calorie, coulomb, etc.). One unfortunate consequence of this older no- menclature is that it does hot keep clearly before the mind the fact that the forms of energy are but phases of the general concept of energy, and hence are mutu- ally capable of transformation. To accentuate this fact we shall always give the value of these arbitrary units, SOME FUNDAMENTAL PRINCIPLES. 13 Le., those which are applicable to but one definite form of energy, in terms of the general standard. The factors of energy. Experience has shown that the total amount of energy of any kind is not the con- dition upon which its transfer from one place to another depends. We often observe, for example, that energy is transferred from a system containing a smaller amount of energy to one containing a greater. It has been found possible, however, to express the amount of each form of energy as a product of two factors, one of which, the so-called intensity factor, conditions the transfer of that form of energy. The other factor, the capacity factor, although having nothing to do with the transfer, is dis- tinguished by the fact that for the whole system it remains constant in value, i.e., the transfer of the energy from one system to another leaves the capacity factor of the whole system just as it was in the beginning, viz. equal to the sum of the component capacity factors. Whenever a transfer takes place, then, there has previously existed a difference in intensity; and in order to cause a transfer of energy to take place, a difference in intensity is essential. Examples of this division of energy into factors are very numerous. Thus for heat energy we have the tem- perature T as the intensity factor, entropy being the capacity factor; for electrical energy (^X^o) we have the electromotive force and amount of electricity; for volume energy (pXv) the pressure and volume; while for kinetic energy (%mc 2 ) we may have either velocity and momentum, or the square of the velocity and the mass. Two bodies at the same temperature, two amounts of electricity, two volumes of gas, etc., will always remain in equilibrium, then, i.e. will give rise to no transfer of 14 PHYSICAL CHEMISTRY FOR. ELECTRICAL ENGINEERS. energy so long as the intensity factors (i.e., the tempera- tures, the pressures, or the electromotive forces) are the same. It is evident since the intensity factor causes the trans- fer of energy, that its value must change during such a transfer, i.e., after the transfer its value must lie be- tween the two original intensity values. Thus two gases separated by a movable partition will cause the partition to move one way or the other, according to which body of gas has the greater pressure, and the final pressure will lie between the two initial pressures. The volumes occupied by the gases, however, will have nothing to do with the transfer of energy, and the final volume of the system will be unaltered, i.e., will be equal to the sum of the two initial volumes. An especially striking exam- ple of the lack of influence of the capacity factor in the transfer of energy is given by a system made up of water in contact with air. So long as the temperatures of the two components (water and air) are the same, the capacity factors may be altered to almost any extent by decreasing the amount of air and increasing that of water; but alter them as we will, no heat will go from the water to the air or vice versa, so long as the tempera- tures remain the same. If the temperatures be different, however, and the temperature of the gas be higher than that of the water, heat will immediately go from the air to the water, notwithstanding the fact that the amount of heat in the air is infinitesimal as compared to that in the water. Expressing these facts mathematically, designating the intensity factor by the letter i, and the capacity factor by c, the value of the total amount of energy,_E, is given by the equation SOME FUNDAMENTAL PRINCIPLES. i$ Naturally, when both c and i are allowed to vary, the change in is given by the expression dE=cdi+idc, or, retaining one constant and allowing the other to vary, by either AE=cAi (c is constant), or AE=iAc (i is constant), where A represents a finite increase. When there are two forms of energy active in a system, and a change in one produces a corresponding change in the other, we have .e., c\ i = c^i. This equation is exceedingly valuable hi deriving the relation existing between the two kinds of energy (heat and electrical energy, for example) in a system, and we shall have occasion to make use of it in our later work. CHAPTER II. THE GENERAL PROPERTIES OF GASES. The gas laws. In spite of the intangibility which characterizes the gaseous state, and the consequent dif- ficulty in the investigation of it, our knowledge of the laws governing the behavior of gases is far more com- plete than is that of the laws regulating the behavior of substances in the other, more tangible, states. It is not to be assumed from this statement, however, that we know why gases behave as they do, for that is just what we do not know. The gas laws simply state how substances in the gaseous state will behave under those conditions which can influence them; and have, and can have, nothing to do with the question as to the cause of this behavior. Indeed, the difference between our hypothesis-free standpoint and one that retains hypotheses is very well illustrated by the difference in meaning of these two words, how and why. To say how a thing will behave we need only be familiar with the thing, and the more familiar we are with it the more accurate will be our prediction. In other words, we can say how a thing will behave under any condition by citing facts that have been observed, or an experimental law, i.e. a gener- alization of such facts. To be able to say why a thing behaves as it does, on the other hand, it is necessary to 16 THE GENERAL PROPERTIES OF GASES. 17 go beyond the facts themselves, and, as the facts form the total of our knowledge, assume that a certain struc- ture, for example, is responsible for the behavior. But, since in our assumption we have gone beyond the facts, we cannot prove it to be either correct or false by aid of the facts. It is true, of course, that the assumption may satisfactorily account for the facts as we know them at the time, but we have no reason to believe that it is the only one that will account for them, or even that it will account for those which are yet to be dis- covered. In other words, the hypothesis at best enables us to see how the facts, or a certain number of them, might be explained ; but it does not add anything of value to what has already been deduced from the facts them- selves, nor does it lead more than they do to the discov- ery of new facts.* This repetition of our standpoint and its advantages is quite necessary here, for no portion of chemistry is richer in hypothetical assumptions than that which includes the gaseous state, and in no place is a distinc- tion between hypothesis and fact more essential. In order that we may always retain our standpoint, and devote ourselves exclusively to facts, therefore, we shall have to continually inquire as to how things occur, avoid- ing any assumption as to why they should occur as they do. Although the gaseous state as such has but little im- portance in electrochemistry, it is absolutely essential that the electrochemist obtain a clear idea of its laws, if he is to understand the laws which have been found to govern the behavior of substances in solution. And certainly no one can question the value of these laws to * See Ostwald's Vorlesungen iiber Naturphilosophie, I.e. 18 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. the electrochemist, for, with but few exceptions, all electrochemical processes take place in solutions. Since gases are characterized by an absence of form, i.e. occupy entirely any space in which they are present, their condition is dependent solely upon external influences. Of the variations which a gas may undergo, as the result of a change in these influences, the most important are those of volume, pressure, and temperature. Experiment shows, however, that when the temperature is retained constant, the volume of the gas is the greater the smaller its pressure, and vice versa; and, so long as the volume (pressure) remains unchanged, the pressure (volume) is the greater the higher the temperature, and vice versa. These laws are the generalization of the facts observed by several generations of investigators, and an exception to them is yet to be found. In quantitative work, however, such purely qualitative laws are of little value, for they simply indicate the direction of the variation without at all showing its extent. And when altered to show the extent, they lose the absolute generality which distinguishes them, and hold true with accuracy only between certain limits. Quantitative experiments on gases have resulted in the following conclusions: When the temperature is retained constant, the volume is inversely proportional to the pressure (Law of Boyle). And retaining the volume (pressure) constant the increase in pressure (volume) per degree centigrade is 1 /273 of its value at o centigrade (Law of Charles). Starting with unit volume at o, and constant atmos- pheric pressure, then, and decreasing the temperature to 273 centigrade, the volume of the gas will be reduced to zero if the law of Charles holds at such a temperature. THE GENERAL PROPERTIES OF GASES. 19 In other words, for each decrease of i the volume will be reduced 1 /2?z of its original value, and at 273 the loss in volume will be 273 /273, i.e., i. If we consider 273 centigrade as the zero of a new scale (the abso- lute zero), and employ centigrade degrees, calling the temperatures absolute temperatures, we can say at o absolute (i.e., 273 centigrade) the volume of a gas (as a gas) is zero, and its volume will increase per degree by 1 /273 of the value it would have at 273 absolute (i.e., o centigrade), and that increase will always be the same, independent of the actual tempera- ture. Expressing these relations mathematically we have v a (T is constant), and voc , or v-j, and pv=kT. But at 273 absolute (o centigrade) =k2 73, and by combination, eliminating the common constant k, we obtain 273 20 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. which gives the value of the product pv at any tempera ture (absolute) in terms of p and v at 273 absolute. The term - - is a constant, however, which depend 273 for its value only upon the weight of gas occupying th volume v , and the units of volume and pressure chosen Considering v as the volume of i gram of gas, we hav where, although the value of r, the specific gas constani depends upon the units, it is constant for any one gas. Regarding VQ as the volume occupied by i mole of gas on the other hand, and designating it by F , we fm< - RT, 273 where R, the molecular gas constant, is the same for al gases, for, by definition, page 10, i mole of gas alway occupies the normal volume under standard conditions. The difference in meaning of these two constants, and R, accentuates the difference in standpoint of physic and physical chemistry. While the results of physic are always given for a specific quantity of substance without regard to the similarities which chemistry ha discovered to exist between the formula weights of al substances, physical chemistry employs physical method: from the chemical point of view, i.e. strives to disco ve: general laws by applying both physics and chemistry tc the facts observed. Since a mole of any substance in the gaseous stat< THE GENERAL PROPERTIES OF GASES. 21 occupies the same volume as 32 grams of oxygen, i.e. 32X699.8=22393.6 cubic centimeters, or nearly 22.4* liters, at o and 76 centimeters pressure, we can find the value of R= for the various units in which pressure 273 and volume may be expressed. As the specific gravity of mercury is 13.6, the pressure of 76 centimeters is 1033.6 grams per square centimeter (i.e. 13.6X76), and we have o 1033^6X22400 273 273 = 84800 (F in c.c., p in grams per sq. cm.) = 848ooX 980.1 = 8.3 X io~ 7 (F in c.c., p in dynes) I X22.4 273 = 0.0821 (F in liters, p in atmospheres). Another law which is of great value in considering the gaseous state is that of Dalton, according to which each component of a mixture of gases exerts the same pressure in the system as it would exert were it alone present in the volume of the mixture. In other words, the pres- sure of a system composed of several gases is an addi- tive property. Summarizing the gas laws in their mathematical forms, then, we have * As the factor to transform grams to ounces (av.) is the same as that for the transformation of liters into cubic feet, i mole of gas in ounces (av.) occupies 22.4 cubic feet at o and 76 cms. of mercury pressure. 22 PHYSICAL CHEMISTRY FOR ELECTRICAL EMG1HEERS. Vi:v 2 : :p 2 :pi, or p iVi = p 2 v 2 = const. (T is constant), pi : p 2 : :Ti:T 2 (v is constant), vi : v 2 : : T\ : T 2 (p is constant) , pV=RT (where R is constant for all gases and depends in value upon the units of pressure and volume chosen), and where P is the total pressure of a mixture of gases and the terms p are the partial pressures of the components of the mixture. As already mentioned, to secure quantitative results it is necessary to restrict the limits of the general laws holding for gases. Both these laws, viz. that of Boyle and that of Charles, hold rigidly, then, only between certain definite limits. In fact, stated as above, they hold rigidly only for ideal gases. It has been found, however, that the further a gas is removed from its lique- faction point, the more nearly ideal it is; i.e., the more accurately is its behavior represented by the laws. In- deed, the law of Boyle, as applied to the so-called perma- nent gases, gives very satisfactory results, except when the pressure is very high; as does the law of Charles, so long as the temperatures are not excessive.* * For hydrogen the product pV (or pv) at constant temperature increases steadily and regularly with an increase of pressure, and is expressed very accurately by the altered form of the equation p(Vb) = const., where b is a constant. All other gases, on the other hand, starting with atmospheric pressure and constant temperature, give a value of pv, which first decreases with increased pressure, then passes through a minimum, and finally increases steadily. The rela- tions, in such a case, can be followed by aid of Van der Waal's equation, THE GENERAL PROPERTIES OF GASES. 2 3 Dissociation. By aid of the relation pV=RT the so-called equation oj state jor gases, it is possible to obtain a new form of definition for the formula (molecular) weight in the gaseous state. Instead of defining it as the weight which under like conditions will occupy the volume of 32 grams of oxygen, we may say the formula pV weight of a gas is the weight which will give V, in -=- = R, such a value that R is approximately equal to that cal- culated for oxygen. Neither of these methods, however, is adapted to labora- tory requirements. In practical work it is much simpler to take advantage of the fact (p. 10) that equal volumes of gas contain the same number of formula weight (moles), and to determine the density of the gas in terms of oxygen. Since the formula weight of oxygen is 32, the formula weight of the gas can then be obtained directly by multiplying the density ratio by 32; i.e., weight of i c.c. of gas M = 32 X : TI 1 > weight of i c.c. of oxygen the conditions of temperature and pressure being the same. For certain substances, however, the formula weight so determined is found to vary with the temperature and pressure, and in some cases to be equal to but one-half the sum of the combining weights of the con- i.e., ( p + y- 2 J ( V b} = const., where a and b are constant values depend- ing only upon the nature of the gas. In the case of ethylene, for example, the results by this formula agree very accurately with the experimentally determined ones up to a pressure of 400 atmospheres. ("Elements," pp. 34-38.) 24 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. stituents. Such a result at first glance would natu- rally lead one to object to the process of reasoning em- ployed on page 9, and,, indeed, without further knowl- edge of the substances behaving in this way, it would be impossible to defend its use. It is obvious, then, that either our definition of a formula weight in the gaseous state (p. 10) is incorrect in some cases, or that some process common to these is responsible for the abnormal results observed. Further investigation of these substances, however, shows conclusively that our definition is correct in all cases, and that the abnormal results are due to a more or less complete decomposition of the substance, a dis- sociation, as it is called. Of the substances for which such abnormal results are observed we may mention ammonium chloride (NH 4 C1), phosphorus pentachloride (PC1 5 ), and nitrogen tetroxide (N2OJ as typical examples, although many others might be cited. The formula weights (by defi- nition) of these substances have each been observed to vary from the normal value represented by the formula to a minimum value which is one-half of this, the amount of the variation depending upon the temperature and pressure. In general, the higher the temperature under constant pressure the lower the value, while at constant pressure the formula weight is the lower (one-half of the formula weight being the limit) the lower the pressure. The very fact that a general rule of this sort exists for substances behaving abnormally is evidence that it is not the definition of the formula weight which is at fault, but that some process occurring similarly in all these cases is responsible for a change in the formula weight itself. Indeed, later work, by which the actual THE GENERAL PROPERTIES OF GASES. 2$ presence of the products of the decomposition was shown, proved that the definition of a formula weight even in these cases is correct and that it was only our ignorance of the effects of temperature and pressure upon these substances which led us to infer otherwise. Instead of destroying the usefulness of one of our fundamental principles, then, these " abnormal " results have simply introduced to our attention a very important and com- mon process, viz., that of dissociation. In the three cases mentioned above the dissociation has been found to take place according to the following schemes, the symbol + being used to show that the reaction is reversible, i.e. that it goes in one direction or the other, depending upon the conditions: NH 4 C1^NH 3 +HC1, N 2 4 ^N0 2 +N0 2 . The presence of the NH 3 , HC1, Cl, and NO 2 in these cases can be proven without difficulty. The gas which is evolved from ammonium chloride by heat, for example, can be shown to contain NH 3 and HC1 by allowing it to diffuse through a porous diaphragm. Here the NH 3 being the lighter diffuses more rapidly than the HC1, and an excess of NH 3 is found on one side of the par- tition, while an excess of HC1 remains on the other. The presence of chlorine in PC1 5 , and NO 2 in N 2 O 4 , is even easier to show, for the dissociation can be followed by the eye, the chlorine imparting a green, the NO 2 a brown- ish red, color. These methods, together with the many others, although 26 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. proving conclusively that a decomposition (dissociation) does take place, are of little value in determining the extent to which it takes place, for at best they are but qualitative. For this purpose the simplest and most accurate method is based upon the density. It will be observed that in each of the cases above complete dis- sociation would transform one formula weight (mole) into two. The volume occupied by the two formula weights, however, will be double that which would contain, under like conditions, the formula weight of the undissociated substance, and the density (i.e. the weight of unit volume) after complete dissociation will be one-half what it would be without dissociation. When the dissociation is not complete we can also follow it and determine its extent from the two densities, i.e. with and without dissociation, although the relation is not so simple. Assume in a case such as the above that we start with i mole of the undissociated state, and that this dis- sociates to the extent of oc%. From the one mole before dissociation, then, we obtain i +a moles after dissocia- tion, for we have i a still undissociated, and a each of the two products, i.e. i a + 2a = i+a moles in total. Under like conditions, the volumes occupied by the substance before and after dissociation, then, will be related as III+CK; and the densities before and after will be related as i +a : i, for the greater the volume produced by the dissociation of the original mole the smaller will be the density of the system. In general, consequently, for all substances forming 2 moles from i by dissociation we have the relation THE GENERAL PROPERTIES OF GASES. 27 where d d is the observed density after dissociation, and d u is the density without dissociation, both being meas- ured under the same conditions of pressure and tem- perature. The term d u can, naturally, be obtained from the formula weight of the undissociated substance, i.e. is approximately one-half the formula weight, when based upon hydrogen, or one thirty-second when based upon oxygen. When one mole of substance produces three moles by complete dissociation, for example the case of ammo- nium carbamate, i.e. NH 2 CO 2 NH 4 ^C0 2 + 2 NH 3 , and a is the degree of dissociation, we have (i a) -f- 3a:i::d u :d d . And, in general, where i mole falls into n moles by dissociation, I-OL + na:i: :d:d ud , or a (n-i)d d 9 from which a may be calculated without difficulty. It is not to be assumed because we have restricted ourselves to these few typical examples that the process of dissociation is simply a scientific curiosity, for it is not only exceedingly important in itself, but is abso- lutely essential to things we shall have to consider later. Without knowing it, in fact, the reader has probably made use of the process of gaseous dissociation in quali- tative analysis, for the Marsh test for arsenic depends solely upon it. The gaseous arsine, which is dissociated by its passage through the red-hot tube, re-unites in the colder portions, but, since the hydrogen diffuses more 28 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. rapidly than the arsenic, an excess of the latter is left behind, and crystallizes upon the walls. In addition to the influence of pressure and temperature upon .the degree of dissociation of a gaseous substance, it has been found that the introduction of one of the prod- ucts o) the dissociation into the system, either before or after the process takes place, always decreases the degree of dissociation of the substance. Although the calculation of the effect of these three influences upon the dissocia- tion can only be considered later, it may be mentioned here that a formula can be derived by which both the effect of pressure and that of the addition of a definite amount of a product may be calculated at constant temperature, and that another formula enables us to follow the influence of temperature. The following tables will enable the reader to gather some idea of the effect of pressure and temperature upon the dissociation, and also make clear the calculation of a in the various cases. DISSOCIATION OF NITROGEN TETROXIDE, N 2 O 4 . (Density of N 2 O 4 = 3.18; of NO 2 +NO 2 = 1.59; air=i.)* Temp. Sp. Gr. of Gas. Percentage Dissociation. 26. 7 2.65 19.96 35-4 2.53 25-65 39. 8 2.46 29.23 49. 6 2.27 40.04 60. 2 2.08 52.84 70. o .92 65.57 80. 6 .80 76.61 90. o .72 84.83 100. I .68 89.23 in . 3 65 92.67 121. 5 .62 96.23 135- .60 98.69 154. o .58 I OO . OO * The densities of hydrogen, oxygen, and air are related as : : 15.88: 14.45, from which the densities in other units may be calculated. THE GENERAL PROPERTIES OF GASES. 29 DISSOCIATION OF PC1 6 . (Density PC1 5 = 7.2; PC1 3 + Cl,= 3.6; air=i.) Temy. Density. Percentage Dissociation. 182 5.08 41-7 190 4.99 44-3 200 4.85 48-5 230 4.30 67.4 250 4 . oo 80 . o 274 3-84 87.5 288 3.67 96.2 3 3-6S 97-3 DISSOCIATION OF N 2 O 4 . (Equal Temperatures, Varying Pressures.) Temp. Pressure. Density (air = i). i8.o 279.0 mm. 2.71 17.3 i8.s 136.0 " 2.45 29.8 20. o 301.0 " 2.70 17.8 20. 8 153.5 " 2 -46 29.3 When i mole of gas is formed from a solid or liquid, at the constant pressure p, the work done is equal to p times the increase in volume. Since the volume of the solid or liquid is negligibly small, as compared to that of a gas, however, we may regard the total volume occupied by the gas as equivalent to the increase of volume. The work done, then, will be equal to pV. But the product pV at constant temperature is constant, inde- pendent of the countless values of p and V from which it might be made up ( for />oc j, and is equal in value jor i mole to RT, in which R is a constant, energy quantity. In calculating amounts of work of this kind, then, we shall always employ the right side of the equation pV=RT, for it shows the relations much more definitely and 30 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. clearly than the other. Thus to form i mole of gas at any temperature against any pressure always requires the work RT, which will be given in the units employed for R, since T is a pure number. If instead of forming one mole, as we suppose, the gas dissociates into others, the work involved will still be RT units for each mole, but iRT in total, where i is equal to the number of moles formed from one original mole. This value i may also be expressed in terms of the degree of dissociation, for, if this is a, and n moles are formed from one, i, the total number of moles, will be (i a) +na, and the work of formation from a liquid or solid will be [(i a) +na]RT units. Naturally, we can also define a formula weight, in the gaseous state, in such terms. The formula weight is then that weight of gas which in forming from a liquid or a solid (i.e., from a volume which is negligible) will perform approximately the same amount of work as 32 grams of oxygen would, i.e., RT units, at the tempera- ture T, against any pressure. We could also determine dissociation in this way, i.e., by determining the work done and finding the value of i as a difference, from which a, when n is known,' can be found; but the method based upon the density is far simpler and decidedly more practical. Partial pressures and concentrations. For the descrip- tion of a gaseous system composed either of a single gas or a mixture of gases it is essential that we have a convenient method of representing the amount of a gas present. This is not only necessary for the descrip- tion of such a system, and for use in the formula (mentioned above) which shows the dependence of the degree of dissociation upon the pressure, and upon the amount of THE GENERAL PROPERTIES OF GASES. 31 one of the dissociation-products that has been added to the system, but will also serve to simplify some of our later work. Thus far we have used either the density of the gas or the volume (F) which contains i mole for this pur- pose, but these are not the only forms of expression, nor are they even the most convenient. We shall therefore consider briefly the other, better mehods for the definition of the amount of gas present in a system. Since under definite conditions i mole of gas (by defi- nition) occupies a definite volume, and i liter of this vol- ume will contain a definite fraction of a mole, under those conditions, it is evident that any change in the system due to an alteration of the conditions can be accurately described by a statement of the change in concentration, i.e., the change in the number of moles per liter. But, at constant temperature, the concentration is propor- tional to the pressure, i.e., the greater the concentration the greater the pressure, and vice versa, so that it is obvious that a change in any gaseous system can also be accurately described by a statement of the change in pressure (or partial pressures in case the system is com- posed of several gases). As these two terms, concentration and partial pressure, are to be used constantly in our later work, it will be necessary here to find the exact quantitative relation which exists between them. From the definition of the formula weight, however, remembering the laws of Boyle and Charles (p. 18), this relation follows directly, provided the laws continue to hold. For at o C. the concentration i, i.e.. j mole per liter, is equivalent to a pressure of 22.4 atmospheres [P^-y], an d at any other 32 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. absolute temperature T, the unit of concentration gives T the pressure 22.4 atmospheres. In order that the reader may become familiar with the use of these terms, we shall now consider a specific case of the system formed by dissociation. At 190 C., for example, we find that PC1 5 is 44.3% dissociated (p. 29), i.e., the reaction goes toward the right until 44.3% of the PC1 5 originally present is decomposed and equilibrium is established. We have then a system composed of the three gases, PC1 5 , PC1 3 , and chlorine. Since the data in the table (p. 29) are given for atmospheric pressure, the total pressure after dissociation remains atmospheric and the volume increases. Starting with i mole of PCls, assum- ing no dissociation to take place, the volume would be 22.4 - - liters. But by the dissociation we lose 0.443 m l e f PC1 5 , an d gain -443 of a mole of each of the products, PCla and chlorine, hence the total volume after dissociation is 1.443X22.4 - liters, for the mole of PC1 5 has been transformed into (i -0.443) + 2X0.443 = 1.443 moles, the temperature and pressure remaining unchanged. The partial pressures in atmos- pheres, then, will be - ~' 443 for PC1 5 , and each x -443 1-443 for PCla and chlorine, i.e., the number of moles of each, divided by the total number of moles in the system, THE GENERAL PROPERTIES OF GASES. 