GIFT OF INTBODUCTION HYSICAL CHEMISTEY BY JAMES WALKER, D.Sc., Pn.D. PROFESSOR OF CHEMISTRY IN UNIVERSITY COLLEGE, DUNDEE ILontJon MACMILLAN AND CO., LIMITED NEW YORK : THE MACMILLAN COMPANY 1899 All rights reserved PEEFACE THIS book makes no pretension to give a complete or even systematic survey of Physical Chemistry ; its main object is to be explanatory. I have found, in the course of ten years' experience in teaching the subject, that the average student derives little real benefit from reading the larger works which have hitherto been at his disposal, owing chiefly to his inability to effect a connection between the ordinary chemical knowledge he possesses and the new material placed before him. He keeps his everyday chemistry and his physical chemistry strictly apart, with the result that instead of obtaining any help from the new discipline in the comprehension of his systematic or practical work, he merely finds himself cumbered with an additional burthen on the memory, which is to all intents and purposes utterly useless. This state of affairs I have endeavoured to remedy in the present volume by selecting certain chapters of Physical Chemistry and treating the subjects contained in them at some length, with a constant view to their practical application. In choice of subjects and mode of treatment I have been guided by my own teaching experience. I have striven to smooth, as far as may be, the difficulties that beset the student's path, and to point out where the hidden pitfalls lie. If I have been successful in my object, the student, after a careful perusal of this introductory text-book, should be in a position to profit by the study of the larger systematic works of Ostwald, Nernst, and van 't Hoff. As I have assumed that the student who uses this book has already taken ordinary courses in chemistry and physics, I have devoted little or no space to the explanation of terms or elementary notions which 237362 vi. INTEODUCTION TO PHYSICAL CHEMISTRY are adequately treated in the text-books on those subjects. I have throughout avoided the use of any but the most elementary mathematics, the only portion of the book requiring a rudimentary knowledge of the calculus being the last chapter, which contains the thermodynamical proofs of greatest value to the chemist. Since it is of the utmost importance that even beginners in physical chemistry should become acquainted at first hand with original work on the subject, I have given a few references to papers generally accessible to English-speaking students. J. WALKER. August 1899. CONTENTS CHAPTER I PAGE UNITS AND STANDARDS OF MEASUREMENT ... 1 CHAPTER II THE ATOMIC THEORY AND ATOMIC WEIGHTS . 8 CHAPTER III CHEMICAL EQUATIONS . 22 CHAPTER IV THE SIMPLE GAS LAWS . 27 CHAPTER V SPECIFIC HEATS ... 30 CHAPTER VI THE PERIODIC LAW ...... 38 viii INTRODUCTION TO PHYSICAL CHEMISTRY CHAPTER VII PAGE SOLUBILITY 50 CHAPTER VIII FUSION AND SOLIDIFICATION . ... 60 CHAPTER IX VAPORISATION AND CONDENSATION . . . .73 CHAPTER X THE KINETIC THEORY AND VAN DER WAALS'S EQUATION . 84 CHAPTER XI THE PHASE RULE . . . .97 CHAPTER XII THERMOCHEMICAL CHANGE . . 117 CHAPTER XIII VARIATION OF PHYSICAL PROPERTIES IN HOMOLOGOUS SERIES . 127 CHAPTER XIV RELATION OF PHYSICAL PROPERTIES TO COMPOSITION AND CONSTITUTION 136 CONTENTS CHAPTER XV PAGE THE PROPERTIES OF DISSOLVED SUBSTANCES . 148 CHAPTER XVI OSMOTIC PRESSURE AND THE GAS LAWS FOR DILUTE SOLUTIONS 158 CHAPTER XVII DEDUCTIONS FROM THE GAS LAWS FOR DILUTE SOLUTIONS . 169 CHAPTER XVIII METHODS OF MOLECULAR WEIGHT DETERMINATION . 176 CHAPTER XIX MOLECULAR COMPLEXITY . . . . . .193 CHAPTER XX ELECTROLYTES AND ELECTROLYSIS . . .201 CHAPTER XXI ELECTROLYTIC DISSOCIATION . . . .217 CHAPTER XXII BALANCED ACTIONS . . 234 x INTRODUCTION TO PHYSICAL CHEMISTRY CHAPTER XXIII PAGE RATE OF CHEMICAL TRANSFORMATION . 254 CHAPTER XXIV RELATIVE STRENGTHS OF ACIDS AND BASES 266 CHAPTER XXV EQUILIBRIUM BETWEEN ELECTROLYTES . 283 CHAPTER XXVI APPLICATIONS OF THE DISSOCIATION THEORY . . 296 CHAPTER XXVII THERMODYNAMICAL PROOFS . . . .311 o , 18x733 .121x27 found to consist of ^r g. of water vapour and - ^-^ g. of nitrobenzene vapour. The ratio of the weight of water to nitro- benzene in the vapour is then 18x733 to 121 x 27, or roughly 4 to 1 ; and this is the ratio of the weights in the distillate. Thus, ix VAPOKISATION AND CONDENSATION 83 although nitrobenzene has only 1/27 of the vapour pressure of water at the boiling point of the mixture, one-fifth of the liquid collected is nitrobenzene. This, of course, is due to the molecular weight of the water being so much smaller than that of the nitrobenzene. If an organic substance is unaffected by water and has a vapour pressure of even 10 mm. at 100, distillation with steam for purposes of purifica- tion will in general be repaid. For although its vapour pressure may be only an insignificant fraction of that of water, the higher molecular weight makes up for this, and appreciable quantities come over and condense with the steam. It is thus the low molecular weight of water which renders it so specially suited for vapour distillation. CHAPTEE X THE KINETIC THEORY AND VAN DER WAALS'S EQUATION IN the preceding chapters we have seen that a consideration of the composition and properties of substances, and the changes which they undergo, has led to the conception of atoms and molecules ; but as yet we have not dealt with the mechanical constitution of these substances, in other words, we have not considered how the molecules go to build up the whole whether they are at rest or in motion, or whether in the different states of matter there is a difference in the state of the molecules. It is plain that the kind of matter most suitable for study from this point of view is matter in the gaseous state, for in this form substances obey laws which in point of simplicity and extensive application are not approached by substances in either of the other states of aggregation. We have the simple laws of Boyle, Gay- Lussac, and Avogadro, which connect in a perfectly definite manner the pressure, temperature, volume, and number of molecules in all gaseous substances, whatever their chemical nature or other physical properties may be. These laws point to great simplicity in the mechanical structure of gases, and to the sameness of this structure for all gases. Various hypotheses have from time to time been put forward to explain the behaviour of gases, but only one has been found to be at all satisfactory, and to some extent applicable to the other states of matter. This hypothesis is called the kinetic theory of gases, and is, in its present form, chiefly due to the labours of Clausius and Maxwell. According to this theory, the particles of a gas which are identical with the chemical molecules are practically independent of each other, and are briskly moving in all directions in straight lines. It frequently happens that the particles encounter each other, and also the walls of the vessel containing them ; but as both they and the walls are supposed to behave like perfectly elastic bodies, there is no loss of their energy of motion in such encounters, merely their directions and relative velocities being changed by the collision. CH.X KINETIC THEORY AND WAALS'S EQUATION 85 The total pressure exerted by a gas on the walls of the vessel con- taining it is due to the impacts of the gas molecules on these walls, and is measured by the change of momentum experienced by the particles on striking the walls. Suppose a particle of mass m to be moving with the velocity c, and let it impinge on the wall at right angles. The particle will rebound in its line of approach with a velocity equal to its original velocity, but of course with the opposite sign. The original momentum was me, the momentum after collision is - me, so that the change of momentum is 2mc. If we calculate now the change of momentum suffered by all the particles of a quantity of gas in a given time by collision with the walls, we are in a position to give the total effect on the walls, and thus the pressure. For the sake of simplicity, we imagine that the vessel is a cube, the length of whose side is s, and that all the molecules have the same mass m, and the same velocity c. Let the total number of molecules in the gas be n. The molecules, according to the original assumption, are moving in all directions, but the velocity of each may be resolved into three components parallel to the edges of the cube, the components being related to the actual velocity by means of the equation x 2 + y 2 + z 2 - c 2 . Consider a single molecule with respect to its motion between two opposite sides of the cube. Its velocity component in this direction is x, and the number of /> impacts on the sides in unit time will be -. The change of momentum on each impact is 2mx, so that in unit time the total change of momentum o* of the molecule caused by impacts on the walls considered is 2mx. . The action of the molecule on these walls, therefore, is , and on s the other two pairs of walls it will be ^-, and - . Thus the s s action of the molecule on all the walls in unit time is 7 2 + z? 2we 2 s s Now in the whole quantity of gas there are n molecules, so ^nmc^ that the total action of the gas on the walls of the cube is - S The surface of the six sides of the cube is 6s 2 , and the quotient -g-j- therefore gives the action per unit surface. But s 3 is equal to v 9 the volume of the cube, so that we have finally p -^ or pv = fyimc 2 . All the magnitudes on the right of this equation are constant at constant temperature, hence the product of the pressure and volume of the gas is constant, and thus from the assumptions of the kinetic theory we deduce Boyle's Law. In the above deduction the vessel was supposed to have the 86 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. cubical form ; but any space may be considered as made up of a large number of small cubes, and as the impacts on the opposite sides of all faces common to two cubes would exactly neutralise each other, the presence of these internal partitions does not affect the impacts on the outer faces of the external cubes, which in the limit constitute the walls of the containing vessel. Since Jmc 2 is the kinetic energy of a single molecule, the expres- sion ^nmc 2 or \n . Jmc 2 may be read as two-thirds of the total kinetic energy of the gas, and we may say that the product of the pressure and volume of a gas is equal to two- thirds the kinetic energy of its molecules. We learn that systems of moving particles, such as gases are imagined to be on the assumptions of the kinetic theory, are in equilibrium with each other when the kinetic energies of their particles are equal ; and we know that gases are actually in physical equilibrium when their pressures and temperatures are equal, i.e. they may then be mixed without the pressure or temperature undergoing alteration. Let us consider a number of gases at the same pressure and at the same temperature. If the temperature of the gases is in every case altered in the same degree, the pressure remaining constant, the gases are still in physical equilibrium, and consequently the kinetic energies of their particles must have altered in an equal degree. But the product of pressure arid volume of a gas is proportional to the kinetic energy of its particles, and this product has therefore been altered to the same extent for each gas. Since, however, the pressure remained constant throughout, the volume of each gas has thus undergone ^fee same rela- tive change. Thus the kinetic theory enables us to deduce that the volume of different gases is affected equally by the same change of temperature if the pressure remains constant, i.e. that all gases have the same coefficient of expansion. Avogadro's Law may also be deduced from the kinetic theory by making use of considerations similar to the above. Take equal volumes of two gases at the same temperature and pressure. Since p =p', and v = v', pv =pv', and consequently But the kinetic energies of the two gases must also be equal, since they are in mechanical equilibrium, i.e. whence, dividing the first equation by the second, n n. Equal volumes of different gases, therefore, at the same temperature and pres- sure contain the same number of molecules. We may write the equation pv = ^nmc 2 in the form c= l-^t or nm since - - is the density of the gas, being its weight divided by its KINETIC THEOEY AND WAALS'S EQUATION 87 > 0. fe * \J~d~' volume, c = I --. It appears, then, that the speed of the molecule of a gas is inversely proportional to the square root of the density of the gas, a result which is in harmony with the experimental result that the velocity of diffusion (and the velocity of transpiration) of a gas is inversely proportional to the square root of its density. It is possible to obtain a value for the speed of the molecules of a gas by substituting the known values in the equation for the velocity given above. Thus for 32 g. of oxygen under standard conditions we havejp= 1,013,000 dynes 1 per square centimetre, mn = 32 g., and ,= 22,380 c,, so that ^3 x 1,013,000 x^O = ^ IQQ ^ metres per second. The molecule of oxygen therefore at C. moves at the rate of 46, 1 00 cm. per second, or nearly 1 8 miles per minute. The speed of the molecules of any other gas at any temperature may be got from the / d T formula c x = c / ^ ^^ in which d l is the density of the gas, d Q the ^u dt . 2tto density of oxygen, and T the temperature in the absolute scale. In the foregoing we have spoken of the velocity of the particles of a gas as if all the particles had the same velocity. This, however, can- not be the case, for even though the particles had the same velocity at the beginning of any time considered, the velocities of the individual particles would speedily assume different values owing to their encoun- ters. It must be understood, therefore, that the velocity in the above formulae means a certain average velocity, some of the particles having a greater and some a smaller speed than corresponds to this value. The bulk of the particles have velocities in the neighbourhood of this mean velocity, and the farther we diverge from the mean, the fewer particles we find possessing the divergent values. If two different gases are brought together, their particles in virtue of their rapid motion in straight lines will soon leave their fellows and mix with the particles of the other gas. This process of intermixture we are acquainted with practically as gaseous diffusion. Two gases, no matter how different their densities may be, will mix uniformly if brought into the same space, but the rate of intermixture is very much slower than what we should expect from the rate at which the particles move. This discrepancy, however, may be easily explained. The particles in a gas at ordinary pressure are comparatively close together, and consequently encounter each other frequently; so that, though their rate of motion between individual encounters is very great, their path between points any distance apart is, owing to these encounters, a very long and irregular one, and the rate of mixing is therefore com- paratively small. After the mixture of two gases has attained the same composition in every part, there is no further apparent change ; 1 One gram weight is equal to 981 dynes. 88 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. but the motion of the particles, and thus the mixing process, is sup- posed to go on as before, only now the further mixing does not alter the composition. The kinetic theory may be applied in a general way to the study of the processes of evaporation and condensation. In a liquid the par- ticles have not the same independence and free path as gas particles, although they are in general identical with the gaseous molecules and have equal velocities. A gaseous substance, in virtue of the freedom of its molecules, can expand so as to fill any space offered to it. A liquid does not do so at low pressures, but retains its own proper volume, al- though its molecules still possess sufficient independence to move easily between collisions, and thus enable the liquid under the influence of gravitation to accommodate itself to the shape of the vessel containing it. In spite of the clinging together of the liquid molecules, it happens that some of them near the surface have sufficient motion to free them- selves from their neighbours, and leaving the liquid altogether, to be- come free gas molecules. If these gas molecules move away unhindered, other molecules from the liquid will take their place ; and so the liquid will go on giving off gas molecules until all has evaporated. If, however, the liquid is kept in a closed space, the gas molecules which leave its surface will be able to proceed no farther than the walls of this space, and must eventually return in the direction of the liquid. It will consequently happen that some of them will strike the surface of the liquid again and be retained by it. But the liquid molecules still continue as before to become gas molecules and leave the surface of the liquid, so that at one and the same time there are molecules entering and molecules leaving this surface. When in^a given time as many molecules leave the liquid as are reabsorbed by it, no further apparent change takes place the relative quantities of the liquid and the vapour remain the same. A stationary state of balance or equilibrium has thus set in, and we may now look at what determines this state. The number of molecules leaving the liquid depends on the temperature, for it is only those molecules which attain a certain velocity that will succeed in freeing themselves ; and the motion of the molecules of a liquid, like those of a gas, depends directly on the temperature. The number of the molecules reabsorbed by the liquid depends on the number of gas molecules striking the surface in a given time, i.e. on the number of molecules contained in a given space and on their speed. As we have seen, this number and the speed together determine the pressure exerted by a gas, so the number of molecules reabsorbed depends on the pressure. Temperature thus regulates the number of molecules freed, and gaseous pressure the number of molecules bound in a given time ; consequently for each state of equilibrium when these two numbers are equal, a definite temperature will correspond to a definite gaseous pressure of the vapour in contact with the liquid or vapour pressure of the liquid, x KINETIC THEORY AND WAALS'S EQUATION 89 as it is shortly termed. Every liquid, therefore, has at each tempera- ture a definite vapour pressure ; and this vapour pressure increases as the temperature rises, for more molecules at the high temperature will have the speed necessary to free them. It may be noted that as it is the molecules with the greatest speed, i.e. with the highest temperature, that first leave the liquid, the average temperature of the liquid must sink as evaporation goes on, unless heat is supplied from an external source. The kinetic theory not only gives us the ordinary gas laws, which are, strictly speaking, obeyed only by ideal gases and not by any actual gas, but also when properly applied affords us a probable explanation of the deviations from the gas laws which are experi- mentally found. So far, we have considered the gas molecules as mere physical points occupying no volume whatever ; but certainly if gaseous particles are supposed to exist at all, they must be supposed to possess finite though small dimensions. It is evident that the volume in which these particles have to move is not the volume occupied by the whole gas, but this volume minus at least the volume of the particles. So long as the volume occupied by the gas is great and the pressure small, the volume of the particles vanishes in comparison with the total volume, and the gas laws are closely followed; but when the pressure is great and the total volume small, the volume of the particles themselves bears a considerable proportion to this whole, with the consequence that the divergence from the gas laws is great. Owing to this cause, the pressure would increase in a greater ratio than the volume would diminish, as the following reasoning will serve to show. Suppose a molecule to be oscillating between two parallel walls in a direction at right angles to them, and suppose the distance between the walls to be equal to 100 times the diameter of the molecule. It is evident that the molecule from its contact with one wall has to travel, not 100 diameters before it comes in contact with the other, as it would have to do if it were a point without sensible dimensions, but only 99. It will therefore hit the walls oftener in a given time than if it were without sensible dimensions, and that in the ratio of 100 to 99. Now suppose the distance between the walls to be reduced to 10 molecular diameters. The particle has now only to travel 9 times its own diameter in order to pass from contact with the one wall to contact with the other. It will therefore in a given time hit the walls oftener in the ratio of 10 to 9, or 100 to 90, than if it were a mere point. By diminishing the distance of the walls to one-tenth, therefore, we have increased the pressure not to ten times the original value, but to this value multiplied by 99 : 90, i.e. the pressure has increased to eleven times its former magnitude. We might now write the gas equation pv RT in the form p(v-b) = ET, where b is a constant for each gas depending on the magnitude of the 90 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. molecules of the gas. But there is still another influence at work which interferes with simple obedience to the ideal gas laws. The particles of a liquid undoubtedly exercise a certain attraction on each other, and this attraction must still persist when the liquid particles have become particles of vapour, only in the latter case the particles are in general so far apart that the effect of the attraction is incon- siderable. If the gas is compressed into a smaller volume, however, the influence becomes felt more decidedly owing to the comparative proximity of the particles. Van der Waals has assumed that this attraction is proportional to the square of the density of the gas, or reciprocally proportional to the square of the volume. The effect of the mutual attraction of the particles is the same as if an additional pressure were put upon the gas, so that the correction is applied by adding it to the value of the external pressure. If a denotes the coefficient of attraction, i.e. the value of the attraction when the gas occupies unit volume, then the correction for any other volume is -g. The equation for the behaviour of a gas under all conditions is, therefore, according to van der Waals, (p + This equation not only gives the behaviour of the so-called permanent gases very accurately up to high pressures, but even that of a comparatively compressible gas like ethylene. The following table gives the values of pv for ethylene actually found by Amagat, and those calculated from the equation p + - s - ) (v - G'0024) = constant, p being expressed in atmospheres, and v being taken equal to 1000 when p = I . p pv pv (observed). (calculated). 1 1000 1000 45-8 781 782 84-2 399 392 110-5 454 446 176-0 643 642 282-2 941 940 3987 1248 1254 The agreement between the observed and calculated values is very satisfactory. If the gas were an ideal gas the values of pv would remain the same for all pressures. We see, however, that this constancy is far from being fulfilled. The gas is at first more compressible than corresponds to Boyle's Law, and then at higher ,x KINETIC THEOEY AND WAALS'S EQUATION 91 pressures less compressible, the minimum value of the product occurring when the pressure is about 80 atmospheres. From the form of the equation it may be seen that the two corrections act in opposite ways, the value of the product pv being increased by the attraction, and diminished by the finite dimensions of the molecules. At low pressures the effect of the attraction greatly overweighs the volume correction which in its turn becomes preponderant when the pressure reaches a high value and the total volume becomes small. With ethylene at the temperature considered, the two corrections balance each other at about 80 atmospheres, and here the gas within narrow limits of pressure obeys Boyle's Law, for the product pv then remains sensibly constant. All gases hitherto investigated, with the single exception of hydrogen, give similar deviations from Boyle's Law : the product of pressure and volume at first diminishes, afterwards to increase as the pressure rises. At higher temperatures the deviations are. of the same kind, but not so marked. This may be seen directly from the formula, the constants a and b being independent of the temperature, and the value of the expression on the right-hand side increasing in direct proportionality with the absolute temperature. In the case of hydrogen there is no preliminary diminution of the value of pv, on account of the constant of attraction a being so small that its effect is counterbalanced from the first by the effect of the constant b. The equation of van der Waals is especially interesting in its application to the continuous passage from the gaseous to the liquid State, as it holds good not only for gases, but also in many I ways for liquids. If we rearrange the equation so as to give co- efficients of the powers of i\ we obtain p J p p This cubic equation has in general three solutions, so that for each value of p we have in general three corresponding values of v. The graph of the equation for constant values of a and b, and for various values of T, is given in Fig. 15. The isothermal curves thus obtained would represent the behaviour of a substance at various temperatures. The curves for the lower temperatures are wavy in form, and are cut by horizontal lines of constant pressure, sometimes in three points and sometimes in one. When the curve is cut by the horizontal line only once, the point of intersection gives the real solution of the equation, the other two solutions being imaginary. The resemblance of these curves to the curves on p. 77, which roughly express the result of actual experiment, is at once evident. In the case of the theoretical curves f we have no sudden breaks such as we have in the actual discontinuous passage of vapour into liquid by increasing pressure. Van der Waals's i equation assumes a continuous passage from the liquid to the vaporous 92 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. state, and vice versa, such as we find when we take the temperature and pressure above their critical values for the substance under considera- tion. When the passage be- tween the two states is dis- continuous, as it usually is, we proceed along a horizontal line from one part of the theoretical curve to another, this line of constant pressure cutting the curve in three points. The least of the volumes corresponding to the constant pressure is the volume of the substance as- liquid; another volume, the greatest, is the volume of the vapour derived from the liquid; the substance in a homogeneous state occupying the third intermediate volume is unknown. By studying supersaturated vapours and superheated liquids, we can advance along the theoretical curve for short distances beyond B and C without discontinuity, but the sub- stance in these states is com- paratively unstable. In the neighbourhood of the third volume the state of the substance is essentially unstable, increase of pressure being followed by increase of volume, and so we cannot hope to realise it. Van der Waals has pointed out, however, that in the surface layer of a liquid, where we have the peculiar phenomena of surface tension, it is possible that such unstable states exist, and that the passage from liquid to vapour may after all in the surface layer be really a continuous one. It will be noticed that as the temperature increases, the wavy portion of the curve gets continually smaller, and the three volumes get closer and closer together, finally to coalesce in a single point. Here the three solutions of the equation become identical, the volume of the liquid becomes equal to the volume of the substance as gas, and there is no longer any discontinuity or distinction between the liquid and gaseous states. In short, the substance at this point is in the critical condition : the curve is the curve of the critical temperature, the pressure is the critical pressure, and the volume is the critical volume. FIG. 15. x KINETIC THEOKY AND WAALS'S EQUATION 93 When the three roots of a cubic equation become equal, certain :elations exist between this triple root and the coefficients of the powers of the variable. If the equation is ind if the triple root is represented by , the following relations hold good : In van der Waals's equation there are only the pressure and volume of the gas, the constants a and 5, and the gas constant R. Now we can express the constant R in terms of the constants a and I is follows. Under normal conditions, Ietjp = 1, = 1, and T Q = 273. Then the equation becomes (!+)(! -&) = 27372 whence xJ" 273 or, if we make 27-5 = ft t ^ ne coefficient of expansion, we have Van der Waals's equation then becomes (cp. p. 91). p / p p If we denote now by <, TT, and 6 the critical values of v t p, and T respectively, we obtain for the critical equation, in which the three roots become identical, whence <=35 (Critical volume), TT = 9 |r- 2 (critical pressure), 94 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. Here we have general expressions for the critical values of a substance in terms of the constants which express the deviation of the substance from the laws for ideal gases ; and conversely we can give the numerical values of these constants from the values of the critical data, viz. These relations have been tested in several cases, and a fair approxi- mation of the calculated to the observed values has been found. If in van der Waals's equation we express the values of the pressure, temperature, and volume as fractions of the corresponding critical values, and at the same time express these latter in terms of the deviation constants, the equation becomes . p v T in which * ') n = ~ ; U Here everything connected with the individual nature of the substance has disappeared, and we have an equation which applies under certain restrictions to all substances in the liquid or gaseous state, just as the gas equation holds good for all gases, independently of their specific nature. The chief point to be noted here is that whereas for gases the temperature, pressure, and volume may be measured in the ordinary units without impairing the validity of the comparison of different gases, it is necessary in the case of liquids to effect the comparison under "corresponding " conditions, the temperatures of the two liquids to be compared being, for instance, not equal on the thermometric scale, but being equal fractions of the critical temperatures of the two substances. It is one of the tasks of physical chemistry to compare the physical properties of different substances, and to trace, if possible, some connection between their magnitude and the chemical constitution of the substances considered. Now we know that most physical properties of substances vary with the temperature and pressure. The question therefore arises : At what temperature and pressure are we to compare the properties of different substances ? It is evidently purely arbitrary to make the comparison at the so-called standard conditions of and 760 mm., for these conditions have no relation whatever to the properties of the substances themselves, and are merely chosen for convenience sake as being easily attainable in the circum- stances in which we work. Van der Waals answers the question by x KINETIC THEORY AND WAALS'S EQUATION 95 saying that the properties ought to be compared at corresponding temperatures and pressures, meaning thereby at temperatures and pressures which are equal fractions of the critical values in the absolute scale. Suppose, for example, we were to compare ether and alcohol with respect to some particular property. The critical temperature of ether is 194 C., or 467 absolute; that of alcohol is 243 C., or 516 absolute. Let the property for the alcohol be measured at 60 C., then we shall have as the corresponding temperature x for ether 273 + a; 273 + 60 _ 00 ~ _, .. , ... = K . . or x = 28 C. The pressure when small has not, as 4o7 olb a rule, a great effect on the properties of liquids, so that in general we may make the comparison at the atmospheric pressure without committing any serious error. It should, however, be stated at once that the data for the comparison of different substances under corresponding conditions are for the most part still wanting, so that it is not known whether the theoretical conditions would lead to sensibly greater regularities than those observed among the properties when measured under more usual conditions. The kinetic theory affords also some account of the phenomena of solution. If we take, for example, the case of the solution of a gas in a liquid, we can easily see that the gas molecules impinging on the surface of the liquid may be held there by the attraction of the molecules of the solvent. When, however, a number of the gas molecules have accumulated in the liquid, some of them, in virtue of their motion, will fly out from the surface of the solution, and this will happen the more frequently the more molecules there are dissolved in the liquid. But as the number of gas molecules striking the surface of the liquid remains constant at constant pressure, it will at last come to pass that the number of molecules entering and leaving the liquid will be the same. There is then equilibrium, and the liquid is saturated with the gas. As the number of gas molecules striking the liquid surface is proportional to the pressure, the number of molecules leaving that surface when the liquid is saturated, and consequently the number of molecules dissolved in the liquid, is likewise proportional to the pressure. This is Henry's Law, and Dalton's Law also follows at once ; for in a gaseous mixture the number of molecules of each gas striking the surface is proportional to the partial pressure in the mixture, and independent of the other components. It will be seen from this explanation that there is a great similarity between the solution of a gas in a liquid and the phenomena of evaporation and condensation. The same analogy appears when we consider the solution of a solid. When a soluble crystalline substance is introduced into a solvent, some of its particles become detached and enter the solvent. After a time certain of these detached particles come into contact with the solid again, and are retained by it. This give-and-take 96 INTRODUCTION TO PHYSICAL CHEMISTRY CH. x process goes on until the same number of particles leave the solid and return to it in a given time. No further apparent change then takes pla(5e, and the solution is saturated. The number of particles which return to the solid evidently depends on the number of them in unit volume of the solution, i.e. on the strength or concentration of the solution. If the solid is brought into contact with a stronger solution than the above, more particles will enter the crystal than will leave it, and so the crystal will increase in size. Such a solution is supersaturated with regard to the solid. In a weaker solution, fewer particles will come into contact with the solid and be retained by it than will leave it, i.e. the solution is unsaturated and the crystal will dissolve, in part at least. In the chapter on evaporation and condensation we had occasion to refer to the vapour tension of liquids, meaning thereby the tendency of the liquids to pass into vapour under the specified conditions. There is equilibrium when the vapour tension of the liquid is balanced by the gaseous pressure of the vapour above the liquid. A similar term has been employed to express the tendency of a substance to pass into solution, the substance having a definite solution tension for each solvent it is brought into contact with. When the pressure of the dissolved substance in the solution is equal to the solution tension of the solid there is equilibrium. Here a new conception is introduced, namely, the pressure of a substance in solution. What this pressure is, and how it may be measured, will be seen in Chapter XVI. A brief account of the Kinetic Theory will be found in CLERK-MAXWELL, Theory of Heat, chap. xxii. The student is also recommended to read, in connection with this chapter, J. P. KUENEN, on " Condensation and Critical Phenomena," Science Pro- gress, New Series, 1897, vol. i. p. 202 and p. 258. CHAPTER XI THE PHASE RULE A SUBSTANCE is in general capable of existing in more than one modification. For example, water may exist as ice, as liquid water, or as water vapour. Sulphur exists as vapour, liquid, and as two distinct solids, namely, as monoclinic and as rhombic sulphur. Para- azoxyanisoil, as we have seen, forms not only solid and gaseous modifications, but can also exist as a crystalline liquid distinct from the ordinary non- crystalline liquid. All such modifications, when they exist together, are mechanically separable from each other, and are in this connection called phases. A single substance may assume the form j of many different phases, but these phases cannot in general all exist ! together in stable equilibrium, being subject to certain restrictions regulat- ;: ing their coexistence, which may be stated in the form of definite rules. As a familiar example we shall take the substance water in the three phases ice, water, and vapour. The physical conditions determining the equilibrium of these phases are temperature and pressure. We know that at the pressure of 1 atmosphere, water is in equilibrium with ice at the temperature of zero centigrade, and with water vapour at the temperature of 100 centigrade. For a given pressure, then, there is a definite temperature of equilibrium between such a pair of phases ; and we shall also find that for a definite temperature there is a definite equilibrium pressure. Consider the two phases, water and water vapour. To each temperature there corresponds a fixed vapour pressure, which is the pressure of water vapour, or the gaseous phase, which is in equilibrium with the water ~or liquid phase. By drawing the pressure - temperature diagram, therefore, of a substance, we are enabled to study conveniently the equilibrium between its phases. In Fig. 16 the line OA represents the vapour-pressure curve of water, each point on the line correspond- ing to a certain pressure measured on the vertical axis, and to a certain temperature measured on the horizontal axis. For the sake of clearness, the curves in the diagram have all been drawn as straight lines. Ice, like water, has a vapour-pressure curve of its own, and H 98 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. this has been represented in the diagram by the line OB. It will be observed that the two vapour -pressure curves have not been repre- sented as one continuous line, but as two lines intersecting at a point 0. If we inquire into the meaning of this intersection, as interpreted from the diagram, we find that at a certain temperature t ice and water have the same vapour pressure, for the point corresponding to this temperature belongs both to the vapour-pressure curve of water and the vapour-pressure curve of ice. It is easy now to show that there is in fact a temperature at which the vapour pressures of ice and water are identical. Water at its freezing point is in equilibrium with ice, i.e. ice and water can coexist at this temperature in any proportions Solid Q f Temperature FIG. 16. whatever, and these proportions will remain unchanged if the mixture is surrounded only by objects of this same temperature. Now let th water and ice coexist, not, as is usual, under the pressure of one atmosphere, but under the pressure of their own vapour. Th temperature of coexistence will no longer be exactly 0, but temperature very slightly higher ; otherwise the conditions arc unchanged. If the ice, at this equilibrium temperature between ice and water, have a vapour pressure higher than that of water at the same temperature, diffusion will tend to bring about equalisation of pressure, i.e. the pressure of vapour over the ice will become less than its own vapour pressure, so that ice will evaporate ; and the pressure of vapour over the water will become greater than its own vapour pressure, so that water will be formed by condensation. Ice will have therefore been converted into water, which is contrary to our original xi THE PHASE RULE 99 assumption that the proportions of water and ice present are not under the circumstances subject to alteration. Similarly, if water at the equilibrium temperature have a greater vapour pressure than ice, ice would be formed indirectly through the vapour phase at the expense of the liquid water, so that our assumption in this case also would be contradicted. There only remains, then, the alternative that the vapour pressures of ice and water are equal when the ice and water are in equilibrium, which is in accordance with the representation of the diagram. At any point on the line OA water and water vapour can coexist in equilibrium, at any point on the line OB ice and water vapour can coexist. At the point 0, where these two lines intersect, all three phases can exist together in equilibrium, and such a point is therefore called a triple point. When a substance can only exist in three phases, only one triple point is possible. The triple point in the case of water is not quite identical with the melting point of ice, because the melting point is, strictly speaking, denned as the temperature at which the solid and liquid are in equilibrium when the pressure upon them is equal to one atmosphere. At the triple point the pressure is not equal to the atmospheric pressure of 760 mm. but to the vapour pressure of ice or water, which is only 4 mm. Now, it has been shown, both theoretically (cp. Chap. XXVII.) and experimentally, that pressure lowers the equilibrium temperature of ice and water by about 0'007 per atmosphere, so that the freezing point under atmospheric pressure is about 0*007 lower than the triple point. The effect of pressure on the melting point of ice may be represented in the diagram by the line OC, inclined from the triple point towards the pressure axis. At any point on this line, ice and water are in equilibrium with each other, the temperature of equilibrium falling with increase of pressure. The diagram for equilibrium of the three phases of water consists therefore of three curves meeting in a point, the triple point. At any other point on the curves, two phases can coexist in equilibrium : (a) Water and water vapour on OA ; (b) Ice and water vapour on OB ; (c) Water and ice on OC. At 0, the common point of intersection, all three phases are in equilibrium together. The three lines divide the whole field of the diagram into three regions. At pressures and temperatures represented by any point in the region AOB water can only exist permanently in the state of vapour. At any point in the region AOC it can only exist as liquid water, and at any point in the region BOC it only exists as ice. The curve OA separates the region of liquid from the region of vapour, but the separation is not complete. The curve is the curve of vapour pressures, and, as