33 and multiplied by the total pressure. And the concen- trations, i.e., number of moles per liter, will be i -0.443 273 + 190 Pa ' 1.443X22.4-^^ Q-443 chlorine> 1.443X22.4 i.e. the number of moles divided by the total volume in liters. In case the volume remained constant, i.e., that the pressure increased by the dissociation (assuming the increase of pressure to have no effect upon the degree of dissociation) the final pressure would be 1.443 atmos- pheres. The partial pressures hi atmospheres, then, would be i 0.443 f r PCls, and 0.443 eacn ^ o and chlorine. And the concentrations would be i -Q.443 *"* and 0-443 273 + 190 22.4 273 ~~ C PC1 3 ^chlorine- CHAPTER III. HEAT AND ITS TRANSFORMATION INTO OTHER FORMS OF ENERGY. The first law of thermodynamics. As has already, been mentioned, energy may be manifested in various forms. An amount of energy, then, may be expressed in either the so-called absolute units (erg, dyne, etc.), or in any of the units employed exclusively for one specific form of energy (coulomb, calorie, etc). This conception of energy is the one which has been commonly accepted since the time of J. R. Mayer (1841), who was the first to determine the factor necessary to transform energy expressed as heat into mechanical units, i.e., the so-called mechanical equivalent of heat. It is to be remembered here, however, that the word equivalent is used only in the sense that if heat is transformed into mechanical work we shall always obtain a definite number of mechan- ical units from each unit of heat (calorie) transformed, and does not at all imply that heat is always transformed into work under all conditions, or that all the heat present will be transformed into work. The principle governing the transfer of heat, and the relation existing between the heat transferred and that transformed into work, is to be considered below; here we shall only discuss the relation of . the amount of heat actually transformed into mechanical work to the amount of work which results from the transformation. 34 HEAT AND ITS TRANSFORMATION. 35 Of the various methods for determining the mechanical equivalent of heat (Joule's, Rumford's, etc.), the most interesting in principle for our purposes is that of J. R. Mayer, which was the forerunner of all the others. This depends upon the difference observed when the specific heat of a gas is determined for variable or for constant volume. Since by definition the specific heat of a sub- stance is the heat necessary to raise i gram of it i C. (from some one standard temperature), it is obvious that for gases two values will be found, one when the gas is allowed to expand under constant pressure, the other when the volume is retained constant. Natu- rally, d priori, there is no means of deciding whether these two values will be experimentally identical or not, unless we understand the difference in the process in the two cases. Experiment, however, shows the values to be different when work is done by the expan- sion, the one for constant pressure (i.e. where the volume increases) being the larger; while no difference from that at constant volume is observed when the expansion takes place without involving mechanical work, e.g. into an exhausted space. Mayer was the first to recognize that the only difference possible in these two experimental values is the amount of heat which is used in over- coming the resistance offered to expansion. From actu- ally observed values, then, it was possible for him to cal- culate the exact value of a calorie in mechanical units. By experiments with air, for example, it is found that c p c w = 0.0692 cal., i.e., the amount of work necessary to expand i gram of air by the 1 / 2 73 d part of its volume at o C., against 36 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. the pressure of the atmosphere, is equivalent to 0.0692 cal. Since i gram of air occupies 773.3 c.c., it would occupy 773.3 cm. of a tube having a cross-section of i square centimeter. An increase in temperature of i, then, would involve an increase of 2.83 ( i.e. ^-^ \ cm the work done 46 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. will be equal to / pdv, and expressing both in the t/Vi rT same units we have, since p= (p. 20), dQ^rT v or ^ Jlie gas is next allowed to expand adiabatically away from the heating bath until the temperature falls to T 2 . For this, the new volume being 1*3, we have the relation (p. 39) *-: 3. Next the pressure is increased until the volume is decreased to v^ heat to the amount Q 2 being removed, so that the temperature remains constant at TV The work done here by the gas is - [pdv, and we have t/v Q.-iT.log.g. 4. Finally the gas is compressed adiabatically until the original volume Vi, and the original temperature J"i, are reached. For this we have the relation T 2 Ti HEAT AND ITS TRANSFORMATION. 47 We have thus carried the gas through a series of ideal changes, and have finally the same conditions as those with which we started. The amount of heat Qi has been absorbed at the higher temperature TI, and a smaller amount Q 2 has been evolved at a lower temperature T 2 , and the rest has been transformed into work; i.e., Q, the amount of work produced (in terms of heat), is equal to Qi (?2> for the amount of heat Q 2 has simply been transferred from TI to T 2 . The relation between Q\ and Q 2 , then, is obviously the following: but hence *>2 ^3 . , V2 , #3 ---, ..e.,log.-=log,-; i.e., the amounts of heat absorbed and liberated are pro- portional to the absolute temperatures of the processes. Since Q = Qi Q 2 is the heat transformed into work, we have, then, Qi-Q* Tt-T* /^ T 1 > and 48 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. The heat transformed into work by any reversible process is to that transferred from the higher to the tower tem- perature as the difference in temperature is to the lower absolute temperature. Or, the heat transformed into work is to that absorbed as the difference in temperature is to the higher absolute temperature. Or, from the other stand- point, the work in calories necessary to transfer a certain amount of heat from one temperature to a higher one by a reversible process is to the amount of heat as the temperature interval is to the final high absolute temperature* It will be observed from the above that it is only at the absolute zero that all heat will be transformed into work, for when T 2 = o, Q 2 will be zero, and Qi-Q2 = Q\- In its second form this general rule is that which is used in engineering work to find the efficiency of heat-engines. Thus, at best, an engine working with a boiler tem- perature of 200 C., the temperature of the condenser being 50 C., will have an efficiency of 15 %73 = o.3i7 '-^= - j, i.e., assuming no heat to be lost either by radiation or by the work used in overcoming friction, such an engine would transform 31.7% of the heat absorbed into work. It will be observed that in general we may write this relation in the form From this form it is possible for us to find the capacity factor of heat energy, for J T expresses the variation of * The reader who does not follow the mathematical reasoning is advised at least to become thoroughly familiar with the laws in italics, as well as the three last formulas, for they are vital in some of our later work. HEAT AND ITS TRANSFORMATION. 49 heat energy, i.e., shows the amount of heat transformed into work when the heat Q has its temperature reduced AT . Since the temperature is the intensity factor of heat energy (p. 13), the capacity factor must obviously be the quantity 7p. This factor is called the entropy, and is very important in rinding when a reaction of any kind will take place. By the first law of thermodynamics we find how a reaction will take place, but have no means of deciding whether or not it will take place. By the second law we find that only that reaction can take place of itself by which the entropy of the system will increase, for the denominator of the expression = must always decrease if a reaction is to take place of itself, i.e., heat can only go from a warmer to a cooler body of itself. It is in only the third form (p. 48) that we shall use the second law of thermodynamics for the derivation of formulas, but too much stress cannot be laid upon the capacity factor of heat energy, the entropy, which we have deduced from the second law, for wherever we have to consider heat energy we shall have to employ its factors. CHAPTER IV. SOLUTIONS. The formula (molecular) weight in the liquid and solid states. Most of that which we have found to hold true for gases is directly applicable to substances in the state of solution; and even a cursory glance at the con- tents of this chapter will convince one of it, and justify the time we have devoted to the discussion of gaseous systems all unnecessary, as this may have appeared at first sight. Before discussing this similarity of behavior, however, it would first seem necessary to consider some- thing of the general relations of the other two states (liquid and solid) in which the substances composing the solution may have previously existed, i.e., of the components of the system. But experience shows that such a knowledge is not essential, for much more is known of the laws governing the behavior of solutions than of those regulating that of either liquids or solids. In fact, the behavior of a solution differs so utterly from that of the pure, liquid solvent, that it may be followed without any further knowledge of the behavior of the solvent than that comprised in the few and well-known physical facts with which we assume familiarity. The only purely chemical relation to be considered, indeed, for either liquids or solids, is the definition of the formula weight, and this is only necessary that we may be able 5 SOLUTIONS. 51 to follow tne cnanges in the formula weight as the sub- stance passes through the various possible phases of its existence, i.e., the gaseous, liquid, solid, and dissolved states. The formula weight in the gaseous state, as we have already observed, may be denned hi several ways; and all definitions will lead to the same result, for all are based upon the same fundamental fact, although ex- pressed in other terms. Thus we may say, the formula weight of a substance in the gaseous state is that weight which will occupy, under like conditions, the same volume as 32 units of weight oj oxygen: or, is the weight which when multiplied by the difference in the specific heats under constant pressure and at constant volume (i.e., c p -c^) will give the value 2 calories; or, is the weight oj the gas, occupying any volume under any pressure, which will do the external work RT when its tempera- ture is raised i C., i.e., from Ti to T. And still other definitions could be given. In short, then, the formula weight in the gaseous state is a very definite conception, and can be readily determined experi- mentally. In the liquid state,* on the other hand, we have only one general definition for the formula weight, and this depends upon the so-called surface tension of the liquid, i.e., upon the force in dynes necessary to form a liquid surface with an area of i square centimeter. The sur- face tension of a liquid can be found indirectly from the height to which the liquid ascends in a capillary tube of known radius, and is equal to one-half the product * For further information as to the liquid state, see "The Elements of Physical Chemistry," 3d ed., 1905, pp. 61-90. 52 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. of the height (in centimeters) into the radius (also in centimeters) into the specific gravity, i.e., the surface tension = l / 2 hrs. According to this definition, the formula weight of a liquid is that weight in grams which gives such a surface that its increase, due to a heating of 1 C., involves the surface work of 2.12 ergs; in other words, the temperature coefficient of the formula (molec- ular) surface tension (i.e., the temperature coefficient of the tension of the surface of i mole) is always the same, independent of the nature of the substance. This defi- nition, it will be observed, is very similar to one of those given above for gases, except that surface energy is involved in place of volume energy. There is one question which may arise here, which, if not answered, can lead to confusion; it is, How was this definition obtained originally? Naturally, we may not go very deeply into this matter here, but at any rate the empirical origin of the definition may be pointed out. By assuming the formula weight to be the same in the liquid as in the gaseous state, the factor 2.12 ergs was found to remain constant for a large number of substances. The natural inference, then, was that this value (2.12) is typical of the formula weight, and that any variation from it, when the gaseous formula weight is assumed, shows that the formula weight changes as the substance goes from the gaseous to the liquid state. Using this same plan, other methods decidedly more restricted in their scope have been developed, and as the same discrepancy for any one substance is shown by all, our conclusion is considered to be justified. It must be acknowledged that our definition of for- mula weight in the liquid state is very much less sat- isfactory than that for the gaseous state, but unfortu- SOLUTIONS. 53 nately we find the relations for the solid state * still less satisfactory, for as yet it has been impossible to find any definition for the formula weight in the solid state. It is true that at first glance it would seem that the law of Dulong and Petit that the combining (atomic) weight of an element (excepting carbon, boron, and silicon) when multiplied by its specific heat always gives the approximately constant value 6.34 would suffice for this purpose, since the sum of the combining weights, i.e. the formula weight, multiplied by the specific heat of the compound should give approximately the same value as is obtained when 6.34 is multiplied by the num- ber of combining weights in the formula weight. But a moment's thought shows that this relation is only of value in that it insures uniformity in the choice of the combining ratio to be used as the combining (atomic) weight; in other words, at best, it is only another method for fixing the combining (atomic) weight of an element without the necessity of knowing or studying the com- pounds which it may form (see pages 4 and 40), and has nothing to do with fixing the formula weight of an elementary compound. Thus analysis shows the exist- ence of two chlorides of mercury, the simplest formulas, using at least one combining weight of each element, being HgCl and HgCl 2 , but the application of the law of Dulong and Petit does not enable us to state whether the formula weight in the solid state is HgCl or Hg 2 Cl2,t etc., any more than the analytical result does. And the same is true for the other chloride. It is only because the * For further information on this state, see " Elements," pp. 91-103. t For HgCl we have (200+35.5)0.052=12.25; for Hg 2 Cl 2 (400 + 71)0.052=24.5; while 2X6.34=12.68, and 4X6.34 = 25.36. 54 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. formulas Hg 2 Cl 2 and HgCl 2 (and not H^CU) are found under certain conditions in the gaseous state that they are assumed in the solid state. In all, reactions between solids, then, it is only the actual weight of the reacting substances which is essential, and the very fact that we cannot define formula weight in the solid state shows its utter lack of chemical or physical influence. If at any time this should be changed, and it appear that the formula weight does exert an influence experimentally, naturally all our difficulty would disappear, for then a definition could at once be derived from observations of this influence. The molecular weight of a dissolved substance, in contrast to that of a liquid or a solid, plays a very impor- tant role indeed, and is just as definite in its meaning as that in the gaseous state. Indeed, it will almost appear to the reader, after his perusal of the following pages, that the formula weight is the significant and fundamental conception in the consideration of dissolved substances; and certainly its paramount practical importance is decidedly striking, especially when contrasted with the slight importance of, and impossibility of defining chem- ically, the formula weight in the liquid state, and its apparent utter lack of either chemical or physical influ- ence in the solid state. Osmotic pressure. It is a well-known fact that, when a solution is carefully superimposed upon another con- taining a different amount of the substance, diffusion takes place, and that this diffusion is so directed that the entire system finally becomes homogeneous. There is, then, a tendency for the substance in solution to become uniformly distributed throughout the volume of liquid accessible to it. Following the plan we have adopted, SOLUTIONS. 55 i.e. considering only the facts and avoiding all hypothesis, we shall study the facts of this phenomenon, and the general relations to be derived from them, without at the same time trying to picture their cause, or to explain their origin, by hypothesis. The facts of diffusion may be summed up by the following statement: A system composed of a substance dissolving in a solvent, or of two solutions of different strength, one being superimposed upon the other, behaves as though there existed an attraction between the sub- stance and the solvent, which ultimately produces a uniform concentration of the dissolved substance through- out the entire system. It is to be noted here that this is not in any way an assumption, but simply a description of the behavior of the system as we observe it. The difference in concentration of the two liquids in contact has been observed to influence very largely the force of diffusion and in such a way that the greater the concentration-difference the greater will be the force of diffusion. To find the quantitive relation existing be- tween the amount of dissolved substance in the solution and the force of diffusion into the pure solvent, however, it is necessary for us to find a method of measuring the force of diffusion. Naturally, such a measurement would not be difficult if we could obtain a partition, to be placed between the solution and the pure solvent, which would allow passage to the solvent and not to the solute, for then we could simply measure the force with which the pure solvent goes through the partition. At first glance the difficulty of obtaining such a partition, i.e. one which in the case of water and an aqueous sugar solution, for example, will allow water, but not the dis- solved sugar, to pass through, seems absolutely insur- $6 PHYSICAL CHEMISTRY FOR. ELECTRICAL ENGINEERS. mountable. It is not so, however, for such semi permeable films, as they are called, are very widely distributed in nature, and can also be readily prepared by artificial means. In the first class, for instance, we find the pro- toplasm of which living organisms are made up, while of the second, a good example is a film of copper ferro- cyanide, as formed by the reaction of copper sulphate and potassium ferrocyanide. By aid of a copper ferrocyanide film, permeable to water but not to sugar and many other substances, Pfeffer was able to measure the apparent attraction between substance in solution and the pure solvent, the osmotic pressure, as it is called. The principle of Pfeffer's appa- ratus is illustrated by the figure above. The cylinder A, closed at the bottom, is of porous clay and is intended to support the semipermeable film. This film is prepared by filling the porous cup with a solution of potassium ferrocyanide and allowing it to stand for a day in a solution of copper sulphate. In this way the pores of the cup are filled with the precipitated copper ferrocyanide, which, although permeable to water, is not so to the dissolved substance. SOLUTIONS. 57 To make a measurement with this apparatus the cup so prepared is filled with the solution to be studied and the rubber stopper CC inserted in such a way that the solution rises a short distance in the measuring-tube. The cell is next immersed in water and retained in position by the cork BB. The liquid is riow observed to rise very slowly in the tube until equilibrium is finally attained, i.e., until the weight of the liquid in the tube just counter- acts the pressure with which the water enters the cell. .Since the entrance of water into the cell decreases the concentration of the solution within it, the actual measure- ments are usually made by aid of a mercury manometer, so arranged that the change in volume is negligible. In this way the pressure observed is that which just pre- vents the entrance of water into the original solution, diluted only to an exceedingly small extent, and not, as in the above description, into one diluted by an amount of water equal in volume to the liquid which rises in the tube.* From the experiment, then, we know that a certain definite pressure is necessary to prevent water going into the porous cell to dilute the solution contained in it. This is what we shall call osmotic pressure. When- ever we use this term, consequently, it is without any assumption as to the cause of the pressure, and is merely expressive of the experimental fact that it is necessary to exert a pressure in order to prevent pure solvent flow- ing through a semipermeable film to dilute the solution, which is surrounded by it. It will be noted here that we do not consider what * For experimental details of very accurate measurements, where the pressures rise as high as 25 atmospheres, see Morse and Frazer, Am. Chem. Jour., 34, i, July 1905. 58 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. osmotic pressure really is, but what we mean by the word osmotic pressure. This again emphasizes our standpoint. By defining each concept in terms of experi- ment, and studying the facts of the phenomenon, asking only how it takes place and never why, we may always be certain that our knowledge is based upon facts alone, and is perfectly free from any trace of hypothesis. Pfeffer's results showed that this osmotic pressure increases with the amount of substance dissolved, and for any one concentration is proportional to the abso- lute temperature of the solution. The numerical values for various sugar solutions at 15 C., as found by Pfeffer, are given below, c being the percentage of sugar, and p the pressure, in centimeters of a column of mercury, which will just preserve equilibrium between the water and the solution, i.e. will just prevent water flowing through the semipermeable film. p/c 53-5 The principal difficulty experienced by Pfeffer was the breaking down of the film under the pressure exerted, which, naturally, allowed the solution enclosed to escape and so led to a smaller result than would otherwise have been obtained. Such an action can always be detected, however, by testing the pure solvent for sugar; and, indeed, the presence of this was always observed when the stronger solutions were measured, and was the reason why the 6% concentration was the highest employed. From Pfeffer's observations, as given above, it is c P p/c c P I 53-8 53-8 4 208.2 I 53-2 53-2 6 30-75 2 101.6 50.8 i 53-5 2-74 151.8 55-4 SOLUTIONS. 59 quite evident that, within the experimental error, there exists a general relation between concentration and osmotic pressure, for the term p/c is practically con- stant, at constant temperature. But c, the number of grams of solute in 100 grams of solution, is obviously the reciprocal of the volume of solution, v, containing i gram of solute, i.e., c= . We have, then, however, in place of p/c = constant, the relation pv= constant, and further, since p is proportional to the absolute tem- perature, T, i.e., p=o, when r=o, we may write pv= constant XT. This equation is so strikingly similar to the one already derived for gases (pp. 19 and 20) that it at once suggests an analogy between the behavior * of a substance in solu- tion and one in the gaseous state, and makes the deter- mination of the value of the constant a point of extreme interest. Since for the gaseous state we have found the constant R (in pV = RT) to be a constant for all sub- stances, when the volume V is that occupied by the for- mula weight (i.e., the combining weight of an element multiplied by some small whole number, or the sum of the combining weights of the elements composing a compound), it is but natural to think that some such analogous result may be found for substances in solu * It is to be remembered here that experiment shows the analogy in behavior, and does not justify the assumption that the reason for the behavior is the same. Although it has been possible by aid oj hypoth- esis to formulate a kinetic theory (i.e., in one sense of the word a hy- pothesis) of gases, all efforts to do likewise with liquids or solutions have been futile. 60 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. tion. Such a result, however, would lead immediately to a definition of formula weight in solution, if such a conception exist, for it would show that the formula weight exerts an easily observed influence upon the physical behavior of the solution. And its importance would probably not be restricted to osmotic pressure alone, for presumably then the physical behavior of a solution in other respects would also be influenced by the formula weight, and we could employ the defini- tions for formula weight so obtained not only as defi- nitions, but, knowing the formula weight, could also calculate the exact behavior in these other respects of any substancce dissolved in a solvent. In order to find whether the constant in our equation has any definite relation to that for gases, however, it is necessary to calculate it for one formula weight in the dissolved state. Here, naturally, arises the question as to what the formula weight in solution is, for as yet we have not derived any definition of it. Let us assume temporarily that sugar has the same formula weight in solution as it is generally assumed * to have in the * It will be observed after a survey of chemical compounds that many formulas are employed which cannot be proven experimentally. Under the atomic hypothesis it was usually assumed that the formula of a compound was that simple relation which would give at least one atom of the substance present to the smallest extent, and would lead to a whole number of atoms of the other constituents. Or, from the hypothesis-free standpoint, would involve a whole number of combining weights of each constituent. This, naturally, is but following out the consequences of the choice of hydrogen (or a / 16 oxygen) as the basis of the combining weights. But in either case the assumption of a formula weight without experimental foundation is not justified. In the case of sugar the structure, the products of decomposition, etc., have led to the formula C l2 H 22 O n , and we assume without direct ex- perimental evidence that this would also be the formula in the gaseous state if no decomposition took place; SOLUTIONS. 6l other states, viz., C^H^On, and calculate the constant on the assumption that the v in our equation is the volume of solution in which this formula weight (342 grams) is dissolved. Pfeffer found for a i% solution of sugar (i.e., 342 grams in 32,400 c.c.) at o an osmotic pressure of 49.3X13.6 = 671 grams per square centimeter; hence 671X34200 constant = - = 84200, 273 273 i.e., assuming that sugar has the formula Ci 2 H 2 2Oii in solution we find that the osmotic constant is identical, within the experimental error, with the gaseous constant (p. 21) which would be obtained if sugar in the gaseous state had the same formula. This same constant has also been obtained for many other organic substances, using the customary formula weights (p. 60) (many of which can be confirmed by aid of the gaseous density), and we may say in general that it enables us to find the formula weight from the observed osmotic pressure, or, knowing the formula weight, to calculate the osmotic pressure. It is true that we find in many cases that this constant is only obtained when the formula weight in solution is taken as some small multiple of that in the gaseous state, but in every such case it can be experimentally shown, by a method given below, that the formula weight undergoes a change when the substance goes out of that solvent in which its behavior seems abnormal, into another in which it is normal, i.e., in which the assumption of the gaseous formula weight in solution leads to the above constant. Naturally, direct osmotic pressure observations with 62 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. solvents other than water cannot be made with the ferrocyanide film, but it has been possible to find other substances which are permeable to other solvents and yet not to dissolved substances. Thus vulcanite and rubber have been found permeable to ether and not to alcohol, so that it is possible to measure the pressure with which ether goes through the film to dilute a solution of alcohol in ether. Pressures of this kind have been observed up to 50 atmospheres, but no very accurate measurements have been made.* From the fact that the osmotic constant is identical with the gas constant, both calculated for the volume containing i mole, it is evident that the equation pV=RT is applicable to both the dissolved and gaseous conditions, p being the osmotic pressure in the one case and that of the gas in the other, while in both cases V is the volume occupied by i mole and R is a constant depending in value only upon the units chosen. This fact may also be expressed in other ways. For example, we may say that the pressure necessary to just prevent pure solvent flowing through a semipermeable film to dilute a solution is the same as would have to be exerted upon the amount of substance contained in the solution if it were in the gaseous state, at the same temperature and occupying the same volume, to just prevent it expanding; provided, of course, that the formula weight is the same in both states. Or, with the same proviso as to formula weight, as van't Hoff originally announced the law, the osmotic pressure oj a substance in solution is the same pressure as that amount 0} substance would exert were it in the gaseous state at the same temperature and occupying the same volume. Pos- * See Raoult, Zeit. f. phys. Chem., 27, 737, 1895. SOLUTIONS. 