we have already seen, there is a limiting pressure beyond which the vapour pressure of a liquid 100 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. cannot rise. This is the critical pressure of the substance, and it is attained at the critical temperature. The curve OA, therefore, ceases abruptly at a point A, the values of the pressure and temperature at which are the critical values. Beyond A there is no distinction between liquid and vapour ; the two phases have become identical. It is possible to pass from a point M in the liquid region to a point N in the gaseous region in an infinite number of ways, which may be represented on the diagram by straight or curved lines. If these lines cut the line OA, there is discontinuity in the passage, for at the pressure and temperature represented by the point of intersection the two phases will coexist. For example, we may pass from M to N, by means of lines parallel to the axes, along the path MLN. The line ML represents increase of temperature at constant pressure ; the line LN diminution of pressure at constant temperature. The pressure at L is greater than the vapour pressure of the liquid at the constant temperature considered, and so the substance exists at this point as liquid only. As the pressure is gradually released, a point is at last reached at which it is exactly equal to the vapour pressure of the liquid. The liquid now begins to evaporate, and the two phases exist together, the point at which this occurs being the point of intersection of LN with OA. No further reduction of pressure can be effected until all the liquid has been converted into vapour, after which the pressure may be diminished until it attains the value represented by the point N. If, on the other hand, we follow the line MPQN, which does not cut the line OA, we can pass from the state of liquid at M to the state of vapour at N without any discontinuity whatever. We first increase the temperature, following the line MLP, to a value above the critical temperature, the pressure being all the time above the critical value. This takes us into the region where there is no distinction between liquid and vapour, so that by first reducing the pressure and then, lowering the temperature we pass without any break to a sub- stan6e in the truly vaporous state at N, the substance at no time having been in the state of two distinct phases. In what has been said above as to the condition of the substance in the various regions of the diagram, it has been assumed that stable states only are under consideration. If we disregard this restriction, then we may have, for example, liquid water in the region BOC, for water may easily be cooled below its freezing point without actually freezing, and exist as liquid at points to the left of 0. Such super- cooled water has a vapour pressure curve which is the continuation of the curve OA, and has been represented in the diagram by the dotted line OA'. This curve lies above the vapour-pressure curve for ice, so that at any given temperature below the freezing point the vapour pressure of the supercooled liquid is greater than the vapour pressure of the solid. This rule holds good for all substances, and we find in general that the vapour pressure of the stable phase is less than ,the xi THE PHASE fekE? 101 vapour pressure of the unstable phase. It should be noted that the instability in such examples is only relative. A supercooled liquid may be kept for a very long time without any solid appearing (cp. Chap. VIII), but as soon as the smallest particle of the sub- stance in the more stable solid phase is introduced, the less stable, or, as it has been called, the metastable phase is transformed into it. That the metastable substance should have a higher vapour pressure than the stable substance is not surprising, if we consider that the phase of higher vapour pressure will always tend to pass into the phase of lower vapour pressure when the two substances are allowed to evaporate into the same space, although they are not themselves in contact. The vapour of the substance of higher vapour pressure will on account of that higher pressure diffuse towards the substance of lower vapour pressure and there condense. More of the metastable phase will then evaporate in order to restore equilibrium between itself and the vapour, with the result that there will again be diffusion and condensation on the stable phase until all the metastable phase has been thus indirectly converted by evaporation into the stable phase. The vapour at pressures and temperatures represented by points on the line OA' is in an unstable state with regard to the -solid ice, being supersaturated, although it is only saturated with regard to the supercooled liquid. It is likewise possible to supersaturate vapour at temperatures above the freezing point, i.e. to have the substance in the state of vapour in the region COA ; and also to have a liquid substance in the region ACM by superheating. Water, for example, if free from dissolved gases, may be heated to 200 or over at the atmospheric pressure without boiling. It has always been found impossible, on the other hand, to heat a solid above its melting point. Water in the form of ice has never been observed in the region COA. We shall next proceed to the consideration of a substance capable of existence in more than three phases, taking sulphur as our ex- ample. Here we have not only the liquid and vaporous phases, but the two solid phases of rhombic and monoclinic sulphur. Rhombic sulphur is the crystalline modification usually met with, and this on heating rapidly melts at 115. If we keep it at a temperature of 100, however, for a considerable time, we find that it becomes converted into the other modification, monoclinic sulphur. This latter on heating does not melt at 115 but at 120, in accordance with the general rule that each crystalline modification of a substance has its own melting point. If the monoclinic sulphur be cooled to the ordinary temperature, it gradually passes again into the rhombic modification. We should be inclined, therefore, to say that at the ordinary temperature rhombic sulphur is in a stable state, while monoclinic sulphur is in a metastable condition. At 100 the reverse is the case : here monoclinic sulphur is the stable variety, and rhombic 102 'INTteODUCTiON *0 PHYSICAL CHEMISTRY CHAP. sulphur the metastable variety. We have seen that in the case of solid and liquid there is a temperature at which both phases are stable together, namely, the melting point. Above or below this temperature only one of the phases is stable. We should therefore expect by analogy that there is a temperature at which the two solid phases of sulphur should be equally stable, i.e. should be able to coexist without any tendency of the one to be converted into the other. Careful experiment has revealed such a temperature. At 9 5 '6, the transition or inversion temperature, both rhombic and monoclinic sulphur are stable, and can exist either separately or mixed together in Rhombic 95-6 115" 120 131 Temperature FIG. 17. i any proportions. Below this temperature the monoclinic phase gradually passes into the rhombic phase ; above it, the rhombic phase gradually passes into the monoclinic. A transition temperature of this sort is then quite comparable to a melting point, the chief difference being that while a solid can never be heated above its melting point without actually fusing, a substance like rhombic sulphur may be heated above its transition point without undergoing transformation. It is thus possible to investigate the properties of rhombic sulphur up to its melting point, 115, although between 95*6 and that temperature it is in a metastable condition, and is apt to suffer transformation into 1 The point C in the actual diagram would lie very much higher than is here represented. xi THE PHASE RULE 103 the stable monoclinic modification. The transition point, like the melting point, is affected by pressure, and in the case of sulphur increase of pressure has the effect of raising the transition temperature. All these phenomena may be represented diagrammatically by means of temperature-pressure curves (Fig. 17). The line OB in the figure represents the vapour-pressure curve of rhombic sulphur ; OA is the vapour pressure curve of monoclinic sulphur. These vapour pressure curves must meet at the transition point, for at that temperature both modifications are equally stable and must have the same vapour pressure. Below that temperature the vapour pressure of the metastable monoclinic phase must be greater than that of the stable rhombic phase. The line OA 7 , therefore, which is the prolongation of the line OA, represents the vapour-pressure curve of monoclinic sulphur below the transition point. Above 9 5 -6 rhombic sulphur is the metastable phase, and consequently has the greater vapour pressure. This is represented in the diagram by the dotted line OB', which is the continuation of the line OB. The line OC gives the effect of pressure on the transition point, sloping upwards away from the pressure axis in order to represent rise of transition point with rise of pressure. The point thus corresponds very closely to the point of Fig. 16, being like it a triple point at which there is stable equilibrium of three phases, viz. rhombic, monoclinic, and gaseous. The lines OA, OB, and OC diverging from represent as before the conditions of equilibrium between pairs of phases, and the dotted lines OA' and OB 7 similar conditions in metastable regions. It has been already stated that monoclinic sulphur melts at 1 20. At this point also a triple point must exist, for here monoclinic sulphur, liquid sulphur, and sulphur vapour are in equilibrium. The melting point of monoclinic sulphur is raised by pressure, instead of being lowered, as is the case with water. We have therefore three curves intersecting at A, namely, OA, representing the vapour pressure of monoclinic sulphur ; AD, representing the vapour pressure of liquid sulphur ; and AC, representing the influence of pressure on the melting point. It happens that the lines OC and AC, representing the effect of pressure on the transition point and on the melting point of monoclinic sulphur, although both sloping upwards from the pressure axis, meet at a point C, corresponding to a temperature of 131. This point C is also a triple point, for at the pressure and temperature which it represents, three phases rhombic, monoclinic, and liquid sulphur can coexist in equilibrium. At pressures above this, monoclinic sulphur has no stable existence at any temperature whatever. We have seen that rhombic sulphur may be heated above its transition point to a temperature at which it melts, viz. 115. The rhombic is here a metastable phase, and so also is the liquid formed by its fusion. We are therefore now dealing with a triple point in a 104 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. metastable region, represented in the diagram by the point B', which is the intersection of the prolonged vapour -pressure curves OB and DA for rhombic sulphur and liquid sulphur respectively. The dotted curve from B' to C represents the effect of pressure on the " metastable " melting point of rhombic sulphur, this curve being continued in the curve CE for the effect of pressure on the " stable " melting point of rhombic sulphur, the possibility of transition into monoclinic sulphur ceasing at C. The chief features of the diagram may thus be represented as follows, metastable conditions being enclosed in brackets : REGIONS DIVARIANT SYSTEMS. BOCE Rhombic EGAD Liquid BOAD Vapour OCA Monoclinic CURVES MONOVARIANT SYSTEMS. BO, (OB') . . Rhombic, vapour OA, (OA') Monoclinic, vapour AD, (AB') Liquid, vapour OC Rhombic, monoclinic AC Liquid, monoclinic CE, (CB') Rhombic, liquid TRIPLE POINTS NONVARIANT SYSTEMS. O Rhombic, monoclinic, vapour A ...... Monoclinic, liquid, vapour C ...... Rhombic, monoclinic, liquid (B') (Rhombic, liquid, vapour) Monoclinic sulphur offers the peculiarity that its range of exist- ence is limited on all sides. It can only exist in the stable condition between certain temperatures and certain pressures, the extreme limits being given by the temperature and pressure values at O and C. Diagrams similar to the sulphur diagram can be drawn for most other substances that exist in more than one crystalline modification. For example, para-azoxyanisoil yields a similar figure, although one of the crystalline modifications is in this case a liquid. The two chief triple points are here the points at which the solid crystal passes into the liquid crystal, and where the liquid crystal passes into an ordinary liquid. The first point is usually spoken of as the " melting point " of the substance, and the second as its transition point. These points therefore occur in the reverse order to the corresponding points for sulphur, but otherwise the diagram is much the same. In the case of a single substance, there is only one point, the triple point, at which any three phases can exist together. On this account, a system consisting of three phases of a single substance is called a xi THE PHASE RULE 105 nonvariant system, for if we change any of the conditions here temperature or pressure one or more of the phases will cease to exist. When the system consists of two phases, it is said to be monovariant, there being for each temperature one pressure, and for each pressure one temperature, at which there is equilibrium. When the system consists of only one phase, it is said to be divariant, for within certain limits both the temperature and the pressure may be changed arbitrarily and independently. The regions in the diagram therefore correspond to divariant systems ; the curves to monovariant systems ; and the triple points to nonvariant systems. When the systems considered contain two distinct components, say salt and water, and not one, as in the preceding instances, the phenomena become more complicated ; for here, besides temperature and pressure, we have a third condition, viz. concentration, entering into the determination of phases. The liquid phase, for example, may be pure water, or it may be a salt solution of any concentration up to saturation. The phase rule developed by Willard Gibbs furnishes us, however, with general methods for treating such systems theoreti- cally. It states, for instance, that if the number of phases exceeds the number of components by 2, the system is nonvariant. As we have seen, this is true for one component, and it is equally true for two components. With the components salt and water we have a nonvariant system when the four phases, salt, ice, saturated solution, and vapour, coexist. There is only one temperature, one pressure, and one concentration at which the equilibrium of these four phases can take place ; the point at which these particular values are assumed is called a quadruple point, and it coincides practically with what we have hitherto called the cryohydric point (cp. p. 64). Again, the phase rule states that if the number of phases exceeds the number of components by 1, the system is monovariant. If, therefore, there are three phases with the components salt and water, a monovariant system will result. Suppose the phases are salt, solution, and water vapour. If we fix one of the conditions, say the temperature, the other conditions adjust themselves to certain definite values. At the given temperature, the salt solution assumes a definite concentration, viz. that of the saturated solution. This solution of definite concentration has a definite vapour pressure, less than that of pure water. By fixing the temperature, therefore, we also fix the concentration and the pressure. Suppose, again, that the three phases are ice, salt solution, and water vapour. Let the concentration of the solution be fixed, and it will be seen that the temperature and pressure adjust themselves to definite values. First, a solution of the given concentration can only be in equilibrium with ice at a certain temperature fixed by the rule for the lowering of the freezing point in salt solutions (cp. p. 63). At this temperature the solution being of a fixed concentration will have a vapour pressure defined by the 106 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. law of the lowering of vapour pressure in solutions. By fixing the concentration, therefore, we likewise fix the temperature and pressure of equilibrium. If the number of phases is equal to the number of components, the phase rule states that the system is divariant. Let the two phases in our example with two components be salt and solution, and let the temperature be fixed. The pressure and concentration are no longer fixed as in the last case, but may vary in such a way that a given variation in the pressure produces a concomitant variation in the concentration of the saturated solution. It is necessary, of course, that the pressure should be above a certain limiting value in order that the third phase, of vapour, should not appear. That the concentration of the saturated solution, i.e. the solubility of the salt changes with the pressure, has been experimentally ascertained in a number of cases. Sometimes the solubility increases with pressure, sometimes it diminishes, according as the volume of the solution is less or greater than the volume of the solvent and dissolved sub- stances separately. If the two phases considered be salt- solution and vapour, it is obvious that although the pressure is fixed, the concentration and the temperature are not thereby defined. An increase of concentration will counteract the effect of a rise in the temperature, so that concentration and temperature may be made to undergo concomitant variations even though the pressure remains constant. If the number of phases is less than the number of components by 1, the system is then, according to the phase rule, trivariant. In the case of two components, the trivariant system has only one phase, and with our example of salt and water, the salt solution may be taken as the most representative phase, since it contains both components. If the temperature and pressure are both fixed, we are still at liberty to vary the concentration as we choose, i.e. a change of pressure at the fixed temperature causes no concomitant change in the concentration. Here, then, we meet with the greatest degree of freedom in varying the conditions in the case of two components, as we cannot reduce the number of phases further. With two components we sometimes get diagrams for melting and transition points which closely resemble those obtained for one component, when instead of pressure we substitute concentration and neglect pressure altogether. Thus with the two components paratoluidine and water, we may draw the following diagram (Fig. 18). On the vertical axis concentrations are measured instead of pressures, and on the horizontal axis temperatures are plotted as before. The line BO represents the concentrations of the solutions in equilibrium with solid paratoluidine at different temperatures, i.e. it is the solubility curve of solid paratoluidine in water. Under water, XI THE PHASE KULE 107 paratoluidine melts at about 4 4 '2, slightly lower than the temperature of fusion of the dry substance. 1 If we heat the system above this temperature, the solid phase disappears, and a liquid phase takes its place. Now, the liquid phase has its own solubility curve LO, and this must cut the solubility curve of the solid at the point at which the solid melts. This can be shown in the same way as that adopted to prove that water and ice have the same vapour pressure at the melting point, by substituting in this case solubility for vapour pressure. In general, we may say that if two phases are in equilibrium with each other, and one of them 260 2-20 1-80 1-40 1-00 0-60 20" 30 40 50 Temperature FIG. 18. 60 70 is in equilibrium with a third phase, then the second of the original pair will also be in equilibrium with the third phase. We see that there is thus considerable resemblance between the melting point of a substance under its own saturated vapour and the melting point of a substance under its own saturated solution. If we bear in mind * that concentration of a solution corresponds to pressure of a gas (Chap. XVI.), the reason for the resemblance of the solubility and pressure diagrams becomes apparent. 1 The reason for the lower melting point of paratoluidine under water is evident. Any substance soluble in water dissolves, when melted under its aqueous solution, a portion of the water with which it is in contact. The solid paratoluidine is thus not in equilibrium with pure fused paratoluidine, for which the fusing point is highest cp. p. 69), but with a solution of water in paratoluidine. 108 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. The similarity is also observable in the case of transition points. If we consider the two components, sulphur and an organic liquid capable of dissolving it, the general rule referred to in the preceding paragraph teaches us that at the transition point of rhombic and monoclinic sulphur the solubility of rhombic and monoclinic modifica- tions in the solvent must be the same. For if a certain solution is in equilibrium with one of the modifications, it must be in equilibrium with the second also, since at the transition point the two modifications are in equilibrium with each other. We may say shortly, therefore, that the vapour- pressure curves and the solubility curves of two modifications of the same substance cut at the transition point. This actually gives us in some cases a practical method of determining the transition point. Sometimes the transition of one modification into the other proceeds with such extreme slowness that it is almost impossible to observe the transition temperature directly. If, however, we investigate carefully the vapour pressures or the solubilities of the two modifications at different temperatures, we can construct curves of vapour pressure or solubility ; these curves will be found to intersect, and the point of intersection may be taken as the point of transition. Hydrated salts present many interesting aspects when viewed from the standpoint of the phase rule. The components here are the anhydrous salt and water, and the number of phases which they form may be very great, each solid hydrate being a phase distinct from the others. Let us take as our first example sodium sulphate in the form of the decahydrate Na 2 S0 4 , 10H 2 0, and the anhydrous salt Na 2 S0 4 . The solubility curves of these solids have been already given on p. 51, the concentrations being referred to the two components, anhydrous salt and water. The two solubility curves intersect at 33, i.e. the two solids are at that temperature in equilibrium with the same solution. They must, therefore, according to the rule already given, be in equilibrium with each other, i.e. 33 is the temperature of transition of the decahydrate phase into the anhydrous phase. This may be confirmed directly by heating the decahydrate alone. At 33 it melts, but the fusion is not complete, for besides the liquid phase, a new solid phase, the anhydrous salt, comes into existence. We have therefore at 33 the four phases of decahydrate, anhydrous salt, saturated solution, and water vapour, all in equilibrium. This point is thus a quadruple point, and as the system consists of two components and four phases, it is nonvariant. Consequently, if we alter the temperature, pressure of vapour, or the concentration of the solution, the equilibrium will be disturbed. If the alteration is only slight and temporary, the equilibrium will re-establish itself ; if the alteration is permanent, some of the phases will disappear. Many instances like the above are known. The essential feature is that one hydrate loses water, forming a solution and a lower hydrate or anhydrous salt. When this is the case there is a definite transition XI THE PHASE KULE 109 temperature from one hydrate to the other, the higher hydrate, i.e. that with the greater amount of water of crystallisation, existing below the transition temperature, and the lower hydrate above this point. Sometimes a hydrate on being heated melts without separation of a new solid phase. An example of this kind is to be found in the ordinary yellow hydrate of ferric chloride, Fe 2 Cl 6 , 12H 2 0, already referred to on page 68. This hydrate on heating melts completely at 37, the liquid having the same composition as the solid. Here, then, we are dealing with a melting point in the ordinary sense, the three phases of solid, liquid, and vapour existing together. If the system consisted of only one component, the number of phases at the melting point would exceed the number of components by 2, and the system would be nonvariant. But the number of components is 2, and the number of phases at the melting point exceeds the number of components by 1 only, so that the system is monovariant according to the phase rule. This is as much as to say that we are not fixed down to absolutely definite values of temperature, pressure, and concentration for the equilibrium of the solid, liquid, and vaporous phases, but may alter any one of these conditions within limits, the alteration of one being attended by concomitant alterations in the two other factors. For example, we may change the concentration of the liquid by adding one or other of the com- ponents to it. Such a change in the composi- tion of the liquid phase will bring about a certain definite change in the temperature of equilibrium and in the vapour pressure. If, on the other hand, we alter the temperature, it will be found that the vapour pressure and the concen- tration of the liquid phase will undergo corresponding varia- tions. The curves of tem- perature and concentra- tion for the hydrates of ferric chloride are given in Fig. 19. In this Mols. Fe 2 C! 6 to WOMols.H.,0 diagram pressure is not FlG 19 considered, and the curves represent equilibrium curves between solid and liquid phases. 110 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. The line on the left represents the temperatures at which ice is in equi- librium with solutions of ferric chloride of various concentrations ; it is, in short, the freezing-point curve of ferric chloride solutions (cp. p. 68). The curve CAC' gives the equilibrium of solid dodecahydrate with ferric chloride solution ; it is the solubility curve of the dodecahydrate. The point C, where it intersects the ice curve, is the cryohydric point, and lies at 55. At a temperature of 37 the liquid with which the dodecahydrate is in equilibrium has the same composition as the dodecahydrate itself. This temperature may therefore be called the melting point of the dodecahydrate, and it is the maximum temperature at which the hydrate can exist either by itself or in contact with any solution of ferric chloride. Addition of either water or ferric chloride to the liquid will lower the temperature at which the dodecahydrate will separate from the solution. If we follow the curve for the dodecahydrate to greater concentra- tions, we find that another hydrate may make its appearance at C', which lies at about 27. It will be seen that this point resembles the cryohydric point C, inasmuch as it is a quadruple point, the four phases being the dodecahydrate, the new heptahydrate Fe 2 Cl 6 , 7H 2 0, the saturated solution, and aqueous vapour. The only difference between the equilibrium here and at the cryohydric point is that the two solid phases are both hydrates, while at the cryohydric point one of the solid phases is ice. The point C' corresponds to the point of intersection in the sodium sulphate diagram (Fig. 4, p. 51). It is the intersection of the solubility curves of the dodecahydrate and the heptahydrate, and therefore represents the transition point of these two phases. An investigation of the curve of the heptahydrate shows that it is of the same nature as the curve of the dodecahydrate. It reaches a maximum as before, the concentration of the solution and the temperature there being equal to the composition and melting point of the hydrate. This curve for the heptahydrate finally cuts the curve of a lower hydrate, and similar curves are repeated until at last the solubility curve of the anhydrous salt is reached. The curve for each hydrate reaches a maximum temperature, which is the melting point of the hydrate, and cuts the curves for other hydrates at temperatures which are transition temperatures. In the preceding instances we have seen how water may be removed from hydrates by continually raising the temperature, solu- tion being at the same time present. Now, it is possible in many cases to remove the water of crystallisation from a hydrate without any solution being formed at all. This can be done most conveniently by placing the hydrate in an evacuated desiccator over a substance such as sulphuric acid or phosphorus pentoxide. The hydrate is at a given temperature in equilibrium with a small, and in most cases measurable, pressure of water vapour, i.e. it has a vapour pressure just as a solution has. If the pressure of water vapour above the hydrate XI THE PHASE RULE 111 30mm. is kept beneath this value, the hydrate will lose water and be con- verted into a lower hydrate or the anhydrous salt. In an evacuated desiccator containing phosphorus pentoxide the pressure of water vapour is practically kept at zero, so that the loss of water by the hydrate goes on continuously. Copper sulphate in the form of the pentahydrate CuS0 4 , 5H 2 0, for example, gradually loses water under these conditions, and is converted into the greenish-white monohydrate CuS0 4 , H 2 0. The vapour pressure of this hydrate is so small at the ordinary temperature that it remains practically unchanged in the desiccator. In this mode of dehydration of a hydrate, we have only three phases coexisting, viz. the higher hydrate, the lower hydrate, and aqueous vapour. The liquid phase is entirely wanting. Now, the system consists of two components, the anhy- 47mm. drous salt and water, so that the number of phases exceeds the number of com- ponents only by one. The system is therefore mono- variant, i.e. we can change | one of the conditions with- out destroying the equili- brium altogether, the other conditions at the same time undergoing concomitant alterations. It should be 4-5 mm. noted that the condition of concentration is here practi- G H 2 3 H 2_ . . 1 H 2 o o H 2 o cally absent, for there is no phase present in which the concentration varies continuously as it does in a solution. The effect of passing from one hydrate to another at constant temperature is seen in the accompanying diagram (Fig. 20), which represents the dehydration of copper sulphate pentahydrate at 50. The dehydration does not proceed in one step from the pentahydrate to the anhydrous salt, but in three stages, two intermediate hydrates being formed. Each of these hydrates has its own vapour pressure ; and where two hydrates coexist, the observed vapour pressure is the vapour pressure of the higher hydrate. Pressures have been tabulated on the vertical axis, and composition in molecules on the horizontal axis. Until the molecule of copper sulphate has lost two molecules of water, the vapour pressure remains constant at 47 mm., after which there is a sudden drop to 30 mm. The first of these values is the pressure of the pentahydrate ; the second is the pressure of the trihydrate formed as 3 H 2 1 H 2 O Composition FIG. 20. 112 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. the first step in the dehydration. This value of the pressure is retained until two more molecules of water have been lost, when it sinks suddenly to 4 '5 mm. This indicates that a monohydrate, with a vapour pressure of 4'5 mm. has been formed. Further dehydration produces no diminution of the pressure until all the water has been lost, when, of course, the pressure altogether disappears. This method of systematic measurement of vapour pressure during the dehydration of a hydrate at constant temperature can be used to ascertain the existence of intermediate hydrates, which may not be easily prepared in other ways. h W Ice Anhydrous salt Temperature FIG. 21. If the dehydration were conducted at another temperature than 50, a similar diagram would result ; the values of the pressures for the different temperatures would, however, be all higher or all lower than before. As the system with three phases is a monovariant one, the temperature and the pressure may be altered without the equilibrium being destroyed, but to a given alteration of the one there corre- sponds a definite alteration of the other. Each hydrate, therefore, has a vapour-pressure curve precisely like that of liquid water. These curves are represented in the temperature-pressure diagram of Fig. 21 by the lines OM, OT, OP, for the monohydrate, trihydrate, and pentahydrate respectively. The vapour-pressure curve of ice is repre- sented by the line OA, and that of water by the line AW, the point A where these curves intersect being the freezing point, or more xi THE PHASE RULE correctly the triple point. SC is the curve of vapour pressures of solutions saturated with the pentahydrate at different temperatures. This curve has a smaller vapour pressure than that of pure water, and consequently cuts the curve for ice at a temperature below the freezing point. The line SC is the curve for the equilibrium of the three phases, pentahydrate, solution, and vapour. At the point C, where it cuts the ice curve, the three phases are also in equilibrium with ice, so that C represents the cryohydric point for copper sulphate. As the lower hydrates cannot exist in contact with ice or an aqueous solution in stable equilibrium, these curves do not cut the lines CS or OC at all, unless, indeed, we represent them as all meeting OC at O, the point at which the pressure becomes zero, as has been indicated in the diagram. If the pressure of water vapour does not reach the vapour pressure of the monohydrate, copper sulphate will exist as the anhydrous salt. It can exist then as anhydrous copper sulphate in contact with water vapour at any point in the region beneath MO. If the pressure of water vapour is equal to the vapour pressure of the monohydrate, this salt can exist in presence of the anhydrous salt and water vapour at points on the curve MO. If the pressure is greater than the vapour pressure of the monohydrate, the anhydrous salt ceases to exist, and passes into the monohydrate. The region of existence of the monohydrate is MOT, the lines MO and TO bounding this region indicating the pressures at which it can coexist with the anhydrous salt and the next higher hydrate respectively. Similarly, TOP is the region of the tri- hydrate, OP giving the pressures at which it can coexist with the pentahydrate. The region of this, the highest, hydrate is POCS. The form of this region is different from that of the previous regions, because the pentahydrate phase can at certain pressures and tempera- tures coexist with ice as is represented by the line OC. If the pressure of water vapour is increased to values above those given by the vapour- pressure curve of the pentahydrate CS, some of the vapour will con- dense with formation of a new phase, viz. solution. The region of the existence of solutions is SCAW, bounded by the vapour-pressure curve of the saturated solutions, of ice, and of pure liquid water respectively. The diagram throws some light on the behaviour of hydrated salts when exposed to an atmosphere containing the ordinary amount of moisture. The pressure of water vapour in a well-ventilated labora- tory in this country is about 8 to 10 mm. on the average. If the vapour pressure of a hydrate is greater than this amount at the atmospheric temperature, the hydrate will lose water, i.e. will effloresce. This is the case, for example, with common washing soda, Na 2 C0 3 , 1 OH 2 0, which, when exposed to the atmosphere, loses water in the form of vapour, with production of a lower hydrate. If, on the other hand, the pressure of water vapour in the atmosphere is greater than the vapour pressure of the hydrate, the water vapour I 114 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. may condense, and a higher hydrate or a solution may be formed. Thus, if anhydrous copper sulphate, or one of the lower hydrates of this salt be exposed to the atmosphere, the water vapour will be slowly absorbed with ultimate formation of the pentahydrate, for all the hydrates of copper sulphate have a lower vapour pressure than the pressure of the water vapour usually found in the atmosphere. If calcium chloride or its common hydrate, CaCl 2 , 6H 2 0, is exposed to the air, it deliquesces, i.e. forms a liquid phase. Here the vapour pressures of the hydrate and of the saturated solution are lower than the pressure of water vapour commonly in the atmosphere, amounting to only 2 or 3 mm. at the ordinary temperature. The result is that a solution is formed which will become more and more dilute by absorp- tion of water vapour, the process coming to an end when the vapour pressure of the solution is equal to the pressure of water vapour in the atmosphere. Formation of New Phases. Under conditions where a new phase may appear, it does not necessarily follow that it must appear. When a crystalline solid is heated to its melting point, it invariably melts if any further heating is attempted. Here we have the new phase, the liquid, making its appearance as soon as the conditions are such that its stable existence becomes possible. If we cool the liquid, on the other hand, we may easily reach temperatures considerably below its freezing point without any solidification actually taking place. Here the new phase, the crystalline solid, does not appear when its existence becomes possible, but may remain unformed for an indefinite period. We meet with the same reluctance to form new phases at transition points. Rhombic sulphur can exist at temperatures above 9 5 '6, the transition point into monoclinic sulphur, and the latter may remain for a long time unchanged even at the ordinary temperature, which is far below the transition point into the rhombic modification. New hydrates of well-known substances are constantly being dis- covered since investigations have been directed to their formation, although it is practically certain that conditions compatible with their existence must have previously been encountered in actual work with these substances. Although sodium sulphate in the form of the decahydrate is efflor- escent under ordinary atmospheric conditions, its vapour pressure being greater than the pressure of water vapour in the atmosphere, yet it may, if perfectly pure, remain for a long time in the air without a trace of efflorescence being observable. On the other hand, there are many salts which are under the atmospheric conditions capable of taking up moisture to form a higher hydrate, or even a solution, and yet remain quite unaffected in the air. In each case, removal or absorption of water would result in the formation of a new phase, but the tendency to the formation of the new phases is so small that their formation may be delayed or never occur at all. It must be borne in mind that xi THE PHASE RULE 115 the reluctance to the production of new phases only applies to the first appearance of the new phase. As soon as the smallest particle of it appears, or is introduced from without, its formation goes on steadily, land in many cases very rapidly. Supersaturated solutions of sodium i thiosulphate, for example, may be kept for years without showing any j| tendency to crystallise, but if the merest trace of the solid crystalline ; phase is introduced, the whole mass becomes solid in the course of a few seconds. Similarly water, if perfectly air-free, may be heated to a temperature much above its boiling point, but in such a case, when iithe smallest bubble of vapour is formed in the interior of the liquid, ! jthe whole passes into the new vaporous phase with explosive violence. When ice and a salt (or other substance soluble in water) are brought together at a temperature below the freezing, point, there is s the possibility of the formation of a new phase the solution and this i; phase generally forms, the temperature then under favourable condi- tions falling to the cryohydric point. It is questionable, however, if this is invariably the case ; and it seems quite possible that two sub- stances in the solid state might be brought together at a temperature ibove the cryohydric point without liquefaction taking place. It is evident from what has been said in this chapter that it is (lot always the phase most stable under the given conditions which ictually exists. A metastable phase may exist for an indefinite time r,vithout passing into the most stable phase, provided that this latter phase is entirely absent. As soon as the stable and metastable phases \re brought into contact, however, the former begins to be produced at ihe expense of the latter. The transition from the metastable to the ".table phase takes place as a rule fairly rapidly, but in some instances [ihe transformation is so slow as to be practically unobservable. Strongly overcooled liquids, for example, crystallise with extreme lowness, even after they have been brought together with the stable Tystalline phase (cp. p. 62). The crystalline modifications of silica quartz and tridymite), are at ordinary temperatures so far beneath heir temperature of transformation, if such exists, that they show no endency to reciprocal transformation, and both forms must be accounted table. The same holds good for the two forms of calcium carbonate, .rragonite and calcspar. Yellow phosphorus is at ordinary tempera- ures metastable with regard to red phosphorus, which is the stable orm, yet in the dark it keeps for an indefinite time without under- ling much alteration, even although it may be in contact with red >hosphorus. When we come to inquire what phase will be formed when there 3 the possibility of formation of several different phases, we find that ; is not, as we might be inclined to expect, always the most stable hase that is formed, but rather a metastable phase, which may there- lifter pass into the stable phase. A substance, then, in passing from n unstable to the most stable phase very frequently goes through 116 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP, xi phases of intermediate degrees of stability, so that the transformation does not occur directly, but in a series of steps. Liquid phosphorus, for instance, which is itself metastable with regard to red phosphorus, does not on cooling pass into the latter, most stable, modification, but into the metastable yellow phosphorus. Molten sulphur, again, when quickly cooled by pouring into cold water, does not pass directly into the stable rhombic sulphur, but into the comparatively unstable plastic sulphur, which then in its turn undergoes transformation into more stable varieties. It is a common experience in organic chemistry to obtain substances first in the form of oils which afterwards crystallise, sometimes only after long standing. The alkali salts of organic acids, for example, on acidification with a mineral acid in aqueous solution, very frequently do not yield the free acid in the most stable solid form, but as an oily liquid, which crystallises with, more or less rapidity. Thus if solid paranitrophenol is dissolved in caustic soda solution, and the solution then acidified with hydrochloric acid, the