63 sibly a better form of this law, and certainly one that is more general, is as follows: The osmotic pressure exerted by I mole of substance in solution is the same as the gase- ous pressure exerted by I mole, provided the conditions of temperature and volume are identical. Based upon experiments with comparatively weak solutions, then, we may define the formula weight in the state of solution. And naturally the possible forms of expression are similar to those for the gaseous state. The formula weight in the dissolved state is that weight which in the volume o) approximately 22.4 liters of solvent will exert the osmotic pressure o[ i atmosphere at o, or a corresponding pressure at another temperature or volume; or is the weight which occupies such a volume oj solvent pV that R in the equation R=^=r will have approximately the value of 84,800 when p is expressed in grams per square centimeters and V in cubic centimeters. One other thing has been observed in working with these comparatively weak solutions, i.e., that the osmotic pressure is independent of the nature of the semipermea- ble film. We may say, then, from this, and the fact that the nature of the solvent has no influence upon the pressure, provided the formula weight is the same in all solvents, the osmotic pressure exerted by I mole in any solvent is a constant so long as the temperatures and the volumes of the solvents are alike. All our conclusions thus far have been drawn from experiments upon comparatively weak solutions, and, indeed, up to a very recent date all our knowledge oi osmotic pressure was derived from such experiments as have been mentioned (together with very many others), where the concentration never exceeded a certain small 64 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. value. Naturally, this left much uncertainty as to our quantitative law for osmotic pressure, and to the limits within which the observed values of formula weights held. One point in particular here is the question as to which volume, that of solution or of solvent, is to be considered as V in pV= RT. For experimentally it has been impossible to tell which should be chosen owing to the fact that the two volumes would be practically identical for such dilute solutions as we have considered. All doubt on these and many other points connected with osmotic pressure, however, have been removed very recently by the brilliant work of Morse and Frazer,* who have measured directly, and with very great ac- curacy, the osmotic pressures of sugar solutions up to a concentration of 342 grams of sugar in i liter of water. Their results, a summary of which is given in the follow- ing table, show that everything said above for dilute solu- tions holds true for the more concentrated solutions of sugar, if the volume V in pV=RT is taken as that of the pure solvent.^ This interpretation of volume simplifies the laws of osmotic pressure very considerably, for, according to it, the osmotic pressure of a mole of substance in a liter of any solvent is the same, independent of the possible expansion or contraction caused by the solution; while if the volume of the final solution were the significant conception, the osmotic pressure would only be the same when the changes in volume caused by solution are the same in extent and direction. Whether other substances will lead * Am. Chem. Jour., 34, i, July 1905. "j" In all of the above laws and definitions, then, this volume is to be taken when the solution in question is so strong that its total volume is appreciably different from that of the pure solvent. SOLUTIONS. OSMOTIC PRESSURES AND FORMULA WEIGHTS. SUGAR SOLUTIONS AT ABOUT 2O. Weight- Volume- Pressure at Same molar. Moles in 1000 Gr. H 2 O (WO. molar. Moles per Liter. Temperature. ** ^(22.4+0.0824/1 Gaseous. Osmotic (P). P 0-05 o . 04948 I. 21 1.26 327-5 O. IO 0.09794 2.40 2-44 336.9 0.20 0.19192 4.82 4-78 345-2 0.25 0.23748 6.06 6.05 342.9 0.30 0.28213 7.22 7-23 342.0 0.40 0.36886 9.68 9.66 343 - 1 0.50 0.45228 12.07 '12.09 341-7 0.6O 0.53252 14.58 14.38 347-1 0.70 0.60981 17.16 I7-03 344-8 0.80 0.68428 19.17 19.38 338.5 0.89101 0.75000 21. 48 21.21 346.5 0.90 0.75610 21-73 21. 8l 340.0 1. 00 0.82534 24.27 24-49 339-2 Mean 341-2 to a similar law, and whether still more concentrated solutions of sugar will continue to follow this law (contrary to gases, where high pressures fail to give a constant when multiplied by the volume), are questions for the future. At any rate, the authors have so perfected their method that after a short time we should have a very complete quantitative knowledge of osmotic pressure, and be able to state exactly the limits within which the laws that we know will hold. Until that time, then, we may consider the laws above as binding, for they can be indirectly con- firmed by other methods. Since our definitions show the osmotic pressure of i mole of substance in a certain volume, it is simply a matter of calculation to find the formula weight from the osmotic pressure observed for a solution containing 66 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. a known amount of substance. In general, we have the proportion T where 22.4 - is the osmotic pressure in atmospheres of a solution of M grams in i liter of solvent,* and W is the number of grams in i liter of solvent which gives an osmotic pressure of P atmospheres. For example, at o C., a 2% solution of sugar, i.e., about 20 grams to i liter of water, gives an osmotic pressure equal to 101.6 cms. of Hg, hence the formula weight of sugar can be found from the proportion .e., =335. Just as we found a definition of formula weight in the gaseous state from the work done by expansion (pp. 29- 30), we can also find one adapted to the dissolved state from the work necessary for the removal of solute, against the osmotic pressure, from the solution. Imagine a cyl- inder provided with a semipermeable piston, water being above it and a solution below. If work is done upon the piston, i.e. if it is lowered into the solution, pure solvent will be removed from below it to the mass of solvent above. And since pV=RT, the work necessary to remove" the amount of solvent which has previously * Since i mole in 22.4 liters of solvent gives an osmotic pressure of i atmosphere at o, and p=,i mole in i liter of solvent would give a T pressure of 22.4 atmospheres, or at T 22.4 atmospheres. SOLUTIONS. 67 contained i mole will be equal to RT. In other words, the formula weight in solution is that weight which can be separated from the solvent by the work RT, or, what is the same thing, is the weight which was previously dis- solved in the volume of solvent requiring RT units of work to remove reversibly from the solution. Any method, then, by which solvent can be removed from a solution will serve as a method of denning formula weight (as freezing, boiling, etc.). In addition to the method of measuring osmotic pres- sure which was described above, there is one that is so simple and at the same time so striking that a short description of it will possibly make the conception more clear in the reader's mind. If in a moderately strong solution of copper sulphate we place a drop of a strong solution of potassium ferrocyanide, it is immediately sur- rounded by a semipermeable film of copper ferrocyanide. We have, then, a semipermeable film of copper ferrocy- anide surrounding a strong solution of potassium ferro- cyanide. Since the ferrocyanide is stronger than the copper sulphate (i.e., contains a greater number of formula weights per liter), water will flow into the bubble with a greater force than it will flow outward, and this can be proven by the swelling of the bubble and the formation of dark streaks in the copper sulphate solution, which is con- centrated by the removal of water. If the copper sulphate drop is placed in the potassium ferrocyanide, the oppo- site effect is observed, i.e., water flows outward from the bubble, and this decreases in size. By this method, then, it is always possible to show an equal number of formula weights to the liter, for, under such conditions, no change in the size of the bubble can result. And, naturally, the formula weights need not be of the same substance, 68 PHYSICAL CHEMISTRY FOR. ELECTRICAL ENGINEERS. i.e., other things may be added to the weaker solution until no change in the size of the bubble is observed. After such an addition, then, we can conclude that the sum of the original number of moles to the liter of the weaker solution and those added is equal to the original number of moles in the stronger solution. Vapor pressure. It has been known for many years that the vapor pressure of a liquid is always depressed when substance is dissolved in it. It was not until 1887, however, that Raoult applied chemical conceptions to the physical facts, and obtained general results. Pro- ceeding in a way similar to that used in finding a defini- tion of formula weight based on osmotic pressure, and always using the formula weight of the solvent as found in the gaseous state, Raoult found that the vapor pressure of a solution is related to that of the pure solvent as the number of moles of solvent is to the total number of moles in the system, i.e., of solvent plus solute. Our definition of formula weight in the dissolved state by aid of vapor pressure, then, is not quite as simple as that based upon osmotic pressure, for it necessitates a knowledge of the formula weight of the solvent when in the gaseous state. The formula weight in the dissolved state is that weight which when dissolved in pp moles of any solvent depresses its vapor pressure 1%. And, so far as we know, the formula weight by this definition agrees in each case with that found from osmotic pressure. Expressed in the form of an equation, the above rela- tion between the vapor pressures and the number of moles may be written f^ N p~N+n' SOLUTION. 69 where p' is the vapor pressure of the solution, p that of the pure solvent, (=) is the number of moles of \ m l I W\ dissolved non-volatile substance, and Nl = 77 ) is the number of moles of solvent calculated from the gaseous formula weight. This relation can also be written in other forms, i.e., can readily be transformed into P-V _ p N+n' V ~N- This latter form is very useful for the determination of the formula weight of a dissolved substance, the other forms being better adapted for the calculation of vapor pressure from known concentrations. Thus experiment shows that a solution of 2.47 grams of ethyl benzoate in 100 grams of benzene has a vapor pressure of 742.6 mm. of Hg, while pure benzene shows 751.86 mm., both at 80 C. Since ^=2.47, Af =78 (i.e., gaseous C 6 H 6 ), W=ioo, ptf n p> = 742.6, and p= 751.86, we find from . =j^ that ^=154, while the formula in the gaseous state, , leads to the value 150.* * For the vapor pressure of a system of two non-miscible liquids, as well as for those cases where both solvent and solute are volatile, see "Elements," pp. 119, 158-161. For the theoretical deduction of these empirical relations from the conception of osmotic pressure, see pp. 161-167. 70 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. Boiling-point. Since the vapor pressure of a solution is lower than that of the pure solvent, and the boiling- point is that temperature at which the vapor pressure becomes equal to the atmospheric pressure, the boiling- point of a solution must be higher than that of the pure solvent. Just as general relations were found for the vapor pressure and osmotic pressure, so they have been found for the boiling-point. In few words, it has been found that I mole oj a non-volatile substance dissolved in 100 grams oj a solvent gives a definite, constant increase oj the boiling-point, which depends in value only upon the nature oj the solvent. This so-called molecular in- crease oj the boiling-point is usually designated by the letter K. The formula weight oj any substance in the dissolved state, then, is that weight which in 100 grams of the solvent will increase its boiling-point K. The value of K for a solvent must consequently be known before it is possible to define a formula weight in that solvent. Although this value K can be found by direct observa- tion when the formula weight is known, i.e. by finding the effect on the boiling-point of a small amount in 100 grams of solvent, and then calculating this weight to the formula weight, it can also be found by calculation, i.e., by using the conception of osmotic pressure. In this case we separate, as vapor, the solvent from the solution, and the amount of work necessary for this must be equal to that necessary for an osmotic separation (p. 66). Since i mole in a liter of water would give an osmotic pressure of 22.4X^-^ atmospheres at 100, the removal of the amount of solvent containing i mole would necessitate SOLUTIONS. 71 the expenditure of 22.4X-; liter- atmospheres * as osmotic work. The separation of this weight of solvent as vapor, however, since the latent heat of evaporation of i gram of water is 535.1 cals., would require thermal work equal to 1000X535.1 cals. By the second law of thermo- dynamics (p. 48, third statement), then, we have the relation. Work done _ Increase of temperature Heat during it, in terms of work High abs. temperature ' 22 . 4 xfglit.at (1000X535.1) 0.041 lit. at. 373+ K' 1 from which K=ioK'=$.2, for K refers to 100 grams of solvent.f Some other values o'f K, as found by experi- ment (they agree with the calculated ones), are as fol- lows: Benzene, 26.70; chloroform, 36.60; carbon disul- phide, 23.70; ether, 21.5, etc. Since the increase of the boiling-point is known for a solution containing i mole in 100 grams, the molecular weight of a substance in solution, or the increase of the boiling-point caused by the solution of any amount of substance, can be found by aid of a simple proportion. * A liter-atmosphere is the work done when the pressure of the atmos- phere on i square decimeter is overcome through i decimeter, since a liter is a cubic decimeter. t A very general relation for this purpose is K= , where T is the boiling-point of the pure solvent and w is its latent heat of evap- oration for i gram. (See " Elements," pp. OF THE 72 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. We have, in general, M\K\:W:M, where M is the formula weight, K is the increase when M grams are present in 100 grams of solvent, and Jt is the increase for W grams in 100 of solvent. Know- ing any three terms, then, the other one may be readily calculated. Freezing-point. The fact that the vapor pressure of a solution is lower than that of the pure solvent necessi- tates the depression of the freezing-point of a solvent in which substance is dissolved, i.e., causes pure solvent t to separate from the solution by freezing at a lower tem- perature than is observed for the pure solvent. This relationship, perhaps, is not so obvious as is the one for the boiling-point, but a glance at the figure above will make it quite clear. Here ww is the vapor-pres- sure curve for water, ss that for the solution, and I that for ice: At the point /=o C., ice and water have the same vapor pressure, and consequently are in equilib- rium. The solution and ice, however, will only be in SOLUTIONS. 73 equilibrium at the temperature corresponding to the point of intersection of their curves, so that the freezing- point of the solution must always lie below that of the pure solvent, if its vapor pressure does. And the more substance there is in solution the lower will be the curve ss, and the lower the freezing-point, i.e., the point of intersection. Exactly as with the boiling-point, it has been found that I mole of substance dissolved in 100 grams 0} any solvent will depress the freezing-point of this k, where the value of k depends only upon the nature of the solvent. And just as the boiling-point law holds only when pure solvent and no solute separates, i.e., where the solute is non- volatile, so here this law only holds when it is pure solv- ent which separates in the solid state* One thing is to be observed especially in regard to freezing. If the liquid is overcooled and solid is caused to separate by stirring, it is to be remembered that the freezing-point observed is not that of the original solution, but of the stronger solution which is produced by the loss of the solvent solidifying, for the freezing-point of a solution is that temperature at which it exists in equilibrium with the solid solvent, f The value of k can be determined here in a similar way to that used for K in the boiling-point, and can also be cal- culated by an analogous method of reasoning. Since i mole in a liter of water at o exerts an osmotic pressure of 22.4 atmospheres, the osmotic work necessary to remove the solvent which has previously contained i mole will be * In case these conditions are not fulfilled, it is still possible to get results, but the calculations are necessarily more complicated. f For the correction to be made here, see " Elements," pp. 185-186. 74 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. 22.4 liter- atmospheres. The heat involved in the freezing- out of i liter of water, however, would be 1000X80 cals., where 80 cals. is the latent heat of solidification of i gram of water. By the second law of thermodynamics (p. 48, second statement), then, we have Work done Lowering of temperature Heat during it, in terms of work High abs. temperature 22.4 lit. at. k' * * 80000X0.041 lit. at. "273' from which &=io&' = i8.6.* Other values of k are as follows: Acetic acid, 38.8; benzene, 49.0; phenol, 75.0, etc. Here, also, we have the general relation where M grams in TOO grams of solvent produce a depres- sion of k in the freezing-point and W grams in the same weight leads to a depression of At. And, just as above, when any three of these are known the fourth may be calculated. Coefficient of distribution. One exceedingly impor- tant relation has been observed as to the distribution of a substance between two non-miscible solvents. This relation, indeed, is the one mentioned above (p. 61), by which the change in the formula weight during transi- tion from one solvent to another is detected. When a * A general relation, corresponding to the one for the boiling-point, is &= , where w, here, is the heat of solidification of i gram of solvent and T is its freezing-point. (See " Elements," pp. 177-179.) SOLUTIONS. 75 solution is agitated with an equal volume of another solvent, which can dissolve the solute, but does not form a homogeneous mixture with the first solvent, it is found that the distribution in the two layers is either the same for all original concentrations of solute in the first solvent, or differs with that concentration. Thus when, by our definitions, the formula weight is the same in the two solvents (succinic acid dissolved in water and shaken with ether, for example) the ratio of the concentrations hi the two layers is independent of the original amount dissolved in the first solvent. When the formula weights, by definition, differ in the two solvents, however, the ratio depends upon the original concentration, and in such a way, as we shall see later, that it is possible for us to calculate the relation of the formula weight in one to that in the other. This relation is not restricted to a solution containing one solute, for it has been observed that when several solutes are present each is distributed as if it alone were present, and its behavior is entirely unaffected by the others. Since all the above-mentioned properties of a solution, i.e. the freezing-point, boiling-point, vapor pressure, and osmotic pressure, depend upon the concentration of substance dissolved, and since by definition the effect of i mole in the dissolved state is known, it is possible to go from a result by one method to the results by the others when we know the concentration of the solution in moles per liter. Thus, suppose a water solution gives an osmotic pressure of i atmosphere at o, and, assuming the formula weight to be independent of temperature, we wish to find the freezing-point, boiling-point, and vapor pressure of the solution. Since i mole dissolved in i liter of water 76 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. or any other solvent at o gives an osmotic pressure of 22.4 atmospheres, in a solution with an osmotic pressure of i atmosphere we must have x moles per liter, the value of x being determinable from the proportion 22.4:1 ::i:x. The boiling-point of this solution, then, if the solvent be water, will be ( X5.2J+ioo, and the freezing-point ( Xi8.6j; while the vapor pressure will be p' in 1000 i f o T = ~~ > where p is the vapor pressure of pure P 1000 * I /yt 18 water at o. Or, if the vapor pressures of solvent and solution are given, we can first find the ratio of moles of solute to moles of solvent (based on the gaseous formula weight) from , =-r-=,- and then calculate, for example, the number of moles dissolved in i liter of water / AT I0 (i.e., where N= ^- = 55 \ I O Electrolytic dissociation or ionization. In obtaining our definitions of formula weight, as given above, we have assumed in each case, as a temparory supposition, that the formula weight in the dissolved state is identical with that in the gaseous state. In this way a constant relation has been observed for a large number of substances and our temporary assumption is justified. Sooner or later, in all cases, however, apparently abnormal results (the SOLUTIONS. 77 formula weight being greater than that in the gaseous state) are observed, and it becomes necessary to decide whether the definitions so obtained are incorrect; or whether in these abnormal cases there is not some specific influence which has not been considered. Since as a rule these abnormalities disappear when the substance is dissolved in another solvent, the relation then being the same as for substances which behave normally in all solvents, the only conclusion possible is that the definitions for formula weight in all these cases are correct, and that the apparently abnormal results are really due to a change in the formula weight. This conclusion, indeed, is the only one possible in such cases, for all definitions of formula weight in solution, independent of the principle upon which they are based, give sensibly the same formula weight when carried out with the same degree of accuracy. In other cases, and these form a very large class composed of inorganic salts, bases, and acids, the same question as was considered above also arises, but here the formula weight (by definition) appears smaller than the gaseous formula weight. Again the abnormality may be due to incorrect definition, or to the fact that a specific action causes the formula weight to change. And again here we find that the solvent has a great influence, i.e., in some solvents abnormal results are observed, while in others the behavior is quite normal (according to definition). Thus Arrhenius observed that all those substances, and only those, which give abnormally large osmotic pressures in solution are capable of conducting the electric current, and ij they are dissolved in other solvents, in which they behave normally, they lose this power. Arrhenius determined the electrical conductivity of such solutions in terms of molecular conductivity; the ?8 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. molecular conductivity of a solution being denned as the reciprocal of the resistance (in ohms) of the volume of liquid which contains one formula weight of the sub- stance, i.e., the weight of the generally accepted formula, the electrodes being i cm. apart and large enough to contain between them the entire amount of solution. This value, naturally, is not found directly, but is calcu- lated from that value found for a centimeter cube of the solution. (See Chapter VII.) In this way, always having i mole (according to the accepted formula) between the electrodes, he found that the more dilute the solution the greater is the molecular conductivity. In many cases, indeed, he was able to reach such a dilution that the molec- ular conductivity attained a maximum value, which is unaffected by further dilution. This molecular conduc- tivity at infinite dilution, as it is called, is designated by the term /z^, that value for any dilution V being desig- nated by fjL v From this it is apparent that the solution undergoes some kind of a change as the result of dilution; and the investigation of such solutions at various dilutions shows, indeed, that the formula weight (according to definition) also changes with the dilution, the formula weight de- creasing to a minimum, constant value, which for binary electrolytes is one-half the formula weight of the substance dissolved. We may conclude, then, that the breaking down of the formula weight of a substance in a solution is very intimately connected with the power it possesses of conducting the electric current. These facts formed the starting-point of what is known at present as the "theory of electrolytic dissociation." As this theory to-day is much misunderstood by many, and is the subject of much speculation on the part of SOLUTIONS. 79 others, it will be necessary for us to consider carefully just what is fact and what assumption, and to see clearly which portions are hypothetical and which are destined to remain under any hypothesis or lack of hypothesis; in other words, which are experimental facts. It may be said, however, that that which is hypothesis in this theory is unessential, as far as the use of the data is concerned, and the only hypothesis present, as we shall consider it, is that inherent in the terminology, which is a relic of the atomistic hypothesis and utterly beyond our power either to prove or disprove. The salient facts which have been grouped in this theory, for it is a theory in the sense that it is a law of nature holding between certain limits, although these are not as yet definitely fixed, are as follows: (1) The molecular conductivity of certain substances in water is found to increase up to a maximum, constant value, and this increase is the result of dilution. (2) Those solutions which conduct the current also give abnormal osmotic pressures, freezing-points, boiling- points, and vapor pressures ; in other words, the formula weight (according to the above definitions) decreases with increased dilution, and finally reaches a minimum value, which, for binary electrolytes, is one-half the accepted formula weight of the substance. (3) Those substances which in water conduct the current and give abnormal osmotic pressures, depressions of the freezing-point and vapor pressure, and increases of the boiling-point, give normal values when dissolved in other solvents in which they do not conduct. (4) The nearer the value of fi v is to that of u^ the more abnormal the value of the osmotic pressure, etc. (formula weight); of the solution. And the solution for 8o PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. which fjL w is found also gives the maximum osmotic pressure, i.e., the minimum formula weight. (5) The molecular conductivity of a solution at infinite dilution is an additive value, i.e., is equal to the sum of the conductivities of the substances of which it is com- posed. The meaning of this is as follows: The molecu- lar conductivity at infinite dilution of, for example, potassium chloride plus that of nitric acid minus that of potassium nitrate is found to be equal to that of hydro- chloric acid. In other words, ^ooKCl + /*ooHN0 3 - /"ooKN0 3 = /*ooHCl For this to be true, and it is true in general for all sub- stances, it is necessary that the molecular conductivity of such a substance in solution be the sum of two values which are independent each of the other. Chlorine, for example, as the constituent of an electrolyte, at the dilution giving p^, has the same conducting effect when part of a compound with one element as it has when combined with any other. It is possible, ihen, to find the value of p^ for any binary electrolyte when the values for the elements composing it are known. In other words, the conductivities of the solution as pro- duced by the presence of any element can be calculated; and from these values, by summation, the value of /^ for any binary electrolyte can be found. (6) When a solution is electrolyzed, the products of electrolysis appear instantaneously at the electrodes so soon as the circuit is completed. This indicates (since the solvent, water, does not conduct beyond a very small extent) that whatever does carry the current through the liquid is charged with electricity even before the current is applied, for the conduction is due to the dis- SOLUTIONS. 8 1 solved substance, and the speed of movement of the sub- stance can be measured, so that it is no question of matter being electrically charged at one electrode before carry- ing this charge bodily through the solution to the other. (See Chapter VII.) Further, it is observed that the same amount of electricity, 96,540 coulombs, is necessary for the separation of one equivalent weight (in grams) of any element; in other words, that 96,540 coulombs of electricity are transported through the liquid with each equivalent weight (in grams) of an element. (Faraday's Law, see Chapter VII.) (7) The properties of electrolytes are found to be the sum of the properties of the products observed during electrolysis. Thus any solution giving off chlorine on electrolysis, excluding secondary reactions, will precipi- tate silver from its solution as the chloride. And if chlorine cannot be produced in any way by the elec- trolysis, silver will not be precipitated as chloride from its solutions. And, on the other hand, silver is only precipitated by chlorine when contained in a solution from which silver can be deposited by the current by primary action. The catalytic effect of acids on the inversion of sugar as well as on the decomposition of methyl acetate is found to be proportional to the ratio for the acid; /*oo and when a large amount of a salt of this acid is added to the acid this effect is decreased. But this is only true when the salt added is an electrolyte. All copper solutions, when very dilute, show the same blue color, and this also depends upon the ratio , "oo and can also be changed, as the effect of acids was 82 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. above, by the addition of a large amount of an elec- trolyte which contains the same acid radical as the cop- per salt in question. Further, when the colored copper solution is super- imposed upon a colorless solution of another salt, the blue color boundary is observed to move with the current, i.e., to the cathode, where copper is deposited. Hence the substance which is moved in this direction contains only copper, the negative radical separating at the anode. In other words, copper in solution, when it conducts the current, is blue. 8. Observation shows that when an element is sepa- rated on one electrode, anode or cathode, it is always separated on that one by primary action ; in other words, the sign of the electricity transported by an element is always the same. And unless an element in the pure state, when dissolved in water, reacts with the water it does not conduct the current. This circumstance is assumed to be due to the fact that only one kind of electricity could be carried by the substance, and hence it pro- duces no conduction. The question now arises as to what theory can be found to correlate these facts and observations so that the generalization thus obtained may be employed to foresee other facts, and applied to other observations, that they, in their turn, may be elucidated and general- ized. By the word theory, then, we do not mean a hypothesis, in which something not observed is added to the facts to "explain " them, but only a generalization of observed facts. In other words, what law of nature, hold- ing within definite, if small, limits, can be obtained from the above experimental facts when considered together? SOLUTIONS. 83 The generalization which has been made from these facts is known as the theory of electrolytic dissociation, and, considering those portions which are free from hypothesis and fulfil the above conditions, in other words, omitting the hypothetical portions which it has attained since the time of its inception, we find in it, within cer- tain limits, a definite law of nature. The principal points of this theory are summarized below in brief form, and will each be expanded in the later portions of the book. A substance in solution, which conducts the electric current, is dissociated or ionized into its constituents, and these constituents, when secondary actions are ex- cluded, appear at the electrodes during electrolysis. The extent of the ionization or dissociation in any solution being given at the dilution V (number of liters in which i mole, according to the accepted formula, is dissolved) by the ratio = a. /JflO These products of ionization or dissociation are charged with electricity, 96,540 coulombs being carried by the gram equivalent of any element (see (6) above). A further proof of this charged state of ionized matter is given by the fact that not only is the current carried by a solution dependent upon the number of gram equiva- lents transported, but, as we shall see later, any other means of depositing the constituents of the solution upon the electrodes liberates an amount of electricity which depends also upon the number of gram equiva- lents deposited. And all cells in order to give a current must contain electrolytes, i.e., solutions which are ionized. Since a solution which by conductivity is shown to be completely ionized, or practically so, leads to a formula 84 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. weight, by osmotic pressure or any of the other methods, of one-half the value expressed by the formula weight, then, from the case of hydrochloric acid in solution, where we can designate the process by the equation the formula weight of the hydrogen and chlorine in the ionic state, according to our definition of formula weight, must be synonymous with the combining weight. The ionic state, then, is an allotropic form of the or- dinary state of the constituents, and differs from that in being charged with electricity, in having less energy than when in the gaseous state, and in always being trans- formed into the ordinary state on the loss of its charge of electricity. Since the constituents in the case already mentioned, and in general in all cases, show a formula weight (by the definitions) which is the same as the combining weight, it is possible to determine a, the degree of ionization, by osmotic-pressure, etc., measurements, or from the average formula weight of the substance in solution, as determined by osmotic pressure or any of the other methods. If, for example, we start with one formula weight of hydro- chloric acid in a solution, and a moles of it are ionized, the total number of moles will consist of (i a) of un- ionized HC1 and a moles each of H' and Cl' (where the dot indicates positive electricity as the charge and the accent negative). The total number of moles in the volume of the solution will go then from i to (i a) + 2a, i.e. i +a, and the ratio of osmotic pressure when entirely un-ionized to that when partially ionized will be the same as this. In other words, if the formula weight in a certain volume should give the osmotic pressure p', it will give, SOLUTIONS. 85 when ionized to the extent a, the pressure p=(i+a)p'. Since the number of moles (by definition) shown by the same weight is thus increased, the formula weight will be smaller, and the relation between the two values of the for- mula weight will be M'(L +a)=M, where the M refers to substance if it were un-ionized, i.e., is the accepted formula weight of the substance, and M f is the formula weight (by definition) observed in the dissolved state. Just as with gaseous dissociation, the ionization of a substance in solution is affected by the presence of one of the products of the ionization, and later, when we consider the quantitative effects for gases, we shall study the quantitative effect for substances in solution. Owing to the fact that the constituents produced by the ionization of a substance in solution are called ions (in the Faraday sense of charged atoms) it is usually assumed that ionized matter also has an atomic structure. As this is hypothesis, if we are to follow our plan, we must either use the word with an altered meaning or employ another word representing the same facts in its place. We shall use the word ionization here only in the sense that it is expressive of the experimental relation -, and employ the expression ionized matter to designate all that is ever legitimately included in the word ion, i.e., all the facts and none of the hypotheses. Summarizing our argument, the application of the experimental definitions of formula weight in solution (derived as given above) indicates that certain substances are decomposed in certain solvents, the fraction decom- posed being a in the expressions M'(i+a) = M (accord- u ing to any of the methods) and a = , and increasing S6 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. with the dilution up to the value which gives M=2M'. This is, of course, only true for substances giving at the maximum dilution a formula weight of one-half the generally accepted one; in general the fraction decom- posed can be found from [(i a)+na]M' = M ) where M at the maximum dilution n= r That this is really the result of a decomposition, and not merely the failure in these cases of the definitions of formula weight in solution, is evidenced by the above facts and many others, given later. And that this ionized matter which is formed is electrically charged is also not to be doubted, as well from the above facts as from the general agree- ment of the results by electrical and other methods. It is always to be remembered, then, that when we speak of ionization we mean something which can be defined in terms of experiment, and is free from hypothesis. And the same is true of ionized matter, so long as we do not assume for it a certain structure such as is naturally assumed in the impression made by the expressions " an ion " or " the ions." Later we shall find that starting with this conception for a simple substance we can derive other experimental definitions, not only for ionization, but also for the amount of any one definite kind of ionized matter which is present with any number of other kinds. One fact may be mentioned here which indicates what a very marked difference dilution makes in the behavior of a substance, and which decidedly supports the con- clusions we have just drawn. Although hydrochloric acid is more volatile than hydrocyanic acid, it has been observed that from a mixture of the dilute acids (o.i molar of HC1) it is possible to distil the HCN quantita- SOLUTIONS. 87 lively (provided the dilution of the HC1 is retained at about this value by the frequent replacement of the water lost). In the light of the above theory the difference between the two acids in solution is that while is 000 nearly equal to i for HC1, it is very small for HCN. In other words, HC1 is composed principally of the ionized constituents H* and Cl', which cannot produce HC1 gas without going through the state HC1 in solution, and that is prevented by the nearly constant dilution which is retained during the distillation. Any gaseous substance, then, which in solution is largely ionized is more difficult to distil from the liquid than an un-ionized or less ionized one. The HCN, being dissolved and retained in this state in solution, can be expelled readily just as any other gas which undergoes no great change in solution. This method, indeed, was discovered as the result of such theoretical reasoning, and it is but one example of the many practical applications of the above generalization.* It is not to be imagined that the facts mentioned above are the only ones leading to these conclusions, for later, throughout our work, we shall find occasion to consider other things which will confirm each of the steps leading to the final conclusion. In other words, it is not to be thought that the whole theory has been described in this place, or that, because some of the points mentioned are not clear, the theory itself is to be condemned, for many of the points can only be brought out after considering certain other methods which will enlarge our horizon. It may be said, however, that these further aids but * For details of the separation see Richards and Singer, Am. Chem. J., 27, 205, 1902. 88 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. confirm and make more evident the truth of the con- clusions we have arrived at. At the same time we must not forget that we have been speaking of this subject as lying within certain limits, and so cannot expect our con- clusions to hold outside of them, nor to condemn them because they do not. The relation of substances in non- aqueous solvents to a certain extent is different, and con sequently these conclusions could not be expected to hold. As a matter of fact, the conduction relations for these solutions are so utterly different from the aqueous ones that it would be impossible to attempt to consider them together in the light of our present knowledge. All of these points will be discussed more fully later, however, and the limits stated, within which our conclusions in general will hold. It is to be remembered, though, that simply because our theory does not hold for solutions in certain non-aqueous solvents (solutions which show no similarity in behavior to the aqueous ones, and which may or may not be solutions as we consider them, but may involve an entire rearrangement of the composition of the solvent, or solute, or both), it should not be considered as false and of little use, for the two kinds of systems are so different that it would be impossible to imagine from our pres- ent knowledge that both are subject to the same laws. The values for a, the degree of ionization, for a few electrolytes are given below for varying conditions. These are the values as found from the ratio of molecular con- ductivities, since that method is apparently the most delicate one which we possess for this purpose. Naturally, instead of first finding the formula weight in solution, by aid of one of the practical definitions, and then calculating a from the relation of this value to the generally accepted formula weight, we can find SOLUTIONS. 89 4 S 16 16 64 16 64 5 12 HBr V2 5 a 0.897 0.932 0.950 0.965 DEGREES OF IONIZATION. AgN0 3 AgNO 3 25 V 60 a a 0.828 16 0.841 0.899 64 0.909 0.962 5 12 0.964 40 HC1 0.832 a 0.904 2 0.876 0.965 16 o-955 HI 25 KC1 V 25 NaCl 25 NH.Q LiCl 25 a a a a a 0.895 2 .... o-737 0.926 10 0.86 0.842 0.852 0.803 0-945 100 0.94 o-937 0.94 0.907 0.963 1000 0.98 i oooo o . 99 3 0.982 0.979 16667 0.006 directly the number of moles present, when we have started with i formula weight (the accepted value) in a certain volume of liquid. If the commonly accepted formula weight in a certain volume should give the osmotic pressure P, and the substance is ionized, the observed osmotic pressure would be [(i a)+wa]P, and the vapor pressure would be that calculated for w[(i a) + wa], where n is the number of moles which would be present with- out ionization. In a corresponding way, also, the boil- ing-point of a solution containing i accepted formula weight in 100 grams of solvent would be [(i a)+na]K, higher than the pure solvent, and the freezing-point would be depressed by [(i ct)+ na]k. The freezing- point depression produced by the dissociation of i ac- cepted formula weight in 100 grams of solvent is usu- ally designated as the molecular depression of the sub- stance. Thus a 0.0107 m l ar solution of KOH (assum- 90 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. ing this formula) depresses the freezing-point o.0388. Since I mole in solution depresses the freezing-point 1 8. 6 when dissolved in 100 grams of water, the molecu- lar depression of our solution is 0.0388X10 = 36.261; w.wiwy ~lf\ r>f\t hence i.e., The thermal relations of electrolytes. Two salt solutions which are so dilute that the ratio =i (p. 83) do not evolve or absorb heat when mixed, provided no chemical reaction takes place between them. This fact was first observed by Hess and has been con- firmed by all observers since. Another experimental fact observed to hold for solu- tions of electrolytes is as follows: When an acid is neu- tralized by a base, both being in so great a dilution that for each = i, which is also true for the salt formed, the heat evolved is equal to 13,700 cal. and is independent of the nature of the base and acid used or the salt formed, so long as this latter at that dilution fulfills the condition ^ = i. These facts, taken in connection with those mentioned above (pp. 76-90) and the conclusions arrived at there, are not so startling as one might imagine at first glance. Since for the acid and base we have the relation /Wid SOLUTIONS. 91 and, since the salt is observed to have a formula weight (by definition) equal to one-half the generally accepted formula weight, i.e., is completely ionized according to all the possible methods of measurement, it is quite cer- tain that it is made up of the substances previously com- posing the acid and base in the same state as that in which they existed in them. In other words, expressing the chemical equation in accord with the experimental facts above, we have where n represents the number of moles of water present in the system before the reaction. Since the conductivity shows the constituents of the salt (the two kinds, + and , of ionized matter) to be present in the same form they were in originally, the only portion of the reaction which could possibly involve heat is the formation of water from ionized hydrogen (H") and ionized hydroxyl (OH'). As we know that hydrogen and hydroxyl in the ionized state can exist together to but an infinitesimal extent (for pure water conducts only very slightly), the following conclusion is certainly justified : When an acid unites with a base (at any rate in the con- dition in which we have assumed them) the cause of the reaction is the inability o) ionized hydrogen to exist in the presence of ionized hydroxyl beyond an exceedingly small amount, and the heat of the neutralization (for this case) is that heat which is evolved during the formation of water from its constituents in the ionized state in this way, i.e., 13,700 cols, for each mole of H' and OH' (by definition) forming one mole of H 2 O. By a method which we shall consider later (Chapter VI) 92 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. it is possible not only to show the presence of, but to calculate accurately, the heat involved in the ionization of a substance. When the acid and salt are completely ionized, for example, and the base but slightly, it is pos- sible to show just how much extra heat (either positive or negative) is involved by the further ionization of the base. For the partly ionized base must increase in ionization as its ionized OH' is used up, since the more dilute the solution of the base the greater, up to a certain point, is its ionization. If both the acid and the base are but partly ionized the result will differ still more, for heat will be absorbed or evolved by the further ionization of both of these. In general, we shall have, then, if the salt, also, is not completely ionized, i.e., if more heat is liberated by its undissociated product being formed, -0:3) - where a 1= ionization of the acid, wi=its heat of ionization; a 2 = " " the base, w 2 = " " " 3 = " " the salt, w 3 = " " " # =heat of association of i mole of ionized H* with i mole of ionized OH' to form i mole of un-ionized H2O; i.e., the heat generated by the neutralization of an acid by a base is equal, for each mole of water formed, to 13,700 col. plus the product of the heat of ionization of the salt into its un-ionized portion minus the same products for the acid and base. SOLUTIONS. 93 Naturally, the negative value of the heat of association of ionized H* with ionized OH' is the heat of ionization of water, i.e., the heat necessary to form i mole of ionized H* and i mole of ionized OH' from water. Later we shall consider this relation more in detail, i.e., after we have studied the method to be used for the measurement of the heat of dissociation. It is obvious from the above that the thermal prop- erties of electrolytes are additive when they are in such a dilution that they fulfill the condition - = i ; and when r* not in this condition the change in the thermal effect depends upon the amount of heat involved in causing them to attain this state. When a precipitate is formed in such a solution (i.e., when a chemical reaction takes place, which was excluded above) it is often possible to find its heat of formation just as we found that of water- above. An example of this is the following : Ag' Aq + NO 3 'Aq + Na' Aq + Cl'Aq = AgClAq+Na'Aq+NO 3 'Aq + i5,8oo cals. or Ag' Aq + Cl'Aq = AgClAq + 1 5,800 cals. i.e., when i mole of un-ionized AgCl is formed in a solu- tion from the ionized silver and ionized chlorine 15,800 calories are evolved. Conversely, if i mole of AgCl were dissolved, this amount of heat would be absorbed, i.e., the heat of solution of a substance is equal to the nega- tive value of the heat of precipitation. 94 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. Although it is not always possible, we can find the heat of formation in solution in still another way. The princi- ple of this is as follows: By electrical measurements it has been possible to find the amount of heat involved when 2 grams of gaseous hydrogen form 2 grams of ionized hydrogen in solution. This value is approxi- mately equal to 4 J,* but since there is some uncertainty about its exact value, it is usual to assume it equal to zero. Later, then, when this value has been accurately determined, the results found in this way can be readily recalculated. From this value, by dissolving a metal in a completely ionized acid, i.e., by the substi- tution of metal in the ionized state for the hydrogen, which is evolved as a gas from that state, we can observe directly the heat of formation of the ionized metal from massive metal. By then determining the heat of solu- tion of a completely ionized salt of this metal, the heat due to the negative radical in the ionized state can be determined readily, for the heat of solution of the salt is equal to the sum of the heats of ionization of the con- stituents, of which we assume that of hydrogen to be zero. In this way the table given below has been prepared by Ostwald. In order to find the heat of formation of a salt it is only necessary to obtain the sum of the heats due to the kinds of ionized matter into which it decomposes, taking into account the valence of the ionized matter as indicated by the dots for the electro-positive and the accents for the electro-negative substances. * One joule (j)=io 7 ergs=o.239i cal., i.e., i cal. = 4.i83 j. A unit a thousand times as great as j is designated by J, and we find I J= 239.1 cal.= io 10 ergs, i.e., i cal. = 0.004183 J. SOLUTIONS. 95 Cathion Matter. J = joules X i o 3 . Anion Matter of J = joules Xio 3 . Hydrogen IT + o Hydrochloric acid Cl' + 164 Potassium Sodium K- Na' + 259 + 240 Hypochlorous acid Chloric acid CIO' + 109 CIO/ + 98 Lithium Li' + 263 Perchloric acid CIO/ - 162 Rubidium RV + 262 Hydrobromic acid Br 7 + 118 Ammonium NH/ + 137 Bromic acid BrO/ + 47 Hydroxylamine NH,O + i57 Hydriodic acid I' + 55 Magnesium Mg" + 456 lodic acid IO/ + 234 Calcium Ca" + 45(?) Periodic acid IO/ + 195 Strontium Sr" + 501 Hydrosulphuric acid S" - 53 Aluminium AT" + 506 HS' + 5 Manganese Iron Mn" Fe" + 210 + 93 Thiosulphuric acid Dithionic acid S 2 O/' +581 S 2 0/' +1166 Fe- 39 Tetrathionic acid S 4 0/' +1093 Cobalt Co" + 7i Sulphurous acid SO/' + 633 Nickel Ni" + 67 Sulphuric acid SO/' +897 Zinc Zn" + 147 Hydrogen selenide Se" - 149 Cadmium Cd" + 77 Selenious acid SeO/' + 501 Copper Cu" - 66 Selenic acid SeO/' + 607 Or - 6 7 (?) Hydrogen telluride Te" - 146 Mercury Kg' - 85 Tellurous acid TeO/' + 323 Silver Ag- 106 Telluric acid TeO/' + 412 Thallium Tl- + 7 Nitrous acid NO/ + 113 Lead Pb" + 2 Nitric acid NO/ + 205 Tin Sn- + 14 Phosphorous acid HPO/ + 603 Phosphoric acid PO/" +1246 HPO/' -f 1277 Arsenic acid AsO/"+ 900 Hydroxyl OH' +228 Carbonic acid HCO/+ 683 CO/' + 674 These numbers hold only for the case that the ionized matter is in very dilute solution, i.e., Aq should be added to the symbol of each kind. For stronger solutions, in which the ionization is not complete, other amounts of heat are involved which, unless allowed for, will lead to incorrect results. The equations J and 96 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. mean that by the transformation of the accepted formula weight of metallic sodium into the ionized state 240 J are evolved, and for the change of the formula weight of chlorine gas into two formula weights of ionized chlorine (by definition, p. 84) 2X164] are liberated. CHAPTER V. CHEMICAL MECHANICS. The law of mass action. In considering such a reaction as or any other reversible, gaseous process which finally attains a state of equilibrium, the question naturally arises, Iri which direction, and to what extent, will the reaction go, when we start, for instance, with a certain amount of the three gaseous constituents, HI, I, andH? From the purely chemical point of view the above equation simply provides that if we start with i mole of hydrogen and i mole of iodine, and if these unite completely, 2 moles of hydriodic acid gas will be formed; or if we start with 2 moles of hydriodic acid gas, and this is completely decomposed, we shall obtain i mole each of hydrogen and iodine. As to what portion of the hydrogen and iodine will unite to form hydriodic acid, or what portion of a definite original amount of hydriodic acid will decompose to form hydrogen and iodine, or what will take place if all three substances are mixed together, we are utterly ignorant, failing further information than that contained in the chemical equation. 97 98 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. The answers to these questions can only be obtained by the application of a very general law which was first announced by Guldberg and Waage in 1864. The qualitative form of this law of mass action is as follows * Chemical action, at any stage of the process, is propor- tional to the active masses of the substances present at that time, i.e., to the amounts of each present in the unit of volume. In this form, however, the law of mass action is of but little practical use. It will be necessary, then, for us to derive a quantitative expression of it, and thus to obtain it in such a form that it may be applied to our needs in answering questions similar to those above. Imagine a reaction of the type + n 2 A 2 + n\A / + nJA 2 ' having taken place in a closed vessel and to have attained a state of equilibrium in which we have the partial pres- sures pi, p 2 , pi and p 2 f . Assume, further, that it is possible to insert each of the substances on the left against its gaseous or osmotic pressure, pi, p 2 , and to remove each of the products, as they are formed, from the gaseous or osmotic pressure pi, p 2 to the original external pressure po, and that this insertion and removal is isothermal and reversible. Since by such a series of operations we would do work on one side ( ), and obtain work ( + ) from the other, the sum of the two amounts (regarding the signs) would give us an expression for the work (+ or ) which is done by the system itself during the transformation, CHEMICAL MECHANICS. 99 at constant temperature, of n\ moles of A\ and n 2 moles of A 2 to n\' moles of A\ and n 2 moles of A 2 , the initial and final pressures being the same, viz., pQ. And this in its turn would lead to the expression of the quan- titative relation existing between the active masses of the constituents at equilibrium, i.e., to the relation we seek. Since the work required to change the osmotic or gaseous pressure of i mole of substance from p to pi is given by the expression RT\og ,* that for n\ moles Pi will be n>iRT log. . For n 2 moles of A 2 we have, then, Po p2 the corresponding expression n 2 RT\og e . The sum of these two terms is the work done, i.e. lost, by us in the process. The gain of work for us then for this stage is By the removal, as they are formed, of n\' moles of A\ and n 2 moles of A 2 , the amount of work (a gain for us) is In total, then, our gain in work in transforming n\ moles of AI and n 2 moles of A 2 into n\ moles of A\ loo PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS and n 2 moles of A 2 at constant temperature, the initial and final pressure being p^ is or W = RT(ni log, po+n 2 log, po-ni' log, p Q -n 2 ' log, p ) I +RT(ni' log, pi' +n 2 ' log, # 2 ' -i log, #1 - 2 log, #2). But, as we simply wish to get the relation which depends upon the pressures in the reaction at equilibrium, and the pressure po has nothing to do with this, we can assume po to be i, and obtain, since the first term is equal to zero, W = RT(m' log, pi'+nj log, p 2 '-ni log, p v -n 2 log, p 2 ). As the processes of insertion and removal are assumed to be isothermal and reversible, this work, W, must be the maximum work which can be done by the reaction, and hence must be a constant at any one temperature. We have, then, anc since if the logarithm is a constant the expression itself must be a constant, and since T and R are also constants, _ Constant = K = or, since pressure and concentration are proportional, CHEMICAL MECHANICS. 101 when the values of K and K' may or may not be alike, according as we have the same number of moles on each side of the chemical equation, or a different number. The constant (K or K') is known as the constant of equilibrium. We may express the law of mass action as follows, then : At equilibrium the product 0} the pressures (concentra- tions) of the substances on the right (final ones), each raised to a power equal to the number of formula weights reacting, divided by the product of the pressures (concen- trations) of the substances on the left (initial ones), each raised to a corresponding power, is a constant for any one reaction at any definite temperature* The variation of this constant, K or K', with the tem- perature is to be considered later, after we have studied the application of this most important and general law. Equilibrium in gaseous systems. For gases we can most conveniently use the form of the law of mass action which refers to partial pressures. We have, then, for the equilibrium of a gaseous chemical system An illustration of the application of this formula is 'given by the gaseous reaction 2 HI=H 2 +I 2 . * In case the reader cannot follow this derivation he should at any rate memorize this law and thoroughly understand the meaning of the two mathematical forms of it. 102 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. The reaction, whatever the original amounts, will only progress until the pressure of H is pi that of / is p 2 , and that of HI is p, where K being a constant depending only upon the temperature and the nature of the reaction. This reaction has been studied by Bodenstein, who found by experiment that K at 444 C. is equal to 0.02012. If we heat hydriodic acid, then, to this temperature it is possible to calculate, from the original quantity the amounts of hydrogen, iodine, and undecomposed hydriodic acid present at equilibrium when the tempera- ture is 440, for under these conditions pn = pi- .If we have a mixture of H, I, and HI, of which the partial pressures are respectively a, b, and d, and wish to find in which direction and to what extent the reaction will go at 440, we proceed as follows: Let x represent the unknown partial pressure of H lost to form HI, where x may be positive or negative; then, according to the chemical equation 2HI=H 2 +l2> we shall have at equilibrium, as the partial pressure of HI, as that of H uncombined, CHEMICAL MECHANICS. 