paranitrophenol is liberated as an oil which crystallises after a few minutes. If we consider that the least stable phase of a substance has always the greatest vapour pressure, or, if we are dealing with solutions, the ' greatest solubility, the formation of intermediate metastable phases is not perhaps so unaccountable as at first sight appears. In the above instance of paranitrophenol, the system before acidification consists of ' a liquid phase only, since for our present purpose we may neglect the ] vapour phase altogether. On acidification, the new phase may not make its appearance for some moments, owing to the general reluctance exhibited in the formation of new phases, and the new phase which . eventually does make its appearance is that which entails least altera- > tion in the system, i.e. that which leaves most in the solution. In other words, the most soluble and least stable phase is formed first, the less soluble and most stable phase only appearing as a product of the transformation of the former. A very complete, non-mathematical, exposition of the subject dealt with in this chapter is given by W. D. BANCROFT in his book The Phase Rule (Ithaca, New York). CHAPTER XII THERMOCHEMICAL CHANGE A CHEMICAL change is almost invariably attended by a heat change, the latter being generally of such a nature that heat is given out during the progress of the action. Vigorous reactions are accompanied by con- siderable evolution of heat ; in feeble reactions, on the other hand, the heat evolution is comparatively small as a rule, and in some cases gives place to heat absorption. In special circumstances there might be neither evolution nor absorption of heat, but instances of this are rare or altogether wanting. From the fact that vigour of chemical action frequently goes l hand in hand with heat evolution, it was at one time thought that measuring the amount of heat evolved in any given action was tanta- mount to measuring the chemical affinity of the substances taking part r| in the action ; but this point of view has of late years been entirely given up owing to practical difficulties in reconciling it with the facts, and to a general advance in our theoretical knowledge of the subject. If heat evolution were to be taken as an accurate measure of chemical affinity, there is an obvious difficulty in explaining why certain changes take place with absorption of heat, since this would correspond , to a negative chemical affinity, and there would therefore be no reason, chemically speaking, why the action should take place at all. By introducing the heats of attendant physical changes in a somewhat arbitrary way, it was found possible to explain away the exceptions, but the explanations were in many instances so laboured that it became expedient to drop the rule altogether, in the strict sense, and be con- tent with the recognition of a general parallelism between the amount of heat evolved in an action and the readiness with which it takes place. The amount of heat change attendant on a chemical change is perfectly definite in ordinary circumstances, and is easily susceptible of exact measurement. A gram of zinc when dissolved in sulphuric acid will always occasion the same heat development if the conditions of ! the chemical action are the same. If the conditions are different, the 118 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. thermal effect will also be different. Thus it is necessary in the first place to ensure that in each case exactly the same chemical action occurs. The action of zinc on sulphuric acid differs according as the acid is concentrated or dilute. In the former case, zinc sulphate and sulphur dioxide are the chief products, in the latter case zinc sulphate and hydrogen. These are essentially different chemical actions, and evolve different amounts of heat for a given quantity of zinc dissolved. But even if we ensure that the only products are zinc sulphate and hydrogen, there will still be a difference in the heat development if the sulphuric acid in two cases is at different degrees of dilution. The difference here, however, will be slight, and may for most purposes be neglected. Again, a difference in the temperature at which the action takes place will occasion a difference in the heat evolution ; but in this case also the difference is comparatively slight,, and negligible for small variations of temperature. Lastly, if the zinc and sulphuric acid form part of a voltaic circuit, as in a Daniell or a Grove cell, the heat evolution is then very different from what it is if the chemical action is not accompanied by the generation of an electric current. From the standpoint of the conservation of energy, these phenomena are easily understood. Each substance, under given conditions, pos- sesses a certain definite amount of intrinsic energy, so that if we are dealing with a system of substances, a definite amount of energy is associated with that system as long as it remains unchanged. If it changes now into another group of substances, each of these will have its own intrinsic energy, and the new system will in general have a different amount of energy from that of the original system. Suppose the \ second system has less energy than the first. From the law of con- ! servation, it is plain that the difference of energy between the two systems cannot be lost, but must be transformed into some other kind of energy. Now the energy difference between two systems is usually accounted for as heat, and in our example the heat evolved during the solution of zinc in dilute sulphuric acid measures the difference of the ^ intrinsic energy of the zinc and dilute sulphuric on the one hand, and ; hydrogen and dilute zinc sulphate on the other. If pure sulphuric is taken instead of a mixture of sulphuric acid and water, the second system is now sulphur dioxide and zinc sulphate, mostly in the solid anhydrous state a system which has quite a different amount of intrinsic energy associated with it from hydrogen and dilute aqueous solution of zinc sulphate, so that the energy differences (and therefore the heat evolved) are widely divergent in the two cases. When the zinc and sulphuric acid form part of a galvanic cell, the initial and final systems are the same as above, so that there is the same energy difference as before ; but now all the energy does not pass into heat, some of it being transformed into electric energy, which takes the shape of an electric current passing outside the system. The consequence xii THERMOCHEMICAL CHANGE 119 is that much less heat is obtained by the solution of the zinc in this case than was obtained when no electric current was generated. Dealing now with smaller heat effects, we find that the intrinsic energy of a system is not the same at one temperature as it is at another; for if we wish to raise the temperature we must supply energy in the form of heat to the system, the quantity supplied depend- ing on the heat capacity of the substances which compose the system. In changing from one system to another, therefore, at different tempera- tures, different amounts of heat will be evolved, for in general the heat capacities of the two systems will be different. If we dilute a solution of sulphuric acid or of zinc sulphate, we find that a heat change accompanies the process, and as this heat of dilution is not as a rule the same for two substances, the total thermal effect depends on the concentration of the solutions employed. From the principle of the conservation of energy we see that if we have in a chemical change the same initial system and the same final system, the same thermal effect will always be produced no matter how we pass from the first system to the second, provided that no other form of energy than heat is concerned in the transformation. Hess, who originally worked this out experimentally, gives the follow- ing numerical example. Pure sulphuric acid was in one experiment neutralised with ammonia in dilute aqueous solution ; in other experi- ments it was first of all diluted with varying amounts of water before neutralisation, the heats of dilution and the heats of neutralisation being noted in each case. The experiment resulted as follows : Mols. Water Heat of Dilution. Heat of Neutralisation. Sum. 595-8 595-8 1 77-8 518-9 5967 2 116-7 480-5 597-2 5 155-6 446-2 601'8 The first column gives the number of molecules of water which were added to one molecule of sulphuric acid ; the second gives the number of heat units evolved on the addition of the water ; and the third gives the number of heat units evolved on the neutralisation of the resulting solution by dilute ammonia. It will be noticed that the sum of the two heats is very nearly the same in the four cases, for in each the starting-point is from pure sulphuric acid and dilute ammonia, and the product is dilute ammonium sulphate. This constancy of the total heat evolved is frequently made use of in thermochemistry for the determination of heat changes not easily accessible to direct measurement. Yellow phosphorus, for example, on conversion into red phosphorus is known to give out a considerable amount of heat, but the direct determination of this amount is a matter of some difficulty. An indirect determination, on the other hand, may be made with the greatest ease. Favre found that when a gram atom of yellow phosphorus is oxidised to an aqueous solution of phosphoric \ 120 INTRODUCTION TO PHYSICAL CHEMISTEY CHAP. acid by means of hypochlorous acid, the oxidation is attended by the disengagement of 2386 heat units. A gram atom of red phosphorus in similar circumstances yields 2113 heat units. If now a gram atom of yellow phosphorus were first converted into red phosphorus, and this then oxidised to phosphoric acid, the total heat evolution would be 2386 heat units, since the sum must be equal to the heat evolved in the direct oxidation. But the second part of the action, viz. the oxidation of the red phosphorus, yields 2113 units, so the first stage of the action, viz. the transformation of the yellow into the red phosphorus, must yield 2386 - 2113 = 273 units. Of the heat units referred to on p. 6, the most convenient for thermochemical purposes is the centuple unit, denoted by K, which is the quantity of heat required to raise the temperature of 1 gram of water from C. to 100 C. With this unit, which has the advantage of being easily determined practically, the ordinary heats of reaction are represented by numbers such as those in the preceding paragraph, not inconveniently large and not requiring the use of fractional values, since the degree of accuracy in the experimental determinations corre- sponds to about one unit. We have no means of determining the amount of intrinsic energy in any substance; we can only measure differences between the intrinsic energies of certain substances, or systems of substances. If all substances were mutually convertible, directly or indirectly, we might take one substance as standard, and refer all intrinsic energies to it by means of numbers stating the quantity of energy possessed by the substance in excess of the standard. But chemical substances are not mutually convertible without restriction. In particular, the elements cannot be converted into each other by any means in our power. We are therefore unable to compare the intrinsic energies of the elements together, and so for purposes of calculation we may adopt any value for them that we please. The easiest system is to make the intrinsic energies of all the elements equal to 0, and refer all other intrinsic energies to this value for the elements. If in the equation Pb + I 2 = PbI 2 , we take the ordinary chemical symbols of the elements and compounds as signifying the amounts of intrinsic energy in the substances, as well as the quantities of the substances themselves, the equation does not balance, for in the conversion of lead and iodine into lead iodide there is heat evolution, viz. 398 K for one gram atom of lead, so that the equation to be an accurate energy equation should read The intrinsic energy of a gram molecule of lead iodide is 398 K less than the sum of the intrinsic energies of the atoms from which it is formed, and is therefore equal to - 398 K, since the sum of the intrinsic energies A xii THERMOCHEMICAL CHANGE 121 of the elements is zero. If we actually write the amounts of energy associated with the various substances, we have the equation Pb + I 2 = PbI 2 + = -398 K + 398 K. Now, the heat given out on the production of lead iodide, or any other substance from its elements, is called the heat of formation of the substance, and we see from the above instance that this must be equal to the intrinsic energy of the substance with the sign reversed ; for on the left-hand side of the equation the sum of the energies is always equal to zero, being the intrinsic energy of elements alone, so that the sum of the energies on the left-hand side must also be zero, and the intrinsic energy of the compound thus equal to its heat of formation with the sign reversed. The heats of formation of compounds from their elements are for this reason very important in thermochemical calculations, their practical use being as follows. If from the sum of the heats of formation on the right hand of an ordinary chemical equation we subtract the sum of the heats of formation on the left hand, we obtain the heat given out or absorbed during the reaction. If the difference has the positive sign, the heat is evolved ; if it has the negative sign, the heat is absorbed. If we reverse the signs of the heats of formation, i.e. if we write the values of the intrinsic energies, and subtract the sum on the right hand from the sum on the left, we arrive at the same result. As an example, we may take the displacement of copper from copper sulphate by metallic iron according to the equation Fe + CuS0 4 , Aq = Cu + FeS0 4 , Aq ; where Aq indicates that the substance to whose formula it is attached is in aqueous solution. The heat of formation of copper sulphate in solution is 1984 K, and of ferrous sulphate under the same conditions 2356 K. The two metals have of course no heats of formation. If we subtract, therefore, the heat of copper sulphate from that of ferrous sulphate, we get the heat of reaction required, viz. 372 K. Writing the equation with the values of the intrinsic energies, we have Fe + CuS0 4 = Cu + FeS0 4 - 1984 K = - 2356 K + 372 K. If we know the heat of a reaction and the heats of formation of all the substances but one concerned in the action, we can calculate the heat of formation of that substance directly from the energy equation. For example, the heat of neutralisation of hydrochloric acid by caustic soda, when both substances are in aqueous solution, is 137 K, the heats of formation of dissolved hydrochloric acid, dissolved caustic soda, and liquid water respectively being 393 K, 1118 K, and 683 K. If we let 122 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. x represent the unknown heat of formation of sodium chloride in aqueous solution, we obtain the following equation : HC1, Aq + NaOH, Aq = NaCl, Aq + H 2 -393 K- 1118 K = -x -683K + 137K, whence x= 965 K. When an element like sulphur exists in more than one modification it is necessary to specify which modification we assume to have zero intrinsic energy, as there is always a heat change in passing from one modification to another. As a rule, the commonest or most stable variety is taken as the standard as convenience dictates. Heats of formation of sulphur compounds are generally referred to rhombic sulphur ; those of phosphorus compounds to yellow phosphorus. In the case of carbon compounds we seldom deal directly with heats of formation, but rather with heats of combustion, on account of their practical importance, and also on account of the ease with which they can be determined. The heat of formation, however, can easily be calculated from the heat of combustion. We find, for example, that methane has a heat of combustion equal to 2138 K, the products of combustion being carbon dioxide and water. Now, the heat of forma- tion of carbon dioxide from carbon in the form of diamond is 943 K, and of water 683 K. For the heat of formation of methane we have therefore the following equation : CH 4 + 20 2 = C0 2 + 2H 9 -x + = -943 K - 2x683 K + 2138 K. whence x = 1 7 1 K. We see from this that the combustion of methane gives out less heat than we should get by burning the same quantity of carbon and hydrogen as the free elements, and this is true of most carbon compounds. The difference between the heat of combustion of a hydrocarbon and that of the carbon and hydrogen composing it is not as a rule very great, so that a calculation of the latter gives an approxi- mate value for the former. Thus the heat of combustion of the carbon and hydrogen in amylene, C 5 H 10 , would be5x943K + 5x683K = 8130 K. The heat of combustion of amylene vapour was found by direct experiment to be 8076 K, a number differing only slightly from the preceding one. When a carbon compound contains oxygen as well as hydrogen, its heat of combustion may be roughly determined by means of "Welter's rule." According to this rule, the oxygen is subtracted from the molecular formula together with as much hydrogen as will suffice to convert it completely into water, the heat of combustion of the carbon and hydrogen in the residue then giving an approximate value of the heat of combustion of the whole compound. As an example we may take propionic acid, C 3 H 6 2 , whose heat of combustion has been found by direct experiment to be 3865 K. If we subtract xii THEEMOCHEMICAL CHANGE 123 2H 2 from the molecular formula, we are left with the residue C 3 H 2 , the elements of which have the heat of combustion 3 x 943 + 683 K = 3502 K. It is evident that the approximation is here by no means close, the error in this case being about 10 per cent. A better result may usually be obtained by subtracting the oxygen, not with the cor- responding quantity of hydrogen, but with the corresponding quantity of carbon, and then estimating the heat of combustion of the elements in the residue. In the above example we subtract C0 2 from the formula C 3 H 6 2 , and have C 2 H 6 left as residue. This gives the heat of combustion 2 x943 + 3x683 = 3935K, a much better approximation to the experimental value. As another instance of the two methods of calculation, we may take cane sugar, C 12 H 22 O n . By Welter's rule we subtract 11H 2 from the molecule, and for the residue C 12 get the heat of combustion 12 x 943 = 11,316 K. By the other method we subtract 5'5C0 2 in order to dispose of the 11 atoms of oxygen, and obtain the residue 6 - 5C and 11H 2 . The heat of combustion of these quantities of the elements is 6 '5 x 943 + 11 x 683 = 13,642 K. The value experimentally found is 13,540 K, a number much closer to the second calculated value than to the first. Some hydrocarbons have a greater heat of combustion than that of the carbon and hydrogen contained in them. The heat of combustion of acetylene, for example, is 3100 K; the heat of combustion of the two atoms of carbon and the two atoms of hydrogen contained in its molecule being 2x943 + 683 = 2569K. This corresponds to a heat of formation of - 531 K, i.e. this amount of heat is absorbed on formation of acetylene from its elements. We have here, then, an example of an endothermic compound formed from its elements with heat absorption, in contradistinction to the bulk of compounds, which are exothermic, i.e. are formed from their elements with evolution of heat. Other common examples of endothermic compounds are carbon disulphide, which is formed with a heat absorption of 287 K, and gaseous hydriodic acid, which is formed with a heat absorption of 6 1 K. Endothermic compounds like these are comparatively unstable, and give out heat on their decomposition. Hydriodic acid gas, for instance, is decomposed by gentle heating ; carbon disulphide can be split up into its elements by mechanical shock ; and carbon and hydrogen may be regenerated from acetylene by the passage of electric sparks through the gas. The substance hydrazoic acid, or azoimide, N 3 H, has a large negative heat of formation, which is no doubt closely associated with its extremely explosive properties. Such endothermic compounds are formed directly from their elements with difficulty, if they can be formed at all. At ordinary temperatures their direct formation does not take place, but if the elements are brought into contact at a very high temperature, then 124 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. combination may occur. Thus carbon disulphide is formed by passing sulphur vapour over red-hot carbon. Acetylene is produced when carbon and hydrogen are brought into contact at the very high temperature of the electric arc. This behaviour is exactly opposite to what we find with the common exothermic compounds, which are stable enough at ordinary temperatures, but are frequently decomposed by high temperatures. It has already been stated that for thermochemical purposes it is necessary to specify exactly the condition of each of the substances concerned in the action under discussion. This is not only true for the chemical condition, but also for the physical state of the substances. We must know whether the substances are in the solid, liquid, or gaseous states, or, if they are in the state of solution, in what solvent, and at what dilution. This is so because change of physical state is accompanied by heat change, which must be taken into account in thermochemical investigations. Liquid sulphur, on combining with oxygen to give sulphur dioxide, will not give out the same amount of heat as rhombic sulphur, for the latter on melting absorbs about 3 K, which must therefore be added to the heat of combustion of the rhombic sulphur. The correction for the difference between the solid and liquid states is often small, and never amounts to more than about 50 K. If the substance is in the state of vapour, the heat of vaporisa- tion must be added to the thermochemical data for the liquid. This correction is often considerable, amounting approximately to one-fourth of the value of the boiling point of the substance on the absolute scale (Trouton's rule). Thus the correction for water according to this rule would be 0'25 x 373 = 93 K, the actual heat of vaporisation at 100 being 97 K. The heat of formation of liquid water at the ordinary tem- perature from oxygen and hydrogen is 683 K, the number we have used throughout in the above calculations. At 100 the heat of formation of liquid water is somewhat less, viz. 676 K. If, now, we want to find the heat of formation of gaseous water at 100, we must subtract the heat of vaporisation of the liquid, viz. 97 K, and thus obtain 579 K as the heat of formation of water vapour. There is still another circumstance which must be taken into consideration when a chemical action is accompanied by the disappearance or formation of gases; or, in general, when the action is accompanied by a great change of volume. Each gram molecule of gas generated performs an amount of work equal to Q-Q2T K, for, as we have seen, the equation pv = RT becomes pv- 2T for the gram molecule, and R has the value 2 in small calories, or 0'02 in centuple calories (see p. 29). This amount of heat, then, is absorbed on production of the gas. If, on the other hand, a gram molecule of gas disappears, a corresponding amount of heat is pro- duced in the action. At 27 the actual amount per gram molecule is 0-02 x (27 + 273) = 6 K, the value being the same for all gases. This correction is of importance in the case of carbon compounds, which, xii THERMOCHEMICAL CHANGE 125 under ordinary circumstances, are burned at the atmospheric pressure, the volume increasing considerably during the combustion. The actual thermochemical measurement is usually made, on the other hand, in a closed " calorimetric bomb," the volume thus remaining constant. If we consider the combustion of benzene, for instance, we have the following volume relations : C 6 H 6 +90 2 -6C0 2 +3H 2 0; 9 vols. 6 vols. or, if the substances are all in the gaseous state, C 6 H 6 +90 2 =6C0 2 +3H 2 1 vol. 9 vols. 6 vols. 3 vols. Each volume in the above equations is the gram-molecular volume, the volume of the liquid substances being negligible. If both the benzene and the water formed by its combustion are in the liquid state, there is a shrinkage of three volumes on completion of the combustion. If all the substances are in the gaseous state, there is a shrinkage of only one volume. On the supposition that the combustion takes place in the calorimetric bomb at 27, with evolution of m centuple calories, then if the liquid benzene is burned at constant pressure, we shall have a heat evolution of m+ 18K. Suppose now that the benzene is burned as vapour at 27 by passing a stream of air or oxygen through the liquid and igniting the mixture at a jet, the water vapour being carried off without condensing ; and suppose further that the heat of vaporisa- tion of benzene and water at this temperature are b and w respectively, then we can calculate the heat of combustion under these conditions as follows. To vaporise the gram molecule of benzene, b heat units are absorbed, and this heat is given out again when the benzene ceases to exist as such, so to the heat of combustion of the liquid we must add this heat of vaporisation. But the water obtained in the previous case was liquid water, in the formation of which from vapour there was evolved w heat units per gram molecule. This amount of heat is not given out if the water remains in the gaseous state, so that from the heat of combustion given above we must now subtract 3w. Finally, there is now a shrinkage of only one volume, so that there is to be added 6 K to the heat of combustion at constant volume. The heat of combustion under the circumstances is therefore m + b 3w + 6K. The apparatus used to measure heats of chemical change is essentially the same as that used in physics for measuring heat quantities, and in particular the water calorimeter is universally employed. The chemical action is allowed to take place in a chamber immersed in a known amount of water of known temperature, and the change of temperature brought about in this water by the chemical action is noted. As the apparatus itself, viz. vessels, thermometers, stirrers, etc., is heated along with the water it contains, its water equivalent, i.e. the quantity of water which has the same heat capacity 126 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP, xn as the apparatus, must be determined and added to the quantity of water actually employed in the experiment. This can be done by adding a known quantity of heat to the apparatus and ascertaining the resultant change of temperature in the water of the calorimeter. The chief source of error in such experiments lies in heat exchange with external objects by conduction and radiation. To reduce this error to a minimum, the chemical action must be made to go as fast as possible, and the temperature of the calorimeter must never be allowed to depart greatly from the temperature of the room in which the experiment is made. Conduction is avoided by having the calorimeter surrounded by one or two vessels with stagnant air spaces between them, contact between the vessels being made by a bad heat conductor, such as cork, and reduced to as few points as possible. If the calorimeter is constructed to contain half a litre of water, the heat capacity of the apparatus is small in comparison, and by the use of a thermometer which can measure differences of temperature to a thousandth of a degree, very accurate results can be obtained with a relatively small expenditure of material. The most convenient form for the calorimeter is that of a cylinder, whose height is one and a half times to twice its diameter, so that in many cases an ordinary beaker serves the purpose very well, a larger beaker with a cover being the surrounding vessel. For further information concerning the methods and results of thermo- chemistry, the student may consult MUIR AND WILSON, Elements of Thermal Chemistry. CHAPTEE XIII VARIATION OF PHYSICAL PROPERTIES IN HOMOLOGOUS SERIES IN the homologous series of organic chemistry, for example the series of the saturated alcohols, there is a close resemblance in chemical properties amongst the members, so that it is possible to give general methods for the preparation of the substances and general types of action into which they enter. The actual readiness with which the substances are formed or are acted on by other substances, may, and usually does, differ from case to case, there being a gradation in chemical activity as successive members of the series are considered. Sodium, for instance, acts on the alcohols with formation of sodium alkyl oxides and hydrogen according to the equation 2ROH + 2Na = 2RONa + H 2 , but the vigour of the action is very different according as the alcohol is one high or low in the series. With methyl alcohol (CH 3 . OH) and with ethyl alcohol (C 2 H 5 . OH) the action is brisk ; with amyl alcohol (C 5 H n . OH) it is already sluggish at the ordinary temperature. Corresponding to this gradation of chemical activity within the ; series we have a gradation in physical properties, and here, on account of the accuracy with which these physical properties can be measured, the differences are more readily observed and more readily brought under general rules. We may take first for consideration the specific gravities in the series of normal primary saturated alcohols, which ;are exhibited in the following table (p. 128). The values of the specific gravity are for 0, and are referred to the specific gravity of water at 0. It will be seen that as the molecular weight of the alcohol increases, the specific gravity increases likewise. The difference in composition from step to step is one atom of carbon and two atoms of hydrogen ; and to this constant difference, CH 2 , there corresponds a continually diminishing difference in the values of the specific gravities as the series is ascended. 128 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. Specific Gravity. Difference. Methyl alcohol CHg.OH 0-812 -006 Ethyl C 2 H 5 .OH 0-806 + 011 Propyl C 3 H 7 . OH 0-817 + 006 Butyl C 4 H 9 .OH 0-823 + 006 Amyl C S H U .OH 0-829 + 004 Hexyl C 6 H 13 .OH 0-833 + 003 Heptyl C 7 H 15 .OH 0-836 + 003 Octyl C 8 H 17 .OH 0-839 + 003 Nonyl ,, C 9 H 19 .OH 0-842 An exception is found in the first member of the series. Reasoning by analogy from the other members, we should expect methyl alcohol to have a considerably lower specific gravity than ethyl alcohol, but instead of this it has a higher specific gravity, the value being inter-; mediate between those for the second and third members of the series.; The exceptional behaviour of the first member of a series is not confined to this series, or to this property, being of frequent occurrence amongst organic compounds. If instead of considering the specific gravities of the compounds (i.e. the weights which occupy unit volume) we consider the specific volumes (i.e. the volumes which are occupied by unit weight), or still better, the molecular volumes (i.e. the volumes occupied by the molecular weights), we are enabled to bring greater regularities to light. The specific volume v is the reciprocal of the density d, and the molecular volume V is the product of the specific volume and the molecular weight, or the molecular weight divided by the density. 1 M The values of v -=, and V are contained in the following table : CL Cl/ M V Difference V Differenc Methyl alcohol CH 3 OH 32 1-231 39-4 + 10 + 17.7 Ethyl C 2 H 5 OH 46 1-241 57.1 -17 + 16-3 Propyl ,, C 3 H 7 OH 60 1-224 73-4 - 9 + 16-5 Butyl ,, C 4 H 9 OH 74 1-215 89-9 - 9 + 16-2 Amyl ,, C 5 H U OH 88 1-206 106-1 - 5 + 16-4 Hexyl C 6 H 13 OH 102 1-201 122-5 - 5 + 16-2 Heptyl C 7 H 15 OH 116 1-196 1387 - 4 + 16-2 Octyl C 8 H 17 OH 130 1-192 154-9 - 4 + 16-2 Nonyl , , C 9 H 19 OH 144 1-188 171-1 ' xin HOMOLOGOUS SERIES 129 The regularity exhibited by the molecular volumes is much more striking than that displayed by the specific volume or by the specific ji gravity. Here the difference between neighbouring members, instead [ of continuously diminishing, remains practically constant throughout 1 the series. The volume occupied by the molecular weight of the I alcohol is increased by 16*2 units for every addition of CH 2 to the i molecule of the alcohol. The value 16 '2 may therefore be looked upon as the " molecular " volume of CH 2 under the given conditions, I and in this particular homologous series. Under other conditions the I value for CH 2 may be, and is, different. The volume of organic I compounds is usually affected greatly by temperature ; the coefficient | of expansion of ethyl alcohol being, for example, some twenty times j as great as that of water at the ordinary temperature. It is therefore of importance to determine under what conditions the volumes of different compounds are to be compared, more especially when they belong to different series. In the above instances the specific gravities were measured at (and compared with water at 0). This choice of temperature is evidently arbitrary, bearing no relation to the properties of the compounds themselves, but as far as we have seen it has the merit of leading to regular results. Kopp found, by studying a great many liquid substances, that if the molecular volume of each was determined at its own boiling point, not only were the regularities within each series preserved, but the same regularity held good for practically all series. No matter what homologous series ! was studied, Kopp found that a difference of composition of CH 2 corresponded to a constant difference in the molecular volume, the value being in his units 22. It must be noted that this volume is not absolutely constant, but is merely an average, the actual differences being liable to slight fluctuations about the mean. The value is greater than that obtained when the homologous compounds are all measured at the same temperature, because, as we shall see, the boiling points in homologous series rise as the series is ascended. Thus the molecular volumes of two neighbouring compounds measured at their f respective boiling points will show a greater difference than if they were measured at the same temperature, for the molecular volume of the compound with greater molecular weight is ascertained at a higher temperature than the molecular volume of the substance with lower molecular weight, and there is therefore the expansion between the two temperatures to be added to the value that would be obtained if both were measured at the boiling point of the lower compound. Besides this regularity others come to light. It was found by Kopp that the densities of isomeric compounds (measured at their boiling points) were equal, and consequently that their molecular volumes were also equal under these conditions. For example, he found the following numbers for compounds having the formula !C 6 H 12 2 :- * 130 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. Molecular Volume. Methyl valerate 149'2 Ethyl butyrate 149 '3 Butyl acetate 149 '3 Amyl formate 149 '8 This constancy is not displayed if the measurements are made at the same temperature if the boiling points of the isomeric substances are widely different. Thus for the butyl alcohols we have, when the densities are all measured at 20 (against water at 4) Boiling Point. Density. Normal primary C 2 H 5 . CH 2 . CH 2 OH 117 O'SIO Iso-primary (CH 3 ) 2 . CH . CH 2 OH 107 0'806 Tertiary ' (CH 3 ) 3 .C.OH 83 0786 The alcohol with highest boiling point has the greatest density, i.e. the smallest volume when the measurements are all made at one temperature. Its higher boiling point, however, allows of greater expansion, so that when the determinations are made at the boiling points, the increase of volume due to the higher temperature to some extent compensates for the smaller original volume at 20. The choice of the boiling points of the compounds as the temperatures at which the comparisons are to be made, although it involves the properties of the compounds themselves, is still to a certain extent arbitrary, inasmuch as they are the temperatures at which all the substances have an arbitrary vapour pressure, viz. 76 cm. The justification of this choice lies in the fact that the observed regularities under these conditions are great; and although a better selection of conditions might conceivably be made, it is unlikely that any fresl regularities would be brought to light. The heat evolved by the complete combustion of an organic com- pound (the carbon becoming carbon dioxide, and the hydrogen becoming water) is an example of a property exhibiting constant differences between neighbouring members of a homologous series when molecular quantities are compared. The following table contains the heats of combustion of gram-molecular weights of the fatty acids expressed in centuple calories (p. 122) : Acid. Difference. Formic CH 2 2 590 K 1543 Acetic C 2 H 4 2 2133 1546 Propionic C 3 H 6 2 3679 ,, 1548 Butyric C 4 H 8 2 5227 1540 Valeric C 5 H 10 2 6767 1545 Caproic C 6 H 12 2 8312 For each difference in composition of CH 2 there is a difference in the xm HOMOLOGOUS SEEIES 131 molecular heat of combustion amounting on the average to 1543 K. This difference is found to be practically the same in all homologous series. Thus for the alcohols we have Alcohol. Difference. Methyl CH 4 1685 K 1561 K Ethyl C 2 H 6 3246 1565 Propyl C 3 H 8 4811 1565 Butyl C 4 H 10 6376 1558 Amyl C 5 H 12 7934 A property which in general varies regularly in homologous series is the boiling point. As we ascend a simple series the boiling point invariably rises, but the rise at each succeeding step in general grows smaller and smaller as the molecular weight increases. The following table gives the boiling points of some of the normal saturated hydrocarbons. In the second column under t is the boiling point at 76 cm. in the centigrade scale ; under T we have the boiling poin the absolute scale, i.e. t + 273. Hydrocarbon. t t (calculated). T Difference. C 7 H 16 100-5 100-8 373-5 25-0 C 8 H 18 125-5 1261 398-5 24-0 CgH^ 149*5 149-9 422-5 23-5 C 10 H22 173-0 172-5 446-0 21-5 C n H 24 194-5 193-8 467-5 20-0 C 12 H 26 214-5 214-2 487-5 19-5 CjjjHsg 234-0 234-3 507-0 18-5 C 14 H 30 252-5 253-0 525-5 18-0 C 15 H32 270-5 271-1 543-5 17-0 C 16 H 34 287-5 288-9 560-5 The boiling points of most series exhibit a regularity similar to the above, but the differences are not usually so great as is the case with the hydrocarbons. The boiling points of most members of a homo- logous series can be expressed by a fairly simple formula. If M is the molecular weight of the compound, T its boiling point in the absolute scale, and a and b constants for the series, then in general The values under " t calculated " in the above table were obtained by means of a formula of this kind, the constants for the series being a = 132 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. 3 7 '3 8 and b 0'5. In some series (e.g. the alcohols, the alkyl bromides, and the alkyl iodides) a formula of this type cannot success- fully be applied, but in most cases it gives accurate results. The student must again be reminded that the boiling point of a series of compounds is a magnitude arbitrarily selected in so far as the pressure under which the compound boils is itself quite arbitrarily chosen equal to the average pressure of the atmosphere. It is found, however, that a formula of the above type is capable of expressing the relation between boiling points and molecular weights under any pressure. In the same series the constant a has different values for different pressures, whilst the constant b retains the same value for all the pressures. Thus the boiling points of the above series of hydrocarbons under 3 cm. pressure may be expressed by the formula r=2 instead of which is valid for 76 cm. The constancy of b for different pressures leads to the following results. If we take two substances belonging to the same series, we have for their boiling points at a certain pressure p At another pressure p' we have the boiling points T = aW, and TJ = a'Mf. M, M' and b remain the same throughout; so, by dividing each equation of the first pair by the corresponding equation of the second pair, we obtain T a T, a T T, ' = That is, if b remains constant for different pressures, the ratio of the boiling points (expressed in the absolute scale) at any two given pressures will remain the same for all members of the homologous series. Transposing the last equation, we have 'L -L rp ' rr, J M 2 l i.e. the ratio of the absolute boiling points of two substances belonging to the same homologous series is independent of the pressure. The boiling points of isomeric substances are in general not the same, as may be seen, for example, in the case of the butyl alcohols given in the table on p. 130. We usually find, as here, that the isomers containing the longest carbon chain boil at a higher temper- ature than those with branched carbon chains. As a rule, the boiling point of the first member of a homologous xni HOMOLOGOUS SERIES 133 series is considerably higher than that calculated from the formula which includes the other members of the series. This abnormally high boiling point is displayed still more markedly when, instead of one characteristic group, the first member of the series has two. Thus, for example, in the simplest series of the dicyano-derivatives, the first member has a boiling point which is actually higher than those of the three succeeding members : Malonic nitrile Methyl-inalonic nitrile Ethyl-malonic nitrile Propyl-malonic nitrile (CN) 2 CH 2 (CN) 2 CH . CH 3 (CN) 2 CH . CH 2 CH 3 (CN) 2 CH . CH 2 CH 2 CH 3 218 197 206 216 Difference. -21 + 9 + 10 - 9 + 4 The glycols behave similarly : Ethylene glycol CH 2 (OH) . CH 2 (OH) 197 Methyl-ethylene glycol CH 2 (OH) . CH(OH) . CH 3 188 Ethyl-elhylene glycol CH 2 (OH) . CH(OH) . CH 2 CH 3 192 The melting points in homologous series often show the peculi- arity that the substances with an even number of carbon atoms form a regular series by themselves, and those with an odd number of carbon atoms form a regular series by themselves. The following table contains the melting points of the higher fatty acids : Melting Point. Acid (Odd). - 1 '5 C 7 H 14 2 (Enan thylie C 9 H 18 2 Pelargonic C n H220 2 Undecylic C 13 H 26 2 Tridecylic C 15 H 30 2 Pentadecylic C^H^Oa Margaric C 19 H 38 2 Nondecylic If we take the successive members of the series, we find an alternate rise and fall in the melting point as we pass from one acid to the next. If, however, we separate the acids into those with an even and those with an odd number of carbon atoms, we have a rise in the melting point in each series. It will be observed here again that the differences decrease as we ascend the series. In some homologous series the difference between the homologues with even and those with uneven numbers of carbon atoms is so Acid (Even). Caproic C 6 H ]2 2 Caprylic C 8 H 16 2 Capric Ci H 20 2 Laurie OiAA Myristic C^HaA Palmitic CieH 32 2 Stearic CisHgeOa Arachic C 20 H4o0 2 -10-5 + 16-5 + 12-5 + 31-4 + 28 + 44 + 40-5 + 54 + 51 + 62 + 60 + 68 + 66-5 + 75 134 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. marked that in the one case we may have the melting point rise, and in the other case fall, as we ascend. The normal saturated dibasic acids afford an instance in -point : Acid (Odd). C 5 H 8 4 Glutaric C 7 H 12 4 Pimelic Acid (Even). Succinic C 4 H 6 4 C H 10 4 C 8 H 14 4 Adipic Suberic Sebacic Decane-dicarboxylic C 12 H220 4 Dodecane-dicarboxylic C 14 H 26 4 Melting Point. 