103 and of / uncombined, And x must have such a value (i.e., the reaction must go so far) that at equilibrium it will just satisfy the equa- tion of the law of mass action for the reaction at 440, (a-x)(b-x) K= 0.0201 2 = Knowing the values of a, b, and d, in this system it is then possible to solve the equation for #, and to find how and how far the reaction will go. For example, in this case, if x is found to be positive in value, the reaction will go toward the left, as we have assumed; if negative, in the opposite direction. There is one thing to be said of the solution of such equations. There are two possible values of x\ which is to be taken ? It will be found in this case, as, indeed, in all others, that only one value is hi accord with the existing data, so that it alone could be taken. For instance, if the positive value of x is larger than a or b, it would lead to an absurdity, for it would show a nega- tive value for H or /, and the other value is the correct one. In cases of equations of a higher degree, where more than two roots exist, this same rule is to be fol- lowed. A possible case here is to have two values of the same sign, but one smaller than the other. There can be no question in such a case, however, for if the reaction would be in equilibrium after the smaller change had occurred, it could not go out of this state to attain the equilibrium shown by the greater value, hence the lower value is to be taken as the correct one. 104 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. Here we have used the partial- pressure form of the law of mass action; we could use the other just as well, however, for it will be observed that the constant factor which would transform pressures to concentrations, P c =r, is eliminated, since we have the same 22-4X 273 number of formula weights (2) on each side of the equa- tion 2HI=H2+l2> For this reason the constant, K, for this reaction, as for all others with the same number of formula weights on both sides, has the same value for concentrations, pressures, volumes under standard con- ditions, or any other terms in which the amounts may be expressed, so long as they are proportional to the formula weights. A further effect of this condition of equal volume on the two sides, is that the direction of the reaction is per- fectly independent of pressure (Le Chatelier's theorem, p. 42) ; and Lemoine has shown this to be true for the decomposition of HI for pressures ranging from 0.2 to 4.5 atmospheres. In using concentrations in place of partial pressures it is always to be remembered that the concentration (i.e., moles per liter) is the actual number of moles present divided by the total volume (see pp. 32-33). An exam- ple will perhaps make this clearer. In the reaction A = 2B + D, at equilibrium, we have o.i mole of A, 0.3 of B, and .05 of D in 10 liters at atmospheric pressure and o. Starting with 0.5 mole of A, o.i of B, and 0.4 of D) in 22.4 liters, find direction and extent of the reaction. Here we must first find the constant of equilibrium for the data given at equilibrium. Since we have o.i mole of A in 10 liters, the concentration of A, at equilib- CHEMICAL MECHANICS. 105 rium, is , of B , and of D - ; 10 10 10 /0. 3 W0.05\ fe/ hence \ I0 Assuming that x moles of A are formed by the reac- tion, the final volume will be [(o.$+x) +(o.i 2x) + (0.4 x)]22>4 liters, the temperature remaining con- stant at o, i.e., (1-2^)22.4 liters. The concentrations at equilibrium, then, will be 7 -^-r moles per liter (i- 2^)22.4 of A, - '- r of By and - '- r - of D\ hence the (12^)22.4 (1-2^)22.4 value of K, as found above, is to be equated to these values in the following way: / O.I 2X \ 2 / 0.4 # \ \(l-2X)22.4/ \(l-2>r)22.4/ "5+^ \ -2X)22.4/ K and the sign of x will show the direction of the reaction, and the numerical value its extent (p. 103). Since according to the law of mass action the con- centration is to be raised to a power, it is the whole frac- tion representing it which is to be so treated, i.e., the num- ber of moles per liter. When applied to the equilibrium resulting from a gaseous dissociation the constant of the law of mass action is usually designated as the constant of dissocia- lo6 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. tion. From it, it is possible, just as above, to calculate the degree of dissociation from a certain amount of the dissociating substance, or how much of the products, when present alone, or with the substance, will unite to form the substance itself. And, conversely, we can calculate K for each of the substances for which data was given on pages 28 and 29; and the values will be dependent only upon the temperature, the units employed (i.e., c or p), and nature of the substance. When, as is the case here, we know a, the degree of dissociation of the substance, we can proceed as follows: For the reaction PC7 5 <=tPC/ 3 + C7 2 , for example, we have for concentrations the relation K= - - - '. Starting with i mole of PC/s, which, if undissociated, would occupy V liters at atmospheric pressure, with a as the degree of dissociation, the concentrations, where V is the final volume, i.e., (i+a)F / , at the same tem- V _ xy perature and pressure, are as follows: For PCl$ y ? for PC/a -y, and for chlorine -77-, all at equilibrium; hence K, which we wish to determine, is to be found from _ a 2 = /. a a i a\ V' 6 " F" X F"" F /' At 250 for PC/s a = 0.8 (p. 29) and, since the pressure is atmospheric, i mole must be present in 22.4 - - / O liters; this is equal to the term V above. F, then, is equal to (i +0.8) (22.4 - - j, and we have as the dis- CHEMICAL MECHANICS. sociation constant for PC1 5 at 250 fo.8) 2 OF From the value thus obtained we could then calculate the direction and extent of the reaction at 250 when we start with definite amounts of the three constituents, or the value of a for a different V, i.e., when the pressure is other than atmospheric. This value will only hold for the temperature of 250, however. A physical idea of the dissociation constant, as found for concentrations, can be obtained by aid of the formula a 2 K=- ( ry. Assuming that a is equal to 0.5, i.e., that the degree of dissociation is 50%, for a reaction by which i mole is transformed into 2, we find that K= , . T ,, or (o-5) F 2K= -y. The dissociation constant of such a reaction, then, when multiplied by 2 is equal to the reciprocal of the final volume resulting from the dissociation of i mole into 2 to the extent of' 50%. This volume is that in which i mole of the original substance must be placed in order that at that temperature it may dissociate to the extent of 50% into two others. Since -^, the reciprocal of the volume produced by the dissociation of i mole, is equal to C, the concentration in moles per liter, we also have 2K=C. An example of the use of this relation is given by the reaction N 2 C>4+ NO 2 +NO 2 , for which a 2 K= , _ ^=0.0138 (calculated from ^=183.69 mm., io8 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. ^ =1.894, ^ = 3.18, i.e. a = 0.69 and F=m, all at 49. 7). Nitrogen tetroxide, then, should be 50% disso- ciated at a concentration of 2X0.0138 = 7^=0 moles per liter, or at a dilution of i mole in 36.3 liters. Experi- ment shows that at this temperature 01 = 0.493 at the dilution i mole in 40 liters, which, considering the single value from which K is determined, and the evident small error in the experimental observations, is a satis- factory agreement. A similar definition could also be deduced for the a 3 reaction A = 2B + D, where K= . _ ^ 2 , although the above simpler one suffices for a physical idea of the dissociation constant. It is to be noted here that the product of the substances on the right of the equation has always been placed in the numerator of the fraction giving the value of K (p. 100). This arrangement, of course, is optional, so long as it is retained the same. In the one case the value of K will simply be the reciprocal of that of the other. In speaking of dissociation (p. 28) it was mentioned that the addition of one of the products of dissociation to the system, or their previous presence over the disso- ciating body, decreases the extent of the dissociation. That this must be true according to the law of mass action is made obvious by the consideration of any defi- nite case. Supposing, for example, in the case of phosphorus pentachloride, the space over it contains chlorine prior to the dissociation. Since the ratio ^^ must be C PCl s a constant, less of the PCl$ will dissociate, for less CHEMICAL MECHANICS. 109 of it, with the chlorine already present, will suffice to cause the ratio to attain the value it must possess at that temperature. Or to take another case, suppose that o.i mole of B (p. 104) were introduced into a vacuum, and the substance A allowed to dissociate into this, arrangement being made by a movable piston, for example, so that the final pres- sure would be atmospheric. What would be the effect of this o.i mole of B upon the dissociation of A, the tem- perature being o? If the amount of A when undisso- ciated were i mole, the volume occupied by it and the o.i mole of B would be i.i (22.4). Assume the disso- ciation to give rise to xf moles of .5, then the number of moles of B in the final volume would be (o.i +#0i and since the final volume will be o.i + ( i ) +-oc f , i.e. i.i \ 2/ 2 times the original one, we have O.I + :*/ 2 (I. I+*022.4' (LI (1.1+^)22.4' and K _ Ui-i+y (l.I +^22.4 no PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. where the value of K was found above (p. 105). This yf is smaller than the value (x) which would be obtained from i mole of A in the pure state, occupying the same volume (i.e., (i.i+#0 22.4 litres) at the same tempera- ture. And the difference between them is the depression of the dissociation, in terms of B y due to the addition. Since for every mole of A lost, two of B are formed, x otf gives the decrease of the dissociation of A (in moles) due to the presence of the o.i mole of B. The addition of an indifferent gas, either before or after the dissociation, to a system composed of a disso- ciating substance and its products, has no effect upon the degree of the dissociation, so long as the total 'volume is unchanged, for then the partial pressures (and con- centrations) remain unaltered. An increase of volume, on the other hand, such as was allowed to take place above, no matter what its cause, results in an increase in the degree of dissociation. When due to the addition of an indifferent gas, this is the only effect, and the nature of the gas is without influence. When due to the addi- tion of one of the constituents, however, it is partly com- pensated by the depressing effect of this upon the disso- ciation,* according to the law of mass action, and it is possible to cause the one influence to just compensate the other, so that no change in the dissociation is to be observed as the result of the addition with an increase of volume. A somewhat more complicated case of the application * The value of x' above is larger than it would have been if the volume were not allowed to increase, as can be seen by substituting 22.4 in the denominators in place of (1.1 + ^)22.4. The very expansion of volume, without the presence of B, would cause the dissociation to increase, CHEMICAL MECHANICS. HI of the law of mass action to homogeneous gaseous sys- tems is given by the dissociation of carbon dioxide, ac- cording to the scheme If at equilibrium at any definite temperature the partial pressures are p for CO 2 , p\ for oxygen, and p 2 for CO (where these come from the CO 2 ), then for that tempera- ture and if oxygen is already present from an exterior source to the pressure a, the decrease in the pressure of carbon monoxide due to its effect upon the dissociation can be readily calculated. We have, then, (P + 2X) 2 from which x can be calculated (p. 103). p 2 2X will then give the partial pressure of carbon monoxide, and p + 2X that of carbon dioxide, in the presence of a of oxygen, the constant remaining as above. For carbon dioxide at atmospheric pressure and 3000, a = 0.4, i.e., 0.5 of the total pressure (0.5 of an atmosphere) is due to CO 2 , 0.33 to CO, and 0.17 to oxygen, conse- quently ^=0.704. The constant for COg may also have a different value; H2 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. it is that which is obtained from the formula and is equal to K' = - , which, with the above data, leads to the value K f = 0.272. Naturally, what was said of the arrangement of the ratio expressing K also holds here. Either constant may be used for this temperature, provided we always use the same form of relation. This is also true for the reaction or which may be written by the a formula either as 2 _ a . a a i a = 4a 2 (. /2\2 i- a \ ~i-a)7 V' 6 " \V) TV' or And one form must be selected and retained. Thus far in our consideration of gaseous equilibrium we have applied the law of mass action only to systems composed exclusively of gases. Other systems exist, however (i.e., those made up of a liquid or a solid which evolves a gas or a number of gases), to which the law of mass action can be applied with great success. And the application is usually much simpler in such cases, for it is a well-known experimental fact that a liquid or a solid gives a definite, constant gaseous pressure at any CHEMICAL MECHANICS. 113 one temperature, and that the amount of liquid or solid has no further effect so long as it produces sufficient gas to cause the pressure to be attained in that volume. A similar action is observed when we saturate a solution with a substance, for so long as sufficient solid is present to saturate the solution an excess will not supersaturate it. The active mass of the solid, in applying the law of mass action, then remains constant hi value, and its effect can be included in the constant of equilibrium. Thus in the reaction solid CaCO 3 ^ solid CaO+gaseous CO 2 , although the pressures n\ and 7r 2 , due to the gaseous CaCO 3 and CaO, p being that of CO 2 , can be given in the equation a constant value will also be found by employing the simpler form For since at any one temperature n\ and 7r 2 remain constant p must also be constant, i.e. the equilibrium at any one temperature depends only upon the pressure of CO 2 produced, and this is shown to be true by experi- ment. In the same way it has been shown that equilibrium in the reaction solid NH 4 HS < H4 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. is present when K'n=pip2i or, in the simpler form, when K=pip 2 . And in the reaction solid NH 4 OCONH 2 when K'x=pi 2 p 2 , or when K=pi 2 p 2 . And both of these results are confirmed by experiment. These relations, just as those for homogeneous sys- tems, hold also after the addition of one of the products of dissociation, or when one of the products is initially present, in the space into which the solid is to sublime and dissociate. Contrary to the case of homogeneous equilibrium (p. no), however; an increase of volume has no effect upon the equilibrium, so long as the solid (or liquid) phase is present, for the dissociation pressure is dependent, in such a system, only upon the tempera- ture and nature of the system. Equilibrium in liquid systems. The reaction CH 3 COOH + C 2 H 5 OH < CH 3 COOC 2 H 5 +H 2 O, it has been observed, reaches the state of equilibrium when we have present J mole of acid, J mole of alcohol, f mole of ester, and f mole of water, provided we start with i mole of each of the two constituents (either acid and alcohol, or ester and water). This reaction goes very slowly at ordinary tempera- tures, but when it reaches the above final state it remains in it indefinitely. If we designate by v the volume (in liters) of the system, and start with i mole of acid, m moles of alcohol, and n moles of ester (or water), then in the state of equilibrium, after x moles of alcohol have CHEMICAL MECHANICS. 115 been decomposed, we shall have moles per liter i-x . x / n+x\ , n + x of alcohol, of acid, - ( or I of ester, and - V V \ V / V f or - 1 of water. And, applying the law of mass action, we obtain (n+x)x K (i-x)(m-xY In the special case of equilibrium above, however, m=iy n=o, x=%; hence This value of K is one of the few which are practically independent of temperature. At 10 it is found that 65.2% of the acid and alcohol undergoes change, while at 220 the decomposition is but 66.5%. This equation has been tested by experiment with very satisfactory results. // has been found, also, that by using a large amount of acetic acid to a small amount oj alcohol, or vice versa, the formation oj ester and water is almost complete, as it should be by the law of mass action. In the same way a large amount of water upon a small quantity of ester causes the latter to be almost completely transformed.* Amylene in contact with acid forms an ester, accord- ing to the equation * For results of experiments, see " Elements," p. 248. Ii6 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. If x is the amount of ester formed when equilibrium is established, v is the volume of the system, and i mole of acid is used for a moles of amylene, at equilibrium we shall have moles per liter of amylene and - - of v v 'Y* acid left, while moles per liter of the ester will be formed ; hence K (a-x)(i-xY The value for K has also been experimentally determined in this case and was found to be - , the value of 0.001205 the constant in the reciprocal form K' = ocv being 0.001205.* When a solid goes into solution its action is apparently analogous to its transformation into the gaseous state. A saturated solution, thus, in contact with the solid at any temperature will still be saturated. We have, then, by the law of mass action, for any one temperature, or where c is the concentration of solid in solution and varies with the temperature. If the solid in going into solu- tion dissociates (non-electrolytically) into other substances, an addition of one of these should cause less substance * See "Elements, "p. 250. CHEMICAL MECHANICS. II? to dissolve. This has been proven by Behrend for a solution of phenanthrene picrate in absolute alcohol, in which a decomposition into phenanthrene and picric acid takes place to a large extent. By the law of mass action where c = undissociated phenanthrene picrate, Ci=free picric acid, and 2 = free phenanthrene all expressed in moles per liter. For any one temperature c must be constant, since the solution is saturated; hence = constant. It is obvious that this will also enable us to find the conditions of the equilibrium attained when a substance is distributed between two non-miscible solvents (p. 74). When the formula weight is the same in both solvents, 2 1^2 W\ we shall have K = , where, since ^2 = irr> an d i = ~77> c\ MM I ^2\ the ratio of the concentrations in grams ( ) per liter must also remain constant, independent of the original concentration of substance. In case the formula weight in one solvent is n times that in the other (Ai=nA2), we shall have Kc\ =c 2 n , where Ci would be equal to nc 2 -if all A i were transformed into A 2 . Since Ci = irr> CI = ^TI and Mi=nM 2 , Kci=c 2 n MI M2 A/2 n W-2 1 can also be written in the form ~^-K = -- , where Wi MI Wi Ii8 PHYSICAL CHBM/SfRY FOR ELECTRICAL ENGINEERS. and w 2 SLTQ the weights in grams in a certain volume. But, as -r-T" is a constant so long as the ratio of the Mi formula weights does not change with the dilution, we M 2 n may include the constants ~^j and K in a new constant, MI and say, when the formula weight in one solvent is always n times that in the other (at the dilutions in question), the W2 n ratio of distribution in the form - will remain constant, Wi independent oj the original dilution. An illustration of this is given by the distribution of benzoic acid between benzene (w) and water (w 2 ), the values of - , for various original concentrations, being 0.062, 0.048, and Wo 0.030, while those of , (which must be constant if w 2 2 \ is), for the same dilutions, are 0.0305, 0.0304, and / 0.0293, the differences lying well within the experimental error. The formula weight of benzoic acid in benzene, then, must be twice that in water, a fact which has been confirmed by aid of the definition of formula weight in solution, as based upon freezing-point depression. The effect of temperature upon the equilibrium- constant. Wherever a change in temperature changes the equilibrium, and does not alter the nature of the equilibrium, i.e., does not cause the disappearance of any of the constituents of the previous equilibrium, it is possible, knowing certain factors, to find the effect upon the constant of equilibrium. This relation, which can be readily derived from the law of mass action and the CHEMICAL MECHANICS. 119 second law of thermodynamics,* we shall regard as given, and simply consider its typical applications. The differ- ential form of the equation obtained is (log, JQ = where q is the heat evolved or absorbed as heat by the reaction. To integrate this expression it is necessary to assume that q itself is independent of the temperature. This will undoubtedly be practically true for small tempera- ture intervals; for larger ones, however, we must be satisfied to obtain q as the value for the temperature which is the mean of the two extreme temperatures. By integration, under the above assumption, we find which, using ordinary logarithms and solving for q, is transformed into 2X2.306 (log K' -log K) TT q= T-T' -cals.,t where 2.306 is the reciprocal of the modulus of the system of logarithms. This formula enables us to determine the variation of the equilibrium constant K with the temperature (p. 101). * See " Elements," pp. 452, 253. fThis second form of the equation is to be preferred for the actual calculations, although the other is simpler in form. 120 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. q here only expresses the heat of the reaction as it would be if no external work were done by or upon the reaction. The allowance for this, in comparing the results of the formula with observed results, must be made in each case, as is illustrated in the applications given below. One consequence of this formula is of special impor- tance. If q is zero, the value of K does not change as the result of a variation in the temperature. Thus the re- action between acid and alcohol, mentioned above (p. 115), the mutual transformation of optical isomers, and a number of others, are found neither to absorb nor gen- erate heat, nor to suffer a displacement of equilibrium, i.e. a change in the value of K, by a change in tempera- ture. We shall now consider the method of applying this equation for various purposes to various equilibria. Vaporization. The condition regulating the equilib- rium between a liquid and its vapor is the pressure or concentration of the latter (pp. 112-113), and this de- pends upon the temperature. We have then K- L JL ~V~RT And if p and $ refer to the two temperatures T and T r we may write, since R is constant, P' , P Regnault found for water that at ^ = 273, ^ = 4.54 mms. of Hg, and at T f = 273 + 11.54, p f = 10.02 mms, of Hg, hence q is equal to 10100 cals. for i mole (18 CHEMICAL MECHANICS. 12 f grams) of water. Direct experiment shows this value to be 10854 cals., but here external work to the amount fT+T'\ 2! - ) = 557 cals. (p. 44) is done and, consequently, must be subtracted before any comparison is possible. We find in this way that where P' refers to the temperature J 1 ', and it follows that / P' P\ log, K' - log, K = 2 (log - log ^ j = Since at 273 + 9.5, P=iy5 mms. of Hg, and at i, P r = 5oi mms. of Hg, =21550, while 122 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. by direct experiment the value is 21639 (i- e -> 22800 T+r>\ noi, where 1161=4 - ) The general form of the equation for a dissociation of this kind, where HI moles of one gas and n^ moles of another, etc., are evolved, is log, '-log, <==(! + 2 + . . . )log. -log, R\T T f This formula may also be applied to the dissociation of salts containing water of crystallization into gaseous water and the dehydrated or partially dehydrated salt. Solution o) solids. In this case the equilibrium depends only upon the concentration of solid substance in the solution, and the temperature ; we have, then, K=c, where c is the concentration of a saturated solution at the temperature T. If c f is the concentration of such a solution at another temperature 7 1 ', then it is possible for us to calculate the heat of solution of the solid, the increase of volume being so small that the external work is practically equal to zero. van't Hoff found by experiment with succinic acid in water that for 7^ = 273, = 2.88 moles per liter; and for 7^ = 273 + 8.5, ' = 4.22 moles per liter. For i mole, then, ^=6900 cals., while Berthelot found 6700 cals. by direct experiment. lonization of solids in solution. If a substance is very slightly soluble, and the solution consists principally of ionized matter with very little undissociated substance, CHEMICAL MECHANICS. 123 the heat of dissociation must be the same as the heat of solution, i.e., equal to the negative value of the heat of precipitation from the two kinds of ionized matter. Thus for AgCl we have If the solubility at T = c, and at T / = c f t in moles per liter, then, since 2 moles of the ionized matter are formed from i mole of the salt, we have, as on page 122, logX-log.= 4Vr T For T = 273 + 20, c=i.ioXio~ 5 , and for ^ = 273+30, c' = i.73Xio~ 5 ; hence q= 15900= 159^." For the negative heat of precipitation we found (p. 93) i$SK, which is an excellent agreement. We have assumed the ionization to be complete here, and the fact that the heat results agree cannot but be considered as confirmatory of our assumption. Dissociation of gaseous bodies. When a substance A dissociates according to the scheme the equation of equilibrium is where c, c\, c^ . . . are the concentrations of A, A\, AI, . . . in moles per liter. If the mole occupies the volume V at T and the 124 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. volume F' at T 1 ', then for the dissociation of N 2 O4 we have N 2 O 4 +=* 2NO 2 , hence (p. 112) K' = ,, "'* and K (i -OF' (i-a)F a 2 q/i i Since the density of undissociated N 2 O4, from its for- mula weight (96), is 3.179 (based upon air), a for any temperature can be found from the relation a = i. 3-179 Since one mole of gas at any temperature and atmospheric pressure occupies 0.0819 ^ liters (from j, the volume occupied by i mole, which has undergone dissociation to the extent a is (i +a) 0.0819 T liters, where a is the dissociation at the temperature T. From the results observed, viz., 7^ = 273 + 26.1, ^ = 2.65; T' = 273 + 111.3, A' = 1.65; we find a = 0.1986, F = 359.i, #' = 0.9267 and F' = 67i, hence = --> where the values of K are for the same dilution and represent the change of ionization, even though the values are not the same as other dilutions, and the values V cancel and need not be considered, q is then the heat liberated when a mole of substance is formed in solution by the union of the kinds of ionized matter composing it. It will be observed here that these values differ from those calculated from the table given on page 95, for these refer to the heat of forma- tion of ionized matter from substance already in solu- tion, while those refer to the compound process of solu- tion and ionization, i.e., the difference in energy between the ionized state and the solid or gaseous state. Some of the results as found by Arrhenius are given below, the unit being the small calorie. HEATS OF IONIZATION. Substance. Temperature. Calories. Propionic acid .................. | ^ o ^ ~ 557 Butyric acid... ..{35^ ~ 935 Phosphoric acid Hydrofluoric acid ............... 33 3549 Potassium chloride .............. 35 362 1 ' iodide ............... 35 916 bromide .............. 35 425 Sodium chloride ................ 35 454 ' ' hydrate ................. 35 1292 " acetate ................. 35 391 Hydrochloric acid ............... 35 1080 * Zeit. f. phys. Chem., 4, 96, 1889, and 9, 339, 1892. EQUILIBRIUM IN ELECTROLYTES. 14.7 HEAT NECESSARY TO COMPLETE THE IONIZATION, (i a)w (i mole in 200 moles of water). Substance. Temperature. Calories. Potassium bromide 35 58 " iodide 35 132 " chloride '. 35 - 56 Sodium hydrate 35 180 chloride 35 - 81 Hydrochloric acid 35 136 Hydrofluoric acid 33 3304 Phosphoric acid 21. 5 1682 For the temperatures of 35 in the table, 2^ = 273 + 18, ^273 + 52; for 2i. 5 , r = 2 7 3 + i8, r = 273 + 25 in 177 the log*"^ formula (p. 119). From data such as the above it is possible to calculate the heat of neutralization of an acid by a base. The formula for this (p. 92) is q=x+w 3 (i a 3 )w 2 (i a 2 ) Wi(i i), where the figures i, 2, and 3 refer respectively to acid, base, and salt, and x is the heat of formation of i mole of water from ionized H* and ionized OH', i.e., 13,700 cal. In the table below the calculated values of q at two temperatures are given, together with the observed values at one of the temperatures. HEAT OF NEUTRALIZATION OF ACIDS WITH NaOH. (i mole of acid + 1 mole of NaOH 4-400 moles of H,O.) At 35. At 21.5. Calc. Calc! ObsT" HC1 12867 13447 13740 HBr 12945 13525 13750 HNO 3 12970 13550 13680 CH 3 COOH 13094 13263 13400 C^COOH 13390 13598 13480 CHC1 2 COOH 14491 14930 14830 H 3 PO 4 14720 14959 14830 HF 16184 16320 16270 148 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. It is obvious from the results above that the value oj the heat 0} neutralization oj an acid by a base cannot be considered as indicative of the strength of the acid. The two latter are relatively weak acids and yet they give rise to the greatest amount of heat. This formula of van't HofFs, as was mentioned above can only be used to calculate the heat of solution of sub- stances which are completely ionized (or practically so) or completely un-ionized. And, naturally, until our knowledge of the conditions governing equilibrium in such systems is considerably broadened, we cannot expect to find a formula that will hold. Solubility or ionic product. Although, as we have seen, the law of mass action cannot in general be applied to the equilibrium of the ionized and un-ionized portions of a substance in solution (except to organic acid and bases), it can be applied with considerable accuracy to a very large number of saturated solutions. An example of such an equilibrium is a saturated solution of silver chloride, which is found to be practically completely ionized according to the scheme AgCl=Ag' Applying the law of mass action to this we obtain or = (i-a)F' when c is the concentration of un-ionized AgCl, c\ that of ionized Ag", and c^, that of ionized Cl'. Since the solu- tion is saturated, the value of c at any temperature must remain constant, for if the solution were unsaturated, solid would dissolve, or if supersaturated, solid would pre~ EQUILIBRIUM IN ELECTROLYTES. H9 cipitate. In a saturated solution of silver chloride, at any one temperature, then, we have the relation Kc = constant = CiC 2 ; i.e., in a saturated solution of a binary electrolyte (of this kind) the product oj the concentrations oj the two kinds 0} ionized matter must remain constant, with unchanged tem- perature. Expressing this in a more general form, we have for the reaction nA in a saturated solution, the relation ci" l c 2 H * = constant = s, where 5 has been called by Ostwald the solubility product of the substance. This solubility product is of para- mount importance in analytical chemistry, for a precipi- tate (when due to an ionic reaction, and most of them can be shown to be due to this) is always and only formed when its solubility product is exceeded. This, of course, presupposes that no supersaturation phenomenon is pos- sible ; if it is, then the so-called metastable limit * must first be exceeded. Just as we found a decrease in the dissociation of a gas or an organic acid, by the addition of one of the products of dissociation from an exterior source, so here the addi- tion of a substance with a kind of ionized matter in common causes the formation and separation in the solid state of the un-ionized substance. In other words, the term 5 still *" Elements," p. 128. ISO PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. retains its constant value, and consequently the kinds of ionized matter composing the substance unite to form more of the un-ionized portion, which, since the solution is already saturated with it, separates out as solid. This has been found to be true by experiment, but only true quantitatively for those substances which are difficultly sol- uble* The effect may be observed most easily by dissolv- ing the difficultly soluble substance in a solution of the salt with ionized matter in common; but it can also be attained by adding to the saturated water solution of the substance a strong solution of the salt, when a pre- cipitation of the substance, usually in the crystalline state, will be observed. Thus if we add to one portion of a saturated solution of silver acetate a strong -solution of sodium acetate containing x moles of ionized CH 3 COO', and the same amount of a solution of silver nitrate containing x moles of ionized Ag* to the liter to another equal portion, we observe an equal precipi- tation of solid silver acetate in the two solutions. The examples below will serve to show how the solu- bility product of a substance can be found, and how when once found it can be employed to foresee the solu- bility of the substance in a solution already containing a common kind of ionized matter. *Noyes and Abbott (Zeit. f. phys. Chem., 16, 138, 1895) have found for those substances which are largely dissociated, and this is general, that the concentration of the un-ionized part of the salt has always the same value when the product of the concentrations of the kinds of ionized matter it produces has the same value, whatever may be the values of the two separate factors of that product. In other words, if the strong electrolyte AD, ionizing into A' and D', has a concentration of AD in a saturated solution equal to y, when A'XD' has the value x, it will also have the value of y whenever A ' X D' has the value of x, whether it be produced by (A' z)(D'+z} or (A' +z)(D'-z), etc. EQUILIBRIUM IN ELECTROLYTES. 15* Silver bromate is soluble at 25 to the extent of 0.0081 moles per liter. Assuming the ionization in this state to be practically complete, and it certainly is nearly so, the concentration of the ionized Ag* and ionized BrO 3 ' will be the same, and equal each to 0.0081 mole per liter. The solubility product at this temperature, then, will be (0.0081 ) (0.0081 ) = 5 AgBr0l . The solubility in a solution of silver nitrate containing o.i mole of ionized Ag* (or in potassium bromate con- taining o.i mole of ionized BrO 3 ') can now be found readily by aid of the relation (0.0081 ) 2 = (0.0081 +.1 -;y)(o.oo8i -y\ and is equal to (0.0081 -y), for that is the concentration of ionized Ag* and ionized BrO 3 ' now existing in the solution, and coming from the salt; the amount o.i of one being due to the other salt, and y being the AgBrO 3 remaining undissolved owing to the presence of this o.i mole of Ag- or BrO 3 '. This is true for all binary salts when they can be as- sumed to be completely ionized, or practically so. Where the substance dissociates into more than two kinds of ionized matter, and can be assumed to be com- pletely ionized, the relation is quite similar. Suppose the salt to dissociate according to the scheme As solubility product we shall have, if c is the solubility of the completely ionized salt MA& c X (^c) 3 = s M A 3 > or for we must nave three times the num- IS 2 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. her of moles per liter of A' as we have of M'" accord- ing to the chemical equation, i.e., c = C M -~ and 3^ = c^. In case of solution in the presence of o. i mole of one of the kinds of ionized matter, we have, then, either (%+ o. i - or CXC +0.1 - from which it is apparent that the effect of equal ad- dition is not the same for the two kinds of ionized matter, i.e., that x and y, the decreases in the solubility, are not equal. In the case the substance is not completely ionized, the solubility product is not so directly related to the solu- bility of the substance as in the above cases, i.e., to the square in one case and twenty-seven times the fourth power in the other. Consider the case of uric acid, which at 25 is soluble to 0.0001506 mole per liter, and is ionized in that condition to 9.5% into H* and the ionized negative radical which we shall designate as V. The solubility product here is naturally (0.0001506X0.095) (0.0001506X0.095) = SHU = #HU(O.OOOI 506 Xo.905). The solubility of uric acid in a molar solution of hydro- chloric acid, for which a = 0.78 (i.e. H*=o.78, Cl' = o.78), is to be found in the following way: (o.oooi 506 Xo.o95 + 0.78 x) (o.oooi 506 Xo.095 x) = (0.0001506 Xo.095) 2 , where (0.0001506X0.095 #) represents the present con- centration of H* and U' from the uric acid, and its EQUILIBRIUM IN ELECTROLYTES. 153 total solubility in the hydrochloric acid solution is (0.0001506X0.905) + (0.0001506X0.0953;), i.e., is equal to the sum of that which is un-ionized and that which is ionized. In general, just as for organic acids, an infinite excess of one of the ions will cause the ionization of the sub- stance to become zero. It is to be observed here, however, that this excess will only cause the solubility to become zero in the case that the ionization is complete. In the case of uric acid, an infinite amount of H* or U' at best can only reduce the solubility by 9.5%, the remaining 90.5% being un-ionized and not affected at that temperature by any addition of substance which does not react chem- ically with it. That a substance is always decreased in solubility by the addition of a substance with ionized matter in common is not true, as the well-known behavior of silver cyanide in potassium cyanide will show. In all such cases, howeve^ the equilibrium which has previously existed is altered in some way, so that the relations are not the same- These cases are usually characterized by the formation of a complex kind- of ionized matter the product of which is exceeded. The removal of the kinds of ionized matter necessary to form this complex kind disturbs the equi- librium of the difficultly soluble salt; the un-ionized por- tion then ionizes further, and its loss is replaced by the solid phase. This process continues, dissolving new salt, until equilibrium is attained, i.e., until the solubility prod- uct, whatever it may be, is just satisfied, when solution ceases. The ionization of silver potassium cyanide takes place almost completely according to the scheme KAgCN 2 =K'+AgCN 2 ', 154 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. but it has been found by Morgan * that in a 0.05 molar solution we have ionized Ag* to the extent of 3.5Xio~ n and ionized CN 7 to 2.76Xio~ 3 moles per liter. Knowing the concentration of the ionized metal, for example (which can be determined by methods given in the next chapter), in the complex salt solution and in a water solution of the difficultly soluble salt, we can fore- see the behavior of that salt when in a solution of a salt which might dissolve it to form a complex solution of that strength. In general, i.e., when the concentration oj ionized metal in a water solution of salt is greater than that of a water solution of a complex salt, the simple salt will dissolve in the solution which will produce the com- plex salt in this concentration. If the concentration oj ionized metal is smaller the solid will not dissolve to any greater extent than it does in pure water, jor the ionic product oj the ionized complex cannot be exceeded. By this law it is possible to find the relative solubility of salts of the same metal in water. Thus silver sulphide is the only silver salt which will not dissolve in potassium cyanide solutions; in other words, is the most insoluble salt of silver, and contains less ionized Ag* in a saturated solution than exists even in a solution of silver potassium cyanide, such as that given above. It is not only for substances in solution that we find this constancy of the product of the concentrations of the kinds of ionized matter, for it also exists in our usual solvent, water, where the ionized portion is so small that the un-ionized portion may be considered as con- stant, i.e. i -a does not differ appreciably from i. Expressing the concentrations of ionized H* and ion- ized OH' in a liter of water by Ci and c 2 , and the *Zeit. f. phys. Chem., 17, 513-535, 1895. EQUILIBRIUM IN ELECTROLYTES. 155 un-ionized portion, which is practically i liter, i.e. 1000 =55.5 moles, by c, we have Io 5 5 . 5^H 2 o = SH Z O = constant. The values of c\ = c 2 = ionized H" ( = ionized OH') in water at various temperatures is as follows: Temp. Moles per Liter. Temp. Moles per Liter 0.35 34 1.47 10 0.56 50 2.48 18 0.80 85. 5 6.20 25 1.09 1.00 8.50 The ionic products (we can hardly call them solubility products), then, are as follows: *o=(o.35X IO ~ 7 ) 2 > 5 34 = (1.47X10 -7)2, S W = (0.56X10 -7)2, s 50 = (2.48X10 -7)2, S ls = (0.80 X 10 -7)2, s 85 , 5 = (6. 2 X 10 -7)2, *25 = (1 .09 X 10 -7)2, S 10(J = (8. 5 X 10 -7)2, where the value of s, in each case, is equal to 55.5 times a 2 the value of K= ( _ ., V being 0.0018 liter, i.e., the volume occupied by the formula weight, 18 grams, of water. Knowing the solubility products of two substances with a kind of ionized'matter in common, it is possible to find how much of each will dissolve when a mixture of them is exposed to the action of a solvent; and this, I5 6 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. of course, may be expanded to three or more substances together. Assume we have the two completely ionized, diffi- cultly soluble salts MA and MAi, with the ionized mat- ter M' in common, and that they are dissolved simul- taneously in water. Call the amount of MA, which dissolves, x, and the amount of MAi y. In the solution, then, we must have x+y moles of ionized M', x of ionized A' and y of ionized A\ r \ and, if s is the solubility product of M A and s\ that of MAi, the relations must be so that by solving the simultaneous equations we can find x and y. An example of this is given by dissolving thallium chloride and sulphocyanate together. The solubilities in water, each for itself, are TlCl = o.oi6i and T1SCN =0.0149. Assuming complete ionization, the solubility products are respectively (o.oi6i) 2 and (0.0149)2, an< ^ if x represents the amount of chloride and y that of sulphocyanate dissolving from the mixture, we have Tl', x = Cl', and y = SCN', and x(x+y) = (o.oi6i) 2 , y(x+y) = (0.0149)2, from which we find x = 0.0118 and ;y=o.oioi, while the values # = 0.0119 an d ;y = 0.0107 are found by ex- periment. It will be observed that in the above examples, except the last, we have tacitly assumed that the dissociation EQUILIBRIUM IN ELECTROLYTES. 157 of the added salt, with a kind of ionized matter in com- mon, is not influenced by the same kind of ionized matter from the difficultly soluble salt. As a rule this is true, for the substances are so insoluble that their effect is infini- tesimal; in the last example, this effect has been allowed for, however, and will show the method of treating such cases. In general, then, we can conclude for difficultly soluble salts (and jor ionized complexes) that they are precipitated (formed) when the product of the concentrations of the ionized substances composing them exceeds the solubility (ionic) product. Although this law holds in general for difficultly soluble salts, isolated cases are to be found where the un-ionized portion does not remain rigidly constant, after the addition of a substance giving the same kind of ionized matter in common; and, to a smaller extent, a slight variation is sometimes observed in the solubility product. Since these cases are very few, and are usually observed for the more soluble salts, it would seem probable that they are due to secondary reactions not yet recognized, or to others not properly accounted for.* Hydrolytic dissociation, or hydrolysis. Hydrolysis is the process taking place in a water solution of a salt t which causes the solution to appear alkaline or acid, or results in a neutral equilibrium according to the scheme MA+H 2 = MOH+HA. If the acid formed is insoluble or un-ionized, the base being ionized, the reaction will be alkaline (action of ionized OH'). When the base is insoluble or un-ionized, * See foot-note, p. 150. 158 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. and the acid ionized, the reaction is acid (action of ionized H')- And, finally, if both acid and base are insoluble or un-ionized, the salt is completely transformed into base and acid, and, as there will remain no excess of either OH' or H', the reaction will be neutral. In other words, then, hydrolysis is the name by which we designate the process resulting from the removal of either H' or OH' (or both) from the water by the ionized A' or the ionized M' of the salt, to form un-ionized or in- soluble substances ; in short, since this removal causes the further ionization of the water, hydrolysis is the chemical process observed to take place between a salt and water. Examples of this process are most common. For instance, all mercury, copper, zinc, etc., salts are acid, for un-ionized basic substances (for which the ionic products are exceeded) are formed by the reaction with water, leaving free, ionized acid; and -potassium cyanide is alkaline, due to the formation of un-ionized hydro- cyanic acid, and ionized potassium hydrate. Since we know the conditions under which insoluble or un-ionized substances will form, i.e. by the exceeding of their solubility products or analogous values, it is possible to find the conditions necessary to produce a hydrolytic dissociation, and to calculate the extent of this when it does take place, i.e., to find the equilibrium which is finally attained in the solution. We recognize at once that if the product of the con- centrations of the ionized M' and the ionized OH' is larger than that which can exist in pure water, un-ionized substance must form. By this formation, however, the equilibrium of H* and OH' will be disturbed, and a further ionization of water must take place, until at length the ionic product is just attained. If the H' and A' at this EQUILIBRIUM IN ELECTROLYTES. 159 point do not unite to form un-ionized acid, the further ionization of water will be unlike what it would be in the absence of this excess of ionized H*, for, since ' must at tne same time be equal to s H2O , we can only have ^ moles per liter of ionized OH' present, when C H - is the total concentration of ionized H" at that time. The process due to the formation of un-ionized or insoluble acid, when no un-ionized base is formed, or forms but slightly, is exactly analogous to the above. In both cases water is decomposed, owing to the removal of one of its two kinds of ionized matter, and the further ionization of water and the formation of the insoluble or un-ionized base or aeid continues until the equations representing equilibrium are fulfilled. For the sake of simplicity we shall first consider sepa- rately the cases that the reaction is caused by the base or by the acid. Case I. The process is due primarily to the formation 0} base. Here it is obvious that ^M-X^OH'>(^MOH C MOH) or >%OH where the terms c refer to the ionic concentrations. Call- ing c the original concentration of salt, and d s the ioni- zation of the salt, the concentration of ionized OH' in water at 25 being LopXio" 7 moles per liter, we shall have d s c X 1.09 X 10 - 7 > (^MOH^MOH) or > J MOH . After equilibrium has been established, i.e. when the degree of hydrolytic dissociation is a, (#HA^HA) or > <> HA - Just as above, since Cn- = i.ogXio~ 7 at 25, we shall have, when c is the concentration of salt, and d$ its ionization, or And if d B is the ionization of the base, c the original concentration of salt, and a its hydrolytic dissociation, then 1 IT* ^HeO and (orig. A' -loss of Ao( tot ^Q H ,) = # H AXHA formed, EQUILIBRIUM IN ELECTROLYTES. 161 or And here again the solubility product may be used on the right if the acid is difficultly soluble. The formulas above, in both cases, may also be written in another form, which, although it does not illustrate so well the principles involved, is more useful in many ways, for it enables us to obtain a constant of hydrolytic dissociation, from which the value of a at any other dilu- tion and that temperature. From page 160, by trans- formation, we obtain Cd s (l -a)s H2 Q = #MOH 2 ^A> or <* 2 _ % 2 o d_ ' (i-a)V'd s '* and from the above equation, in the same way, _ (i-a)V' d s d - *HA' (i-a)F In dilute solutions where d s , d A , and d-Q may be regarded as unity, these equations are simplified to the form __ "nyd. / T ,v M7 V (ia)V A HA We have the following law, then, governing hydrolysis. The expression for the hydrolytic dissociation of a salt a 2 in water, r^, is equal, when due to the formation of base (acid), to the ionic product of water multiplied by the degree of ionization of the salt divided by the ionization constant of the base (acid), multiplied by the degree of ionization of the acid (base). 162 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. This law has recently been much used to determine from the experimentally observed hydrolysis of the salt the ionization constant of the acid or base formed. The constant of hydrolytic dissociation, in such a case, can also be defined (see pp. 107 and 135), when d$, d&, or d& are equal to i, as one- half the concentration, in moles per liter, at which the salt is 50% hydrolyzed. And if a is small enough to be neglected in the term a 2 i a, _ \y = K is a l so reduced (p. 138) to KV=a 2 ; in other words, for the same substance, the hydrolytic dissociation, when small, is proportional to the square root of the dilution of the salt, i.e., aoc\/V or >J , where c, the reciprocal of V, is the original concentration of the salt dissolved. Knowing the constant for hydrolytic dissociation it is also possible to calculate the degree of hydrolysis at any \~ K 2 v 2 KV dilution by the formula a = \JKV+ X . The following examples will serve to show the use which may be made of the above relations. What is the ionic product for water at 25? A o.i molar solution of sodium acetate is 0.008% hydrolyzed, the sodium acetate to be considered as completely ionized, as is also the sodium hydrate formed, and the ionization constant of acetic acid is 0.000018. Here CH 3 COOH = OH' = 0,00008 X o. i = 0.000008, EQUILIBRIUM IN ELECTROLYTES. 163 and since 0.000018 Xo.oooooS H -= =1.44X10 * and What is the hydrolysis of a o. i molar solution of potas- sium cyanide (assuming d s = i) ? K for HCN = 13 X 10 ~ 10 and s H2 o = (i-9Xio- 7 ) 2 at 25. a 2 (1.09 X 10 ~ 7 ) 2 hy(L ~(i-a)F~ 13X10-! ' from which, when d% = i and F=io, a =0.967%. a 2 In the table below are given the values of - - r-~ for various equilibria, in which but i mole of water reacts with the substance. HYDROLYSIS OF HYDROCHLORIDES AT 25. Aniline ............... 2.7 2.25Xio~ 5 5.3Xio~ 10 o-Toluidine ........... 7.0 1.62X10-* 7.3 X io~ n m- " .......... 3- 6 4-ioXio- 5 2.9 Xio- 10 p- " .......... 1.8 1.05X10-* 1.13X10- o-Nitroaniline ......... 98. 6 2.1 5-6X io~ 15 m- " ........ 26.6 3.01X10-' 4.0 Xio~ 12 p- " ........ 79.6 9.58X10-* i.24Xio- 13 Aminoazobenzene ..... 18.1 1.25X10-* 9.5 Xio~ 10 Urea .................... 0.781 i.5Xio- u Thus far we have only considered that one mole of water reacts with the salt; in other words, we have only 164 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. employed salts containing monovalent elements. In case the reaction involves more than one mole of water the treatment is the same as in any other application of the law of mass action. It must be said, however, that such cases, so far as we know at present, are not at all common, the salt often reacting with but one mole of water to form a basic salt which still retains some of the original element. There is one reaction, however, which gives a good constant assuming two moles of water to react, and we shall consider it to show how such relations are to be treated. The reaction is A1C1 8 + 2H 2 = A1(OH) 2 C1 + 2 HC1, which has been investigated by Kullgren.* We have, tnen, where c is the molar concentration of AlCla in solu- tion, and a is the fraction of it hydrolyzed, d$ being the ionization of AlCls, and d& that of the HC1, of which the total amount 20.0 is formed, / s \ 2 2 = It is impossible to use this formula in calculations, how- ever, for as yet we know nothing of K A1 ( O H) 2 ci- But by using the equation in the other form (p. 161), we obtain 4 q;3 dj ^H 2 o ' and from this we can calculate the ionization constant of A1(OH)2C1, when a is known, or dispense entirely * Om metallsalters hydrolys, p. 108. Dissertation. Stockholm. 1904. EQUILIBRIUM IN ELECTROLYTES. 165 with it, i.e., using the K hy d . so determined for the cal- culation of other values. It will be observed here that a, instead of being proportional to VV as it is for the reac- tion with i mole of water, is proportional to VF*. The following results will show how well this equilibrium follows the above law, and how it is possible to find the ionization constant by aid of the hydrolytic dissociation, knowing the ionic product for water at that temperature. HYDROLYSIS OF A1C1 3 AT 100 C. A1C1 3 +3H 2 0=A1(OH) 2 C14-2H' 96 0.1488 0.966 0.76 420X10-' 516X10-' 384 0.3629 0.977 0.85 509X10-' 571X10-' 1536 0.7142 i 0.91 541X10-' 594X10-' Average, ^"hyd. = 5 60 X i o~' The concentrations of base in the three cases are 0.00155 0.000945, and 0.000465, respectively, the acid concen- trations being twice these values. The average value of ^hyd. i n tne l ast column may be used to determine Hz T' We obtain ^ d s this way the value ^Ai(OH) 2 ci = 2.33Xio- 1 , where the ionization, presumably, gives A1C1" and 2 OH'. The formation of a substance containing OH as well as the original negative element is very common. In the case of the chloride of bismuth mentioned above the substance separating out by hydrolysis is not the pure hydrated oxide, but an oxychloride, although apparently this is not true in the case of the hydrolysis of ferric chloride. In this case the reaction is not FeCl 3 + 3H 2 - Fe(OH) 3 + 3 H' + 3 C1', 1 66 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. but, according to Goodwin,* must rather be FeCl 3 +H 2 = FeOH" +H* +3C1', where the FeOH" is colloidal. In certain other cases it has been found that hydrolytic dissociation takes place in stages, i.e., first i mole of water reacts, then another, etc. It is quite certain, how- ever, that this does not occur at the dilutions above of AlCla, for if it did, the formula used would not give a constant value, hence in this one case between these limits of dilution 2 moles of water react with i of salt. T1(NO 3 ) 3 , according to Spencer and Abegg,f on the other hand, seems to react directly with 3 moles of water, the T1(OH) 3 having a solubility equal to io~ 13 - 58 moles per liter, i.e., s = io~ 52 - 896 , but as yet this is the only case known. Naturally, any method for determining the concen-' tration of ionized H* or ionized OH', or the undisso- ciated substance formed, will enable us to find the amount of hydrolytic dissociation. One method, which can be used for salts of weak acids with strong bases, or salts of weak bases with strong acids, has been suggested by Farmer, { and this, owing to the importance of the principle involved, is briefly considered below. The method is based upon the coefficient of distribution of a substance between water and another solvent, benzene (pp. 74-75). Thus, hydroxy- azobenzene has a coefficient of distribution between water and benzene equal to 539, i.e., benzene always * Zeit. f. phys. Chem., 21, i, 1896, and Phys. Rev., u, 193, 1900. f Zeit. f. anorg. Chem., 44, 397, 1905. J Trans. Chem. Soc., 79, 63, 1901, and ibid., 85, 1713, 1904. EQUILIBRIUM M ELECTROLYTES. 167 takes up 539 times as much hydroxyazobenzene as the water, when the two solvents are present in equal volumes. If the two solvents are present in unequal quantity, say i liter of water to q liters of benzene, the hydroxyazo- benzene in the water will be distributed between them in the ratio i : 539*7. By shaking a water solution of the barium salt of hydroxyazobenzene with benzene, then, the free hydroxy- azobenzene, if it be formed by hydrolysis, will be partially extracted by the benzene. Finding the amount of this present in the benzene solution, multiplying it by , we find what is left in the aqueous solution. The sum of these two quantities, then, is the concentration of free hydroxyazobenzene which has been formed as the result of hydrolysis, and, knowing the amount of salt initially present, the degree of hydrolytic dissociation is easily calculated. By this method, for numerous dilutions of a 2 the barium salt, the formula J== _ . was found to give a constant value for K, which at 25 is equal to 24.3X10-7. This method has also been applied to the hydrolysis of the hydrochlorides of weak bases, as aniline, etc. (where the coefficient of distribution of the free base is deter- mined), with very satisfactory results. The values in the table on page 163 were found in this manner. In all such determinations constancy of temperature is of paramount importance, for hydrolytic dissociation, as will have been observed from the foregoing, is largely in- fluenced by the temperature. This is due not only to the increased ionization of water (p. 155) with the temper- ature, but also to the decrease in the ionization constants 168 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. of acids and bases. Thus for acetic acid K l8 =^ 18.3 X 10-6, tf ioo . = n. 4 Xio-, K I56 * = 5.6X10 -6, and K 2l8 * = 1.9X10-6, while for ammonium hydrate K l8 =i'j.iX 10-6, loo . = 14X10-*, and # I56 o= 6.6X10 -6. Ionic equilibria. Knowing the solubility or ionic products, and the ionization constants of the constitu- ents, it is often possible to gain an idea of the mechanism of the reaction and the equilibrium that will be produced; or, on the other hand, to calculate some of these factors when the composition of the system at equilibrium is known. Thus it is possible to calculate the ionization constant of an acid (or a base) from its increased solu- bility in a base (or an acid) with a known ionization constant.* In the same manner, also by aid of the law of mass action, it can be proven that magnesium hydrate is not precipitated by ammonia in the presence of ammonium chloride, not because of the formation of a double salt of magnesium and ammonium, but simply because the de- crease in the amount of ionized OH' from NHOH, by the presence of an excess of ionized NH from the NH^Cl, is so great that the solubility product of Mg(OH) 2 cannot be exceeded.^ And this conclusion has been confirmed by Treadwell,| who studied the freezing-points of solutions of MgCl2 and NH 4 C1 separately, and then when present together, and showed conclusively that no such compound exists in solution. Findlay has investigated the reversible reaction * See " Elements," pp. 329-334; Lb'wenherz, Zeit. f. phys. Chem., 15, 385, 1898. t " Elements," pp. 339-342; Loven, Zeit. f. anorg. Chem., 37, 327, 1896; Muhs, Dissertation, Breslau, 1904. J "Elements," pp. 342-345; Zeit. f. anorg. Chem., 37, 327, 1903. "Elements," pp. 345-346; Zeit. f. phys. Chem., 34, 409, 1900. EQUILIBRIUM IN ELECTROLYTES. 