181 98 149 103 141 107 133 127 123 112 C 9 H 16 4 Azelaic l C 13 H 24 4 Brassylic In the even series the melting point falls, in the odd series the melting point rises, as the molecular weight increases. Once more the rise or fall diminishes in magnitude step by step as we ascend the series. The student may have observed that in these melting point tables the lowest members of the series have been omitted. This is so because they do not fall under the general scheme which includes the higher members of the series. For example, the first member of the normal dibasic acids, oxalic acid, C 2 H 2 O 4 , melts at 189, and the second member, malonic acid, C 3 H 4 4 , melts at 133. It is evident that the melting points are not included in the general scheme which suffices for the other members of the series. In the series of the fatty acids the same irregularity is displayed. Acetic acid, C 2 H 4 2 , which, if the fall observable as we descend the series were maintained to the end, should melt many degrees below zero, in reality melts at + 16 '5. In general, we may say that the lowest member (or members) of a series departs from the regular behaviour exhibited by the higher members amongst them- selves. Exceptions to this rule occur, but they are comparatively rare. The separation of the members of a series into those with even and those with odd numbers of carbon atoms sometimes appears in other properties besides the melting points. Thus in the same series of the normal dibasic acids we have the solubility of the odd members in water considerably greater than the solubility of the even members, as the following table shows : Acid (Even). Solubility. Acid (Odd). Oxalic C 2 H 2 4 8-8 140 C 3 H 4 4 Malonic Succinic C 4 H 6 4 6-9 C 5 H 8 4 Glutaric Adipic C 6 H 10 4 1-5 4-5 C 7 H 12 4 Pimelic Suberic C 8 H 14 4 0142 0-12 C 9 H 16 4 Azelaic l Sebacic C 10 H 18 4 o-oi 1 It is still somewhat doubtful if azelaic acid has the normal structure. xm HOMOLOGOUS SERIES 135 The solubilities are given as parts of acid dissolved by 100 parts of water at the ordinary temperature (15 20). It will be seen that in the separate series the solubility in water falls off very rapidly as the number of carbon atoms in the molecule increases. This is in accordance with what was stated in the chapter on solubility. The solubility of an acid in water is probably connected with the presence in it of the hydroxyl ( - OH) or carboxyl ( - COOH) group (cp. p. 56). Other things being equal, the greater the proportion of hydroxyl (or carboxyl) in the molecule, the greater will be its solu- bility. As we go up the series the proportion which the hydroxyl bears towards the rest of the molecule diminishes, and along with this goes on diminution of the solubility in water. It is evident from the above tables that in this particular series there is some fundamental difference between the homologues with an even and those with an odd number of carbon atoms. This difference is probably to be sought for in some property affecting the substance in the solid state, for both the melting point and the solubility are here properties of the solids, i.e. it is the presence of the solid that deter- mines both. The liquid may be supercooled when not in contact with the solid, and the solution may be supersaturated if the solid is not present. But the solid cannot be heated above its point of fusion without melting, and the solution in contact with it is always exactly saturated. It is in this sense that we say that it is the solid that determines the melting point and the solubility not the liquid or the solution. The analogy that we see between solubility and fusing point in the above tables is an example of a rule of fairly general applicability. We usually find that, when we compare similar substances, fusibility and solubility go together. If we consider a set of isomeric substances, for instance, we find that the order of solubility is usually the same as the order of fusibility, i.e. the most soluble isomer has the lowest melting point. The order of the solubility of isomers is frequently independent of the nature of the solvent. Thus, if one of two isomers is more soluble in water than the other, it will still be the more soluble if alcohol, ether, benzene, etc., be the solvents employed instead of water. In some special cases, not only the order but even the ratio of the two isomers remains nearly the same for all solvents. For example, it has been found that meta-nitraniline is, on the average, 1*3 times more soluble than para-nitraniline in 13 different solvents, the ratio of solubility only varying from T15 to T48. It has also been found that the order of solubility of corresponding salts of isomeric acids is also very frequently the order of solubility of the acids themselves. It must be borne in mind, however, that these rules are all liable to well-marked exceptions. CHAPTER XIV RELATION OF PHYSICAL PROPERTIES TO COMPOSITION AND CONSTITUTION THE properties of substances, when studied in relation to their composition and structure, have been divided into three classes. In the first class we have those properties which are possessed by the atoms unchanged, no matter in what physical or chemical state these atoms may exist. Such properties are called additive, and the best instance of an additive property is found in weight (or mass). Each atom retains its weight unaltered whether it exists in the free state or whether it is combined with other atoms. When atoms com- bine, the weight of the compound is the sum of the weights of the component atoms. This is only another way of stating one of the fundamental assumptions of the atomic theory (Chap. II.), the assumption, namely, which takes account of the indestructibility of matter. Weight is the additive property par excellence, no other property being additive in the strict sense, although in the case of some properties there is an approximation to the additive character. We have seen in homologous series that there exists between neighbouring members a difference in molecular volume which is practically constant. For a difference in composition of CH 2 there is the constant difference of 22 in the value of the molecular volume, which retains nearly the same value for all homologous series. We may therefore attribute the value 22 to the group CH 2 , for whenever the methylene group enters a molecule the molecular volume is increased by this amount. Here we are evidently dealing with an additive property, but the additive character is modified by other influences, for the difference 22 is not absolutely constant, but fluctuates slightly about this value. It should be noted, also, that this number only holds good for the liquid state, for the measurements from which it is derived were all made at the boiling points of the liquid substances. By comparing the molecular volumes of liquids differing in composition in various definite ways, Kopp was able to establish a set of approximate rules such as the following : CHAP, xiv PHYSICAL PROPERTIES 137 (a) When two atoms of hydrogen are replaced by one atom of oxygen, there is a very slight increase in the molecular volume. (b) One atom of carbon may replace two atoms of hydrogen without sensible alteration of the molecular volume. From these rules, in conjunction with the preceding one, we may draw the following deductions : If the increase of molecular volume for CH 2 is 22, and if one atom of carbon is equivalent to two atoms of hydrogen, we may assign the value 11 to an atom of carbon, and the value 11 -f- 2 = 5 '5 to the atom of hydrogen. These values, then, are assumed to be the atomic volumes of carbon and hydrogen. Since there is a slight increase in the molecular volume when two atoms of hydrogen are replaced by one atom of oxygen which is attached to the same carbon atom, the atomic volume of oxygen must be somewhat greater than 11. On the average it is 12*1. It is found, however, that when hydrogen is replaced by hydroxyl, the increase of molecular volume is not 12-1, corresponding to the addition of one oxygen atom, but only about 7'8. When oxygen, therefore, is attached to one carbon atom, it contributes more to the molecular volume than when it is partially attached to carbon and partially to hydrogen. Here we come across an influence which modifies the additive character of all properties except weight, namely, the influence of structure or constitution. The molecular volume is not a purely additive property it is in part constitutive, i.e. is dependent not merely on the number and kind of atoms in the mole- cule but also on their arrangement. We must therefore attribute to oxygen two atomic volumes 12*2 when it is attached to carbon so as to form the carbonyl group (CO), and 7 '8 when it is part of the hydroxyl group, or is attached to two different carbon atoms, as in the ethers. We are now in a position to deduce the molecular volume of a compound containing only carbon, hydrogen, and oxygen, by adding together the volume values of its constituent atoms. Thus, a com- pound of the formula C a H & O c O' rf , where 0' denotes oxygen in a hydroxyl group, has the molecular volume if the molecular volume is determined at the boiling point of the liquid. For example, in valeric acid, C 4 H 9 . CO . OH, we have a = 5, 5=10, c=l, d=l, so that V= 55 + 55 + 12-2 + 7-8 = 130. The molecular volume as experimentally ascertained is 130-5. The agreement in the majority of cases is not so good, e.g. the molecular volume of ethyl oxalate found from the formula is 161, and that found by experiment 167. In this connection it must be remembered 138 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. that Kopp's original rules are not strictly accurate, so that the deductions from them are always liable to slight error. From the consideration of compounds containing other elements than those already referred to, values have been deduced for the atomic values of nitrogen, the halogens, sulphur, phosphorus, etc. Sulphur and nitrogen, like oxygen, have different values for the atomic volume according to the mode in which they are combined with other atoms. When an element or radical exists in the liquid state, the mole- cular volume deduced from its compounds in general agrees with that of the free element or radical. Thus the volume of Br 2 deduced from bromine compounds would be 53 '4, and the molecular volume of free bromine is actually 53 '6. The value of N0 2 deduced from compounds containing it as a radical is 31*5, the value for the free oxide being 32-0. No purely additive property can throw any light on the size of the molecule or its .constitution, as each atom retains the numerical value of the property unchanged whether it exists in the free state or whether it is combined with other atoms in any way whatever. As the value is therefore unaffected by the kind and extent of the combination, it can clearly give no indication as to what that mode or extent of combination may be. When the property is modified by constitutive influences, as is the case with the molecular volume of liquids, it may throw light on the constitution of a substance. Thus, if of two isomeric substances it were suspected that one contained an atom of carbonyl oxygen, whilst the other contained an atom of hydroxyl oxygen, that with the larger molecular volume would be the compound containing the hydroxyl oxygen. The refractive power of liquids is a property which, like the molecular volume, is in general additive in character, although modi- fied by constitutional influences. The refractive index itself cannot be taken as a measure of the refractive power when its relation to the chemical nature of the substance is under investigation, for the index varies greatly with temperature, etc. A better measure is found in 711 the specific refractive constant -= , or (n l)v, where n is the refrac- ts tive index, d the density, and v the specific volume. This expression varies very slightly with the temperature, and is little influenced by the presence of other substances, so that it has frequently been used in the comparison of different liquids. Another specific refractive . . . . A . ?1 2 -1 1 71 2 -1 constant is given by the expression -^ - . -3 , or 2 - . v, which was arrived at on theoretical grounds. When the values obtained by its aid are compared with the values of (n - l)v, it is found that they have an advantage over the latter, inasmuch as they are not only independent of the temperature, but also of the state of aggregation. With the xiv PHYSICAL PROPERTIES 139 empirical refraction constant there is considerable divergence between the values in the liquid and gaseous states, whilst the numbers obtained for the theoretical constant are the same in both cases. Thus for water at 10 we have Liquid .... 0'3338 0'2061 Gaseous .... 0-3101 0-2068 The molecular refractive power is the product of the mole- cular weight into the specific refractive power as measured by either of these expressions. Thus we have the " empirical " molecular refrac- tion, Mv(n - 1) } or V(n 1), and the " theoretical " molecular refraction, n 2 - 1 n 2 - 1 - . Mv, or -g . V. It is found that the molecular refraction of liquids as measured by either of the formulae is essentially an additive property modified by constitutive influences, so that an atomic refrac- tion for carbon, hydrogen, oxygen, chlorine, etc., may be calculated. If the refractive powers of each atom in the molecule be added together, the sum gives the molecular refraction of the compound. As is the case with the atomic volumes, different values have to be attributed to the atomic refraction of oxygen, according as it is carbonyl, or hydroxyl, or ether oxygen, a distinction in this case being necessary between the last two kinds which is not required for atomic volumes. When a substance contains an ethylene linkage, its atomic refraction is considerably higher than would be reckoned from the atomic refractions of the elements composing it ; and when an acetylene linkage is known to exist in the molecule, the excess is even higher. " Atomic " refractions have therefore been attributed to "double bonds" and "triple bonds," which must be added to the atomic refractions of the elements themselves when the total molecular refraction is calculated. Some liquids exhibit the phenomenon of optical activity, that is, when placed in the path of a polarised ray they rotate the plane of polarisation in one sense or the other. Substances which rotate the plane of polarisation in the direction of the hands of a watch are said to be dextrorotatory; substances which rotate it in the opposite direction are said to be Isevorotatory. The specific rotatory power is usually denoted by the symbol [a], which is obtained by dividing the actual rotation observed in the polarimeter by the length of the layer of liquid through which the light passes, and by the density I of the liquid at the temperature of observation. To obtain convenient | numbers, the length is usually given in decimetres. The molecular rotation is the product of the specific rotation into the molecular weight, or more usually the hundredth part of this value, i.e. .. Ma a 140 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. where a is the observed rotation, / the length of liquid, and d its density. It has been suggested that a better expression would be [6] = ~ \/ F", since we are obviously here concerned not so much with the molecular volume of the liquid as with the average distance between the centres of the molecules, the ray of light passing in a straight line through the liquid, and meeting a number of molecules proportional to the cube root of the molecular volume. Since the rotatory power varies with the temperature and with the wave length of the light employed, it is necessary to specify both of these in stating the value of the specific or molecular rotation. When we come to inquire into the nature of liquids and sub- stances in the dissolved state which show optical activity, we find that they possess in every case one or more asymmetric carbon atoms, i.e. carbon atoms which are united to four kinds of elements or radicals all different from each other. If we imagine these four different radicals to be at the four corners of a tetrahedron, we find that we FIG. 23. can arrange them in two essentially different ways, as is shown in the accompanying figures. Here the tetrahedra are supposed to be resting with one face on the paper, the summits being towards the reader. If we call the four different groups a, b, c, d, and place the group d at the summit, then in Fig. 22 the order abc is in the direction of the hands of a watch, and in Fig. 23 in the reverse direction. If two of the groups are made the same, the asymmetry vanishes, as we can see from the figures if we make b = c, the two figures then becoming identical. No altogether satisfactory answer has yet been obtained to the question of what determines the value of the molecular rotation in any particular case. As we have seen, the rotation should vanish if two of the groups become identical, and it may also be perceived from the figures that if we interchange the positions of any two of the groups, the sign of the rotation will thereby be changed. If we suppose each of the groups to be en- dowed with some definite property causing the rotation, and denote the value of this function by a, /?, y, 8 for the groups a, b, c, d. respectively, the rotation of the asymmetric carbon atom must be xiv PHYSICAL PROPERTIES HI determined by an expression of the following or similar type: (a - p)(p - y)(y - 8)(a - y)(a - $)(J3 - 8). If the function becomes the same for two of the groups, the expression becomes zero, i.e. the rotation vanishes, and if we interchange any two throughout, the sign of the whole expression is changed, i.e. the rota- tion from dextrorotatory becomes laevorotatory, or vice versa. What the function is we do not know. It seems to be connected with the weight of the radicals, but cannot be the weight itself, since substances are known to be optically active which have two groups of equal weight attached to the asymmetric carbon atom. Certain regularities have been observed among the molecular rotations of members of homologous series, but numerous exceptions occur. Many of the ap- parent divergencies, however, may be accounted for by the assumption that different members of the same series may have different degrees of molecular complexity when in the liquid state (cp. Chap. XIX.). Some crystalline substances, such as quartz, are optically active, but the activity here is not due to the arrangement of atoms within the molecule, but rather to a certain arrangement of the crystalline particles. A consequence of this is that the activity disappears when the crystalline structure is destroyed, i.e. when the substance passes into the fused or dissolved state. As a general rule, substances which are active in the liquid or dissolved state are not active when crystalline, but a few substances are known which exhibit optical activity both as crystals and as liquids. All liquids when placed between the poles of a magnet or in the core of an electromagnet become optically active. The character of the magnetic optical activity, however, is essentially different from that exhibited by liquids which are naturally active. If we place a tube containing a naturally active liquid between the prisms of a polarising apparatus, and so adjust the prisms that no light passes through the system when we place a light-source at one end and the eye at the other ; and if we then reverse the positions of the light-source and the eye, we find that the passage of the light is still obstructed. The sense and magnitude of the activity, then, are independent of the direction of the light. A naturally inactive liquid in a magnetic field behaves quite differently. If we first adjust the prisms so that no light passes, and then reverse the positions of eye and light-source, we find that light now traverses the system quite freely. We find, in fact, on readjusting the prisms to darkness, that a substance which appeared originally dextrorotatory is now to an equal extent laevorotatory. The difference in the nature of the activity may best be made clear by analogy. The action of a naturally active liquid resembles the action of a screw. If a screw is right handed (dextrorotatory) when viewed from one end of its axis, it is right handed when viewed from the other. A naturally active liquid if 142 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. dextrorotatory when viewed from one end remains dextrorotatory though the direction of the light passing through it is reversed. The polarity of a magnetically active liquid resembles that of a muff. If the lie of the hair in a muff when viewed from one end of its axis is in the direction of the hands of a watch, it will be in the opposite direction if we view it from the other end of the axis. A magnetically active liquid is similarly dextro- or laevo-rotatory according to the direction in which the light passes through it, the magnetic field being supposed constant. The specific magnetic rotation is usually given as the ratio of the specific rotation of the substance (determined as in naturally active liquids) to that of water under the same conditions : the molecular magnetic rotation is this value multiplied by the molecular weight of the substance and divided by 18, the supposed molecular weight of water. We find once more that when the molecular magnitudes for homologous substances are compared, there is a constant difference for the group CH 2 , so that to this extent the property is an additive one. The constitutive influence, however, is much more marked here than in any of the previous instances ; and the property is therefore valuable when applied to solving problems of constitution. When a relation has been established between the constitution of well-known substances and the values of a certain property possessed by them, it is legitimate to draw inferences regarding the un- known constitution of other substances from the known value of this property in their case, the same rules being applied as to substances of known constitution. Of course, the worth of the deduction depends entirely on the number and variety of compounds which have been investigated, and on the exactness of the empirical rules established from the investigation. Melting points and boiling points, for example, are occasionally useful in indicating the probable structure of a compound, but it is seldom that the evidence based on them alone is to be treated with any degree of confidence. To take an instance in point, it was assumed that the decane dicarboxylic acid given in the table of melting points on p. 134 has the normal structure, entirely on the strength of its' melting point falling into the regular scheme displayed by the even members of the homo- logous series which are known to have the normal structure ; for if its structure were not normal, it would in all probability have a melting point diverging widely from the scheme which included the members of the series which were known to be normal in reality. This evidence is slight, but for want of anything better it has a certain validity, although any well-established fact to the contrary would at once be conclusive against it. When the connection between the constitution and the value of a physical property is better marked and susceptible of being laid down in more definite rules, as is the case with the molecular refraction or XIV PHYSICAL PROPERTIES 143 magnetic rotation, greater confidence can be placed in the conclusions drawn from the value of the property in a particular compound as to the constitution of that compound. Great care, however, must be exercised in certain cases, for we are liable to draw conclusions which are not warranted by the facts. Thus, for example, from a considera- tion of the molecular refraction of aromatic compounds, it has been concluded that benzene has three ethylene linkings in the molecule, H C HC i.e. that Kekule's formula J is the correct one. Against this we must place the fact that benzene does not behave chemically as if it had a molecule containing three ethylene linkings, and at present the chemical evidence must be held to outweigh the evidence of the molecular refraction. The real difficulty in a case like this is that we can express the general chemical behaviour of the fatty compounds by means of three different kinds of carbon linking simple, ethylenic, and acetylenic, all perfectly well defined chemically ; whilst in the aromatic compounds we meet with something entirely new, and not readily brought into any of the above classes of carbon linking. How we are to represent this new kind of linking we do not at present know. Various attempts have been made, mostly based on the supposed properties of the tetrahedral carbon atom some of them yielding ordinary static formulae, some of them kinetic formulae in which the carbon atoms are assumed to be in a state of constant vibration in a specified manner. These formulae have all their peculiar merits and demerits. None can be held to be entirely satisfactory, in view of the conflict of evidence, and it may perhaps H C still be said that the least definite formula , which makes no assumptions as to the mode of linking of some of the bonds, is also the happiest in representing the known properties of aromatic compounds. As we have seen in Chapter XIII., the molecular heat of combustion is essentially an additive property, though subject to modifications conditioned by differences in constitution. The modifications occa- sioned by changes of constitution are often very slight, so that as long as we are dealing with saturated compounds, it is found that isomeric substances have approximately the same heats of combustion. Thus for propyl alcohol we have 4986 K, and for isopropyl alcohol 4933 K ; 144 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. for anthracene and phenanthrene, which both have the formula C ]4 H 10 , we have 16,943 K and 16,935 K respectively. In other cases the differences are greater, but when substances of nearly the same character are considered, the differences are not of much importance. Unsaturated compounds exhibit great divergence from saturated compounds, and values have been attributed to the ethylene linkage and to the acetylene linkage. Conclusions, too, have been drawn as to the constitution of benzene from the heat of combustion. This amounts for benzene, C 6 H 6 , to 7878 K, the value for the isomeric substances, dipropargyl, CH C . CH 2 . CH 2 . C CH, and dimethyl- diacetylene, CH 3 . C | C . C | C . CH 3 , being 8829 K and 8474 K respectively. Here the differences for the isomeric substances are great, owing to undoubted differences in the constitution especially in the mode of linking of the carbon atoms. The exact nature of the differences in constitution corresponding to the differences in the heat of combustion is, however, as before, uncertain, for we have nothing but analogy to guide us, and are apt to assume that the rules that hold good for the fatty compounds have the same validity for aromatic compounds, which in all probability is not the case. Some properties are exceedingly valuable in certain special cases for giving us an insight into the constitution of chemical compounds. For instance, if a new compound displays optical activity, we know that in all likelihood it contains one or more asymmetrical carbon atoms, i.e. carbon atoms which are combined with four different kinds of atoms or groups of atoms ; for all well-investigated optically active compounds have been proved to contain such asymmetric carbon atoms. Again, for organic acids we have the molecular conductivity in aqueous solution (cp. Chap. XXI). This property, as we shall see, is closely related to the strengths of the acids, and also to their constitution. The molecular conductivity of acids varies with the strength of the solution in which they exist, and the actual variation is different according to the acid which is dissolved. There is a general rule, however, to which the variations for nearly all acids conform, and this enables us to calculate for each acid a constant independent of the strength of the solution whose conductivity is measured. It is this dissociation constant which is useful in the discussion of questions regarding the constitution of organic acids. In it there is scarcely a trace of an additive nature in the sense in which the term has hitherto been used. For example, the dissociation constants K for the fatty acids are K. Formic H.COOH 0'0214 Acetic CHg.COOH O'OOISO Propiomc C 2 H 5 .COOH 0*00134 Butyric C 3 H 7 .COOH 0-00149 Isobutyric C 3 H 7 . COOH 0'00144 Valeric C 4 H 9 .COOH 0-00161 Caproic C 5 H n . COOH '001 45 xiv PHYSICAL PROPERTIES 145 If we except the first number of the series, which as usual diverges from the others, we see that the constants have all approximately the same values, although constant additions are made to the molecule. In the case of the normal dibasic acids of the oxalic series we have a very different table : K. Oxalic (COOH) 2 10 '0 (?) Malonio CH 2 (COOH) 2 0-163 Succinic C 2 H 4 (COOH) 2 0-00665 Glutaric C 3 H 6 (COOH)o '00475 Adipic C 4 H 8 (COOH)2 0'0037 Here the values of the constants sink steadily as the molecular weight increases, although the fall becomes less and less as we proceed. In the normal acids the carbon atoms are supposed to be linked to each other in a continuous chain. The two carboxyl groups are there- fore at opposite ends of the chain. Now, in the fatty monobasic acids, in which there is only one carboxyl group, the addition of CH 2 has little influence on the dissociation constant. We therefore attribute the great diminution of the value in the series of dibasic acids to the increasing distance of the carboxyl groups from each other. In oxalic acid the carboxyl groups are directly attached, and their proximity is assumed to increase the dissociation constant. Each separate group has acid properties, and the two groups reinforce each other's acidity. In a higher member of the series, e.g. adipic acid, it is true that we have still two acid groups, but they are so far apart that we suppose them to have little reciprocal influence in increasing the acid properties. This mode of viewing the relation between constitution and dissocia- tion constant leads on the whole to consistent results where the con- stitution has been well ascertained. When chlorine replaces hydrogen in an organic acid, the strength of the acid, and with it the dissociation constant, is increased. Thus trichloracetic acid is very much stronger than acetic acid from which it is derived, being comparable in point of strength with the ordinary mineral acids. The dissociation constants of the chloracetic acids are shown in the following table : Acid. Formula. K. Acetic CHg.COOH 0*0018 Monochloracetic CH 2 C1.COOH 0'155 Dichloracetic CHC1 2 .COOH 5*14 Trichloracetic CC1 3 .COOH 120 The replacement of hydrogen by chlorine obviously does not correspond to a constant addition to the dissociation constant, but rather to multiplication by a factor, although here the factor at successive stages diminishes. The influence of other replacing atoms and groups is seen in the following derivatives of acetic acid : 146 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. Acid. Formula. K. Acetic CH 3 .COOH 0-0018 Glycollic CH 2 (OH).COOH 0-0152 Thioglycollic CH 2 (SH) . COOH 0-0225 Bromacetic CH 2 Br.COOH 0138 Chloracetic CH 2 C1.COOH 0-155 Malonic CH 2 (COOH) . COOH 0-163 Sulphocyanacetic CH 2 ( SON) . COOH 0-265 Cyanacetic CH 2 (CN).COOH 0-370 From the table it appears that the influence of a chlorine atom is almost as great as the influence of another carboxyl group, although much less than the influence of a cyanogen group. Reasoning on the same lines as those adopted in treating the constants of the normal dibasic acids, we should expect that the farther removed any of these substituting atoms or radicals are from the carboxyl group, the less will be their influence on the dissociation constant. This expectation is in general justified. We have, for example, the following constants for the hydroxyl derivatives of propionic acid : Acid. Formula. K. Propionic CH 3 . CH 2 . COOH '00134 /3-Lactic CH 2 (OH).CH 2 .COOH 0-00311 a-Lactic CH 2 . CH(OH). COOH 0'0138 Glyceric CH 2 (OH) . CH(OH) . COOH -0228 The influence of the hydroxyl group in the a-position is much more marked than the influence of the same group in the /^-position. When there is a hydroxyl group attached to each carbon atom, the influence is greater still. In the aromatic series we meet with peculiarities not easily explic- able. Benzoic acid, C 6 H 5 . COOH, has the constant O'OOGO, whilst the three hydroxyl acids have the values given below : COOH COOH COOH H-Cv '/0--OH H-CC ' ;C-H H-C \ \ H CC - xC H H C^i /C OH H Cxi /C H C C C H H AH 1 Salicylic. Metahydroxybenzoic. Parahydroxybenzoic. K = 0'102 0-0087 0-0029 In the orthohydroxyl acid, where the carboxyl and hydroxyl groups are on neighbouring carbon atoms, we have a great increase over the value of the constant of the parent acid. In the meta acid the effect on the constant is comparatively slight. In the para acid, contrary to expectation, we have an actual diminution instead of an increase. It is obvious from a consideration of these results that very great caution xiv PHYSICAL PROPERTIES 147 must be exercised in drawing conclusions for aromatic bodies from rules derived from the study of fatty compounds. The so-called geometrical or space isomerism is often accompanied with striking differences in the values of the dissociation constants. From the saturated dibasic acid, succinic acid, COOH . CH 2 . CH 2 . COOH, are derived two unsaturated acids, maleic acid and fumaric acid, which both in all probability have the structural formula COOH CH : CH COOH. "VVe often express the difference between these two acids by writing their formulae thus : H C COOH COOH C H II II H C COOH H C COOH Maleic acid. Fumaric acid. K = 117 K = 0'093 the difference corresponding to a supposed difference in the arrange- ment of the hydrogen and carboxyl groups on the carbon tetrahedra. The dissociation constants are much greater than that of succinic acid, and the constant of maleic acid is twelve times that of fumaric acid. The above mode of formulation might lead us to expect such a difference, for in maleic acid the two carboxyl groups are on the same side of the central plane, and in fumaric acid on different sides, and consequently more remote from each other. From the purely chemical point of view, the investigation of the rela- tionship between the value of physical properties and the constitution of chemical compounds is chiefly important as affording a means of ascertaining the otherwise unknown constitution of certain compounds by a determination of their physical constants. Too much reliance, however, should not be placed on this mode of settling chemical constitutions. The interpretation of the results is often doubtful, and that most frequently in cases where the determination of the constitution by chemical methods presents special difficulty. To sum up, we may say that the physical method usually offers valuable indications of the direction in which a chemical solution of the problem is to be sought, rather than a final solution of the problem itself. Besides additive and constitutive properties, we have what have been called colligative properties. The numerical value of these properties depends only on the number of molecules concerned, not i on their nature or magnitude. For example, if we take equal numbers jof gaseous molecules of any kinds whatever, they will always occupy the same volume when under the same conditions (Avogadro's Law), i The gaseous volume then is a colligative property. We make use of these properties in the determination of molecular weights, and they will therefore be further referred to under that heading. CHAPTER XV THE PROPERTIES OF DISSOLVED SUBSTANCES IF we ask ourselves the question : " To which state of aggregation is the state of a substance in solution comparable 1 " we find that there are only two answers admissible, viz. the liquid state or the gaseous state. It is obvious that when a solid is dissolved in a liquid it at once loses the properties which are characteristic of the solid state. Its particles become mobile, and all properties which depend on regular arrange- ment of particles disappear. Thus the solid may be a double-refracting crystal ; its solution exhibits none of the phenomena of double refraction. It may be an optically active solid, and yet its solution may show no signs of optical activity. In such cases the passage of the substance into solution exhibits considerable analogy to the passage of the substance into the liquid state. A double-refracting crystal almost invariably loses its double refraction when it melts, and most substances which are optically active in the crystalline state are inactive after fusion. The analogy which here holds good between the dissolved and the liquid states might, however, be equally well applied to the dissolved and the gaseous states. It is true that the solution of a substance in a liquid solvent is itself a liquid, but it by no means follows that the state of the substance within the solution is accurately comparable to that of a liquid. Indeed, if we look a little more closely into the matter we find that in the case of dilute solutions, at least, there is far more likelihood of the dissolved substance being in a condition comparable with that of a gas. One of the characteristic properties of a gas is its power of diffusion. If its pressure (or what is proportional to its pressure, its density or its concentration) is greater at one part of the space containing it than at another, the gas will move from the region of higher to the region of lower pressure or concentration, until by this process of diffusion the pressure or concentration is everywhere equalised. This process of diffusion goes on independently of the presence of another gas. A coloured gas such as bromine vapour may CHAP, xv PROPERTIES OF DISSOLVED SUBSTANCES 149 be seen to diffuse against gravity into another gas, say air, until the colour of the contents of the cylinder is everywhere of the same depth. The rate at which the diffusion takes place is, however, greatly influenced by the presence of another gas, the rate becoming rapidly less as the concentration of the other gas increases. This may be easily rendered evident by taking two cylinders, one evacuated, and one containing air, and breaking a bulb containing liquid bromine at the bottom of each. In the cylinder containing air the diffusion takes place slowly, an hour perhaps elapsing before the bromine vapour reaches the top of the vessel. In the evacuated cylinder the diffusion is apparently instantaneous. The particles of the foreign gas thus obstruct the movement of the particles of bromine, and render the process of diffusion slower. Now, in the case of a substance in solution we have the same process of diffusion as we have with gases. If we take a solution of a coloured substance, such as bromine itself, or preferably a coloured salt like copper sulphate, and place it in the bottom of a cylinder, afterwards covering it with a layer of pure water, we find that the colour of the copper sulphate solution gradually rises in the cylinder, proving that the copper sulphate is moving from a region of greater concentration to a region of less or no concentration. In this case | also the diffusion is against gravity, for the solution of copper sulphate ; if concentrated has a much higher specific gravity than water, so that .i the centre of gravity of the liquid will be raised as the copper ' sulphate becomes more uniformly distributed throughout it. It is true that the diffusion of a dissolved substance is very much I slower than the diffusion of a gas, months or even years elapsing before .: uniform concentration is attained in a cylinder not more than a foot high. But the difference is only a difference in degree, for it has been shown that gases under great pressures mix with extreme slowness against the action of gravity, many of the characteristic phenomena of the critical point, where the pressure is high, being obscured owing to this cause, unless the substances are mechanically mixed by stirring. When we come to consider the arrangement of the particles of a substance in dilute solution relatively to each other, we again find a i resemblance to the gaseous state, as in the case of chlorine water, for instance. Water at the ordinary temperature takes up about 2 '2 times its volume of chlorine. As the dissolved chlorine is uniformly ; distributed in the solution, the average distance between the chlorine i particles in the chlorine water is not very much less than the average ' distance between the particles in the chlorine gas ; and if the chlorine | water is only half saturated with chlorine, the distance between the j. particles is practically the same as the distance between the particles of the gas. So far, then, as the relative position of the particles of a I gas and of the same substance in dilute solution is concerned, there is 150 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. great similarity between the two states ; and as the particles of a gas at low pressures have very little influence on each other, so we may suppose that the particles of a substance in dilute solution are mutually independent. If we ask what concentration of a solution is comparable to the concentration of a gas under ordinary condi- tions, we find that although the concentration is comparatively small, it is still such as may be frequently met with in ordinary laboratory work. A gram molecule of a gas at and 760 mm. occupies 2 2 '4 litres ; a solution then containing a gram molecule of dissolved substance in 22*4 litres at will be equally concentrated with a gas under standard conditions. This solution in the terminology of volumetric analysis is about twenty-second normal, and solutions twentieth or even fiftieth normal are by no means uncommon in volumetric work. There is therefore an a priori probability that the state of a substance in dilute solution resembles in some respects the state of a gas, and it would not be surprising, therefore, to find that dissolved substances obey laws comparable to the gas laws. In the sequel we shall see that this is actually true. In order to get some idea of the effect of the solvent on a dissolved substance, we shall consider briefly some properties of substances which are easily measurable both for the substances themselves and for their solutions. It is evident that the properties most suited for study in this respect are those which are possessed by the dissolved substance alone and not by the solvent. We get a good example of such a property in the case of optically active liquids dissolved in optically inactive solvents. We can easily measure the specific rotation given by oil of turpentine, say, in the pure state and in various inactive solvents. If the solvent has no influence on the dissolved body, the specific rotation ought to remain the same whether the substance is in solution or whether it is in a state of purity. The specific rotation of laevorotatory oil of turpentine is 3 7 '01. Solutions containing 10 per cent of this substance dissolved in various solvents gave the following specific rotations, as calculated by the formula [a] = =-, where p is the . $ number of grams of active substance in v cubic centimetres of solution (see p. 139): Solvent. Specific Rotation. Alcohol 38-49 Benzene 39 '45 Acetic acid 40 '22 There is evidently here an action of the solvent, for these numbers are all different from the number obtained for the pure substance, and not only are they thus divergent, but they differ also from each other. The optically active liquid alkaloid, nicotine, shows the same behaviour in still more striking fashion. The specific rotation of the pure substance is 16T5 ; the specific rotation of a 15 per cent solution XV PROPERTIES OF DISSOLVED SUBSTANCES 151 in alcohol is 14T6 , and that of a 15 per cent solution in water is only 75*5. Here the specific rotation of the nicotine sinks to less than half its original value when the substance is dissolved in about six times its own weight of water, so that we cannot avoid the inference that the water exercises some powerful influence on the nicotine dissolved in it. In the other instances given above the solvents also play an important part in modifying the properties of the substances dissolved in them, although to a less extent than is the case with nicotine ; and it will be observed that each solvent exerts its own peculiar influence. The extent to which the rotatory power 25 20 Etfiylic Tartrate 100 80 60 40 Percentage of Tartrate FIG. 24. 