169 solid PbSO 4 + dissolved 2NaI = ni-n 2 'C v + vt where n\ and n 2 are the valences of the anion and cathion matter respectively, and c v is a constant for all electrolytes. When c v is known for all dilutions, and also the terms A v , n\, and n 2 , we can find the value of A^, i.e., the equivalent conductivity at infinite dilution. If we designate (HI -n 2 -c v ) by d v , then Below are given the values of d v for different dilutions and values of n\ -n 2 at 25. Valence, i "n z without knowing the solubility of the salt. In case the conductivities of the constituents of the salt for which we are to determine A are not included in the table we can naturally calcu- late them from the values at infinite dilutions of salts for which Joo can be found by experiment. In general, then, we would have Thus for BaSO 4 , we have i.e., Joo(|BaSO 4 ) = 1 15 + 128 - 115 = 121. The influence of temperature upon conductivity is pri- marily the result of temperature upon the speed of migra- tion (for the ionization does not change very greatly with the temperature, see page 144) and is shown in the table on page 184. In general the variation in the equivalent conductivity of largely ionized salts is about 2^% per de- gree, and this, naturally, must always be considered when calculating ionization or solubility, as was done above. For the calculation of the conductivity of a mixture of substances the reader must be referred elsewhere,* for the relations are too complex to be discussed here. * "Elements," pp. 396-399. ELECTROCHEMISTRY. 19* ELECTROMOTIVE FORCE. The chemical or thermodynamical theory of the cell. We shall not consider here either the methods for deter- mining the electromotive force nor the standard cells upon which such measurements are based, but shall devote ourselves exclusively to the consideration of those factors which condition the rise and magnitude of an electromotive force in a system. Since in general it is the chemical energy of a process which is transformed into electrical energy, and since the heat developed by a chemical reaction under certain conditions is proportional to the chemical energy involved, it is possible to derive a formula from which the electrical energy can be calculated when the heat developed by the reaction is known. Such a formula, however, proves to be satisfactory only in isolated cases, and the varia- tion has been shown to be due to the loss or gain of heat during the process, i.e., either less or more than the heat developed by the chemical reaction itself is trans- formed into electrical energy.* Imagine a reversible cell in which the amount of heat q is liberated or absorbed during the passage of i gram equivalent of ionized matter through the solution. As- sume this cell to be in a constant temperature bath so * To obtain electrical energy from a chemical reaction it is usually necessary to so separate the process that it may take place in two por- tions, at points which are spatially separated. Thus zinc dissolves in acid, giving off hydrogen gas and evolving heat. When the zinc is connected by a wire to a plate of platinum, however, and both are placed in acid, the zinc dissolves, but the hydrogen is evolved from the platinum plate and a current of electricity flows through the wire; the process has been separated into two spatially separated portions, and a current is the result. 192 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. that its temperature cannot vary i.e. if heat is absorbed it is replaced, if liberated it is removed thus preventing q from causing any change in the temperature of the cell. If TT, the electromotive force of this cell, is just compen- sated by TT, the process will be in equilibrium. For two energies in equilibrium, however, we have found (p. 15) that where c and i are the factors of the one kind of energy, and ci and i\ those of the other. In this case the two kinds of energy are electrical and thermal, hence or which gives the change in E.M.F. due to the absorption or liberation of q calories during the process. If we now pass 96,540 coulombs of electricity through such a cell, during which to keep the temperature con- stant it is necessary to supply q calories, the electrical energy obtained by the process must be equal to the chemical energy involved plus the electrical energy equiv- alent to the heat q. We have, then, expressing all terms in like units, * According to the second law of thermodynamics (p. 48), the capacity factor of heat energy is equal to ~, for we find AQ=-j=JT, and know that T is the intensity factor. ELECTROCHEMISTRY. 193 or, since E g = ne , TT= i.e., the actual E.M.F., n t is only equal to that calculated from the chemical energy (as heat) of the process when the E.M.F. is independent of the temperature. Other- wise TT is smaller or larger than , according as T-r^ Q A 1 is negative or positive in value. An illustration of the application of this formula is furnished by the Grove gas-cell. Here ;r= 1.062 volts and g, the heat evolved by the chemical reaction, 34200 An is 34,200 calories, hence 1.062= ^ ~AT> Le ' TAir -T=r = 0.418 volts, in place of 0.416 as found by actual experiment. The value 23,110 here is the quan- tity 96,540 coulombs expressed in calories, i.e., is 96, 540X0. 2394 XTT = 23,1 IOTT. In other words, 23, HOT: is the work in calories necessary to separate i gram equiv- alent of any substance at n volts. The osmotic theory of the cell. Considering a cell from the standpoint of our conclusions respecting the nature of electrolytes, it is possible to see more clearly into the cause of the rise of a difference in potential between two solutions, or a metal and a solution. Assume we have two solutions in contact, and that they contain the same kind of monovalent ionized matter in dif- fering concentration. The difference of potential existing on their boundary can now be calculated by aid of the fol- lowing process of reasoning : If U a and U c are the mobilities of the respective kinds of ionized matter, then, by the pas- 194 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. sage of 96,540 coulombs of electricity, the following changes must take place : Assuming that the current enters on the concentrated side and passes through both solutions, C gram equivalents (moles in this case) of posi- tively charged ionized matter will go from the concen- trated side to the dilute, and, during the same time, T , " TJ gram equivalents of negatively charged ionized L/C i U a matter will go from the dilute solution to the other. Let p be the osmotic pressure of the two kinds of ionized matter in the concentrated solution and p f that of those in the dilute solution, the maximum osmotic work to be done by the process, then, will be (p. 99) and this must be equal to TTSQ for the process going in this way, i.e., to the electrical work done at the contact surface of the two solutions. Since R in calories, when divided by 23,110, gives the value in electrical units, we can find in this way the difference of potential of two solutions, and experiment has shown the calculated values to agree with those observed. This same method of reasoning may be applied to the cell Concentrated amalgam of the metal M . Water solution of a salt of the metal M . Dilute amalgam of the metal M. In this case the passage of 99)540 coulombs causes i gram equivalent of the metal M to go from the con- centrated side to the dilute, and, if there are n equivalents to the mole (i.e., the metal is n valent), the maximum ELECTROCHEMISTRY. 195 work per equivalent will be where ci and c 2 are the concentrations of the metal M in the two amalgams. Again, here, the value of TT in volts can be found by dividing the expression by 23,110. And in the same way we can calculate a formula for a cell with electrodes of the same soluble metal in two different concentrations of a solution of a salt or salts of that metal. Consider, for example, the cell Cu dilute CuSO, concentrated CuSO 4 Cu. By passing a current through this cell in the direction of the arrow the following changes will take place: 1. For each 96,540 coulombs of electricity i gram equivalent of copper will dissolve from the electrode in the dilute solution, i.e., will be transformed from the metallic to the ionic state; 2. At the boundary of the two solutions the process described above will take place; and 3. One gram equivalent of ionized Cu" will be de- posited from the concentrated solution upon the electrode. As the result of processes (i) and (3) i gram equiva- lent of ionized Cu" will go from the concentrated to the dilute solution. The maximum work of this process, then, for each 96,540 coulombs, will be RT where n is the valence of the metal and pi and p2 are the 196 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. osmotic pressures exerted by the ionized Cu" in the two solutions. By the second process (contrast with direction of cur- rent in the case above) we have as the work P n Neglecting the difference of potential between the liquids, which is usually very small, we have or, including that, RT 2U _ U c +U a i.e. the sum of the two, where, when the value 2 calories is substituted for R, and 23,100 for , TT is given in volts. < By separating the amount of work -log into ns p 2 two portions, so that each will represent the maximum work at an electrode, we can write, neglecting the dif- ference of potential at the boundary of the two solutions, RT P RT P where P is a constant for any one metal at one tempera- ture in the same solvent, and is called the electrolytic solution pressure. Here, again, it is not so much a question of what electrolytic solution pressure is, as it is of what we mean ELECTROCHEMISTRY. 197 by the word electrolytic solution pressure. It will be seen that this conception leads to a constant value for any one metal, and a value for other metals which can be found in terms of the first by finding the electromotive force when the solutions contain the same quantity of T) ionized metal, for this is then equal to - - log neo ' J^2 We speak of positive electrolytic solution pressure when the metal dissolves, negative when the ionized metal deposits upon it. If a metal has a tendency to dissolve in a solution, i.e. to form ionized metal, the solution must be positive against it, for the metal loses positive electricity. Un- less there is some means of neutralizing this difference of potential, it is quite evident that solution must soon cease, and that all the positively charged ionized matter present (and remaining as such) must be immediately at- tracted back again to the metal. Naturally, if the negative charge on the plate becomes neutralized by a positive charge from without the system, solution will continue until the electrolytic solution pressure is compensated by the osmotic pressure of the ionized metal in the solution, or until the metal is all dissolved. This is the case, for example, with zinc, where the electrolytic solution pres- sure is positive and has a very high value. The other extreme, i.e. where ionized metal from the solution is usually precipitated, i.e. is transformed to the un-ionized metallic state, is illustrated by copper in its solutions. Here ionized metal is precipitated upon the electrode, which thus acquires a positive charge. Naturally, here also, the amount deposited is exceed- ingly small, for it also produces a difference of potential; and when the positive charge is neutralized by a negative 198 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. one from without the system, the process of precipita- tion continues. By combining the electrodes of two such systems, i.e. the zinc and copper by a wire, the solutions by a siphon, the process in each may continue, for the charges upon the electrodes can neutralize and consequently give rise to a current which flows until the zinc is all dissolved or the ionized copper all precipitated, with a loss of its charge, as metal. Further, in such a case, by applying a positive current to the copper electrode, it is possible, if the impressed electromotive force be greater than that of the cell, to reverse the process, i.e., to dissolve copper and precipitate zinc. The action of such a cell, when in operation, then; is to transform metallic zinc into the ionized metal, and ionized copper into un-ionized metal. The condition of the zinc and copper electrodes before they are connected is shown in the figure below. To obtain an E.M.F. from them it is only necessary to con- nect the solutions by a siphon and the electrodes through a wire. From the formula on page 196 it will be seen that it is p the ratio which is of importance. The osmotic pres- sure pj then, has much to do with the size of this ratio. ELECTROCHEMISTR Y. 199 This fact is observed best by the addition of potassium cyanide to the copper solution in which there is a copper electrode. Here experiment (loss of color, for example) shows the formation of the ionized complex CuCN/' from the ionized Cu", and at the same time it is observed that the copper electrode dissolves, i.e. becomes negative in value (all ionized Or* is removed as it is formed), so that P appears positive in value. Indeed, the addition of potassium cyanide to the copper side of such a com- bination of a copper and zinc system, for this reason, reverses the polarity of the cell, and the copper becomes the negative pole. The actual presence of such a layer of ionized matter around the oppositely charged metal (the Helmholtz double layer, as it is called), as we have concluded must necessarily be present, has been shown by Palmaer* with an arrangement which in principle is like that shown in the figure below. Drops of mercury are allowed to fall into a weak solution containing ionized mercurous mercury, metallic * Wied. Ann., 28, 257, 1899. 200 PHYSICAL CHEMISTRY FOR. ELECTRICAL ENGINEERS. mercury being in the bottom of the tube containing the solution. If now the double-layer theory is true, the drops of Hg as they form should have the electricity of the ionized Hg 2 deposited upon them, and these positively charged drops should then attract the negatively charged ionized matter, forming on each a double layer. When such a drop reaches the mercury at the bottom it will unite with that, forming ionized Hg 2 once more and releasing the ionized radical, and the concentration of mercury salt should be greater at the bottom than at the top; and Palmaer's experiments showed this difference in concentration to actually exist. Experiment has shown that the metals Na, K . . . , etc., up to Zn, Cd, Co, Ni, and Fe are always negative against their solutions, i.e., P>p. The noble metals, on the contrary, are positive against their solutions, although in some few cases it is possible to get a solution in which P>p. In general, though, for the noble metals P < p. A negative element has exactly the same action except that, in general, as far as is known, P>p. Here, although P>p, the electrode is positive against the solution, for the negatively charged ionized matter formed from the electrode leaves positive electricity behind. In general, the electrolytic solution pressure depends upon the temperature, the nature of the solvent, and the concentration of the substance in the electrode (see p. 206). Naturally, from this experimental conception of electro- lytic solution pressure it is possible to derive the formula for the E.M.F. given by any combination. When i mole of ionized metal is formed from an electrode against the osmotic pressure p, the osmotic work is ELECTROCHEMISTRY. 201 from which, by integration, we obtain The corresponding electrical work, however, is 7re , where TT is the difference of potential and SQ is the quantity of electricity carried by i gram equivalent of ionized matter. We have then g, RT 1 P - log,-, from which when P = P, n o. Expressing this in elec- trical units we find, in general, where n is the valence of the metal, 0.0002 P 7T=- -T log -VOltS, so that at if For a substance forming negatively charged ionized matter we have, correspondingly, 0.0002 _ P 0.0002 _ 202 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. Combining two electrodes, knowing that ionized matter is formed at one and disappears at the other, we have 0.0002 . PI 0.0002 P 2 u *- <*> -**> = log h ' ~^r T g h volts> Differences of potential. Calculation of the electro- lytic solution pressure. In order to measure the electro- lytic solution pressure, i.e. P in the equation 0.0002 when p, the osmotic pressure of the ionized metal, is known, it is necessary first to determine TT, the difference of potential between the metal and its solution. To do this, naturally, it is necessary to combine the electrode with another, which gives a known difference of poten- tial, and thus, knowing n and n\ (see above), we can readily find 7T 2 . It has been found that when mercury drops into an electrolyte the difference of potential soon becomes zero. Any cell of which this arrangement is one elec- trode, then, gives as its electromotive force the difference of potential existing between the metal and solution at the other electrode. As this dropping electrode is cum- bersome and inconvenient for general use, it is usual to employ the so-called normal (or tenth-normal) electrode for the purpose, its value being determined once for all against the dropping electrode. The electrode in com- mon use is made up of metallic mercury in a molar (or o.i molar) solution of potassium chloride which is saturated with calomel. The value of the normal elec- ELECTROCHEMISTRY. 203 trode * is 0.56 volt at 18, i.e., the solution (to which the sign always refers) is negative by 0.56 volt against the metal. The value of the electrode when o.i molar salt is used is 0.613 at 18, for mercury has a negative electrolytic solution pressure, and the weak solution of KC1 dissolves a greater amount of calomel, i.e., con- tains a greater amount of ionized mercury. Since p, the osmotic pressure of the ionized metal in atmospheres, is equal to the product of the concen- T tration in moles per liter and 22.4 , we can readily 273 calculate P in the equation RT if we know TT, the difference of potential between the metal and its solution at the absolute temperature T. At 17 C., thus, we have from which the value of P in atmospheres can be found. The values of P for the various metals, obtained in this way, are given in the table on the following page. It is to be remembered here that these values are merely symbolical, for the gas laws may not be applied to such an extent. The relation between these numbers, how- ever, are those that would be found if the E.M.F.' were measured under the .condition that the osmotic pressures of the ionized metals were the same. * For details as to this, see " Elements," pp. 417-421. 204 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. ELECTROLYTIC SOLUTION PRESSURES OF THE METALS. Zinc .......................... 9-QX io 18 atmospheres Cadmium ..................... 2 . yX io 8 Thallium ...................... 7 . ;X io 2 Iron .......................... i.aX io 4 Cobalt ........................ i .pX 10 Nickel ........................ i . 3X 10 Lead ......................... i.iXio- 3 Hydrogen ..................... 9-9X io~ 4 Copper ........................ 4.8X io- 20 Mercury ...................... i.iX io~ 16 Silver ......................... 2.3X io~ 17 Palladium ____ ., ................ i . 5 X io~ 38 When the metals acting as electrodes are inert, as in the so-called oxidation and reduction cells, the significance of the conception of electrolytic solution pressure natu- rally disappears. An example of such a cell is given by the arrangement plat. Pt FeCl 3 sol ____ SnCl 2 sol. plat. Pt. Here it is simply a question of the electric charge upon the ionized matter, and the action may be expressed by the equation The heat of ionization. Knowing the difference of potential existing between a metal and a solution, and its rate of change with a variation in the temperature, it is possible from the formula on page 192 to find the heat of ionization of the metal. We found there that fp An or ?r= -- \-T . ELECTROCHEMISTRY. 205 The term E c , here, is the heat produced when i gram equivalent of the metal goes from the metallic to the ionic state, and its value is given by the formula E But (page 174), eon = 96540 X 0.2394 XTT volts = 231 ion cals. ; hence the heat of ionization E C1 for i gram equivalent, can be found from the relation JTT\ cals. For copper in copper acetate (molar) at 17, ;r=o.6, i ^ i -i An . and ~TJ^ 0.0007 74> while -j~ for copper in copper sul- . An phate is 0.000757, i.e. an average value of -j=, of 0.000766 volt, hence E c , the heat of ionization of copper, is 8736 cals. per gram equivalent, or 17,472 cals. per mole. It was in this way that the value for H was de- termined for use in the table given on page 95. Concentration cells. If the electrodes of a cell are amalgams containing different concentrations of the same metal, and the solutions are identical with respect to the ionized metal, our general formula for the electromotive force (p. 202), 0.0002^ PI 0.0002^ P-2, 7T = - -rlog-^-- -T log -= VOltS, tti 3 pi 11 2 P2 206 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. becomes, since n\ = n^ and pi = p 2 , 0.0002 PI where PI and P 2 are the electrolytic solution pressures of the two amalgams with, respect to the dissolved metal. We have, then, a concentration cell in which the electrodes have different concentrations. Since the electrolytic solu- tion pressure due to the dissolved metal is proportional to the amount of this dissolved in the mercury, the for- mula acquires the simpler form TT= - Tlog volts. This equation has been tested experimentally for zinc and copper amalgams and found to express "very accu- rately the relations observed. Zinc in amalgams, as well as most other metals, exists in monatomic form, i.e., the formula weight (by freezing- point definition, for example) is found to be identical with the combining weight. If it were diatomic, i.e. the formula weight contained two combining weights, the above formula would have assumed a different form. For the movement of the same weight of ionized matter (see page 201) the osmotic work would then have been JJ^Tlog , and, since the electrical work would have C 2 been unchanged, i.e. 2X9654071, we should have had 1 0.00002 Ci it = ~T log volts, 2 n b c 2 i.e., the electromotive force would have been one-half what it has been found to be, ELECTROCHEMISTRY. 207 Another example of a concentration cell due to a different concentration in the electrodes is given by cells of the type of the Grove gas-battery in an altered form. The . electrodes, here, are of platinized platinum, in which the gas is absorbed under different pressures, and are placed partly in a liquid and partly in the gas at a corresponding partial pressure. Such an electrode is to be considered as a perfectly reversible gas electrode,* i.e., one- from which the material absorbed as a gas is given up in the ionized state, for the metal acts simply as a conductor, as has been shown experimentally by the use of different metals,, the same result being always obtained. In this way reversible gaseous electrodes of all kinds can be made. Oxygen as an electrode, how- ever, gives off ionized OH', since ionized O" is not known to exist, and forms O and H 2 O when the ionized OH' gives up its charge to it. If we have two electrodes of H, under different pres- sures, in contact with a liquid containing ionized H', we shall obtain a certain E.M.F. This may be calculated in two ways, as we did in the case of amalgams. In the second way, however, the process is slightly different, since one mole of H gas forms two moles of ionized H* T> (p. 84). The osmotic work is equal to RT log^ -^, as * 2 before. The electrical work, however, which corre- sponds to this is 2eox, for H 2 = 2H'; hence RT Pi * The electrolytic solution pressure here of the gas electrode is pro- portional to the wth root of the gaseous pressure, where n is the number of combining weights in one formula weight of gas. 208 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. i.e., we have 2 in the denominator, notwithstanding the fact that the gas is monovalent. When the electrodes of the cell are of the same metal, but the concentration of ionized metal in solution differs on the two sides, we have the typical form of a concen- tration cell. An example of this arrangement is Ag(AgN0 3 cone.) (AgN0 3 dilute)Ag, for which, since P\ = P^ we have the formula ?r=o.ooo2T log -, pi being the osmotic pressure of ionized Ag* on the con- centrated side, and p2 that on the dilute. But, since osmotic pressure is proportional to the concentration, we may also use the formula n = 0.0002 T log . 2 Conductivity experiments show that the concentration of ionized Ag' in a o.oi molar solution is 8.71 times as great as that in one that is o.ooi molar (not 10 times), hence the E.M.F. at 18 of the cell Ag(AgNO 3 o.oi molar) (AgNO 3 o.opi molar)Ag is TT = 0.0002 X 291 X log 8.71=0.054 volt, while direct measurement shows 0.055 volt. Since at 17 we have TT = ' log volts, a concen- tration ratio of ionized matter equal to 10 would give ELECTROCHEMISTRY. 209 0.0575 voit ror a monovalent metal, 0.02875 for a di- valent one, etc. Determination of ionization from electromotive-force measurements. Applying the formula we have used for concentration cell, where it is the concentration of ionized matter in solution that varies, it is very simple to deter- mine the concentration of ionized matter on one side, provided that of the same kind on the other side is known. Naturally, here, we can only apply this to the ionized matter coming from the electrode, but indirectly from the effect that other kinds of ionized matter have upon this (pp. 134, 150, 1 51) 'we can find the concentration of the other kinds. An example of the direct method here, and of course the other is identical so far as the electrical part is concerned, for it simply necessitates after that the application of the law of mass action, is given by Good- win's determination of the concentration of ionized Ag* and Cl' in a saturated solution of AgCl, i.e., the solu- bility of AgCl on the justified assumption that the ioniza- tion of AgCl is practically complete. The E.M.F. at 25 of the cell A g (AgN0 3 ^/io) KN0 3 (AgCl in KO/io)Ag is 0.45 volt. Since a for AgNO 3 m /io is 0.82, and for KC1 m /io is 0.85, we have 0.082 0.45 log = 3 3, i.e. c 2 = 1.04X10 ~ 9 , c 2 0.0002X298 where c 2 is the concentration of ionized Ag* in a saturated solution of AgCl in OT /io KC1. Since the concentration of ionized Cl' in the KC1 is 0.085 molar, the solubility product of AgCl is 1.94 X 10 ~ 9 Xo.o85 = 1.64 X 10 - 10 , 210 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. and its solubility is \/i.64Xio~ 10 =i.28Xio~ 5 moles per liter in pure water, i.e., where the amount of ionized Ag* is equal to that of ionized Cl'. In case we had had a solution of a known concentra- tion of a silver salt in place of the AgCl in KC1 we could have determined the fraction of ionized Ag' in it. Or, in the case as it is, we might dissolve the AgCl in some other chloride, and determine, from the known solu- bility of AgCl in pure water (by conductivity, for ex- ample) and the law of mass action, the concentration of ionized Cl' in that salt. One thing to be noted concerning this method is that the smaller the concentration of ionized Ag' (page 203) on the left side, the greater the E.M.F., and, consequently, the more accurate the determination. Another illustration of the use of this formula in this way is Ostwald's determination of the ionization of water. The E.M.F. of the cell (H in plat. Pt) acid base (H in plat. Pt) at 17 is 0.081, using molar acid and base. Since a for the molar acid is 0.8 (i.e. H'=o.8 mole per liter), we have 0.8 0.81 log = ; i.e., c 2 = o.8Xio~ 14 . 