20 is affected by the solvent depends on the strength of the solution. The less of the active substance there is in the solution the more does its specific rotation diverge from the value for the pure substance. This may be seen in the accompanying diagram (Fig. 24), which gives the specific rotation of tartaric ether alone and in solutions of varying concentrations. The effect of water is again specially great, the specific i rotation in 14 per cent aqueous solution being three times as great as the rotation for the pure ether. A curious regularity appears when we consider the rotations of aqueous solutions of different salts of optically active acids and bases. In order to obtain comparable numbers for the different salts, it is necessary to work, not with specific rotations, but with molecular 152 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. rotations, i.e. the products of the specific rotation and the molecular weight. If we take, for example, the more soluble salts of camphoric acid, it appears that the value of the molecular rotation tends towards the same limit in each case as the dilution increases. This is clearly shown in Fig. 25, where the percentage strength of the solution is plotted on the horizontal axis, and molecular rotations on the vertical axis. The values for the molecular rotations are obtained by multiply- ing the specific rotations by the molecular weights, and dividing by 100 in order to get convenient numbers. The curves, which are wide apart at the higher concentrations, come closer and closer together as 60r 55 46 i 40 35 Camphorates Li in Water 60 40 30 20 10 Percentage of Camphorate FIG. 25. the strength of solution diminishes, and might ultimately meet in the same point at zero concentration. This final concurrence, however, cannot be established directly owing to the difficulty of obtaining readings of sufficient exactness with very dilute solutions. 1 The same regularity appears with the soluble salts of other optically active acids, such as malic acid, tartaric acid, and quinic acid ; and also with the soluble salts of optically active bases, such as the cinchona alkaloids, quinine, cinchonine, etc. We may say then, in general, that the molecular rotation of the salts of an optically active acid or base always 1 The salts of a - bromosulphocamphoric acid have a very high rotatory power, and it is therefore possible to investigate them in centinormal aqueous solutions. At this dilution the molecular rotations of the soluble salts are identical. xv PROPERTIES OF DISSOLVED SUBSTANCES 153 tends to a definite limiting value as the concentration of the solution diminishes. This regularity is known as Oudemans' Law, and we may now attempt its interpretation. Taking the camphorates as our example, we note in the first place that the activity of the salts is due to the acid the bases, potash, soda, etc., with which the acid is combined, being optically inactive. Indeed, it is only when we have such a combination of active acid with inactive base, or active base with inactive acid, that Oudemans' Law holds good. Now the salts of camphoric acid, to judge from the general run of the curves in Fig. 25, would in very strong solution have quite divergent molecular rotations ; and this we know definitely to be the case with the salts of malic acid, some of which in strong solution have a positive and some a negative rotation, although their rotations at extreme dilution are all negative. Where the influence of the solvent is least, therefore, the salts have different molecular rotations; where the influence of the solvent is greatest, they have nearly the same molecular rotation. The water therefore tends to bring the acidic, or active, portion of the salts in some way into the same state, for from the similarity of rotation we must argue similarity of condition, as this is no chance agreement, but a general law. This might come about in several ways, but here we shall consider only one possibility. Suppose that the water decomposed the salts into free acid and free base progressively as more and more water was added to the solution. In the strong solutions only a relatively small proportion of the gram-molecular weight of the salt will be decomposed, so that the total molecular rotation will be made up of a relatively small rotation for the free acid, and a relatively large rotation for the acid still bound up with the base in the form of salt. As the different salts have different rotations, their strong solutions will differ considerably in their molecular rotations, as the total rotation is here due mostly to the salt. It is quite different with dilute solutions in which, on our assumption, the salt is almost entirely decomposed into free acid and free base. Here the total rotation is due almost wholly to the acid in each case, the undecomposed salt contributing very little to the total ; so that no matter what the salt is, the value of the molecular rotation will be the same. The assumption, then, of the progressive decomposition of the salt into acid and base under the influence of the solvent water would thus account for the phenomena of molecular rotation in dilute solutions ; but it is on other grounds an improbable one, and fails to explain the numerical value of the limiting molecular rotation for the following reason : A consideration of the curves of Fig. 25 shows that if they intersect on the line of zero concentration at all, it will be at a value of about 39. This value, therefore, would be on the above assumption the molecular rotation of camphoric acid in dilute solution ; but the value actually found for a solution of camphoric acid of 0'6 per cent strength is 93, 154 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. an altogether different number. We must therefore conclude that the supposition of a decomposition into acid and base by the water is untenable, and endeavour to find another explanation. It will be seen that the essence of the above attempted explanation is the assumption of the independence of the inactive basic arid the active acidic portion of the salt in dilute solution. If on any hypothesis of the action of the water we can suppose that the active negative part of the salt is removed from the influence of the inactive positive part of the salt as the dilution increases, we have a sufficient explanation of the constancy of the molecular rotation in weak solutions, and it only remains to select such a hypothesis as shall conform with the other properties of dilute solutions, and give a rational account of the numerical value of the limiting rotation. We shall meet with such a hypothesis in a later chapter. When we deal with properties which are possessed by solvent and dissolved substances alike, it is more difficult to ascertain definitely if the property of the dissolved substance has been changed by the fact of its having passed into solution. As a rule we find that the sum of the values of the property for substance and solvent is not the value for the solution. For example, the volume of solvent and solute is never exactly equal to the volume of the resulting solution, although in some instances, such as cane sugar and water, this is nearly the case. Very frequently we find that solution is accompanied by con- traction in volume. AVhen water and alcohol are mixed in equal measures there is a contraction of about 3 per cent of the total volume. This shrinkage may be due to a change in the specific volume of the water or the alcohol, or both, so that it is by no means easy to apportion the total volume change between the two substances. Even when we take a solution and dilute it by adding more of the solvent, we generally find that a change of volume occurs, a contraction on dilution being the usual result. According to Isidor Traube, the following general rule regulates the density of aqueous solutions : If we consider a quantity of the given solution such that it contains a gram - molecular weight of the dissolved substance, measure its volume and subtract from it the volume of the water which was added to make the solution, we obtain a residue which would be the molecular volume of the dissolved substance had no contraction taken place. The molecular volume may be determined directly with the pure substance, and the difference between it and the residue obtained as above described is constant and equal to 12*4 cc. There is thus a constant contraction in the aqueous solution when a gram -molecular weight of the substance is considered. Traube attributes the contraction to the solvent. This conclusion cannot be considered as fully established, for according to Traube's rule the whole contraction takes place when the dis- solved substance is first brought into contact with the water, no further shrinkage occurring when the solution undergoes dilution xv PKOPERTIES OF DISSOLVED SUBSTANCES 155 with more water, which is not in accordance with the results of experiment. There is an undoubted regularity in the density of aqueous salt solutions. If we consider, for example, the density of normal solutions of a number of salts, we find that the difference in density between a chloride and the corresponding bromide is constant ; that the difference between a chloride and the corresponding sulphate is constant ; in short, that the difference between corresponding salts of two acids is approximately constant, no matter what the base is with which the acids are combined. On the other hand, we find that the difference in the densities of equivalent solutions of corresponding salts of two bases is always the same, and independent of the acid with which they are united. Examples are given in the following table, where the densities are those of normal solutions : Cl Br I JS0 4 N0 3 K 1-0444 1-0800 1-1135 1-0662 1-0591 NH 4 1-0157 1-0520 1-0847 1-0378 1-0307 Difference 0'0287 0'0280 0'0288 0'0284 0'0284 Cl K Na NH 4 iSr Ba 1-0591 1-0540 1-0307 1-0811 1-1028 1-0444 1-0396 1-0157 1-0667 1'0887 Difference 0'0147 0'0144 0'0150 0'0144 0-0141 From a consideration of this table, it is evident that we can obtain the density of the normal solution of any salt by adding to the density of a salt chosen as standard two numbers, or moduli, one of which is characteristic of the base of the salt, and the other characteristic of the acidic portion of the salt. This regularity is known as Valson's Law of Moduli, Valson choosing ammonium chloride as his standard, because its normal solution had the smallest density of any of the salts he investigated. The moduli for the principal series of salts are given below : NH 4 0-0000 Cl O'OOOO K 0-0296 Br 0*0370 Na 0-0235 I 0'0733 Ba 0-0739 NO, 0'0160 iCa 0-0282 ^S0 4 0'0200 Pig 0-0221 |Zn 0-0410 |Cu 0-0413 Ag 0-1069 If these moduli are added to the density of normal ammonium chloride solution, viz. 1-0153, the densities of the other normal salt solutions are obtained, the values holding good for 18. Thus, if we wish to know the density of an equivalent normal solution of copper sulphate, we add to 1-0153 the modulus of copper plus the modulus 156 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. of the sulphates, viz. 0-0413 + 0*0200, and obtain 1-0766 as the result, in accordance with the experimental number. Not only is the law of moduli valid for normal solutions, but also for solutions of other (moderate) concentrations, the moduli being multiplied by the strength of the solution expressed in terms of a normal solution as unity, and then added to the density of the ammonium chloride solution of corresponding strength. We have the following values for the densities of various ammonium chloride solutions, which can be used as standards : Normality. Density. 1 1-0153 2 1-0299 3 1-0438 4 1-0577 Should we wish to know the density of a thrice normal solution of calcium bromide, we add to 1*0438 three times the sum of 0*0282 and 0-0370, viz. 0-1956, and obtain 1*2394, the value actually found by experiment being 1*2395. It is found that other properties of salt solutions besides the density are susceptible of similar treatment by the method of moduli. One salt solution is taken as standard, and from the value for it the values of the others may be obtained by adding the modulus for the acidic portion and the modulus for the basic portion of the salt under consideration. It will be observed that, strictly speaking, the optical rotations of dilute salt solutions are treated by means of moduli. If we take an inactive salt as standard, we can get the value of the rotation of any salt by adding together a modulus for the acid and a modulus for the basic portion of the salt. Thus if we compare in the case of the camphorates dilute solutions of equivalent strengths, we find that their rotations are all practically the same, and equal to what we may term the modulus for the acidic portion of the salt, the moduli for the different inactive basic portions being all equal to zero. From what has been said above, it will appear already that the properties of a substance in general undergo an alteration when the substance is dissolved in a liquid. The question as to whether the process of solution is to be regarded as chemical or physical has been much discussed, and this alteration in the value of well-defined properties has led many chemists to the conclusion that solution must be classed amongst the chemical processes. This conclusion is supported by the existence of heat changes accompanying solution as they accompany all chemical actions. It must be remembered, how- ever, that heat changes, as well as changes of volume and other properties, accompany the purely physical processes of melting, vaporisation, and the like. There can be no reasonable doubt that the dissolved substance and the solvent react on each other so as to influence each other's properties, but at present we are without any xv PROPERTIES OF DISSOLVED SUBSTANCES 157 satisfactory theory as to the origin or nature of this influence, and even without empirical regularities, except in a few special cases, to enable us to say in any given instance how the influence is most likely to become apparent. On the other hand, if we look upon dilute solutions with respect to volume, pressure, and temperature relations as affecting the dissolved substance, we find we can neglect the influence of the solvent altogether, and still obtain simple laws of the most general applicability, as will be shown in the succeeding chapters. CHAPTER XVI OSMOTIC PRESSURE AND THE GAS LAWS FOR DILUTE SOLUTIONS WE have seen in the preceding chapter that in some respects there is considerable analogy between the state of a substance existing as gas and the state of a substance in dilute solution. In the present chapter it will be shown that the analogy is more than superficial, and that laws exist for substances in dilute solution which are quite comparable with the simple laws for gases. These gas laws (cp. Chap. IV.) connect together the volume, pressure, and temperature of the gaseous substances to which they apply, so that we have first to consider what we are to understand by the volume, pressure, and temperature of a substance in solution. The temperature of a substance in solution is evidently the temperature of the solution itself. The volume of a gas we take to be the volume in which the substance as gas is uniformly distributed. Now the volume in which a dissolved substance is uniformly distributed is the volume of the solution, so that this volume corresponds to gaseous volume. There still remains to find for solutions the analogue of gaseous pressure. In the case of gases, the pressure we consider is that on the walls of the containing vessel, and may be measured directly. With substances in solution it is different. Here the pressure on the walls of the containing vessel is not the pressure of the dissolved substance, but is the gravitational pressure of solvent and solute combined. If we could exactly counteract the force of gravitation, there would be no pressure of the liquid on the walls of the vessel at all. There is therefore for substances in dilute solution no obvious magnitude corresponding to gaseous pressure; and yet, until this magnitude was discovered, no progress was made in the theory of dilute solutions. An experiment with gases will serve to show in what direction the analogue of gaseous pressure might be sought. In the case of the liquid solution we have two substances, and we wish to estimate the pressure of one of them. Can we in the case of a mixture of two gases find a method for measuring directly not only the total pressure of the two gases, but the partial pressure contributed by one of them ? CHAP, xvi OSMOTIC PRESSURE AND THE OAS LAWS 159 There is a theoretical method by means of which we can do this, and experiments have been made which go far to confirm the theory. Suppose that of two gases, A and B, one of them, B, can pass through a certain diaphragm, whilst A cannot. Let the gas A be enclosed in a vessel made of the material through which A cannot pass, and let the vessel be connected with a manometer which will measure the pressure of the gas within it. Suppose that the original pressure of A within the vessel is half an atmosphere, and let the vessel and its contents be immersed in the gas B, whose pressure is maintained steadily at one atmosphere. The gas B by supposition can pass freely through the material of which the vessel enclosing A is constructed, and it will do so until its pressure inside the vessel is equal to its pressure outside the vessel, viz. one atmosphere. For, if there is to be equilibrium between the gas inside the vessel and the same gas outside the vessel, there must be no difference of pressure of B throughout the whole space which it occupies, for the gas A exerts no appreciable influence on B. Inside the vessel there is now a total pressure of one and a half atmospheres, one atmosphere being the partial pressure of B, and half an atmosphere being the partial pressure of A, which has remained constant at its original value, owing to the impermeability of the vessel to this gas. Outside the vessel we still have the pressure of one atmosphere, so that the internal pressure registered when equilibrium occurs is half an atmo- sphere greater than the pressure outside the vessel. The excess of pressure inside is evidently due to the gas A which cannot pass through the diaphragm, so that, by taking the difference in pressure on the two sides of the diaphragm, we obtain the partial pressure of the substance to which the diaphragm is impermeable. It is not an easy matter to get a diaphragm which is quite permeable to one gas, and quite imperme- able to another ; but palladium at a moderately high temperature fulfils the conditions fairly well. Palladium has the property of absorbing hydrogen at ordinary temperatures, and parting with the absorbed gas again when heated in a vacuum to temperatures above 100. It exhibits this behaviour with regard to no other gas, so that at temperatures of about 200 it forms a diaphragm permeable to hydrogen, but impermeable to gases such as nitrogen, carbon monoxide or carbon dioxide. Experiments have been made with one of these gases inside a palladium tube, and an atmosphere of hydrogen at known pressure outside the tube. Theoretically, one would expect the internal pressure to increase by an amount equal to the external pressure of hydrogen. This was found to be nearly but not quite the case, the actual increase amounting in different experiments to from 90 to 97 per cent of the theoretical increase. Still the result is close enough to show that this method of measuring the pressure due to one substance in a mixture might be applied in other cases with success. Let us now deal with a liquid solution, say a solution of cane 160 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. sugar in water. If we could procure a semipermeable diaphragm of the proper kind, i.e. one which would be permeable to water and impermeable to sugar, we might be in a position to ascertain the pressure in the solution due to the presence of the sugar, and find how it varied with the concentration of the solution, the tempera- ture, etc. Pfeffer, the plant physiologist, while working at the osmotic phenomena in vegetable cells, prepared various membranes which proved to be perfectly permeable to water, and impermeable to some substances dissolved by the water. Such membranes had been pre- viously discovered by Moritz Traube, who, however, did not give them such a form as to permit of accurate work being done with them. They are essentially precipitation membranes, and their formation can be easily studied on a small scale. If a solution of copper acetate is added to a solution of potassium ferrocyanide, a chocolate -brown precipitate of copper ferrocyanide is produced ; and if the two solu- tions are brought together very carefully, mechanical mixing being avoided, the precipitate assumes the form of a fine film or membrane separating the two, and impermeable to both the dissolved substances. The experiment may be carried out in the following fashion, which was proposed by Traube. A piece of narrow glass tubing about 6 inches in length is left open at one end, and closed at the other by means of a piece of rubber tubing provided with a clip. Into this tube a few drops of a 2 '8 per cent solution of copper acetate are! sucked up, by compressing the rubber beneath the clip with the fingers and then releasing it. The tube is now lowered into a test- tube containing a few cubic centimetres of a 2 '4 per cent solution of potassium ferrocyanide. If the liquid in the inner tube forms a plane surface at its mouth, which can be secured by a slight movement of the rubber tubing, the copper ferrocyanide is deposited as a fine transparent film which closes the opening of the tube. That diffusion of the dissolved salts is prevented by this membrane is evident from the fact that the membrane remains transparent and of excessive tenuity for a very considerable period, showing that the copper and potassium salts no longer come into contact. A substance such as barium chloride, which is easily recognisable in small quantities, may be added to one of the membrane-forming solutions, best to that which is to be placed in the inner tube, and it will be found that even when gravitation would aid the mixing, none of the barium chloride passes through the septum. Whilst these experiments show the possibility of existence of membranes permeable to a liquid solvent and not to certain substances which might be dissolved in it, they are of no use for an investi- gation into the pressure exercised by dissolved substances, for the films are so delicate as to be ruptured by very slight pressures or mechanical disturbance. Pfeffer solved the problem by depositing such xvi OSMOTIC PEESSUEE AND THE GAS LAWS 161 films in the pores of unglazed vessels of fine earthenware, such as those employed in experiments on the diffusion of gases. This he did by placing one of the membrane-forming solutions in the inside of the porous pot and the other solution outside. The two solutions gradually penetrating the wall from opposite directions, at last meet in the interior of the wall, and there deposit a semipermeable film across the pores in which they meet. The film, although as delicate in this as in the former case, is now capable of withstanding a much higher pressure, on account of the support which the material of the porous cell affords it. If the film is to be exposed to high pressures, great precautions have to be taken in its preparation, so as to avoid all possibility of rupture of the membrane ; but if it is merely desired to show the phenomena qualitatively, it may be done simply as follows. The most convenient form of porous vessel to use is that of a bulb provided with a neck into which a rubber stopper may be inserted. These bulbs are used in gas diffusion experiments, and need no further preparation than washing and soaking for a day in running water. The neck of the bulb is dried and coated inside and outside with melted paraffin wax, which is allowed to solidify. A solution of copper sulphate (2*5 grams per litre) is introduced into the bulb up to a level above the bottom of the paraffin coating, and the bulb is then placed in a beaker, into which is poured a solution of potassium ferrocyanide (2*1 grams per litre) until the bulb is immersed up to the neck. After standing for some hours the bulb H is taken out of the solution, emptied, and rinsed with water. If now i the bulb is filled with a strong solution of sugar, and placed in pure I water, the water will pass into the interior through the semipermeable | membrane. This is best seen by inserting into the neck a well- fitting stopper through which passes a length of narrow glass tubing open at both ends. As the water passes into the interior of the bulb, the solution rises in the narrow glass tube, and the pressure on the inner surface of the semipermeable diaphragm increases. Owing to the resistance the water experiences in passing through the fine pores ||! of the bulb the process is a slow one, but in the course of an hour a : >rise of several inches may be noted, and in twenty-four hours the f solution may have risen from six to ten feet in the tube. As a rule, the membrane prepared in this way without any special precautions being taken, breaks down when the pressure exceeds ten feet of water, and the level of the liquid no longer rises. With a perfect membrane capable of resisting high pressures it i becomes a question when the increase of pressure within the bulb will | come to an end. Pf effer devoted his attention to this question, and attained the following results. For a given solution the pressure rises i slowly until a certain maximum value is reached, after which the pressure remains constant. This maximum pressure, called by Pfeffer the osmotic pressure, varies with the nature of the dissolved M 162 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. substance. The following table contains the maximum pressures in centimetres of mercury observed for one per cent solutions of the undernoted substances : Cane sugar . Dextrine Potassium nitrate Potassium sulphate Gum . 47-1 16-6 178 193 7'2 The pressure was found to be dependent on the strength of the solution, being very nearly proportional to the concentration of the solution, as may be seen from the following table for cane sugar, concentrations being given in percentages, and pressures in centimetres of mercury : Concentration. Pressure. Ratio. 1 53-5 53-5 2 101-6 50-8 2-74 151-8 55-4 4 208-2 52-1 6 307-5 51-3 For potassium nitrate the ratios are less constant : Concentration. Pressure. Ratio. 0-80 130-4 163 1-43 218-5 153 3-3 436-8 133 As the concentration increases, the ratio of pressure to concentration here diminishes, but Pfeffer showed that this was really due to the membrane not being perfectly impermeable to potassium nitrate, a small quantity of the salt escaping, especially at the higher pressures, so that the proper maximum was never reached. Temperature also influences the maximum pressure, as the following results with a one per cent sugar solution serve to show : Temperature. Pressure. 6-8 50-5 13'2 52-1 14-2 531 22-0 54-8 There is obviously here a regular increase of pressure with rise oi temperature. All these experiments were made by Pfeffer in 1877, but it was not until 1887 that van 't Hoff published a complete theory of dilute solutions which takes them as its experimental basis. The semi- permeable membrane furnishes us with a means of directly measuring a pressure due to the presence of the dissolved substance, so that we may take it as the analogue in dilute solutions of gaseous pressure ir gases. It should be noted that this pressure does not necessarily depend for its existence on the presence of a semipermeable diaphragm but is only rendered evident and measurable by its means. xiv OSMOTIC PRESSURE AND THE GAS LAWS 163 We are now in possession of all the magnitudes necessary to enable us to investigate the pressure, volume, and temperature relations of substances in dilute solutions, and to compare the numerical results of the investigation with the corresponding relations for gases. The pressure we consider is the osmotic pressure ; the temperature is the temperature of the solution ; and the volume is the volume occupied by the solution. Pfeffer's results show, in the first place, that the osmotic pressure at constant temperature is proportional to the concentration of the solution, i.e. to the amount of substance in a given volume. In other words, the osmotic pressure of a given quantity of substance is inversely proportional to the volume of the solution which contains it. Here, then, is a law in perfect analogy to Boyle's law for gases : the volume varies inversely as the pressure. Let us now consider the effect of temperature on the osmotic pressure, the volume of the solution remaining constant. As we have seen, the osmotic pressure increases with the temperature just as gas pressure does. To determine the exact amount of the increase, Pfeffer made special experiments, with the following results : CANE SUGAR. Temperature C. Temperature Abs. Osmotic Pressure. 14-2 287-2 51-0 32-0 305 54-4 (54-2) 15-5 288-5 52-1 36-0 309-0 56-7 (55'8) SODIUM TARTRATE. Temperature C. Temperature Abs. Osmotic Pressure. 13-3 286-3 143-2 36-6 . 309-6 156-4 (154-9) 13-3 286-3 90-8 37-3 310-3 98-3 (98'4) From these figures it is evident that there is a close proportionality between the absolute temperature of a given solution and its osmotic pressure. In each pair of experiments the osmotic pressure at the higher temperature has been calculated from the experimental value of the osmotic pressure at the lower temperature, on the assumption that the osmotic pressure is proportional to the absolute temperature, and the calculated value has been placed within brackets alongside the pressure actually measured. The difference between the observed and calculated values is not greater than the error of experiment. Here, then, we have another law exactly comparable to the law for gaseous substances : If the volume is kept the same, the pressure is proportional to the absolute temperature (cp. p. 27, law 3). 164 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. Combining these two laws for dilute solutions, we may now say that the product of the osmotic pressure and the volume is proportional to the absolute temperature, i.e. we may write the equation for substances in dilute solution as well as for gases, and it only remains now to find how the constant R is related to the corresponding constant for substances in the gaseous state. On p. 29 we calculated the value of this constant for a gram molecule of a gas, and we shall now proceed to evaluate it from Pfeffer's data for a gram molecule of sugar dissolved in water. For a one per cent solution of cane sugar at Pfeffer observed that the osmotic pressure was 49 '3 centimetres of mercury. This corresponds to a pressure of 49*3 x 13'59 gram centimetres (cp. p. 3). The gram- molecular weight of cane sugar is 342, and consequently the volume of a one per cent solution containing the gram-molecular weight is 34,200 cubic centimetres. The absolute temperature of the solution is 273, so that we have for the constant _ 49-3x1 3-69 x 84,200 _ K ~~ ~273~ 83,900, a value practically identical with the value obtained for the gas constant, which in the same units we found to be 84,700. So far, then, as pressure, temperature, and volume relations are concerned, the analogy between gases and substances in dilute solution is complete. The identity of the constant in the two cases shows that the osmotic pressure of a dissolved substance is numerically equal to the gaseous pressure which the substance would exert were it contained as a gas in the same volume as is occupied by the solution. In fact, if we imagine that the solvent is suddenly annihilated, we should have the osmotic pressure on the semipermeable membrane replaced by a gaseous pressure of equal magnitude. This similarity between gaseous and dissolved substances is of the utmost importance, for it enables us to transfer to dissolved substances conclusions arrived at from the consideration of the temperature, volume, and pressure relations of gases. For example, it at once enables us to determine the molecular weights of dissolved substances from simultaneous measurements of the temperature, volume, and osmotic pressure of the solution, just as the molecular weights of gases and vapours are determined from similar magnitudes. The accurate measurement of osmotic pressure is an experimental task of the utmost difficulty, and has been attempted by only one or two investigators, so that molecular weights can hardly be determined directly in this way. There are, however, other magnitudes, susceptible of easy and exact measurement, which are known to be proportional to the osmotic pressure, and these are now xvi OSMOTIC PRESSURE AND THE GAS LAWS 165 made use of for molecular-weight determinations, as will be shown in a subsequent chapter. In osmotic pressure we can recognise the cause of diffusion of substances in solution. Just as in gases we have movement from regions of higher to regions of lower pressure, so in solutions we have movement from regions of higher osmotic pressure to regions of lower osmotic pressure. Osmotic pressure, then, we take to be the driving force in solutions, and if we calculate its value, we find it to be very considerable. Thus the osmotic pressure of a normal solution is over 22 atmospheres (cp. p. 150), or 330 Ibs. per square inch. In spite of this high driving power, the process of diffusion in solution is, as we have seen, a very slow one. It has been calculated that the force necessary to drive a gram of dissolved urea through water at the rate of 1 cm. per second is equal to forty thousand tons weight. The resistance, then, which the water offers to the movement of the dissolved substance is enormous. This we must take to be due to the smallness of the dissolved particles. A substance in the state of fine dust may take many days to settle, even in a perfectly still atmosphere, while the same weight of substance in the compact state would fall to the ground in as many seconds. The driving force in the two cases is the same, namely, the gravitational attraction of the earth for the given substance, but the resistance which the air offers to the small particles is incomparably greater than that offered to the compact mass. It should be borne in mind that the osmotic pressure in a solution may be regarded as always present, whether a semipermeable membrane renders it visible or not. The osmotic pressure in the ordinary reagent bottles of the laboratory is of the dimensions of 50 atmospheres. This pressure is of course not borne by the walls of the bottle, nor is it apparent at the free surface of the liquid. Where the liquid comes in contact with the enclosing vessel, there we find a liquid surface, and a consideration of the magnitude of the forces at work in the phenomena of surface tension leads us to believe that the pressure at right angles to the free surface of a liquid, and directed towards the interior of the liquid, is measurable in hundreds and even thousands of atmospheres. Osmotic pressures, then, large as they are in ordinary solutions, are small compared to the surface pressures in liquids, and their existence is consequently not evident at the free surface of liquids. It is only when these surface pressures are got rid of that we can measure osmotic pressures directly. The liquid solvent can easily penetrate the semipermeable membrane, so at the semipermeable membrane there is not in the ordinary sense a liquid surface, and consequently there is no surface pressure of the ordinary type. This continuity of the liquid through the semipermeable partition gives us, therefore, the opportunity of determining differences of internal pressure in the solution and the solvent. Various hypotheses have been put forward to explain the nature 166 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. of osmotic pressure, but none of them can be accounted satisfactory. They are based upon more or less probable suppositions as to the nature of the movement of the dissolved molecules, the degree of attraction between them and the particles of the solvent, the capillary phenomena in the "pores" of the semipermeable partition, and the like. Our ignorance of such matters is, however, so great that no profitable conclusions have hitherto been arrived at. One thing may be said about osmotic pressure which is independent of any supposition as to its nature, and has indeed been already indicated in what has preceded. The osmotic pressure of any solution is in- dependent of the nature of the semipermeable membrane. Pfeffer tested several mem- . =j branes besides mem- branes of copper ferrocyanide. For ex- Soluent p| Solution ^ Solvent ample, membranes of Prussian blue or of tannate of gelatine can be deposited in a por- ous cell in the manner described for copper ferrocyanide, and have a similar action with regard to water and substances held by it in solution. Pfefier found that in general the osmotic pressure of any one solution varied with the nature of the membrane he employed, but this was in reality due to the membranes not being perfectly impermeable to the dissolved substance, so that the maximum pressures recorded did not correspond to the actual osmotic pressure, but fell short of it in a degree dependent on the extent to which the membrane leaked. A theoretical proof may be given that the nature of the membrane does not influence the value of the osmotic pressure provided that it is perfectly impermeable to the dissolved substance. Suppose two membranes to exist, one of which, A, generates with a given solution and solvent a higher osmotic pressure than the other membrane B. If the pressure in the vessel containing the solution is less than the osmotic pressure, liquid will flow through the membrane from solvent to solution ; if it is greater than the osmotic pressure, liquid will flow from the solution to the solvent. Let the solution, solvent, and diaphragms be combined into one working system, as in the figure. If the solution is originally at a greater pressure than corresponds to the value of the osmotic pressure generated by B, solvent will flow out through the diaphragm, and the pressure inside will diminish and tend to reach this value. But as soon as the pressure diminishes to a value less than the osmotic pressure generated by A, solvent will flow through A into the cell. The pressure inside will still be too great for B, and solvent will therefore continue to flow out. There will thus be a continuous flow of solvent xvi OSMOTIC PRESSURE AND THE GAS LAWS 167 through the cell from left to right, and as the conditions are not changed by this transference, the flow might go on indefinitely and the current made use of to perform work, i.e. in this way we could obtain a perpetual motion (cp. Chap. XXVII. ). Since this is impossible, our assumption that the osmotic pressures generated by the two diaphragms are different must be incorrect, and we are forced to conclude that the osmotic pressure of a solution is independent of the diaphragm used in measuring it, provided that the diaphragm is completely impermeable to the dissolved substance. Although the direct measurement of osmotic pressure is surrounded by so many difficulties, it is often possible to tell whether a solution has an osmotic pressure greater than, equal to, or less than another solution of the same or a different substance. This may be done either with the aid of a precipitation membrane such as Traube employed, or by means of a natural semipermeable membrane. When a precipita- tion membrane is formed at the end of a tube as described above, the two solutions which form the precipitate have in general different osmotic pressures. But the solvent water moves through the membrane from the solution with less to the solution with greater osmotic pressure. If the solution outside the cell has the greater concentration it gains water, becomes more dilute in the immediate neighbourhood of the membrane, and rises in the external liquid owing to its lesser specific gravity. This is easily rendered visible by means of a Topler apparatus, which detects very small differences in the refractive power of liquids. If the external solution has a smaller osmotic pressure than the internal solution, water will be transferred inwards through the membrane, and the external solution will become more concentrated in the neighbourhood of the membrane, the change betraying itself by differences in density and refractive power as before. If, finally, the two solutions have equal osmotic pressures, no trans- ference of water takes place, and there is consequently no change in the density or refraction of the solutions. We have here, then, a method for determining when two membrane - forming solutions are isomotic, or isotonic, a term sometimes applied to solutions having the same osmotic pressure. A method making use of natural semipermeable membranes is the following. It is known that the protoplasm of vegetable cells has a sort of skin which serves to a certain extent as a semipermeable membrane, for it keeps dissolved substances in the cell sap from passing outwards, while it admits of the free passage of water. If the proto- plasm of the cell then is brought into contact with pure water or a solution of smaller osmotic pressure than the cell contents, water will pass through the skin inwards to the protoplasm. If, on the other hand, the cell content has a smaller osmotic pressure than the solution with which the protoplasm is brought into contact, water will pass outwards through the skin. Should the external solution finally have 168 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP, xvi the same osmotic pressure as the solution within the protoplasmic skin, there will be no transference of water between the cell and the external solution. In the case of some cells, the passage of water to or from the protoplasm is easily visible on account of the apparent increase or diminution of the volume. Thus when a suitable vegetable cell is brought into contact with a solution of higher osmotic pressure than that of the solution within the protoplasm, the granular or coloured cell contents are seen to shrink away from the cell wall owing to the loss of water and contraction in volume which they experience. By diluting the external solution, it is easy to find a concentration which just ceases to produce this contraction ; then the cell contents are isotonic with this solution. In the same way, a solution of another substance may be found which is isotonic with the same cell. These two solutions are then isotonic with each other. This last statement is proved by experiments which show that two solutions which have been found to be isotonic with respect to one kind of cell are also isotonic with regard to other kinds of cell. In this again we have an indication that the nature of the membrane has no influence on the osmotic pressure, if only it is impermeable to the dissolved substance. In what follows it will practically always be assumed that the osmotic pressure of a solution is strictly proportional to its concentra- tion. This is by no means always an exact relation, and really holds good only for somewhat dilute solutions. The reason is of course not far to seek. As has been already stated above, many of the ordinary laboratory solutions have osmotic pressures of nearly 100 atmospheres. Now, concentration of a solution corresponds to absolute density in the case of a gas. Boyle's law for gases states that the pressure of a gas is proportional to its absolute density, or inversely proportional to its volume, which is the same thing. But this by no means necessarily holds good for pressures as high as 100 atmospheres. We cannot expect, then, that the corresponding law for solutions that the osmotic pressure is proportional to the concentration will be exactly true at similar high osmotic pressures. In general, we can expect no exact proportionality between osmotic pressure and concentration at strengths above normal, and we very often find that much more dilute solu- tions have to be considered in order to get the simple laws to apply in strictness. An account of the preparation of osmotic cells for exact measurements will be found in the following paper : R. H. ADIE " On the Osmotic Pressure of Salts in Solution," Journal of the Chemical Society, 1'xix. p. 344. CHAPTER XVII DEDUCTIONS FROM THE GAS LAWS FOR DILUTE SOLUTIONS IN discussing the evaporation and solidification of solutions in Chapters VIII. and IX., we have met with empirical laws which find a theoretical basis in the conception of osmotic pressure. Such are the law that the vapour pressure of a solution is less than the vapour pressure of the pure solvent by an amount proportional to the strength of the solution (p. 79); that the boiling point of a solution is higher than the boiling point of the solvent by an amount proportional to the strength of the solution ; and that the freezing point of a solution is lower than the freez- ing point of the solvent by an amount proportional to the strength of the solu- tion. The more strict thermodynamical deduction of these relations from the gas laws for dilute solutions is given in Chapter XXVII. , but in this chapter we can show, at least approximately, the relations of the various magnitudes. In the first place, we shall consider the connection between osmotic pressure and the relative lowering of the vapour pressure. In the figure (Fig. 27) B represents a porous bulb with a semi-permeable mem- brane deposited within the wall (p. 1 6 1 ). It is filled with a known solution and im- mersed in the pure solvent. The solvent will enter the bulb until the solution in the tube rises to a height where the difference of level of the liquids inside and outside the bulb causes a pressure equal to the osmotic pressure of the liquid. Let this equilibrium be reached in an atmosphere which consists only of the vapour of the solvent, and let the difference in level be represented FIG. 27. 170 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. by h. The temperature throughout is supposed to remain at the constant value T in the absolute scale. In the first place, we note that the pressure of vapour at the level I of the surface of the solution must be the same inside and outside the tube. If it were not, being, let us assume, greater inside than outside, vapour would pass from the region of higher to the region of lower pressure. This would lower the pressure of vapour immediately over the surface of the solution, and some of the solvent would therefore evaporate in order that the pressure would regain its former value, which is the value in equilibrium with the solution. The original value outside the tube is the value corresponding to the vapour pressure of the pure solvent, so that the accession of vapour from the inner tube would increase the pressure of vapour above its equilibrium value, and consequently some of the vapour would condense at the surface of the pure solvent. The process would be then, in short, that some of the liquid inside the tube would distil over and condense to liquid outside the tube. This would evidently increase the concentra- tion of the solution within the bulb, and so we should no longer have osmotic equilibrium between the solution and the solvent. To restore the osmotic equilibrium, some of the solvent would pass inwards through the membrane and dilute the solution to its original concen- tration. The liquid in the tube would then regain its original height and vapour pressure, and the whole process of distillation would recommence. On the assumption, then, that the pressure of vapour at I is greater inside the pressure tube than outside at the same level, we have a continuous circulation of solvent from the solution to the solvent as vapour, and from the solvent to the solution as liquid through the semipermeable membrane. The current thus generated could theoretically be used to perform work, and we should there- fore have a form of the perpetual motion, which is impossible. Similarly, if the pressure at the level I is supposed to be greater outside the tube than inside, we should have a continuous circulation of solvent in the opposite direction. The conclusion we must adopt, then, is that the pressure within and without the tube at the same level I is the same. If now / is the vapour pressure of the solvent at the given temperature, and/ 7 that of the solution, the difference/-/' is evidently the difference of pressure between the levels at the two liquid surfaces, i.e. at the top and bottom of the height h. This difference in pressure is due to the weight of the column of vapour between the two levels on a surface of one square centimetre, and is equal to the product of the height and absolute density of the column of vapour, i.e. to hd, if d is the density expressed in grams per cubic centimetre. Let us now consider a gram-molecular weight of this vapour. For it we have xvii DEDUCTIONS FROM THE GAS LAWS 171 where R is the ordinary gas constant. But the density d is the weight divided by the volume, i.e. d = M/v, where M is the molecular weight of the solvent in the gaseous state. The pressure of the gaseous solvent is /, so that we obtain v = fiT/f, and We have seen above that f-f = 1id, or, substituting the value of d here found,/-/ 7 = h - -^ whence HI = Now / -/' is the lowering of the vapour pressure of the solvent, and f-f ~ is therefore the proportional or relative lowering, with which we are alone concerned. We have thus obtained an expression for the relative lowering of the vapour pressure in terms of the " osmotic height " h, and constants for the gaseous solvent. It is now an easy matter to express h in terms of the osmotic pressure of the solution, and another constant for the solvent. The osmotic pressure, i.e. the excess of pressure inside the cell over that outside, is equal to the height of the column h into the absolute density of the liquid. This we may denote by s, which is the absolute density of the pure solvent, for if the solution considered is very dilute, its density will not greatly differ from the density of the solvent. We have, then, if p represents the osmotic pressure, T> p = As, or h = ~ . Substituting this value of h in the previous equation, s we obtain Here the relative lowering is expressed in terms of the osmotic pressure of the solution and magnitudes referring to the solvent, which for constant temperature are constant. It appears, then, that the relative lowering of the vapour pressure of a liquid by the solution in it of some foreign substance is at any one temperature proportional to the osmotic pressure of the solution and independent of the nature of the dissolved substance. By making use of the gas laws for solutions we can eliminate temperature from the above expression, and put it in a simpler form. If we express the concentration of the solution in the form that n gram molecules of the solute are contained in W grams of the solvent, then we have for n gram molecules the equation pv = nRT. 172 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. - Now the volume in which these n gram molecules are contained is equal to /F", the weight of solvent, divided by s, the density of the solvent, i.e. v = PT/s, so that nsRT Substituting this value in the former equation, we obtain M = _ f '' W ' sRT The relative lowering is now expressed in terms of the concentration of the solution and the molecular weight of the solvent in the gaseous state, which is constant. The temperature has, according to this result, no influence on the value of the relative lowering, a conclusion which is in accordance with the experimental data. For a given weight of any one solvent, the relative lowering is proportional to the number of molecules of dissolved substance in solution. Consequently, if we find that for certain quantities of different substances, dissolved in the same weight of the same solvent, the relative lowering of the vapour pressure of the solvent is the same, we conclude that these quantities contain the same number of dissolved molecules, and thus we obtain a method for determining the molecular weights of substances in solution. The molecular weight of a dissolved substance might also be calculated in terms of the relative lowering of the vapour pressure by finding the numerical value of p from equation (1), and introducing the osmotic pressure thus obtained into the gas equation. For a given amount of the same substance dissolved in the same weight of different solvents, we find from equation (2) that the relative lowering is proportional to the molecular weight of the solvent in the gaseous state, provided that the molecular weight of the dis- solved substance remains the same in the different solvents. The expression for the relative lowering receives a still simpler form if, instead of the actual weight of the solvent, we introduce the number of gram molecules of the solvent. The weight of solvent may be expressed as the product of the number of molecules into the gram -molecular weight of the substance in the gaseous state, i.e. as MN, if N represents the number of gram molecules of the solvent as gas. We have therefore /-/. v ~T~~MN M> The relative lowering is here given as the ratio of the number of xvii DEDUCTIONS FROM THE GAS LAWS 173 dissolved molecules to the number of molecules which the solvent would produce if it were converted into vapour. It must be emphasised that the number of molecules N in the above equation does not denote the number of liquid molecules in the solvent, but only the number of gaseous molecules derivable from the liquid. This caution is necessary, because it has frequently been supposed that the equation enables us to determine the molecular weight of the liquid solvent, which is not the case. As we shall see, methods exist for determining the molecular weights of liquids, but this is not one of them. In deriving the above equations, we have assumed that the specific gravities of the solutions considered are the same as the specific gravities of the respective solvents. This assumption only holds good for very dilute solutions, but it gives serviceable approxi- mations in practical work, as the following numerical examples will show : A solution of 2^L g. of ethyl benzoate in 100 g. of benzene showed a relative lowering of vapour pressure equal to OM3123, i.e. if the vapour pressure of pure benzene is 1, the vapour pressure of the solu- tion is 1-0*0123. The temperature at which the determination was made was 80 C., and at this temperature the density of benzene is 0'812. The molecular weight of gaseous benzene is 78. If we sub- stitute these values in equation (1), we obtain 78 ' 0123 - 0-812 x 84,700 x (273 + 80)' whence^? = 3830 g. per square centimetre, or over 3*7 atmospheres. If we wish to calculate the molecular weight of ethyl benzoate from these data by means of equation (2), we get by substitution 0-0123 = whence n = 0"0158, i.e. in 2'47 g. of benzoate of ethyl there are 0'0158 gram molecules. There is therefore one gram molecule of dissolved ethyl benzoate in 2-47/0'0158 = 156 g., or ,156 is the molecular weight of ethyl benzoate when it is dissolved in benzene. It must be remembered that this number is only approximate, but still it is sufficient to show that the molecular weight of the dissolved substance is practically that of the gaseous substance, which is 150. Equation (3) leads to the same result, for in order to get the number of molecules J\T we have to divide the number of grams taken, viz. 100, by the molecular weight, viz. 78. The determination of the lowering of vapour pressure is somewhat too difficult and tedious to be of much use in fixing the molecular weights of dissolved substances, and it is therefore preferable to ascertain in its stead the elevation of the boiling point, which 174 INTRODUCTION TO PHYSICAL CHEMISTEY CHAP. is for dilute solutions nearly proportional to it, and susceptible of easy and rapid determination. As a solution of a non-volatile substance at a given temperature has a lower vapour pressure than the pure solvent, it is evi- dent that in order that the solution and solvent may have the same vapour pres- sure, e.g. equal to the standard atmospheric pres- sure, it is necessary to heat the solution to a higher temperature than the sol- vent. The boiling point of the solution is therefore always higher than the boil- ing point of the solvent. A consideration of the accom- T / Temperature panying diagram (Fig. 28) will show that for small changes, the lowering of the vapour pressure and the elevation of the boiling point are nearly proportional. In the figure the three curved lines represent the vapour pressure curves of a pure solvent and of two solutions of different concentrations. At the temperature t, the intercept aa^ represents the actual lowering of the vapour pressure of the solution 1, and the ratio aa^:ta the relative lowering. For the temperature r we have the corresponding magnitudes aa x and aa x : ra. Since for any one solution the relative lowering is independent of the temperature, we have aa : : ta = ao^ : ra. Similarly for the second solution aa 2 : ta aa 2 : ra, so that a^ : aa 2 = ac^ : aa 2 , or aa^ : a^ = aa x : a^. If the curves were straight lines, they would in virtue of these proportions meet in one point, and the intercepts of any straight line cutting them would always bear the same proportions to each other. Consider the line ab 2 parallel to the temperature axis. This is a line of constant vapour pressure, and cuts the curves at points corresponding to the temperatures ^ and t 2 . The intercepts a^ and ab 2 represent the elevations of the boiling point, if the boiling point at the atmospheric pressure is t. Now if the curves were straight lines we should have aa^ : aa 2 = ab l : a& 2 , i.e. the elevation of the boiling point would be proportional to the lowering of the vapour pressure. But for small elevations, that is, for dilute solutions, the curves may be treated as straight lines, so that we have a pro- portionality between the elevation of the boiling point and the lowering of the vapour pressure, and thus between the elevation and the osmotic pressure. Another easily determinable magnitude which is proportional to the osmotic pressure is the depression of the freezing point in dilute XVII DEDUCTIONS FROM THE GAS LAWS 175 solutions. By means of a diagram similar to the above we can show that this depression is approximately proportional to the lowering of the vapour pressure, and thus indirectly establish the connection with osmotic pressure. In the figure (Fig. 29) aQ represents the vapour pressure curve of the liquid solvent, say water, and ftO' that of the solid solvent, ice. The tem- perature t, where these two curves intersect, is the freezing point of the pure solvent (cp. p. 98). The curves 1 and 2 re- present as before the vapour - pressure curves of two solutions. These cut the ice curve at two points, l l and & 2 , the cor- responding temperatures ^ and t 2 representing the freezing points of the two solutions, i.e. the temperatures at which ice and the solutions are in equilibrium, and at which, therefore, they have the same vapour pressure. Now as before we may treat the curves 0, 1, and 2 as three straight lines meeting in a point if we only consider small intervals. We have then aa l : aa 2 = ab l : a\ = tt^ : tt 2 . But U^ is the depression of the freezing point for the solution 1, and tt z the depression for the solution 2. Consequently, we have the depressions of the freezing point proportional to the lowerings of the vapour pressure, aa^ and aa 2 . The freezing-point depressions are thus proportional in dilute solutions to the osmotic pressures of the solutions, and can therefore be sub- stituted for the latter in ascertaining molecular weights. It has now been shown on approximate assumptions that Temperature FIG. 29. and Elevation of boiling point = cP, Depression of freezing point = c'P, where P is the osmotic pressure, and c and c' constants. These constants remain in each case the same for a given sol vent,, and are valid for all dissolved substances. They correspond in their nature to the constant factor on the right-hand side of equation (1) in this chapter. They depend on the properties of the solvent, and their derivation from these properties will be shown in Chapter XXVII. CHAPTEE XVIII METHODS OF MOLECULAR WEIGHT DETERMINATION 1. Gaseous Substances Vapour Density WHEN we can obtain a substance in the gaseous state, the determination of its molecular weight resolves itself, as we have seen in Chapter II., into ascertaining what weight of the vapour in grams will occupy 22'4 litres at and 760 mm., or, if we deal with smaller quantities, what weight in milligrams will occupy 2 2 -4 cc. In general, we cannot weigh these volumes of the vapour under the standard conditions. In the case of water, for example, it is impossible to get a pressure of 760 mm. of vapour at 0, the vapour pressure of water at that temperature being only a few millimetres of mercury. We can, however, make the actual determination under any conditions we please, and then reduce to the standard conditions by means of the gas laws. The practical problem to be solved, then, in vapour - density determinations for the purpose of finding molecular weights is to measure the weight, volume, temperature, and pressure of a given amount of substance in the gaseous state. This may be done in various ways, as the following short description of the principal methods will show. Dumas's Method. In this method the weight of a known volume of gas or vapour is determined, a globe of known capacity being filled with the gas at atmospheric pressure and known temperature, sealed off, cooled, and weighed. The volume of the globe is ascertained by weighing it when empty and when filled with water. Its weight when filled with air minus the weight of air contained in it (which can be calculated from the volume and the known density of air) gives the weight of the empty globe. This, when subtracted from the weight of the globe filled with the gas under investigation, gives the weight of that gas which fills the given volume at the given pressure and temperature. The method, when applied to the vapours of substances liquid at the ordinary temperature, usually assumes the CHAP, xvm DETERMINATION OF MOLECULAR WEIGHTS 177 following form: The bulb is of 50 to 100 cc. capacity, as a rule, and has the shape shown in the figure (Fig. 30). Several grams of the liquid substance are placed in the bulb, which is then immersed in a bath of constant temperature about 20 higher than the boiling point of the liquid. The liquid in the bulb boils and expels the air with which the bulb was originally filled. After the vapour ceases to escape from the narrow neck of the bulb, this is sealed off near the end with a small blowpipe flame. There is then in the bulb a known volume of vapour at a known tem- perature and pressure, so that all that has now to be done is to ascertain the weight of the vapour. This method is somewhat troublesome, and is usually applied to substances which are only vola- tile with difficulty. Hofmann's Method. Here instead of taking a known volume of vapour and measuring its weight, we take a known weight of substance and measure the volume which it occupies as vapour. The apparatus employed for the purpose is shown in Fig. 31. It consists of an inner tube about a yard long and half an inch in bore, which is graduated in cubic centimetres. This is filled with mercury, and inverted in a mercury trough, so that at the top of the tube a Toricellian vacuum is formed. Outside this tube is a wider tube which acts as a vapour jacket, the vapour of a boiling liquid passing in through the narrow tube d, and issuing, together with the condensed liquid, through the side tube just above the mercury. A weighed quantity of the liquid, the density of whose vapour is to be determined, is introduced into the inner tube in a bulb or very small stoppered bottle made for the purpose, and containing only about a tenth of a cubic centimetre. The inner tube is then heated by the vapour from a liquid boiling in a suitable vessel attached to d, the boiling being continued until the vapour issues freely from the lower tube and the level of the mercury in the inner tube no longer alters. The weighed quantity of substance has now been converted into vapour, and occupies the volume above the mercury at N 178 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. the top of the graduated tube, at the boiling point of the liquid used for heating, and under a pressure equal to the atmospheric pressure minus the height of the column of mercury ab. The external liquid used for heating the inner tube to a constant temperature is usually water. This evidently will vaporise any liquid with a boiling point under 100, but in reality it can be used for liquids with much higher boiling points, for the vaporisation takes place in the inner tube under reduced pressure. Thus a liquid such as aniline, with 'a boiling point of 180, is easily vaporised in the Hofmann apparatus by boiling water, if the quantity taken is such that the vapour only occu- pies a small proportion of the total volume of the graduated tube. Victor Meyer's Method. This method is akin to Hofmann's inas- much as a weighed amount of liquid is vaporised, but it differs from the other U| // m methods on account of the I II gas whose volume, tempera- ^ \j ture, and pressure are actually determined not being the vapour under in- vestigation, but an equal volume of air which has been displaced by the vapour. On account of the simplicity of the apparatus and the convenience in manipulation, this method is almost universally adopted in practical work where great accuracy is not required. The apparatus consists of a cylindrical vessel of about 1 00 cc. capacity with a long narrow neck, having a top piece furnished with two side tubes. One of these side tubes acts as a delivery tube, and is connected by means of a piece of thick-walled, narrow-bored rubber tubing with the gas-measuring tube g, which at the beginning of the experiment is filled with water. Through the other side tube a glass rod projects into the neck, connection being made by means of a short piece of rubber tube, permitting a good deal of play. A weighed quantity of the liquid whose vapour density is to be determined is contained in the small bulb s, which is held in place by the rod t. The principal tube is closed at the top by a cork, and contains a little asbestos at the bottom, in order to protect the glass from the fall of the bulb. xvni DETERMINATION OF MOLECULAR WEIGHTS 179 The wide cylindrical portion and a large part of the neck are heated by means of a liquid boiling in an external cylinder with a bulb-shaped end. To perform an experiment, the level of water in the measuring tube is adjusted to zero by moving the reservoir n, the bulb is put into position, and the heating started while the tube is still open at the top. When the temperature of the whole apparatus has become steady, the tube is corked, and the level of the water in the measuring tube is observed for some minutes. If it has altered slightly, it is re- adjusted to zero, and the bulb is let drop by drawing back the rod t for a moment. The liquid at once begins to vaporise, and air passes over into the measuring tube. To keep the gases in the apparatus at the atmospheric pressure, and thus prevent leakage, the water in the reservoir is kept at the same level as the water in the measuring tube by continually lowering the reservoir. The vaporisation is complete in less than a minute, as a rule, and if after two or three minutes the level of water in the measuring tube does not change, the volume is read off, the barometric pressure and the temperature on a thermo- meter near the measuring tube being noted at the same time. The distribution of temperature throughout the whole apparatus is the same before and after the volatilisation of the substance, and therefore the volume of air collected is the same as the volume of vapour formed, after reduction to the temperature and pressure at which the collected air is measured. The temperature is the temperature of the water, the pressure that of the atmosphere minus the vapour pressure of water ; for the vapour pressure of the water over which the gas is collected and the pressure of the gas itself together make up the total pressure, which is equal to that registered by the barometer. In the simplest form of apparatus the gas is collected in a graduated tube over water contained in a shallow dish, the side tube being in this case long, of narrow bore, and bent to the appropriate form. It will be seen that this method of vapour-density determination does not involve a knowledge of the temperature at which the vaporisation takes place, for since all gases are equally affected by changes of temperature, the contraction in volume of the hot air on cooling is the same as the contraction which the vapour itself would experience. We thus, instead of measuring the volume at the temperature of vaporisation, measure the reduced volume at the atmospheric temperature. The liquid used for heating should have in general a boiling point at least as high as that of the experimental substance, and the ebullition should be so brisk as to make the vapour condense two-thirds of the way up the outer tube. The mode of calculation of a molecular weight from the observed data for the vapour density may be seen from the following example. The bulb contained 0'1008 g. of chloroform, boiling point 61, and was 180 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. dropped into a tube heated by the vapour from boiling water. The air collected measured 22*0 cc., the temperature being 16 '5, and the height of the barometer 707 mm. Now the vapour pressure of water at 16*5 is 14 mm., so that the actual pressure of the gas was 707-14 = 693 mm. We have only now to solve the following proportion : If 100'8 mg. of chloroform vapour occupies 22*0 cc. at 16'5 and 693 mm., what number of milligrams will occupy 22*4 cc. at and 760 mm. ? i.e. to evaluate the expression 100-8 x 22-4 x (273 + 16'5) x 760 22'0 x 273 x 693 The result we obtain is 119, the actual molecular weight of chloroform as calculated from its formula being a little over 118. It should be borne in mind that in the determination of molecular weights from vapour densities only approximate results are obtained, for we assume that the vapours obey the simple gas laws exactly, which is by no means the case when the vapour is at a temperature only a little removed from the boiling point of the liquid from which it is produced. As, however, the vapour density is used in conjunction with the results of analysis in fixing the accurate value of the molecular weight, an error amounting to 5 or even 10 per cent of the value is unimportant, the number obtained from the vapour density merely determining the choice between the simplest formula weight and a multiple of it. The molecular weight of chloroform in the above example can from the formula be only 118 or a multiple of 118; and the vapour-density estimation shows conclusively that the simplest formula is here the molecular formula. 2. Dissolved Substances Osmotic Pressure Assuming the complete similarity in pressure, temperature, and volume relations of substances in dilute solution and of gases, we c,an evidently determine the molecular weight of a dissolved substance by simultaneous observations of its weight, temperature, volume, and osmotic pressure. Pfeffer found, for example, that a one per cent solution of cane sugar at 32 had an osmotic pressure of 544 mm. One gram, or 1000 mg., of cane sugar here occupied approximately 100 cc. We have then as before the proportion: If 1000 mg. of sugar occupy 100 cc. at 32 and 544 mm., what number of milligrams will occupy 22*4 cc. at and 760 mm. ? The answer is 1000 x 22-4 x (273 + 30) x 760 100 x 273 x 544 The molecular weight of cane sugar calculated from the formula C 12 H 22 O n is 342. It is evident, then, that the molecular formula of xvin DETERMINATION OF MOLECULAE WEIGHTS 181 cane sugar in aqueous solution is the simplest that will express the results of analysis. Were it not for the extreme difficulty of obtaining a membrane perfectly impermeable to the dissolved substance, this method would be the most suitable and the most accurate for determining molecular weights of substances in very dilute solution. 3. Dissolved Substances Lowering of Vapour Pressure An example of how a molecular weight of a dissolved substance may be estimated by this method has been given in the preceding chapter (p. 173). The method has little practical importance, and is scarcely ever employed. 4. Dissolved Substances Elevation of Boiling Point This is a practical method for determining the molecular weights of substances in solution, and is coming more and more into general use. An essential condition for its success is that the dissolved substance should not itself give off an appreciable amount of vapour at the boiling point of the solvent. It is only applicable, therefore, to substances of comparatively high boiling point, say over 200, and cannot be employed with success for liquids such as alcohol, benzene, or water. Two forms of apparatus may be described, which differ principally in the mode of heating. Beckmann's Apparatus. In this form of apparatus the solution is raised to its boiling point by the indirect heat from a burner. Now in ascertaining the boiling point of a liquid, it is customary to place the thermometer, not in the boiling liquid itself, but in the vapour coming from it. In this way superheating is avoided. The liquid itself may be at a temperature considerably above its true boiling point, but a thermometer placed in the vapour will show very little sign of this superheating. The plan, however, cannot be adopted in ascertaining the boiling point of a solution. The vapour which comes from the solution of a non- volatile substance is the vapour of the solvent, a part of which condenses to liquid on the bulb of the thermometer. Now the temperature at which the condensed and vaporous solvents are in equilibrium on the bulb of the thermometer is the boiling point of the solvent and not that of the solution, so that the temperature registered by a thermometer placed in the vapour from a boiling solution is the boiling point of the solvent, slightly raised perhaps by radiation from the hotter solution. It is necessary then to immerse the bulb of the thermometer directly in the boiling solution if the boiling point of the latter is to be determined, i.e. the temperature at which the solution and the vapour of the solvent are in equilibrium. When, therefore, the source of heat is external 182 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. and necessarily of a higher temperature than the boiling point which has to be measured, precautions of the most rigorous kind have to be adopted in order to prevent superheating of the liquid whose boiling point is in question. In Beckmann's apparatus this is done as follows. The boiling tube A (Fig. 33) is about 2 '5 cm. in diameter, provided with a stout platinum wire fused through its end, and filled for about 4 cm. with glass beads. The platinum wire is for the purpose of conducting the external heat into the solution so as to get the bubbles of vapour to form chiefly at one place and pre- vent super - heating. The glass beads are used in order to split up the large bubbles of vapour into smaller bubbles, so that more intimate mixture of the solution and the vapour of the solvent may be secured. The thermometer, which is divided into hundredths of a degree, and may be read to thousandths, is placed so that its bulb dips partly into the glass beads. A device for rendering such a thermometer available for solvents of widely different boiling points is described on p. 186. This inner vessel is surrounded by a vapour jacket, charged with about 20 cc. of the solvent, which is kept boiling during the experi- ment. This jacket reduces radiation towards the exterior to a minimum, and consequently only a com- paratively small amount of heat has to be afforded to the solution in order to make it boil, whereby the risk of superheating is greatly reduced. Both vessels are provided with reflux condensers, which may either be air condensers, as in the figure, or water condensers of the ordinary type, according to the volatility of the solvent. The small asbestos heating chamber C has two asbestos rings, h and h', which protect the boiling vessel from the direct action of the flame of the burner, and two asbestos funnels, ss, which carry off the products of combustion. The heat of the burners reaches the liquid in the vapour jacket through the ring of wire gauze visible in section at d as a dotted line. FIG. xvni DETERMINATION OF MOLECULAR WEIGHTS 183 To perform the experiment, a weighed quantity of the solvent (15 or 20 g.) is brought into the boiling tube, the heating begun, and the thermometer read off when it has become steady, which may not be before an hour has elapsed. The condensed solvent should only drop back very slowly from the condenser, so that the boiling must not be hastened by using a large flame. The condenser K is then removed, and a weighed quantity of the experimental substance added, best in the form of a pastille if a solid, or from a specially shaped pipette if a liquid. The boiling point will now be found to rise, and the thermometer will after a short time again become stationary. The difference between the first and second readings of the thermometer is the elevation of the boiling point. Another weighed quantity of the substance may now be added, and the temperature of equilibrium again read off. This will give a second value for the molecular weight. Landsberger's Apparatus. Since the boiling point of a solution of a non-volatile substance is the temperature at which the solution is in equilibrium with the vapour of the solvent, we can bring a solution to its boiling point by continually passing into it a stream of vapour from the boiling solvent. As long as the solution is under its boiling point, some of the vapour will condense, and its latent heat of condensation will go to heat the solution until finally the boiling point is reached, when the vapour will pass through the solution without further condensation if no heat is lost to the exterior. Here there is little risk of superheating, since the vapour which heats the solution is originally at a lower temperature than the solution itself ; so that if we surround the solution with a jacket of the vaporous solvent, we have all the conditions for real equilibrium, at least so far as the determination of molecular weights is concerned. Landsberger's apparatus secures these conditions in a very simple manner, and a slight modification of it is shown in Fig. 34. The apparatus consists of a flask F, a bulbed inner tube N, which contains the solution, and a wider tube E, which is connected with a Liebig's condenser C. The vapour is generated in F (which contains the boiling solvent), passes through the solution in N, from which it issues through the hole H, to form a vapour jacket between the two tubes, and finally passes into the condenser. The lower end of the delivery tube R, where the vapour passes into the solution, is perforated with a rose of small holes, so that the vapour is well distributed through the liquid. The bulb prevents portions of the liquid being projected through H if the boiling is vigorous. The boiling point of the solvent is first determined by placing enough of the pure solvent in N to ensure that the bulb of the thermometer is just covered by the liquid when equilibrium has been attained. This quantity usually amounts to from 5 to 7 cc. The parts of the apparatus are then put together, and the boiling of the 184 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. solvent in F begun. To ensure regular ebullition, it is necessary to place in F a few fragments of porous tile, which must be renewed every time the boiling is interrupted. If the ebullition is brisk, the vapour heats the liquid in N to the boiling point in the course of a few minutes, as is evidenced by the reading of the thermometer rapidly becoming constant. The boiling is now interrupted, the tube emptied, and the whole process repeated, with the addition of a weighed quantity of FIG. 34. the substance under investigation to 5-7 cc. of solvent in N. The difference between the temperature now observed and the former temperature is the elevation of the boiling point, and it only remains to determine the weight of solvent employed. This is done by detaching the inner tube N, with thermometer and delivery tube, and weighing to centigrams. If from this weight we subtract the weight of the substance taken, and the tare of the tubes, etc., we obtain the weight of the solvent present when the temperature of equilibrium was reached. xviii DETERMINATION OF MOLECULAR WEIGHTS 185 If great accuracy is not desired, several successive determinations with the same quantity of substance may be made by replacing the tube with its charge and continuing the passage of vapour, interrupting the boiling from time to time to ascertain the amount of solvent present at the moment of reading the temperature. In this case, instead of determining the weight of solution, it is more convenient to read off its volume in cubic centimetres by having the tube N appropriately graduated, as shown in the figure, and removing the thermometer and delivery tube at each interruption in order to read the volume. The successive determinations are less accurate on account of the boiling points being observed under somewhat different con- ditions, the pressure of the column of solution increasing, for example, as the solvent condenses in N. For ordinary rough laboratory work a thermometer graduated into fifths of a degree is sufficiently accurate. The calculation of the molecular weight is carried out as follows. For each solvent we have a constant, which is the elevation produced if a gram-molecular weight of any substance were dissolved in a gram of the solvent. Of course such an elevation is purely fictitious as it stands, but it has a real physical meaning if we take it to be a thousand times the elevation which would be produced if a gram-molecular weight of the substance were dissolved in 1000 g. of the solvent. The hundredth part of this constant, i.e. the elevation caused by dissolving 1 g. molecular weight in 100 g. of solvent, is often spoken of as the molecular elevation. For the solvents ordinarily employed the constants are as follows : Solvent. fc V Alcohol 1150 1560 Ether 2110 3030 Water 520 540 Acetone 1670 2220 Chloroform 3660 2600 Benzene 2670 3280 The constants in the first column refer to 1 g. of solvent, the constants in the second column to 1 cc. of solvent at its own boiling point, which are useful if we measure the volume of the solution instead of ascertain- ing its weight. In the calculation we assume exact proportionality between the concentration of the solution and the elevation of the boiling point. We thus obtain for the molecular weight the expression M S - k M= L A' where A is the elevation, s the weight of substance, and L the weight of solvent, both expressed in grams, or 186 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. ,. s k' where V is the volume of solution in cubic centimetres. As an example of the calculation we may take the elevation produced by camphor in acetone. An elevation of 1'09 was produced by '6 7 4 g. camphor dissolved in 6 -81 g. acetone. We have, therefore, 0-674 x 1670 M - 6-81 x 1-09-= 1B1 - The molecular weight of camphor, according to the formula C 10 H 16 0, is 152. An estimation by volume resulted as follows. An elevation of 1-47 was found for 8'1 cc. of an acetone solution containing 0'829 g. camphor. This gives 0-829x2220 8-1x1-47 The molecular weight of camphor in acetone solution is thus in accordance with the simplest formula that expresses its composition. 