3 <2 0.0575' In other words, the concentration of ionized H' from water in the presence of molar base containing 0.8 mole of ionized OH' is o.8Xio~ 14 . Hence the ionic product for water is o.8Xo.8Xio~ 14 , and since vo.8Xo.8Xio- 14 = o.8Xio- 7 , i liter of water contains o.8Xio~ 7 moles each of ionized H' and ionized OH', which is the same value ELECTROCHEMIS TRY. 2 1 1 as that found by Kohlrausch from the conductivity of pure water. The processes taking place in the cells in common use. We shall now consider, by aid of the things we have found above, the processes which take place in cells. The Clark cell is made up according to the scheme Hg-Hg 2 S0 4 ZnS0 4 -Zn. The Hg 2 SO4 although difficultly soluble goes into solu- tion to a slight extent, so that we have a small amount of ionized Hg 2 " present in the solution. The zinc, owing to its high electrolytic solution pressure, goes into solu- tion and consequently forces positively charged ionized matter (Hg 2 ") to give up its charge. The zinc is thus neg- ative from the loss of ionized zinc, while the mercury is positive owing to the deposition upon the electrode of ionized mercury. The rapid polarization of the cell when short-circuited is due entirely to the removal of the ionized Hg 2 ", but the value is restored as soon as the solution again becomes saturated with the mercury salt, i.e., so soon as the original concentration of ionized Hg 2 " is restored. The Leclanche cell consists of a solution of ammonium chloride, in which we have two electrodes, Zn and C + MnO 2 . The action of the MnO 2 is to prevent polarization, the processes taking place without it and with it being as follows: In the cases without MnO 2 the Zn with its high solu- tion pressure goes into solution, driving before it the other positively charged matter, i.e., the ionized NH 4 . This ionized matter decomposes on losing its change, 212 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. forming NH 3 and H gases. The bubbles of H collecting upon the electrode are absorbed and so given off to the air, but as this process is slow, the ionized matter is prevented from giving up its charge and consequently the E.M.F. decreases. It is to get rid of this action of the H gas that the MnO2 is used. In contact with water we have, to a small extent, a solution of MnO 2 , which ionizes according to the scheme MnO 2 + 2H 2 O = Mn :: + 4 OH'. This tetravalent ionized Mn:: has the tendency to go into the divalent state by giving up two equivalents of electricity, i.e., to form ionized Mn". In consequence of this the ionized Mn:: with the ionized NHi is driven to the electrode by the ionized Zn", and, since it gives up two equivalents of electricity more readily than any other kind of ionized matter gives up its entire charge, the electricity is given up by it without any sub- stance which might cause polarization being deposited. We have, then, MnCl2 (i.e., ionized Mn") formed in the solution, and the process continues so long as there is solid Zn and MnC>2 present. The Bichromate cell is arranged according to the scheme Zn - H 2 SO 2 K 2 Cr 2 O 7 - C. The action of the two substances in solution forms chromic acid (H 2 Cr 2 O 7 ). This ionizes into to a considerable extent, and to a smaller degree as follows: ELECTROCHEM1STR Y. 213 This hexavalent ionized Cr- :: has the tendency to give up three equivalents of electricity and to go into the trivalent state (i.e., into ionized Cr"). Accordingly the ionized Zn", which is forced from the electrode, drives before it the ionized Cr, which gives up three of its equivalents and becomes Cr"*, remaining as such in the solution as ionized matter in equilibrium with ionized SO/' (i.e., as 02(804)3). Finally, then, we have a solution of 02(804)3 left in the jar. This change in the number of electrical equivalents by a change of valence always takes place more readily than the change from the ionic to the elemental state, and is of great value as a means of preventing polarization. Accumulators. The action of the lead accumulator or storage-cell also depends upon a change of valence Any reversible cell can be recharged, after it is used up, by the passage of a current through it in the direction opposite to that in which it goes of itself. The lead cell, however, is generally used for the purpose owing to its high E.M.F. Before charging it consists of two lead plates, one of which is coated with litharge (PbO), in a 20% solution of sulphuric acid. If the current is passed through these plates (the PbO being positive), the PbO is transformed into PbO2, lead superoxide (or supersulphate), while spongy lead is deposited upon the other electrode. The flow of current is now stopped, and the cell is charged. The PbO2 is soluble to a small degree, and ionizes as follows: PbO 2 + 2H 2 O = Pb : : + 4OH'. This tetravalent, ionized Pb" has the tendency to give up two of its electrical equivalents and to go into the 214 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. divalent form. Since this is true, the Pb electrode must have the higher solution pressure, and the ionized matter formed from it will drive the ionized Pb - to the electrode, where it will lose two charges of electricity and become divalent. This will continue as long as PbO 2 is present, i.e., until it is all transformed into the divalent state, PbSC>4. Other theories, by Liebenow and others, have also been advanced to explain this cell, but the reader must be referred elsewhere for them (see Dolezalek and von Ende, The Theory of the Lead Accumulator *). Dolezalek has been able to calculate the value of such a cell from the concentration of acid and the vapor pressure of the solution, and finds an excellent agree- ment between theory and experiment. This proves the process, according to him, to be a primary one such as the theory of Le Blanc and that of Liebenow would make it. ELECTROLYSIS AND POLARIZATION. Decomposition values. We must now consider those processes which are just the reverse of those we have been considering. In place of studying what takes place at the electrodes when a current of electricity is pro- duced, we shall consider the changes at the electrodes, and in the solution, when the current is applied to inert electrodes, as those of gold, platinum, carbon, etc. It is observed that, when a current is passed through a solu- tion for a certain time, using such electrodes, and then shut off, an electromotive force in the opposite direction arises; this is called the electromotive force of polarization. . * A glance at this book will show the value in electrochemistry of the general physical chemical relations we have considered, and will con- vince the reader of the especial impc i tance of the law of mass action. ELECTROCHEMISTR Y. 215 Experiment has shown that every solution requires a definite minimum impressed electromotive force to pro- duce continuous decomposition. Some of the values observed for this are given in the tables below. DECOMPOSITION VALUES FOR THE ACIDS. Dextrotartaric = i . 62 v. Pyrotartaric = i . 5 7 v. Trichloracetic =1.51 v. Hydrochloric =1.31 v. Oxalic = o . 95 v. Hydrobromic = o . 94 v. Hydriodic = o . 5 2 v. DECOMPOSITION VALUES FOR THE BASES. Sulphuric = .67 v. Nitric = .69 v. Phosphoric = .70V. Monochloracetic = .72V. Dichloracetic = .66v. Malonic = -72V. Perchloric .65 v. Sodium hydroxide = Potassium hydroxide = Ammonium hydroxide = Methylamine m/\ = m/2 = m/8 = 69 v. m/4 . 67 v. m/2 . 74 v. m/8 75 68 74 DECOMPOSITION VALUES FOR SALTS. ZnS0 4 =2. 35 v. Cd(NO,) 2 =i.98v. ZnBr 2 = i . 80 v. NiSO 4 = 2 . 09 v. Pb(N0 3 ) 2 =i. 5 2v. Ag(NO) 3 =o. 7 ov. CdSO 4 =2. 03v. CdCLj =i.88v. CoSO 4 =i.94v. CoCLj = i . 78 v. It will be observed that for acids and bases there is a certain maximum value, which is reached by many and exceeded by none, and is equal to 1.70 volts. Further, in all cases where the decomposition point is approxi- mately 1.7 volts it is noticed that the products of de- composition are hydrogen and oxygen, and that those with lower values which usually give off other products also attain this value when so dilute that these gases are 2i6 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. evolved. Thus we find the following values for hydro- chloric acid solution, which, when strong, decompose into hydrogen and chlorine: 2 molar = 1.26, 1 / 2 molar = 1.34, 1 / 6 molar = 1.41, Vie molar = 1.62, 1 /32 = i.69. At the dilution of l /32 molar hydrogqn and oxygen are evolved. Le Blanc found by experiment that when a solution (CdSO4) is decomposing steadily the potential difference existing between the cathode (which was originally of platinum) is the same as that observed when a stick of the metal which is deposited is in contact with a solution. Thus a molar solution of CdSO 4 is decomposed steadily at 2.03 volts, and when decomposing the potential dif- ference between the cathode and the solution is +0.16 volts, which is the same as that given by massive cad- mium in molar CdSC>4. The process up to the decomposition-point, then, can be readily followed. Originally the electrolytic solution pressure of the inert electrode is zero. The small amount of current which passes through the solution, however, will suffice to deposit metal upon the electrode, and thus increase the electrolytic solution pressure from zero to a definite, small value. This increase in P naturally prevents the passage of the current at the original voltage. An increase in the voltage, then, will cause the passage of current and the deposition of more metal, and that will again raise the electrolytic solution pressure and prevent the passage of more electricity at this voltage. This process will continue, each increase of the impressed voltage depositing more metal and raising the electrolytic solution pressure. And only at that voltage which is slightly greater than the counter electromotive force exerted by the deposited metal in ELECTROCHEMISTR Y. 217 that solution will the decomposition be steady and continuous. At the same time that this action is taking place at the cathode a similar one proceeds at the anode, where the negatively charged ionized matter is separated in its un- ionized state. For water two decomposition values are observed. The one with electrodes of platinized platinum has the value 1.07 volts, i.e., is practically the same as the electromo- tive force given by a gas-cell with such electrodes; the other, observed when polished inert electrodes are em- ployed, being 1.68 volts. It will be seen here that in the first case we have a reversible process, while in the second it is irreversible. It has been assumed that water is ionized to a very slight extent in H* and O", as well as into H* and OH', and that the first, reversible action is due to the ionized O", the value 1.68 volts being given when ionized OH' is separated according to the scheme It is quite usual, indeed, to designate the value 1.07 as being due to O", and that of 1.68 to OH'. Although we shall use this designation (see table, page 219) we shall only mean that O" denotes reversibility and, OH' irreversibility at the electrode. Hildburgh * has employed a reversible electrode with an irreversible one in a device for rectifying an alter- nating current. He used a large piece of platinized platinum for one electrode, the other being a small point of polished platinum, while the electrolyte is a solution of sulphuric acid which covers one-half of the large electrode. Before being stoppered the bottle, containing * Jour. Am. Chem. Soc., 22, 300, 1900. ; 218 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. the sulphuric acid solution and the electrodes, is filled with hydrogen gas at a certain pressure and sealed. When the point is the cathode, i.e. the large platinized elec- trode is the anode, it is observed that we get a continu- ous decomposition of water at about i.i volts, hydrogen being evolved from the point and oxygen absorbed in the platinum black. When the point is the anode, however, we get bubbles of oxygen first at 1.68 volts, the hydrogen being absorbed in the platinum black. In this way, by taking enough cells in series, one can rectify an alternat- ing current, i.e., transform it into a series of impulses all in the one direction. Primary and secondary decomposition of water. The electromotive force of decomposition of a substance giving off hydrogen and nitrogen is dependent upon the concentration of the ionized H* and ionized OH', but independent of the nature of the electrolyte. The decomposition value is thus the same for acids and for bases, so long as only hydrogen and oxygen are separated. Since by the law of mass action the product of the con- centration of the ionized H' and the ionized OH' must always be the same in any water solution, it follows that for all electrolytes, since the electromotive force of the cell is the sum of the differences of potential at the two electrodes, the minimum value must be the same for all substances giving off oxygen and hydrogen. With the exception, then, of the solutions of metallic salts which are decomposed by hydrogen, and the chlorides, bromides, and iodides which are decomposed by oxygen, the ionized products of water are the only factors in the decompo- sition of solutions, and not those of the dissolved salt. Excluding these solutions, then, we may say that all solutions when electrolyzed show primary decomposition ELECTROCHEMISTRY. 219 of water. The current is conducted through the solu- tion by all the kinds of ionized matter which are present. At the electrodes, however, that process takes place which involves the expenditure of the smallest amount of work and that is the separation of hydrogen and oxygen. Thus we see in all cases that there must be an accumula- tion of the various kinds of ionized matter around the electrode, but that only that kind is separated which does so most easily. Naturally, if the amount of current con- ducted through the liquid is so great that hydrogen and oxygen cannot be separated with the least work (owing to the small concentration of ionized H* and ionized OH') some other material may be separated instead, which will then decompose the water. But for small currents it is undoubtedly true that the decomposition of water is primary, and not secondary. In the table below are given the values necessary for the separation of the various kinds of ionized matter, on the assumption that that for ionized H* is zero. The values are for molar solutions. Ag- =-0.78 Cu"=-o.34 H' =+o.o Pb"=+o.i7 Cd"=+o.38 r =0.52 Br' =0.94 O" = i. 08 (in acid) Cl' =1.31 OH' =1.68 (in acid) OH' =0.88 (in base) SO 4 " =1.9 HS0 4 '=2.6 The values of O" and OH' are true in the presence of a molar solution of ionized H'. If we have H' and OH' in a base, the above value of H" becomes 0.8 and the value of OH' and O" is decreased by 0.8. CHAPTER VIII. PROBLEMS.* GASES. 1. An open vessel is heated to 819 C. What por- tion of the air which the vessel contained at o remains in it? Ans. 0.25. 2. An open vessel is heated until one-half of the gas contained at 15 is driven out. What is the temperature of the vessel? Ans. 303 C. 3. A volume of gas, measured at 15, is 50 c.c. At what temperature would its volume become 44 c.c. ? Ans. -i9.6 C. 4. A volume of gas at 766 mm. pressure is 137 c.c. What would it be at 757 mm.? Ans. 138.7 c.c. 5. What volume does i mole of gas occupy at 50, the pressure being 760 mm. ? At 100, p being 900 mm. ? Ans. ?; so o=26.5, at *> IOO = 25.8 liters. 6. A volume of air in a bell jar over water measures 975 c.c. The water in the jar is 68 mm. above the water in the trough, and the barometer stands at 756 mm. What would the volume be if exposed to standard pres- sure, the specific gravity of Hg being 13.6? Ans. 963.4. 7. At 14 C. and 742 mm. pressure a volume of gas measures 18 c.c. What will be its volume at o and 760 mm. pressure? Ans. 16.72. * for further problems, see " Elements," pp. 453-485. 220 PROBLEMS. 221 8. A volume of H at a temperature of 15 measures 2.7 liters with the barometer at 752 mm. What would have been its volume had the temperature been 9 and the pressure 762 mm. ? Ans. 2.6 liters. 9. What volume is occupied by 44 grams of oxygen at 70 cm. Hg pressure and 35 C. ? Ans. 37.7 liters. 10. J mole of H, J mole of O, and $ mole of N are mixed in a volume of 10 liters at oC. , What are the partial pressures of H, O, and N? Ans. p H = 1156.96, p = 1156.96, and p N = 771.65 grams per sq. cm. 11. What would these pressures (10) be in atmos- pheres at 10 C. ? Ans. pn=i.i64, p = i.i64, and ^=0.774- 12. i liter of N weighs 1.2579 grams at o and 760 mm. Calculate the specific gas constant, r. Ans. 3007 grams per sq. cm. 13. The specific gas constant, r, for N was found above (12). What is it for H ? The combining weight of N is 14.04, and of H is i .008. Ans. 39,080 grams per sq. cm. 14. How much will 100 liters of chlorine at 74 cm. Hg pressure and 30 C. weigh? Ans. 278.7 grams. 15. A solid gives off a gas which is dissociated to 41%, into two products. What is the work done, in calories, gram-centimeters, and liter-atmospheres, when i mole of solid goes into the gaseous state, the tempera- ture of dissociation being 55 C. ? Ans. 925 cals., 39,410,000 gr.-cm., 37.96 L. A. 1 6. How much work will be done by i kg. of CO 2 when heated 200? Ans. 373.1 L. A., 9088 cals. 17. H is at the partial pressure of 2.136 atmospheres in a space of 10 liters. How many moles per liter are there, the temperature being o? Ans. c= 0.0954. 222 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. 1 8. Starting with i mole of A in 22.4 liters (at o, 760 mm. of Hg), assume the dissociation according to the scheme A =2B + $D (where A, B, and D represent moles) to be 23%. What will be the final volume where pres- sure and temperature remain unchanged ? Ans. 43 liters. 19. What are the final concentrations of A, 5, and D in the above? Ans. A =0.0179, 5=0.0107, an d D = 0.01604 rnole per liter. 20. The formula weights of the above are M A = 1 70, MB = 25, and ^# = 40. How many grams per liter are there of each at equilibrium? Ans. ^4=3.04, 5=0.268, and 12=0.642. 21. Assume 17 grams of A (If = 170) in 2.24 liters (o, 760 mm. Hg). Find concentrations, partial pressures and grams per liter of A , 5, and D where the dissociation of A is 20% (M# = 25, M>=4o), and the volume and temperature remain constant. What is the total pres- sure of the system? Ans. ;! =0.036, 5=o.oi8, Z>= 0.027 mole per liter. ^4=6.o6, 5 = 0.447, Z> = i.o7 gr. per liter. ^4=o.8, 5 = 0.4, D = 0.6 atmosphere. Total pressure = i. 80 atmospheres. 22. When heated, PC1 5 dissociates into PC1 3 and C1 2 . The formula weight of PC1 5 is 208.28. At 182 the density is 73.5, and at 230 it is 62. Find the degree of dissociation at 182 and 230. 23. Sulphur in the form S 8 dissociates under certain conditions into the form S 2 . If this dissociation were complete, what would be the density of the gas in the form S 2 ? Ans. 32. 24. The specific heat under constant pressure for helium PROBLEMS. 223 is 1.25; the formula weight is 4. How many combining weights are there in the formula weight? Ans. i. 25. What is the specific heat at constant volume, i.e., c v ? i gram of the gas at o and 760 mm. occupies 509 c.c., and c p =o.2i. Ans. 0.164. 26. The specific heat at constant volume of a sub- stance is 0.075; it s formula weight is 40. How many combining weights are there to one formula weight? Ans. i. 27. The specific heat, c v , of CC>2 is 0.2094; what is the ratio of that for constant pressure to that at con- stant volume? Ans. ~ = i.22. SOLUTIONS. 28. What is the osmotic pressure of a i% solution of glucose (M = iSo) at oC.? Ans. 94.6 cm. H; obs. =94 cm. 29. The osmotic pressure of a solution of cane-sugar at o is 49.3 cm. of Hg. What percentage of sugar (M = 342) is contained in it ? Ans. 0.99% ; obs. = 1.0%. 30. The osmotic pressure of a sugar solution at 32 CD- is 54.4 mm. What is it at i4.2 ? Ans. 51.2 mm. 31. The osmotic pressure of solution containing 10 grams of sugar to a certain volume is 200 mm. What is that for the same volume containing 13.5 grams? Ans. 270 mm. 32. 10.442 grams aniline in 100 grams of ether give a vapor pressure of 210.8 mm. Ether alone (M = 74) gives 229.6. Find the formula weight of aniline in ether. Ans. 87. 33. Find osmotic pressure at o of aniline in (32) in at- 224 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. mospheres and gram per square centimeter, (s for ether is 0.737.) Ans. 19.82 atmos. or 20,460 gr. per sq. cm. 34. The osmotic pressure of a substance in water solu- tion is 100 cm. at o C. Find the vapor pressure of the solution; that of water at o is 4.57 mm. Ans. 4.56 mm. 35. What is the work, in gr.-cms. and calories, neces- sary to separate 200 grams of a substance (M = 6o) from the solvent at 20 C. ? 10 grams of substance to the liter of solvent. Ans. 1953 cals. ; 83,200,000 gr.-cm. 36. The increase in the boiling-point of 54.65 grams of CS 2 caused by the addition of 1.4475 grams of P is o.486. What is the formula weight of P in CS 2 ? Ans. 129.2. What is the formula, the combining weight being 31 ? 37. Calculate the increase in boiling-point of ether when to 100 grams we add a mole of a substance. The boiling-point of ether is 34.97, the latent heat of evapo- ration is 88.39. Ans. 21. 5. 38. The molecular increase of the boiling-point of H 2 O, as caused by the addition of i mole of substance to 100 grams, is 5.2. Find heat of evaporation of H 2 O. Ans. 535.1 cals. 39. In (36) find the osmotic pressure at 46 of P in the CS 2 solution. (s CS2 = 1.2224.) Ans. 6.84 atmos. 40. Find the vapor pressure in (36) of P in CS 2 solu- tion at o; the vapor pressure of CS 2 at o is 127.91 mm. Ans. 125.9 mm - 41. 10 grams of a substance in 100 grams of a solvent increase the boiling-point by o.87. The formula weight of the substance is 60. Find the molecular increase of the boiling-point. Ans., 5.22. 42. 0.284 gram of the oxime (CHs) 2 CNOH causes a decrease of o.i55 in the freezing-point of 100 grams of PROBLEMS. 225 glacial acetic acid, k for acetic acid is 38.8. Find the formula weight of the oxime in acetic acid. Ans. 71. 43. The ionization of a molar solution is 80%, two kinds of ionized matter being formed. What will the depression of the freezing-point be, water ( = 18.9) being the solvent? Ans. ^.4. 44. The molecular depression of an aqueous solution containing an ionized substance is 22. Find the degree of ionization of the substance in that volume. Ans. a = 16.3%. 45. In (42) find the osmotic pressure of the oxime in glacial acetic acid at 17. (Sp. gr. of acetic acid = 1.056.) Ans. i atmos. 46. What is the relation between the osmotic pres- sures of o.oi mole of substance in 1000 grams of water and looo grams of ether (sp. gr. =0.7370), assuming the same formula weight of the solute in each? Ans. P e = o.j^oP w . 47. In (42) find the vapor pressure of the solution at 40 C., the vapor pressure of glacial acetic acid at 40 being 34.77 mm. Ans. 34.69 mm. 48. A o.i molar solution of acetic acid in water freezes o.i927 lower than H 2 O. Find the degree of ionization of the acetic acid. Ans. a = 2%. 49. A 0.15 molar solution of succinic acid freezes o.2864 lower than H 2 O. Find the ionization of the acid. Ans! a = i%. 50. What is the heat of formation of a very dilute solution of magnesium chloride? (See text.) Ans. 1875 cals. 226 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. CHEMICAL MECHANICS. 51. In the volume of i liter there are 0.14 mole of hy- drogen and 0.081 mole of iodine. At the temperature of 440 C. K = ^j2 =0.02. Find the amount of hydriodic acid formed. Ans. 0.14855 mole. 52. The initial pressure of I is 38.2 cm., the fraction uniting with H is 0.8. What was the original pressure of H, 2 = 440? (K = o.02.) Ans. 40.35 cm. 53. At 440 in 50 liters we have a mixture of 2.74 moles of HI, 0.5 mole of H, and 0.3011 mole of I. (^=0.02.) In which direction and to what extent will the reaction go ? 54. At 440 (K=o.o2) 5.30 c.c. of H are mixed with 7.94 c.c. of I. How much HI will be formed? Ans. 9.475 c.c.; observed, 9.52 c.c. 55. 6.63 moles of amylene with i mole of acid shows that 0.838 mole is formed in the total volume of 894 liters. How much will be formed when we start with 4.48 moles of amylene and i of acid in the volume of 683 liters? Ans. 0.811 1 mole. 56. The coefficient of distribution of acetic acid between water and benzene is as - - and at two dilutions. 0.043 0.071 What is the formula weight of acetic acid in benzene? In water it is 60. Ans. 2.02X60. 57. For the reaction solid NH 4 HS=H 2 S + NH3 K = 62,400 (for pressures in mm.) at 25.! C. In a vacuum at 25.! we introduce NH 3 and H 2 S until we have a partial pressure of the former of 300 mm., and of the latter of 594 mm. Then the reaction is allowed to take place. How much does each gas lose in pres- PROBLEMS. 227 sure? (Here the pressure of gaseous NH^HS is so small as to be negligible.) Ans. 157.2 mm. 58. At i8.4 i mole of BaSO4 dissolves in 50,055 liters; at 37. 7 in 31,282 liters. On the justified assump- tion that the BaSCU is completely ionized, calculate the heat of ionization per mole. Ans. 8511 cals. EQUILIBRIUM IN ELECTROLYTES. 59. To i liter of a molar solution of a monobasic acid (K= 0.000018), a binary salt, with ionized matter in common, having an ionization in that dilution equal to 100%, is added. How much (in mo les) in the dry state must be dissolved in the acid solution to decrease the concentration of ionized H* to o.i of its previous value? Ans. 0.04211 mole. 60. A small amount of base is mixed with an excess of a solution containing an equal number of formula weights of acetic and lactic acids. In what proportion will the corresponding salts be formed? Ans. Lactate : acetate: 10.0117: 0.00424. 61. PbI 2 is soluble to 0.00158 mole per liter at 25.2, and ionizes completely (practically) into Pb" and 2!'. What is its solubility in presence of a o.i molar solution of ionized I' from another salt? Ans. i . 5 8 4- 1 o 6 moles per liter. 62. The solubility product of the substance AC 2 is 0.00621. What is the concentration of ionized A" and C' when the ionization is complete into A" and 2C'? Ans. 0.1157 mole per liter of A", and 0.2314 of C'. 63. MCN (solubility is 0.02 and is ionized com- pletely) is hydrolytically dissociated in solution. K for HCN is i3Xio- 10 , and K for H 2 O (25) is (1.09X10-7)2. 228 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. Find the amount of ionized M* from another salt which must be present to prevent hydrolytic dissociation. Ans. 0.02162 mole per liter. 64. At the dilution of 32 liters a binary substance is 0.9% hydrolyzed. What is the percentage of hydroly- sis at the same temperature when the dilution is 100 liters? Ans. 1.584%. 65. The constant of hydrolytic dissociation at 100 for NH 4 C1 is 337X10-10(1.6., *= ( 7- What is the ionization constant of NH^OH? (s H2 o at 100 is (8.5XIQ- 7 ) 2 .) Ans. 214X10-7. 66. AgCl, AgBr, and Agl are dissolved together. What are the concentrations of ionized Ag*, Bi^, Cl', and T? The solubilities are i.25Xio~ 5 , 86Xio~ 8 , and 0.97 Xio~ 8 respectively. 67. Bromisocinnamic acid at 25 is soluble to 0.0176 mole per liter, and is ionized to 1.76% into H* and a negatively charged radical. What is the solubility product of the acid? Ans. 9.6Xio~ 8 . What amount of the acid must always remain in solu- tion at this temperature, even in the presence of an in- finite amount of ionized H* from another acid? Ans. 0.0173 m l e P er liter. What is the solubility of the acid in the presence of a o.oo i molar solution of ionized H* from another acid? A ns. Solubility = o.oi 73 4- 0.0000883 m ole per liter. 68. Find the heat of neutralization of i mole of acetic acid (in 200 moles (3600 gr.) of H^O) with i mole of sodium hydrate (in 200 moles of H 2 O) at 35. (a for acetic acid is 0.009, its heat of ionization is 386 cals.; a for NaOH is 0.861, its heat is 1292 cals., and a for PROBLEMS. 229 sodium acetate is 0.742, its heat being -391 cals. The heat of ionization of water at 35 is 12,632 cals.) Ans. 13,093 cals. ELECTROCHEMISTRY. 69. An aqueous solution of CuSC>4 is electrolyzed until 0.2955 gram of Cu is deposited, using inert electrodes. The solution at the .cathode before the passage of the current gave 2.2762 grams of Cu, and after the passage 2.0650 grams. Find the mobility of ionized Cu" and of ionized SO 4 ". Ans. tf 0^=0.285, #scy' = o.7i5. 70. A 0.02 molar solution of KC1 (=0.002397) gives in a certain cell a resistance of 150 ohms. What is the factor that will transform conductivity results determined in this cell into specific conductivities? Ans. 0.36. 71. The equivalent conductivity of a solution of Na 2 SO 4 in 256 liters at 25 is 141.9. What is it at in- finite dilution ? Ans. 153.9. 72. The conductivity of a solution of AgCl saturated at i8is 2.4Xio~ 6 ; that of the water used is i.i6Xio~ 6 . Find the solubility of AgCl on the justified assumption that it is completely ionized. (^ oKa =I 3 I - 2 > ^4 is 0.06836 volt at o and 0.0735 at 2 4-45-) Ans. # = 510 cal. per mole of Cd. 74. Zn in a molar solution of ZnSC>4 gives a difference An of potential of 0.51 volt and 7 = 0.00076. What is the heat of ionization of Zn at 17? Ans. 33,740 cals. 230 PHYSICAL CHEMISTRY FOR ELECTRICAL ENGINEERS. 75. A cell with electrodes of the same monovalent metal gives an E.M.F. at 17 of 0.35 volt. The con- centration of ionized metal at the positive electrode is 0.02 mole per liter. What is its concentration at the other electrode? (The two solutions are connected with a siphon to prevent diffusion.) Ans. i. 637 Xio~ 8 moles per liter. 76. What would be the E.M.F. of a concentration cell with electrodes of a divalent metal, the concentration of ionized metal being 0.02 and 1.637 Xio~ 6 moles per liter? Ans. 0.175 v ^- 77. In a hydrogen-gas cell (platinized platinum elec- trodes, one-half in solution, one-half in gas) we have acetic acid on one side and propionic on the other, the concentration being identical, i.e. molar. What is the E.M.F. of the cell at 17? Which is the positive electrode ? Ans. Acetic acid positive, 71 = 0.00369 volt. 78. What is the relation of the electrolytic solution pressure of Zn to that of Cu? (The E.M.F. of Cu in CuSO* against Zn in ZnSC>4 is 1.06 volts when the concentration of ionized Zn" is equal to that of ionized ^ Ans. * SHORT-TITLE CATALOGUE OF THE PUBLICATIONS OF JOHN WILEY & SONS, NEW YORK. LOKDOK: CHAPMAN & HALL, LIMITED. 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Statics 8vo, 400 VoL in. Kinetics , 8vo, 3 50 Mechanics of Engineering. Vol. I Small 4to, 7 So VoL II Small 4to, 10 oo Durley's Kinematics of Machines 8vo, 4 oo Fitzgerald's Boston Machinist i6mo, i oo Flather's Dynamometers, and the Measurement of Power i2mo, 3 oo Rope Driving i2mo, 2 oo Goss's Locomotive Sparks 8vo, 2 oo Hall's Car Lubrication i2mo, i oo Holly's Art of Saw Filing i8mo, 75 James's Kinematics of a Point and the Rational Mechanics of a Particle. Sm.8vo,2 oo * Johnson's (W. W.) Theoretical Mechanics i2mo, 3 oo Johnson's (L. J.) Statics by Graphic and Algebraic Methods 8vo, 2 oo Jones's Machine Design: Part I. Kinematics of Machinery 8vo, i 50 Part II. Form, Strength, and Proportions of Parts 8vo, 3 oo Kerr's Power and Power Transmission 8vo, 2 oo Lanza's Applied Mechanics. 8vo, 7 50 Leonard's Machine Shop, Tools, -and Methods. (In press.) Lorenz's Modern Refrigerating Machinery. (Pope, Haven, and Dean.) (In press.) 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Materials of Machines I2mo, i oo Spangler, Greene, and Marshall's Elements of Steam-engineering 8vo, 3 oo Thurston's Treatise on Friction and Lost Work in Machinery and Mill Work 8vo, 3 oo Animal as a Machine and Prime Motor, and the Laws of Energetics. i2mo, i oo Warren's Elements of Machine Construction and Drawing 8vo, 7 50 Weisbach's Kinematics and Power of Transmission. ( Herrmann Klein. ) . 8vo , 5 oo Machinery of Transmission and Governors. (Herrmann Klein. ).8vo, 5 oo Wood's Elements of Analytical Mechanics 8vo, 3 oo Principles of Elementary Mechanics I2mo, i 25 Turbines 8vo, 2 50 The World's Columbian Exposition of 1893 4to, i oo 14 METALLURGY. gleston's Metallurgy of Silver, Gold, and Mercury: Vol. L Silver 8vo, 7 5 VoL n. Gold and Mercury 8vo, 7 50 ** Iles's Lead-smelting. (Postage 9 cents additional.) I2mo, 2 50 Keep's Cast Iron 8vo, 2 50 Kunhardt's Practice of Ore Dressing in Europe .8vo, i go Le Chatelier's High-temperature Measurements. 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