5. Dissolved Substances Depression of Freezing Point EaouWs Method The apparatus chiefly used for determining molecular weights by the freezing-point, or cryoscopic, method is that devised by Beckmann, and figured in the accompanying illustration (Fig. 35). It consists of a stout test-tube A, provided with a side tube, and sunk into a wider test-tube B, so as to be surrounded by an air space. The whole is fixed in the cover of a strong glass cylinder, which is filled with a substance at a temperature of several degrees below the freezing point of the solvent. The inner tube is closed by a cork, through which pass a stirrer and a thermometer of the Beckmann type. This thermometer has a scale comprising 6 and divided into hundredths of a degree, but the quantity of mercury in the bulb can be varied by means of the mercury in the small reservoir at the top of the scale, and thus the instrument can be adjusted for use with solvents having widely different freezing points. To perform an experiment, a weighed quantity (15 to 20 g.) of the solvent is placed in A, and the external bath is regulated to a few degrees below the freezing point of the solvent. Thus if the solvent is water, a freezing mixture at about - 5 should be placed in the ex- ternal cylinder C. The temperature of A is lowered by taking it out of the air jacket and immersing it in the freezing mixture directly until a little ice appears. It is then replaced in the air jacket, and the liquid in it is stirred vigorously. As there is invariably overcooling before the ice and water are thoroughly mixed, the thermometer rises xviii DETERMINATION OF MOLECULAR WEIGHTS 187 during the stirring until it reaches the freezing point, after which it remains constant. This constant temperature is then read off. The tube A is now taken out of the cooling mixture, and a weighed quantity of the substance under investigation is introduced and dissolved by stirring, the ice being allowed to melt save a small residue. The tube is replaced in the air jacket and the temperature allowed to fall in order that the liquid may become slightly overcooled. Stirring is then recom- menced. The thermometer rises, remains constant for a very short time, and then slowly sinks. The maximum temperature is read off, and taken as the freezing point of the solution. The reason for the subsequent sinking of the temperature of equilibrium is plain. As long as the solution remains in the cooling mixture, ice continues to separ- ate. This results in making the remain- ing solution more concentrated than the original solution, so that its temperature of equilibrium with the solidified solvent will sink (cp. p. 63). The highest tem- perature registered corresponds therefore most closely to the freezing point of the solu- tion whose concentration is expressed by the weights of substance and solvent taken, although even it is evidently not high enough, for some of the solvent has neces- sarily separated out as ice before equilibrium can be attained at all. The calculation is precisely the same as that for the elevation of the boiling point. Each solvent has a constant of its own, re- presenting the fictitious lowering of the freezing point caused by dissolving one gram molecule of substance in one gram of solvent. The hundredth part of this constant, i.e. the depression caused by dis- solving 1 gram molecule of substance in 100 grams of solvent is usually termed the molecular depression. The constants for the most common solvents are as follows : FIG. 35. Solvent. Water Acetic acid Benzene Phenol K 1890 3880 4900 7500 188 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. The formula for calculation is M- S -X ~ L A' where M is the molecular weight of the dissolved substance, s its weight in grams, L the weight of the solvent in grams, and A the observed depression. A solution of 1*458 g. acetone in 100 g. benzene showed a depression of 1'220 degrees, whence we have the molecular weight 1-458 x 4900 100x1-220 = The formula C 3 H 6 requires 58. An indispensable requirement of this method is that the solvent should separate out in the pure state without admixture of the dissolved substance. If this does not occur, the method is worthless as a practical means of ascertaining molecular weights. 6. Pure Liquids Surface Tension A method for determining the molecular weight of pure liquids, as distinguished from substances in liquid solution, was indicated by the Hungarian physicist Eotvos in 1886, but received no attention until it was taken up in a practical manner by Ramsay and Shields in 1893. From theoretical considerations Eotvos reasoned that the expression where y is the surface tension, M the molecular weight, and v the specific volume, would in the case of all liquids be affected equally by the same change of temperature. According to the simple gas laws, the expression p(Mv) is affected equally by temperature for all gases. There is an obvious similarity between these two expressions. For the pressure p in the one we have the surface tension y in the other. For Mv, the molecular volume in the one, we have (Mv)\ the molecular surface in the other. In the case of gases we might calculate the molecular weight from the relation as follows. The ratio of the change in the expression to the corresponding change of temperature is where c is a constant having the same value for all gases. If we determine this constant once for all in the case of one gas taken as xvin DETERMINATION OF MOLECULAR WEIGHTS 189 standard, we can calculate the molecular weights of other gases in terms of the molecular weight of the standard gas. Similarly for liquids we have = k, whence M = \ where k is a constant having the same value for all liquids. If then we determine the numerical value of this constant for one standard liquid, we can calculate the molecular weight of other liquids in terms of the molecular weight of this standard liquid. It should be mentioned that both the above expressions hold good only when the molecular weight does not change with the tempera- ture, and are not applicable to gases like nitrogen peroxide in the one case, or liquids like water in the other, where there is such a change. The method for determining the surface ten- sion adopted by Ramsay and Shields was to measure the capillary rise of the liquid in a narrow tube. The simplest form of apparatus they used is shown in Fig. 36. FG is the capillary tube, open at the top and blown out to a small bulb at the bottom, in which there is a minute opening to admit the liquid contained in the wider tube A. D is a closed cylinder of very thin glass, which contains a spiral of iron wire and is connected with the capillary by means of a fine glass rod E. The capillary tube and liquid under investigation are introduced into A through the tube C before it is drawn out and sealed. After being drawn out at I, the open end of C is connected with the air pump, and the liquid within the tube boiled under diminished pressure, with application of heat if necessary. While the vapour is still issuing from the tube the narrow portion is rapidly sealed off at I. The tube now con- tains nothing but the liquid and its vapour, and is ready for the experiment. In order to maintain the liquid at a constant temperature, the tube is surrounded from L upwards by a mantle through which flows a stream of water heated to the desired point. HH represents the section of a magnet which by its attraction for the iron spiral is made to adjust the level of the capillary so that the liquid within it is always at the same, place G, a few millimetres from the end, where the bore of the capillary has been previously determined by L\ ^ F- FIG. 36. 190 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. means of a microscope and micrometer scale. The difference of level between the liquid within and without the capillary is read off by a telescope and scale attached to the apparatus. The temperature is then changed and a fresh reading made, in order to ascertain the temperature-variation. To obtain the surface tension 7 from the observed capillary rise, we have the approximate formula \grdh = 7, where g is 981, the gravitational acceleration in cm. -f sec. 2 , h the capillary height . in centimetres, r the radius of the capillary tube at ' G in centimetres, and d the density of the liquid at the temper- ature of observation. The value of 7 is then obtained in dynes per centimetre. The following values were observed by Ramsay and Shields for carbon bisulphide : Radius of capillary '0129 cm. Temperature 19'4 = 1 1 7 .. 1 whence - = fc. or -c = fc. it c = 2v 2 v In words, the dissociation constant k is numerically equal to half the concentration at which the substance is half dissociated, the concen- tration being expressed in gram molecules per litre. Thus for acetic acid we have & = 0*000018, whence c=0'000036, i.e. acetic acid is half dissociated when the concentration of its aqueous solution is 0-000036th normal. It is obvious that if in such a solution, which contains only about 2 parts of acetic acid per million, the acid is only half dissociated, the Q 226 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. direct determination of the molecular conductivity for solutions in which the acid is wholly dissociated is an impossibility. Yet the value of the molecular conductivity must be known in order that the degree of dissociation may be calculated. It has therefore to be determined indirectly by means of Kohlrausch's Law. Although weak acids and weak bases are but half-electrolytes, their salts are good electrolytes, and as much dissociated in solution as the corresponding salts of strong acids and bases. Thus for sodium acetate at 25 we have the following numbers : V 32 64 128 256 512 1024 75-5 77-6 79-8 81-6 83-5 85-0 88-0 m 0-858 0-882 0-907 0-927 0-949 0-966 For ammonium chloride, Kohlrausch found at 18 numbers much the same as those for sodium chloride, p. 221, viz. 1 2 10 20 100 500 1,000 5,000 10,000 50,000 907 94-8 103-5 107-8 114-2 118-0 119-0 120-4 120-9 120-9 m 0-750 0-784 0-856 0-892 0-945 0-976 0-985 It is an easy matter, then, to find numbers for the molecular con- ductivity at infinite dilution in the case of salts of weak acids or bases. Now, according to Kohlrausch's Law, there is a constant difference between the maximum molecular conductivities of all acids and their sodium salts, a difference due to the difference in speed of the hydrogen and sodium ions. But strong acids like hydrochloric acid are at equivalent dilutions quite as much dissociated as their sodium salts, so that their maximum molecular conductivities may be determined experimentally. For these acids we can thus get directly the difference between their maximum molecular conductivity and that of their sodium salts. This difference, amounting at 25 to about 275 in the customary units, when added to the maximum molecular con- ductivity of the sodium salt of an acid, which can always be directly determined, gives the molecular conductivity of the acid itself at infinite dilution, and this enables us to give the degree of dissocia- tion of the acid at any other dilution. It is a curious fact, of which no adequate explanation has as yet been given, that good electrolytes do not obey Ostwald's dilution law, xxi ELECTROLYTIC DISSOCIATION 227 which holds so accurately for the half-electrolytes. Certain empirical relations have, however, been found connecting the degree oi dissociation and the dilution, and these have a form similar to that of Ostwald's dilution formula, although they have not the theoretical foundation possessed by the latter. The first of these relations is called Rudolphi's dilution formula, and differs from Ostwald's by the square root of the dilution being introduced instead of the dilution itself. It has thus the form o TZ - = constant. and agrees fairly well with the observed values. Thus for ammonium chloride at 18 we have the following numbers: v m, 10 0-852 20 0-887 33-3 0-906 100 0-940 166-7 0-952 500 0-971 1000 0-979 1667 0-985 5000 0-991 Mean . .1*51 The second empirical dilution formula is that of van 't Hoff, which in some respects is simpler than Rudolphi's, and accords quite as closely with the facts. We may write Ostwald's dilution formula in the form = constant, where - is the concentration of the dissociated portion of the J _ rffl electrolyte, and - - the concentration of the undissociated part. If 0$ and C u represent these concentrations, we have the simple expression n 2 -~ = constant. ^u Rudolphi's formula gives no such simple relation of the concentrations of the dissociated and undissociated portions. Van 't Hoff proposes the expression =1 = constant, 228 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. which may be written in the form i 3 -5 = 77-^ = constant. /l-m\ 2 GV \^r) Here again we have a simple relation between the concentration of the dissociated and undissociated portions, and the constancy of the expression is at least equal to that obtained with Kudolphi's formula, as is shown by the subjoined table for ammonium chloride : 10 0-852 1-68 20 0-887 1-66 33'3 0-906 1-60 100 0-940 1-52 1667 0-952 1-51 500 0-971 1-48 1000 0-979 1-49 1667 0-985 1-61 5000 0-991 1-54 Mean . .1'56 According to the foregoing hypothesis of electrolytic dissociation, aqueous solutions of salts, strong acids, and strong bases have a certain proportion of the dissolved molecules split up into charged ions, and the amount in any given case is determinable from measurements of electrical conductivity. The ions are supposed to be independent of each other, and ought to act as separate molecules if the independence is complete. We should therefore expect that salt solutions, when investigated by the customary method of molecular- weight determination, should exhibit exceptional behaviour, and give for the molecular weights of the dissolved substances, values smaller than those which we should deduce from the ordinary molecular formulae. As has already been indicated, such abnormal values are frequently observed. When the molecular weight of sodium chloride and other similar salts is determined from the freezing or boiling points of their aqueous solutions, the numbers obtained are only equal to little more than half the values given by the formula NaCl, i.e. the depressions of the freezing point and the elevations of the boiling point have almost twice the normal value. A normal solution of cane sugar freezes at -1-87; a normal solution of sodium chloride freezes at - 3'46, Such an abnormally high value for the depression indicates that there are more dissolved molecules in the normal solution of common salt than there are in the normal solution of sugar, although each solution in the usual system of calculation is supposed to contain 1 gram xxi ELECTROLYTIC DISSOCIATION 229 molecule per litre. It should be noted at once that it is only electrolytic solutions which show these abnormally high values, the values given by non - electrolytic solutions being almost invariably equal to, or less than, the normal values. Abnormally small values we have already attributed to molecular association (cp. p. 193), so that we ought, by parity of reasoning, to attribute the abnormally large values for salt solutions to a dissociation of the molecules. Van 't Hoff introduced for salt solutions a coefficient i which repre- sents the number by which the normal value of the freezing point, etc., must be multiplied in order to give the value actually found. Thus for the solution of sodium chloride we have the depression 3*46, instead of the "normal" value 1'87 shown by solutions of non- electrolytes containing 1 gram molecule per litre. In this case i = 3-46/T87 = T85, i.e. the normal value 1-87 has to be multiplied by 1'85 in order to bring it up to the observed value 3'46. In the first place, it is to be noticed that for salts, acids, and bases which, according to the theory of electrolytic dissociation, split up into two ions, the coefficient i is never greater than 2. If dissociation into two ions were complete, the value for the depression of freezing point, etc., would be twice the normal value, since each molecule represented by the ordinary chemical formula becomes two independent molecules by dissociation. Now at common dilutions the dissociation is never complete, so that the value of the depression should be something less than 2, as is actually found by observation. When the dissociation hypothesis admits of a dissociation into more than two charged molecules, the depressions of the freezing point give values of i greater than 2. Thus strontium chloride, according to the dissoci- ation hypothesis, splits up at infinite dilution into the three ions, Sr", Cr, and Cl', the first of which has two charges of positive electricity, and the others a charge each of negative electricity. At moderate dilutions, therefore, we ought to expect a value of i less than 3, but probably greater than 2, as the dissociation of salts is generally high. We find in accordance with this that strontium chloride in decinormal solution gives the depression 0*489, instead of the value 0'187 obtained for non-electrolytes. The value of i is therefore 0*489 4- 0*1 87 = 2*6. It is obvious from the above examples that there is a direct numerical relation between the degree of dissociation in any given case and van 't Hoff's coefficient i, so that if one is given the other can be calculated. If the degree of dissociation of a salt in solution is m, the dissociation being into two ions, then there will be present in the solution 1 m undissociated molecules and 2m dissociated molecules in all 1 + m molecules for each molecule represented by the chemical formula. The depression of the freezing point, etc., will therefore have 1 + m times the normal value, i.e. i = 1 + m. 230 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. Should the original molecule split up into n ions when the dissociation is complete, m again representing the dissociated proportion, there will be present I -m undissociated molecules and nm dissociated molecules, in all 1 + (n- l)m molecules for each original molecule, i.e. we shall have i = 1 + (n l)m. Giving m in terms of i, we have m = i- I for dissociation into two ions, and n-1 for dissociation into n ions. A comparison of the values of i deduced for the same solution directly from the freezing point, and indirectly from determinations of m by means of the electric conductivity, shows very fair accordance between the two methods. In the first place, it is found that in the case of non-electrolytes where m 0, we have i=I, i.e. we obtain the normal value for the freezing-point depression, etc. ; and Arrhenius showed from his own freezing-point determinations that the values obtained for i with electrolytes differ from the corresponding values calculated from the electric conductivity by not more than 5 per cent on the average. Although this difference is comparatively great, better accordance was scarcely to be expected owing to the difficulty in effecting the comparison. It must be borne in mind that the molecular conductivity only gives accurate figures for the degree of dissociation when the solutions are so dilute that further dilution does not sensibly change the speed of the ions. Now this condition necessi- tates a dilution of at least 10 litres, i.e. the solution must not be more concentrated than decinormal, even in the most favourable case. But the normal depression for a decinormal solution is only 0*187, so that to obtain an accuracy of 1 per cent on the value of i t as measured by the freezing-point depression, we must be able to determine depressions with an accuracy of about a thousandth of a degree centigrade. It is by no means easy to attain this degree of accuracy, the error in ordinary careful work with Beckmann's apparatus being nearer a hundredth of a degree than a thousandth. For more dilute solutions the relative error on the depression is of course still greater, and extraordinary precautions have to be taken if the freezing points are to be of any value in calculating van 't HofFs coefficient i, or the degree of dissociation m. The following table gives a comparison of the degree of dissociation of solutions of potassium chloride as calculated from Kohlrausch's determinations of the conductivity, and from the mean value of the best series of observations on the freezing xxi ELECTEOLYTIC DISSOCIATION 231 point by different investigators. In the first column we have the dilution in litres, in the second the percentage dissociation deduced from the freezing points, and in the third the same magnitude calculated by means of the dissociation hypothesis from Kohlrausch's observations of the conductivity : Dilution. Percentage Dissociation = IQOm. v From Freezing Point. From Conductivity. 10 86-0 86-2 20 88-8 89-1 33-3 90-8 91-1 100 95-3 94-4 This is a rather favourable example of the close numerical agreement attained by the two methods of calculation, which, it must be re- membered, are entirely independent of each other. As a rule the divergencies amount to as much as 2 per cent of the actual values, but the parallelism between the series is always very close. From what has been stated above, it is obvious that the hypothesis of electrolytic dissociation affords a satisfactory explanation of the anomalous behaviour of the solutions of salts, strong acids, and strong bases with regard to freezing point, boiling point, and in general all magnitudes directly derivable from the osmotic pressure. Not only does it do this, but it explains also very directly the additive character of most of the properties of salt solutions. In a previous chapter it has been shown that the properties of salts in aqueous solution are most simply explained when we assume that they are additively composed of two terms, one depending on the basic or positive portion of the salt, and the other on the acidic or negative portion. Accord- ing to the theory of electrolytic dissociation, the salt is actually decomposed on dissolution in water into its positive and negative ions, which then lead an independent existence, each conferring therefore on the solution its own properties undisturbed by the properties of the other ion. The total numerical value of any given property in a salt solution will therefore be made up of the value for the positive ion plus the value for the negative ion, at least when the solutions considered are dilute, and so the additive character of the properties of salt solutions is satisfactorily accounted for. The theory of electrolytic dissociation offers, too, a simple explanation of a set of facts that are so familiar that we generally accept them without any attempt at accounting for them. It is well known that salts in aqueous solution enter into double decomposition with the greatest readiness. If we add a soluble silver salt to a soluble chloride, a precipitate of silver chloride is at once produced. The positive and negative radicals here exchange partners, and they do so readily because the positive and negative for the most part exist free in the solution as ions, so that the whole action practically consists of the union of the silver ions with the chloride ions to produce the 232 INTRODUCTION TO PHYSICAL CHEMISTEY CHAP. insoluble silver chloride. If alcohol is used as solvent, the action takes place with equal readiness. If we take now a solution in alcohol of an organic chloride such as phenyl chloride, we find that it is a non- electrolyte, and corresponding with this, it may be mixed with an alcoholic solution of silver nitrate without any double decomposition taking place. Even after boiling for a considerable time there is little or no silver chloride produced. In this case there are no free chloride ions in the solution to unite directly with the silver ions, and consequently the action is much slower. Other organic halogen compounds act much more rapidly than phenyl chloride when in alcoholic solution, but it is very doubtful if any act with a speed even approximately comparable with that of the inorganic chlorides. Reactions between salts in aqueous solution, which do not proceed by a simple rearrangement of the ions, are much slower than the double decompositions where the ions undergo no change except rearrangement. We have instances of this type of reaction in the oxidations and reductions occurring in aqueous solution, such as the conversion of ferrous into ferric salts, or stannous into stannic salts, and vice versa. Although one may find exceptions to these rules, they are yet of a general character such as we should expect on the theory of electrolytic dissociation. It has occasionally been urged that the existence of chlorine in a solution of sodium chloride cannot be accepted even hypothetically, as the solution shows none of the properties of a solution of chlorine. This, of course, rests on a misunderstanding. What we suppose to exist in the solution is not chlorine, but the chloride ion. The molecule of the former consists of two uncharged atoms of chlorine, the molecule of the latter consists of one atom of chlorine charged with positive electricity according to Faraday's Law, i.e. every 35 '5 g. of chlorine as ion has 96,500 coulombs of electricity associated with it. But we know that a charge of electricity profoundly affects the chemical properties of substances. The mere fact of the chemical changes accompanying electrolysis is evidence of this, and other instances exist in plenty. Neither aluminium nor mercury decomposes water at the ordinary temperature, but if the aluminium is coated with mercury the amalgam formed has this power, hydrogen being evolved and aluminium hydroxide produced. The action of the copper-zinc couple is similar. These metals separately are unable to perform chemical reactions which are easily brought about by zinc coated with copper. Another familiar instance is the behaviour of zinc towards sulphuric acid. Commercial zinc readily decomposes sulphuric acid with evolution of hydrogen, whilst pure zinc is almost without action on the dilute acid. This is due to the fact that the commercial metal contains other metals as impurities, and these increase the action of the zinc. If to the solution of sulphuric acid in contact with pure zinc we add a small quantity of a platinum or a cobalt salt, these xxi ELECTEOLYTIC DISSOCIATION 233 metals are deposited on the surface of the zinc and vigorous action ensues. In each of these cases the two metals on coming into contact assume electrical charges which greatly modify their ordinary chemical character. The student who wishes to pursue the subject of Electrolysis and Electro- chemistry from the dissociation point of view may be referred to M. LE BLANC, Electrochemistry ; and R LUPKE, Electrochemistry. CHAPTER XXII BALANCED ACTIONS MANY of the chemical actions familiar to the student in his laboratory experience are reversible, i.e. under one set of conditions they proceed in one direction, under another set of conditions they proceed in the opposite direction. Thus if we pass a current of hydrogen sulphide into a solution of cadmium chloride, double decomposition occurs, according to the equation CdCl 2 + H 2 S = CdS + 2HC1. If the precipitate is now filtered off and treated with a solution of hydrochloric acid of the requisite strength, the action proceeds in the reverse direction, the equation being CdS + 2HC1 = CdCl 2 + H 2 S. What determines the direction of the action in this case is apparently the relative quantities of hydrogen sulphide and hydrogen chloride present in the solution. If hydrogen sulphide solution is added to a solution of a cadmium salt which contains a considerable quantity of free hydrochloric acid, a part only of the cadmium will be precipitated as sulphide, part remaining as soluble cadmium salt. We are here dealing with a balanced action, and we shall find it convenient to formulate actions of this type by means of the ordinary chemical equation for the action with oppositely-directed arrows instead of the sign of equality, thus : CdCl 2 + H 2 S CdCl 2 + H 2 S. As a rule, in analytical work we do not wish to stop half-way in a chemical action, and therefore choose such conditions for the action that it will proceed practically to an end. Cadmium chloride is completely precipitated as sulphide when there is little hydrochloric acid present, and when a considerable excess of sulphuretted hydrogen has been added. CHAP, xxn BALANCED ACTIONS 235 A balanced action of the same sort, but with the point of balance or equilibrium towards the other end of the reaction, is to be found when a solution of hydrogen sulphide is added to a solution of nickel or cobalt chloride. A black precipitate of the metallic sulphide is formed, but even though a very large quantity of hydrogen sulphide is present, the precipitation is never complete. A very moderate quantity of free hydrochloric acid will prevent the precipitation altogether. It is thus possible, by suitably choosing the conditions, to effect a separation of cadmium from nickel or cobalt by means of sulphuretted hydrogen, although in the two cases we are dealing with balanced actions of precisely the same type. If some dilute hydro- chloric acid is added to the solution of the mixed metallic chlorides, hydrogen sulphide will precipitate the cadmium almost completely, and precipitate practically none of the nickel or cobalt salt, even though it is present in great excess. If ammonium hydroxide solution is added to a solution of a magnesium salt, say magnesium chloride, part of the metal is precipitated as magnesium hydroxide, in accordance with the equa- tion MgCl 2 + 2NH 4 OH = Mg(OH) 2 + 2NH 4 C1, but the precipitation is never complete, for the reverse action Mg(OH) 2 + 2NH 4 C1 = MgCljj + 2NH 4 OH occurs simultaneously, and a state of balance results. In analytical practice we have usually excess of ammonium chloride present from the beginning, so that the addition of ammonium hydroxide produces no precipitate in the solution of magnesium salt. The reverse action can be easily studied by shaking up freshly-precipitated magnesium hydroxide with a solution of ammonium chloride, when the magnesium hydroxide will be found to dissolve. We may consider such cases of chemical equilibrium from the same standpoint as we adopted in Chapter X. for physical equilibrium. In the last instance given above, viz. the addition of ammonium hydroxide solution to magnesium chloride solution, if we mix definite amounts of solutions of definite strengths, the precipitation of magnesium hydroxide will come to an end when the reaction has proceeded to a certain ascertainable extent. Equilibrium is then reached, and the system undergoes no further apparent change. We may still conceive the opposed reactions to go on as before, but at such a rate that exactly as much magnesium hydroxide is formed by the direct action as is reconverted into magnesium chloride by the reverse action. Now at the beginning of the action there is no magnesium hydroxide or ammonium chloride present at all ; these are only formed as the direct action proceeds. There is therefore at the beginning no reverse action. It is obvious that if a state of balance 236 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. is to be reached, the rate at which the direct action proceeds must fall off, or the rate at which the reverse action proceeds must increase, or finally both these changes may occur together. As the action progresses, the relative proportions of the reacting substances and the products of the reactions vary, and so we should be disposed to connect the actual rate at which magnesium hydroxide is formed or decom- posed with the amounts of the different substances in solution. Further information on this point may be got by adding ammonium chloride to the solution of magnesium chloride from the beginning. If we add a very small quantity of ammonium chloride before we add the ammonium hydroxide, keeping the relative proportions of the other substances the same as before, we shall find that the amount of magnesium hydroxide precipitated is smaller than before, and if we go on increasing the amount of ammonium chloride little by little, we at last reach a point when no magnesium hydroxide is precipitated at all. Evidently, then, the presence of ammonium chloride favours the reverse action, and that in a manner proportionate to the amount added. Increase in the amount of ammonium hydroxide, on the other hand, favours the direct action, so that on the whole we should be inclined to suspect that the rate of a reaction depended on the amount of reacting materials present. Guldberg and Waage, from a consideration of many experiments, formulated the connection between rate of action and amount of reacting substances in the following simple way. The rate of chemical action is proportional to the active mass of each of the reacting substances. This rule must in the first instance be taken to apply to solutions or gases, for it is in their case only that "active mass" can be properly defined. By active mass Guldberg and Waage understood what we usually term the molecular concentration of a dissolved or gaseous substance, i.e. the number of molecules in a given volume, or in the ordinary chemical units, the number of gram molecules per litre. It is possible, however, as Arrhenius has suggested, that this molecular concentration in solution is not really a measure of the active mass, and that instead of it we ought to substitute the osmotic pressure of the dissolved substance. For our present purpose, we may take the active mass to be proportional to the molecular concentration without risk of committing any serious error, for, as we have already seen, there is, at least in dilute solution, almost exact proportionality between osmotic pressure and molecular concentration. Suppose we are dealing with the following chemical action A + B = C + D, where the letters represent single molecules of the substances as in ordinary chemical formulae. Let the molecular concentrations of the original substances A and B be a and b respectively. The rate of the reaction will then, according to Guldberg and Waage, be proportional xxii BALANCED ACTIONS 237 to a and also proportional to &, i.e. it will be proportional to the product ab. At the beginning of the action, then, we have the expression Eate = ab x constant, if by rate of reaction we mean the number of gram molecules of each of the reacting substances transformed in the unit of time, usually the minute. The constant, therefore, in the above equation, which is generally denoted by k, represents the rate at which the action would proceed if each of the reacting substances were at the beginning of the reaction of the molecular concentration 1, as may easily be seen from the transformed equation Eate ''~' It must be remembered that as the action progresses the concentration of both A and B will fall off, and that the rate at which these substances are transformed into the substances C and D must there- fore progressively diminish. If at the time t the concentration of A has diminished by x gram molecules per litre, the concentration of B will have diminished by a similar amount, and the rate at which these substances are then transformed will be Eate^ = k(a x)(b x). The constant k is still the same as in the preceding equation, and has still the same significance, as may be seen from a consideration of the formula. It is indeed characteristic of the reaction, being inde- pendent of the concentrations of the reacting substances, although variable with temperature, nature of the solvent, etc. It is customary to call it the velocity constant of the action, and the student is recommended to bear in mind that it is the rate at which the reaction would proceed were the reacting substances originally present, and constantly maintained, at unit concentration. If the reaction A + E C + D is not reversible, the transformation of A and B into and D will go on at a gradually decreasing rate until at least one of the reacting substances has entirely disappeared. If the action, on the other hand, is reversible, the transformation of C and D into A and B will begin as soon as any of the former substances are formed by the direct action. If we suppose no C and D to be present when the action commences, we have at the beginning of the direct action c = and d = Q. At the time t, when x of the original products has been transformed, we have c = x and d = x. If k' is the velocity constant of the reverse action, then at the time t Eate t = k'x*. Now after the direct action has proceeded for a certain time, which we 238 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. may call r, a state of equilibrium sets in. Let the diminution of the molecular concentration of A and B have at that time the value f, then the rate of the direct transformation is *(*-&&-& The rate of the reverse transformation at the same time is Vff. But these rates must be equal if the system is to remain in equilibrium, for in a given time as much of A and B must be formed by the reverse action as disappear in the same time by the direct action. The equation consequently holds good, and this when transformed gives (-)(&-) *' -^ = - = constant. 5 Since k and k' are independent of the concentration, their ratio is also independent of the concentration, so that we have for equilibrium the product of the active masses of the substances on one side of the chemical equation, when divided by the product of the active masses of the substances on the other side of the chemical equation, giving a constant value no matter what the original concentrations may have been. If we wish to make the formula perfectly general, we may suppose that the concentrations of the products of the direct action were c and d respectively. At the point of equilibrium the concen- trations of these substances will then be c + and d + , so that the constant magnitude will be When, as very frequently happens, the original substances are taken in equivalent proportions, and none of the products of the direct action are present at the beginning, the constant quantity has the simple form in which a represents the original molecular concentration of both reacting substances. No good practical instance of the application of the above formula is known, although approximations to it are in some cases obtainable. The best is perhaps the equilibrium between an ethereal salt, water, and the acid and alcohol from which the ethereal salt is produced. Thus if we allow acetic acid and ethyl alcohol to remain in contact, they will interact with production of ethyl acetate and water. The xxii BALANCED ACTIONS 239 action, however, will not be complete, because the reverse action will simultaneously take place, the ethyl acetate being decomposed by the water into acetic acid and ethyl alcohol. The equation for the reversible action is CH 3 . COOH + C 2 H 5 . OH ^ CH 3 . COOC 2 H 5 + H 2 0. If the acetic acid and ethyl alcohol are taken in equivalent proportions, the action ceases when two-thirds of the substances have been trans- formed into ethyl acetate and water. Supposing the active mass of the acid and alcohol to have been originally 1, the active masses at equilibrium will be Acetic acid = I - = J, Alcohol = 1-2 = 1, Ethyl acetate - f , Water = f , and the constant quantity will be (i-fl_**i_i f I x I This constant holds good for any proportions of the reacting substances, being the ratio of the velocity constants of the opposed reactions. We can therefore use it to determine the extent to which a mixture in any proportions of acetic acid and ethyl alcohol will be transformed. Thus, if for 1 equivalent of acid we take 3 equivalents of alcohol, at what point will there be equilibrium ? If represents the amount transformed at equilibrium, we have whence = 0'9, i.e. 90 per cent of the acid originally taken will be converted into ethereal salt by 3 equivalents of alcohol against 66*6 converted by 1 equivalent of alcohol. Direct experiment has shown that this amount of the acid is actually transformed. In this instance the substances are merely in solution in each other, but the presence of a neutral solvent, such as ether, in no way affects the equilibrium, although it greatly reduces the speed of the opposed reactions. There is an example of equilibrium in aqueous solution which has been verified with the utmost strictness for a very large number of substances, namely, the equilibrium between the ions of a weak acid or base and the undissociated substance itself. Suppose the substance considered to be acetic acid. If a gram molecule of the acid is dissolved in water so as to give v litres of solution, its active mass is l/v in our units. No sooner is the acid dissolved than it begins to split up into hydrogen ions and acetic ions. The rate at which this action 240 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. progresses is, according to the principle of mass action, proportional to the active mass of undissociated acetic acid present at the time considered. Let the proportion of acetic acid dissociated be m. The proportion undissociated will therefore be 1 - m, with the active mass . The rate of dissociation will thus be v 1 -m where c is the velocity constant of the dissociation. But the action is a balanced one, so that undissociated acetic acid will be reformed from the ions at the same time as the dissociation proceeds by the direct action. When the proportion of the acid transformed into ions is m, each of the ions will have the active mass m/v, and thus we shall have for the rate of reunion if c is the velocity constant of this reaction. Suppose now, in particular, that m is the proportion decomposed into ions when equilibrium has taken place. The rates of the opposed reactions must then be equal, and we thus obtain the equation m 2 _ c _ or r - f /v. (1 - m)v c This is Ostwald's dilution formula referred to on p. 224, and it was first deduced by him from the mass-action principle of Guldberg and Waage. The experimental verification has been given for acetic acid and ammonia (p. 224). What holds good for them is equally true for many hundreds of feeble acids and bases which have been in- vestigated. It has already been indicated that this dilution law does not hold good for salts or for powerful acids and bases, and we must leave it meantime an open question whether the principle of mass action, the active mass being measured by the molecular concentration, applies to them or not. It would appear not to do so, and we are as yet without data to explain the divergence. Dilution laws similar to those of van 't Hoff and Rudolphi can be obtained by assuming that the active mass is measured by a power of the molecular concentration other than the first, fractional powers being admissible. Such assumptions are, of course, purely empirical, and are not applicable to all known cases. Their value is therefore at present rather on the practical than the theoretical side. xxii BALANCED ACTIONS 241 The principle of mass action, which has been found above to hold good for substances in solution, also holds good for substances in the gaseous state. Nitrogen peroxide, when dissolved in chloroform, dissociates into simpler molecules (p. 194), according to the equation N 2 4 : N0 2 + N0 2 . This is quite analogous to the equation for the dissociation of a weak acid or base, the only difference being that the products of dissociation are here of the same kind instead of being of different kinds. If m represents the dissociated proportion, 1 - m the undissociated propor- tion, and v the dilution, we have, as before, the following expression regulating the equilibrium m 2 = constant, and this has been proved to be in accordance with the observed facts. Now nitrogen peroxide is also known in the gaseous state, and dissociates under these conditions precisely in the same manner as when it is dissolved in such a solvent as chloroform. The chemical equation for the dissociation is the same as before, and so is the expression for the amount of dissociation. The dilution v, in this case, is the volume occupied by 1 gram molecule of the gaseous substance. In cases of dissociation proper, i.e. cases of balanced action into which gaseous substances enter, the amount of dissociation is most readily determined by ascertaining the pressure and density. With a given amount of substance occupying a certain volume at a known temperature it is easy to calculate what pressure it will exert, accord- ing to Avogadro's principle, if there is no dissociation of the molecules. If there is dissociation, the pressure will, coeteris paribus, be greater, for in the given space there are more molecules than would be if no dissociation had occurred. With nitrogen peroxide, simultaneous determinations of the pressure and density are made at constant temperature, both being varied by changing the volume, and from these the magnitudes entering into the above formula can be calculated. In this case also there is close agreement between the observed numbers and the numbers calculated from the formula. Considering the matter from the kinetic molecular point of view, it is obvious that the position of equilibrium will in some cases depend on the volume occupied by the system. Dilution increases the degree of dissociation of electrolytes in aqueous solution ; and dilution increases the dissociation of nitrogen peroxide, whether in the state of solution or in the gaseous state. Each molecule of undissociated material decomposes on its own account, and is independent of the presence of other molecules. The number of undissociated molecules, therefore, which will decompose in a given time is entirely unaffected R 242 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. by the volume which they may be made to occupy. Dilution then does not affect the number of molecules which will dissociate in a given time. But dilution does affect the number of molecules reformed from the dissociated products in a given time, for here each dissociated molecule must meet with another dissociated molecule if an undis- sociated molecule is to be reproduced. Now the chance of two molecules meeting depends on the closeness with which the molecules are packed. If the particles are close together, they will encounter each other frequently ; if they are far apart, they will meet only rarely. Suppose that we double the volume in which a certain amount of nitrogen peroxide is contained. Each N0 2 molecule has on the average to travel twice as far as before in order to meet another N0 2 molecule. The number of encounters in a given time will there- fore be reduced to a fourth. The rate therefore of the reverse reaction corresponding to the equation N 2 4 ;> 2N0 2 is reduced to a fourth by doubling the dilution ; the rate of the direct action is unaffected. If therefore there was equilibrium between the c^rect and reverse actions before the system was made to occupy a larger volume, this equilibrium will be disturbed and a new equilibrium will be established at a point of greater dissociation. Here there will be proportionately less of the undissociated substance, and proportion- ately more of the products of dissociation, in order that the rates at which the undissociated nitrogen peroxide is decomposed and reformed may again be the same. In general, we may say that when dissociation is accompanied by an increase of volume at constant pressure, as is almost invariably the case, the extent of the dissociation is increased if we increase the volume in which the dissociating substance is contained. Sometimes the dissociation is not accompanied by change of volume of pressure, and then we find that neither pressure nor volume has any influence on the degree of dissociation. Strictly speaking, perhaps the term " dissociation " ought not to be applied to such cases at all, but in ordinary chemical language we almost always allude, for example, to the decomposition of hydriodic acid as a dissociation. The decomposition takes place according to the equation 2HI H 2 + I 2 , two volumes of hydriodic acid giving two volumes of decomposition products. Here, before hydrogen and iodine can be produced, two molecules of hydrogen iodide must meet, and before the hydrogen iodide can be reformed, a molecule of hydrogen must encounter a molecule of iodine. The chances of each kind of encounter will be equally affected by a change in the concentration, so that the equili- brium established for one concentration will hold good at any other xxii BALANCED ACTIONS 243 concentration. Although, therefore, the velocities of the opposed reactions are altered by alteration in the concentration, they are altered to the same extent, and the position of equilibrium is unaffected. Since with gases we usually effect changes in concentration by changing the pressure, we should expect that change of pressure would have no influence on the dissociation equilibrium of hydrogen iodide. As a matter of fact, it has been found that increase of pressure rather increases the amount of dissociation, especially under certain conditions, but this may be due to the action not taking place strictly according to the above equation, or to the volumes of the substances on the two sides of the equation not being exactly equal, owing to a slight divergence from Avogadro's Law. The above instances of balanced action are all of such a type that the system in which the equilibrium occurs is a homogeneous system either a homogeneous mixture of gases, or of substances in solution. We have now to consider cases of heterogeneous equilibrium, the system consisting of more than one phase. As an example, we may take the dissociation of calcium carbonate by heat, according to the equation CaC0 3 ;> CaO + C0 2 . Here we are dealing with two solids and one gas. The active mass of the gas may either be measured by its concentration, i.e. density, or its pressure, the two being closely proportional. There is evidently a difficulty in expressing the active mass of a solid in a similar way. The pressure of a solid cannot be measured as the pressure of a gas can, and the active mass could scarcely be expected to be proportional to the density of the solid, which is the only direct meaning we can give to concentration in this connection. The most instructive way of looking at such an equilibrium is to imagine it to take place entirely in the gaseous phase, the solids simply affording continuous supplies of their own vapour. Every liquid has, as we have seen, a definite vapour pressure for each temperature. At 360 the vapour pressure of mercury is 760 mm. At the ordinary temperature it is not directly measurable, being too small, but the presence of mercury vapour over liquid mercury may easily be rendered evident at temperatures much below the freezing point. Ice, too, has a vapour pressure of a few millimetres at the freezing point, and there is no very good reason to think that this vapour pressure would disappear entirely at any temperature, however low, although it might become vanishingly small. We may then freely admit the possibility of calcium carbonate or calcium oxide vapour over the respective substances, although the pressure of these vapours is so small that they escape our means of measurement, or even of detection. If we are to assume the existence of these minute quantities of vapour, we must assume that the same laws are followed in their case as in those instances which are acces- 244 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. sible to our measurement. In particular, we must assume that at a given temperature calcium carbonate, for example, will have a perfectly definite vapour pressure, which will remain constant as long as the temperature remains constant. We have thus in the gaseous phase a constant concentration or pressure of the vapour of the solid, the presence of the solid maintaining the pressure at its proper value, although the substance may be continually removed from the gaseous phase by the chemical action. This constant concentration is obviously unaffected by the quantity of solid present, as the vapour pressure of a small quantity is as great as that of a large quantity of the same solid. From the above reasoning, then, we conclude that the active mass of a solid is a constant quantity, being in fact proportional to the vapour pressure of the solid, which is constant for any given tempera- ture. Experiment confirms this conclusion, which was, indeed, arrived at by Guldberg and Waage on purely experimental grounds. For the action CaC0 3 ;> CaO + C0 2 , if P, F and p denote the equilibrium pressures (or active masses) of calcium carbonate, calcium oxide, and carbon dioxide respectively, we have at the point of equilibrium the following equation TcP or P = 7-', in which all the quantities are constant except p. At any given temperature, then, the pressure of carbon dioxide over any mixture of calcium carbonate and calcium oxide has a fixed value quite inde- pendent of the proportions in which the solids are present. This particular pressure of carbon dioxide, which is called the dissociation pressure of the calcium carbonate, is the only pressure of carbon dioxide which can be in equilibrium with calcium carbonate, calcium oxide, and with any mixture of the two. Greater pressures are in equilibrium with calcium carbonate only, smaller pressures are in equilibrium with calcium oxide only. As the temperature increases, the dissociation pressure likewise increases, so that we can get a temperature curve of dissociation pressures resembling that for the vapour pressures of a liquid. Similar phenomena to the above are met with in the dehydration of salts containing water of crystallisation. The hydrated and the anhydrous salts are here the solid substances, and water vapour the gas. For a given temperature each hydrate has a definite dissociation pressure of water vapour over it. There is here, however, the complication that a salt usually forms more than one hydrate, in which case the dehydration often proceeds by stages, the hydrate with most water of crystallisation not passing immediately into water vapour and the anhydrous salt, but into water vapour and a lower xxii BALANCED ACTIONS 245 hydrate. Take, for example, the common form of copper sulphate, the pentahydrate CuS0 4 , 5H 2 0. At 50 this hydrate gradually loses water (cp. Fig. 20, p. Ill), and becomes converted into the trihydrate CuS0 4 , 3H 9 0. As long as any pentahydrate remains, a definite dissociation pressure of 47 mm. of water vapour persists over the solid. If the water vapour is removed, the pentahydrate will give up more water, and the equilibrium pressure will be re-established. If the dehydration still goes on, the pressure will remain constant at 47 mm. until all the pentahydrate has been converted into trihydrate, when the pressure will suddenly fall to 30 mm. This new pressure is the dissociation pressure of the trihydrate, which now begins to lose water and pass into the monohydrate CuS0 4 , H 2 0. As long as there is trihydrate present the pressure of 30 mm. will be maintained, but as soon as all the trihydrate has passed into monohydrate, the pressure again drops suddenly to 4 '5 mm., which is the dissociation pressure of the latter substance. The solid dissociation product is now the anhydrous salt, and when all the monohydrate has been converted into this the pressure of water vapour drops to zero. What here holds good for the dehydration of hydrates also applies to the removal of ammonia from such compounds as AgCl , 3NH 3 , in which the ammonia may be removed in successive stages, with formation of intermediate compounds, e.g. 2AgCl , 3NH 3 . Sometimes dissociation results in the formation of two gaseous substances derived from one solid, e.g. the dissociation of ammonium salts, the solid chloride furnishing both ammonia and hydrochloric acid. In this case also there is a constant dissociation pressure for each temperature. If P is the constant vapour pressure of the undissociated ammonium chloride, and p the gaseous pressure of the ammonia or of the hydrochloric acid, these being equal since the two gases are produced in molecular proportions by the dissociation of the ammonium chloride, we have k'P kP = Jc'p 2 , or p 2 = ~ = constant, K whence the total dissociation pressure 2p is also constant. Balanced actions of this kind have been studied with ammonium hydrosulphide and similar compounds which dissociate at comparatively low temperatures. From the above equation it is apparent that the product of the pressures of ammonia and the acid is constant, i.e. for the balanced action NH 4 HS : NH 3 + H 2 S the constant quantity is the product of the pressures of ammonia and hydrogen sulphide. If these substances are derived entirely from the dissociation of ammonium hydrosulphide, the pressures will be equal, but it is possible to add excess of one or other gas from the beginning, 246 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. in which case they will no longer be equal. Their product, however, will retain the same value as before, though their sum, i.e. the total gas pressure, will have changed. This is evident from the equation k'P kP = Wpp' t or pp = = constant, rC in which p and p' represent the pressures of the ammonia and the sulphuretted hydrogen respectively. Since p and p' enter into the equation in the same way, an excess of the one will have precisely the same effect on the equilibrium as an excess of the other. This conclusion also is verified by experiment, as may be seen from the following table, which contains some of the results of Isambert with ammonium hydrosulphide. Under p is the pressure of ammonia in centimetres of mercury, under p' the pressure of hydrogen sulphide, the other columns giving the variable total pressure and the constant product of the individual pressures. For this series the temperature of experiment was 17 '3. P P' P+P' PP' 15'0 15-0 30-0 225 -p fTTQ/ 10 ' 3 21 ' 4 31 ' 7 22 Excess of H 2 S| ^5 41>g 4? . 2 ^ 37 ' 7 6 ' 43 44 ' 2 243 41'6 5-59 47'2 232 It is true that the numbers in the last column do not exhibit very great constancy, but this is due to experimental error, as comparison with other similar series of numbers serves to show. In the previous cases of balanced action we have had gaseous substances on one side of the chemical equation only. We now proceed to deal with a case where gaseous substances occur on both sides. If steam is passed over red-hot iron, a ferroso-ferric oxide of the composition Fe 4 5 is formed, the water being reduced to hydrogen. For the sake of simplicity we may assume that the product of oxida- tion of the iron is ferrous oxide, FeO, so that we have the equation If we continue to pass steam over the solid, all the iron is eventually oxidised, notwithstanding which, however, the action is a balanced one. For if we take the oxide produced, and heat it in a current of hydrogen, the hydrogen is oxidised to water, and the iron oxide reduced to metallic iron, the reduction being complete if the current of hydrogen is continued for a sufficient length of time. The reason why we in each case have the action completed is that the passage of the current of gas disturbs the equilibrium, which is never completely established. The nature of the equilibrium may be seen from the equation H 2 + Fe :> FeO + H 2 . xxn BALANCED ACTIONS 247 On each side we have one solid substance and one gas. Since the gases are now on opposite sides of the equation, the equilibrium is regulated by the quotient of their pressures instead of by their product, as in the preceding case. Let p, P, p', P' be the equilibrium pressures of the reacting substances in the order in which they occur in the chemical equation. We obtain the equilibrium equation p k'P' kpP = k 'p'P', or ~ = -r-p = constant. P KM The ratio of the pressures of water vapour and hydrogen thus determines the equilibrium. If, as occurs when we pass water vapour over the iron, the ratio of the pressure of water to the pressure of hydrogen is greater than the equilibrium ratio, the action proceeds until all the iron has been converted into oxide. On the other hand, when we pass a current of hydrogen over the heated oxide, the above ratio is always less than the equilibrium ratio, and the oxide is completely reduced. The ratio changes with temperature, and becomes equal to unity at about 1000. At this temperature, therefore, if we take equal volumes of water vapour and hydrogen, and pass the mixture over either iron or the oxide Fe 4 5 , no chemical action will take place, for the pressures are then the equilibrium pressures. Corresponding to the preceding cases of equilibrium with gases we have similar instances with substances in solution. In aqueous solution, however, the equilibrium is very often complicated by the occurrence of electrolytic dissociation. Several cases of this kind will be considered in a subsequent chapter. A solution equilibrium analogous to the dissociation of calcium carbonate is to be found in the action of water on insoluble salts of very weak insoluble bases combined with soluble acids. An organic base such as diphenylamine is so weak that its salts when dissolved in water split up almost entirely into the free acid and free base. Diphenylamine itself is practically insoluble in water, and so is the picrate formed from it by its union with picric acid. When the solid picrate is brought into contact with water, it partially dissociates with formation of insoluble base and soluble picric acid. We have, there- fore, the equation Diphenylamine picrate j" Diphenylamine + Picric acid exactly analogous to the equation Calcium carbonate ^ Calcium oxide + Carbon dioxide, with the exception that the picric acid is in the dissolved state, whereas the carbon dioxide is in the gaseous state. Now in the gaseous equilibrium we found that for each temperature a certain pressure, or concentration, of gas was produced. We should expect, 248 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. therefore, that in the solution equilibrium for each temperature there should exist a certain osmotic pressure, or concentration, of the dis- solved substance which should be necessary and sufficient to determine the equilibrium, independent of the proportions in which the insoluble solid substances may be present. This has been confirmed by experi- ment. At 40'6 a solution of picric acid containing 13*8 g. per litre is in equilibrium with diphenylamine and its picrate, either singly or mixed in any proportions, and if diphenylamine picrate is brought into contact with water at this temperature it dissociates until the concentration of the picric acid in the water has reached this point. If we bring carbon dioxide at less than the dissociation pressure into contact with calcium oxide, no carbonate is formed. Similarly, if we bring at 40*6 a solution of picric acid, having a concentration of less than 13 '8 g. per litre, into contact with diphenylamine, no diphenylamine picrate will be formed. On the other hand, if the solution has a greater concentration than 13 '8 g. per litre, diphenyl- amine picrate will be produced until the concentration has sunk to that necessary for the equilibrium. This may be readily shown experimentally, on account of the different colours of diphenylamine and its picrate. The base itself is colourless, picric acid affords a yellow solution, and diphenylamine picrate has a deep chocolate- brown colour. A solution of picric acid at 40 '6 containing 14 g. per litre at once stains diphenylamine deep brown, but a solution at the same temperature containing 13 g. per litre leaves the diphenyl- amine unaffected. If the weak base were soluble instead of insoluble, we should have an equilibrium for solutions corresponding to the dissociation of ammonium hydrosulphide. Urea is such a base, and the picrate of urea is very sparingly soluble as such in cold water, so that we have the equation Picrate of urea ^ Urea + Picric acid. On the left-hand side of the equation we have a solid ; on the right- hand side we have two substances in solution. The sparingly soluble urea nitrate and oxalate afford similar instances. Phenanthrene picrate, when dissolved in absolute alcohol, dissociates into phenanthrene and picric acid, both of which are soluble. This case has been investigated, and found to obey the law of mass action. The calculation is, however, complicated on account of the picric acid being partially dissociated electrolytically, and part of the phenan- threne being associated in solution to larger molecules than that corresponding to the ordinary molecular formula. There are plenty of instances in solution corresponding to the action of steam on metallic iron. If barium sulphate is boiled with a solution of sodium carbonate, it is partially decomposed, according to the equation xxii BALANCED ACTIONS 249 BaS0 4 + Na 2 C0 3 = BaC0 3 + Na 2 S0 4 . Here both the barium salts are insoluble and the sodium salts soluble, so that there is one solid and one salt in the dissolved state on each side of the equation. The active masses of the barium salts may be accounted constant during the reaction, for although they are gener- ally spoken of as "insoluble," they are in reality measurably soluble in water, cp. p. 30 7. The aqueous liquid in contact with them will therefore be and remain saturated with respect to them, i.e. their concentration and active mass in the solution will be constant. The equilibrium will thus be determined by a certain ratio of the concentrations of the soluble sodium salts, independent of what the actual values of the concentrations may be. Guldberg and Waage found by actual experiment that the concentration of the carbonate should be about five times that of the sulphate if the solution is to be in equilibrium with the insoluble barium salts. Such a solution will neither convert sulphate into carbonate nor carbonate into sulphate. If the proportion of carbonate in the solution is greater than 5 molecules to 1, the solution will convert barium sulphate into barium carbonate ; if the proportion is less than this molecular ratio, the solution will convert barium carbonate into barium sulphate. It should be mentioned that when we are dealing with salts in solution, Guldberg and Waage's Law has only an approximate applica- tion if we use concentration in the ordinary sense as the measure of the active mass of the dissolved substances, for by doing so we entirely neglect the effect of electrolytic dissociation. When the reactions are considered more closely, we find that it is usually the ions that are active, so that we should really deal with ionic concentrations in the majority of cases, instead of with the concentrations of the substance supposed undissociated. It happens, however, that in a great many cases the effect of dissociation is such that it affects the two opposed reactions equally, and in such cases the approximate conditions of equilibrium may be arrived at although dissociation is entirely neglected. In the instance given above of the equilibrium between soluble carbonates and sulphates and the corresponding insoluble barium salts, all the substances are highly dissociated, and approxi- mately to the same extent on both sides of the equation, so here the application of Guldberg and Waage's Law in its simple form gives results in accordance with experiment. If we increase the active mass of any substance playing a part in a chemical equilibrium, the balance will be disturbed, and the system will adjust itself to a new position of equilibrium, that action taking place in virtue of which the active mass of the substance added will diminish. Thus if we have carbon dioxide at a pressure such that the gas is in equilibrium with a mixture of calcium oxide and calcium carbonate, and suddenly increase its active mass by 250 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. increasing the pressure upon it, a new equilibrium will be established by the action CaO + C0 2 - CaC0 3 taking place, the tendency of which is to diminish the active mass, or pressure, of the carbon dioxide. Again, if we have an aqueous solution containing sodium sulphate and sodium carbonate in such proportions as to be in equilibrium with the corresponding barium salts, and add an extra quantity of sodium sulphate to the solution so as to increase the active mass of this salt, the equilibrium will be disturbed, and will readjust itself by means of the action Na 2 S0 4 + BaC0 3 = Na 2 C0 3 + BaS0 4 , which will go on until the concentration of the sodium sulphate has fallen to a value which gives the equilibrium ratio with the new con- centration of sodium carbonate. The principle here enunciated is of special importance in its application to cases of dissociation, whether gaseous or electrolytic. For convenience sake it may be stated for this purpose in the follow- ing form. If to a dissociated substance we add one or more of the products of dissociation, the degree of dissociation is diminished. Thus phosphorus pentachloride, which when vaporised dissociates according to the equation PC1 5 $PC1 3 + C1 2) . gives a vapour density little more than half that which corresponds to its usual formula. If, however, the pentachloride is vaporised in a space already containing a considerable quantity of phosphorus trichloride, the vapour density is such as would correspond to the formula PC1 5 for the pentachloride. Here one of the products of dissociation has been added, viz. PC1 3 , and the dissociation of the pentachloride has been in consequence to such an extent that the vapour density has practically the normal value. In the case of the dissociation of ammonium hydrosulphide, the addition of either ammonia or hydrogen sulphide to the normal dissociation products will diminish the degree of dissociation. Here, however, the undissociated substance exists only to a very small extent in the vaporous state, the consequence being that a smaller quantity of the salt is vaporised. This may be seen by reference to the numbers given on p. 246. At 17*3 the dissociation pressure is 30 cm., half of this pressure being due to ammonia and half to hydrogen sulphide. If, now, we allow the hydrosulphide to dissociate into an atmosphere of hydrogen sulphide having a pressure of 3 6 '6 cm., the increase in pressure will only be 10*7 cm., i.e. this will be the dissociation pressure under these conditions. If the dissociation is allowed to take place in an atmosphere of ammonia of 36 cm. pressure, xxii BALANCED ACTIONS 251 we have a similar diminution of the dissociation to about one-third of its normal value. Suppose two substances are dissociating in the same space, and have a common product of dissociation ; it is evident from what has been said that the degree of dissociation of each will be lower than the value it would possess were the substances dissociating into the same space singly. Take, for example, a mixture of ammonium hydrosulphide and ethylammonium hydrosulphide. These substances dissociate according to the equations NH 3 (C 2 H 5 )HS $ NH 2 (C 2 H 5 ) + H 2 S, NH 4 HS NH 3 + H 2 S, and have hydrogen sulphide as a common dissociation product. When they dissociate into the same space, the effect is that as each supplies an atmosphere of hydrogen sulphide for the other to dissociate into, the dissociation pressure of each is diminished below the value it would have were it dissociating alone. Thus at 26*3 ammonium hydrosulphide has a dissociation pressure of 5 3 '6 cm., and ethyl- ammonium hydrosulphide has a dissociation pressure of 13 '5 cm. If the two substances did not affect each other when dissociating into the same space, the dissociation pressure of the mixture would be the sum of the dissociation pressures of the components of the mixture, viz. 67*1 cm. The value actually found by experiment is much lower than this, viz. 5 2 '4 cm., which is even lower than the dissociation pressure of ammonium hydrosulphide itself. It should be stated that this result is not in accordance with the law of mass action, if pressures are taken as the measure of the active mass of the gases. The theoretical result is 55*3 cm., somewhat greater than the dissociation pressure of the ammonium hydrosulphide. When, as in the above instance, two unequally dissociated substances are brought into the same space, the more dissociated substance has a much greater effect on the degree of dissociation of the less dis- sociated substance than vice versa. This we might expect, for the more dissociated substance increases the concentration of the common dis- sociation product far above the value which would result from the dissociation of the less dissociated substance ; while the latter, even if dissociated to its normal extent, would only slightly increase the concentration of the common dissociation product beyond the value obtained from the more dissociated substance alone. There is thus a great relative increase in the first case and a small relative increase in the second, the effects on the dissociation being in a corresponding degree. It remains now to discuss the effect of temperature on balanced action. A rise of temperature is almost invariably accompanied by acceleration of chemical action. In a balanced action, therefore, both 252 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. the direct and reversed actions are accelerated when the temperature is raised. The effect on the opposed reactions is, however, not in general equally great, with the result that the point of equilibrium is displaced in one or other sense. This displacement is intimately connected with the heat evolved in the reaction. If the direct action gives out a certain number of calories per gram molecule transformed, the reverse reaction will absorb an exactly equal amount of heat. Now rise of temperature always affects the equilibrium in such a manner that the displacement takes place in the direction which will determine absorption of heat. If, therefore, the direct action is accom- panied by evolution of heat, the action will not proceed so far at a high as at a low temperature, for if we start with equilibrium at the lower temperature, and then heat the system to a higher temperature, the heat-absorbing reverse action occurs, and the point of equilibrium moves backwards. If, on the other hand, the direct action absorbs heat, the action will proceed farther at high than at a low temperature. In all cases of gaseous dissociation at moderate temperatures the dissociation is accompanied by absorption of heat, so that the degree of dissociation increases as the temperature is raised. Thus the dis- sociation pressure of calcium carbonate rises with rise of temperature, and so does the dissociation pressure of salts like ammonium hydro- sulphide, as the following table shows : DISSOCIATION PRESSURE OF AMMONIUM HYDROSULPHIDE Temperature. Mm. of Mercury. 77 155 12-2 216 17-6 310 22'4 421 27-6 573 Nitrogen peroxide, again, which at the ordinary temperature is only about 20 per cent dissociated into the simple molecules N0 2 , is at 130 practically entirely dissociated. Since the heat of dissociation into ions is sometimes positive, sometimes negative, a rise of temperature may in some cases be accompanied by increased dissociation, in other cases by diminished dissociation. A considerable diminution of the degree of dissociation with rise of temperature has been proved for hydrofluoric acid and hypophosphorous acid. The rule which has here been applied to the displacement of chemical equilibrium with change of temperature is equally applicable to physical equilibrium. If we take a quantity of a liquid, and enclose it in a space greater than its own volume, a certain proportion of the liquid will assume the vaporous state, heat being absorbed in the vaporisation. If now we raise the temperature, keeping the volume constant, the equilibrium will be disturbed, and the endo- xxn BALANCED ACTIONS 253 thermic action will occur, i.e. more liquid will be converted into vapour, and the vapour pressure thus become greater. A number of cases of balanced action of an interesting type have been recently classified under the name of dynamic isomerism. The two substances, ammonium thiocyanate and thiourea, are isomeric, both having the empirical formula CSN 2 H 4 , and their isomerism at temperatures below 100 does not differ in any special way from the isomerism of other substances. If either substance is fused, however, it is partially transformed into the other isomeride, a balance being attained at a point where the liquid consists of about 80 per cent of ammonium thiocyanate and 20 per cent of thiourea. In this particular instance the substances have no marked tendency at the ordinary temperature to pass into each other when pure, so that the opposed reactions and the balance between them can be studied experimentally. Other instances have been investigated in which a change of one isomeride into the other in solution can be followed in the polarimeter, owing to a difference in the optical activity of the two substances. It is probable that most of the phenomena of " tautomerism " and "desmotropy" met with in organic chemistry are referable to similar causes. Liquids, for example, which are usually written with the group - CH 2 . CO , very frequently act as if they contained the enol group, - CH : C(OH) - , and the liquids themselves often give values for their physical properties which would accord with their being mixtures of the two isomerides. That the liquids may be such mixtures is probable, seeing that other instances of isomeric balance are now well authenticated. Cases of balanced action are discussed in the following papers which may be consulted by the student : J. T. CUNDALL " Dissociation of Nitrogen Peroxide " (Journal of the Chemical Society, lix. p. 1076 ; Ixvii. p. 794). Compare also W. OSTWALD, ibid. Ixi. p. 242. J. WALKER and J. R APPLEYARD " Picric Acid and Diphenylamine " (ibid, Ixix. p. 1341). J. WALKER and J. S. LUMSDEN " Dissociation of Alkylammonium Hydrosulphides " (ibid. Ixxi. p. 428). T. M. LOWRY "Dynamic Isomerism" (ibid. Ixxv. p. 235). CHAPTER XXIII RATE OF CHEMICAL TRANSFORMATION IN the preceding chapter we have had occasion to use the conception of reaction velocity in order to facilitate the discussion of chemical equilibrium, without entering into the question of how such a magni- tude can be practically determined. It is only in comparatively few cases that an accurate determination is possible at all, for the vast majority of reactions either take place so rapidly, or are complicated to such an extent by subsidiary reactions, that no specific coefficient of velocity for the reaction can be calculated. One of the simplest reactions, and one of the earliest to be studied with success, is the inversion of cane sugar. When this substance is warmed with a mineral acid in aqueous solution, it is gradually con- verted into a mixture of dextrose and levulose, and the process of conversion can be accurately followed by means of the polarimeter. The cane-sugar solution has originally a positive rotation, the invert sugar produced by the action of the acid has a negative rotation. If therefore we place the sugar solution in the observing tube of a polari- meter, and read off the angle of rotation from time to time, we can tell how the composition of the solution varies as time progresses without in any way disturbing the reacting system. For example, the original rotation of a cane-sugar solution was found to be 46-75, whilst the rotation of the same solution after complete inversion was - 18'70. The total change in rotation, therefore, corresponding to complete conversion of the cane sugar into invert sugar was 46'75 + 18'70 = 6 5 '45. After the lapse of an hour from the beginning of the reaction, the rotation was found to be 35 '7 5. In that time the rotation had thus diminished by irOO, so that the fraction of the original sugar transformed was 11 "00 -f- 65'45 = 0'168. By a similar calculation the quantity of sugar transformed at any other time could be arrived at. The following table gives the rotations actually observed at different times : CHAP, xxiii KATE OF CHEMICAL TRANSFORMATION 255 Time in Minutes. Rotation. Constant. 4675 30 41-00 0-001330 60 3575 1332 90 30-75 . 1352 120 26-00 1379 150 22-00 1321 210 15-00 1371 330 2-75 1465 510 - 7-00 1463 630 -10-00 1386 cc -18-75 From this table it is evident that the rate at which the reaction proceeds falls off as less and less cane sugar remains in solution. In the first two hours the change in rotation is 20*75 ; in the two hours from 510' to 630' the change is only 3*00. This is in accordance with the principle of Guldberg and Waage that the amount transformed in a given time will fall off as less cane sugar remains to undergo transformation. The chemical equation expressing the reaction is C 12 H 22 U + H 2 = C 6 H 12 6 + C 6 H 12 6 . (Dextrose) (Levulose) As the action progresses, water disappears as well as cane sugar; but if we consider that the action, takes place in aqueous solution, it is obvious that the change in the active mass of the water is very slight, and so for practical purposes of calculation, the active mass of the water may be taken as constant. The action is not a balanced one, but proceeds until all the cane sugar has been transformed. If A is the original concentration of the cane sugar, and x the quantity trans- formed at the time t, the rate of transformation at that time will be, according to the unimolecular formula, dx where dx represents the very small quantity transformed in the very small interval dt starting at the time t, and k is the coefficient of velocity of the action. From this equation the integral calculus enables us at once to find a relation between x and t, the corresponding values for any stage of the reaction in terms of the original concentra- tion and the velocity constant. This relation has the form 1 . A or - log . - = constant. t A -x The values of the expression - log -. -- are given in the last column t A x 256 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. of the preceding table, and it will be seen that they remain tolerably constant throughout the reaction. The agreement between experiment and the theory of Guldberg and Waage is therefore in this case quite satisfactory. 1 The part played by the acid in the inversion of cane sugar is not well understood. The acid itself remains unaffected, but the rate of the inversion is nearly proportional to the amount of acid present if we always use the same acid. In such an action as this, the acid is said to act as an accelerator, and this behaviour on the part of acids is by no means uncommon, although the different acids vary very much in their accelerative power. As we shall see later, the accelerative power of acids furnishes us with a convenient measure of their relative strengths. A similar accelerating action of acids is found in the catalysis of ethereal salts. If a salt such as ethyl acetate is mixed with water, the two substances interact, with formation of acetic acid and ethyl alcohol (cp. Chap. XXII). This is in reality a balanced action, but if the solution of the ethereal salt is very dilute, the transformation is almost complete, and the action becomes of the same simple type as the inversion of cane sugar, the equation being CH 3 . COOC 2 H 5 + H 2 = CH 3 . COOH + C 2 H 5 OH. If no acid is present, the action takes place with extreme slowness, but in presence of the strong mineral acids, such as hydrochloric acid, it progresses with moderate rapidity. Although the progress of the action cannot conveniently be followed by a physical method, as in the sugar inversion, a chemical method may be employed without disturbing the reacting system. At stated intervals a measured portion of the solution is removed and titrated with dilute alkali. As the action progresses, the titre becomes greater, owing to the pro- duction of acetic acid, and from this increase we may deduce the amount of transformation. It is found that at each instant the rate of transformation is proportional to the amount of ethereal salt present, and a velocity constant may be calculated by means of the same formula as was used for sugar inversion. From a comparison of the velocity constants obtained with different acids, it appears that the relative accelerating influences of the acids is the same for both actions. Saponification of ethereal salts by alkalies affords us an example of a bimolecular reaction. The equation for the saponification of ethyl acetate by caustic potash is CH 3 . COOC 2 H 5 + KOH = CH 3 . COOK + C 2 H 5 OH, and the action proceeds until one or other of the reacting substances 1 The inversion of cane sugar was studied, both practically and theoretically, by Wilhelmy, with the result arrived at above, before Guldberg and Waage enunciated their principle, and the numbers in the table are taken from Wilhelmy's work. xxin EATE OF CHEMICAL TRANSFORMATION 257 entirely disappears. If we take both substances in equivalent propor- tions, and represent the active mass of each by a, the general equation for the rate of transformation will be integration of which leads to the equation . -- t a(a - x) Experimental work confirms this conclusion, the expression on the right-hand side of the last equation being in reality constant. The course of the saponification can be easily followed by removing measured portions of the solution from time to time, and titrating them with acid. As the action proceeds, the amount of acid required to neutralise the potassium hydroxide in solution falls off proportion- ally. For a given ethereal salt, equivalent solutions of the caustic alkalies and the alkaline earths effect the saponification at practically the same rate, the rate of saponification for ammonia being very much smaller. For a given base the rate of saponification is greatly affected by the nature of both the acid radical and the alkyl radical which go to form the ethereal salt. Thus at 9*4 the velocity constants of various acetates on saponification by caustic soda are as follows : Methyl acetate Ethyl acetate Propyl acetate Isobutyl acetate Isoamyl acetate 3-493 2-307 1-920 1-618 1-645 While the corresponding numbers for various ethyl salts with the same base at 14*4 are Ethyl acetate Ethyl propionate Ethyl butyrate . Ethyl isobutyrate Ethyl isovalerate Ethyl benzoate 3-204 2-186 1-702 1-731 0-614 0-830 Other instances of bimolecular reactions which have been investigated are the formation of sodium glycollate from sodium chloracetate and caustic soda, according to the equation CH 2 C1 . COONa + NaOH = CH 2 (OH) . COONa + NaCl, and the formation of such salts as tetra-ethyl ammonium iodide from tri-ethylamine and ethyl iodide, in accordance with the equation A bimolecular reaction, which is, strictly speaking, a balanced action, s 258 INTRODUCTION TO PHYSICAL CHEMISTEY CHAP. but proceeds very nearly to an end in aqueous solution, is the forma- tion of urea from ammonium cyanate. According to the ordinary equation, NH 4 CNO = CO(NH 2 ) 2 , we should expect the action to be unimolecular, but experiment has shown that it is a bimolecular reaction in dilute solutions, the reacting substances being the ions of the ammonium cyanate, with the equation In decinormal solution at ordinary temperatures, about 95 per cent of the ammonium cyanate is converted into urea. Trimolecular reactions are, comparatively speaking, rare, amongst those which have been most thoroughly investigated being the reduction of ferric chloride by stannous chloride, viz. 2FeCl 3 + SnCl 2 - 2FeCl 2 + SnCl 4 , and the reduction of a silver salt in dilute solution by a formate, according to the equation 2AgC 2 H 3 2 + HC0 2 Na = 2Ag + C0 2 + HC 2 H 3 O 2 + NaC 2 H 3 2 . For such reactions we have the following expression for the rate : where the reacting substances have each the original active mass a. On integration this becomes , _ 1 x('2a - x) ~ t ' 2a*(a - xf ' and it has been ascertained experimentally that the expression on the right-hand side of the equation is actually a constant. No action has hitherto been investigated which follows the equation expressing the rate for a higher number of molecules than three. At first sight this is surprising, for in our ordinary chemical equations we are familiar with well-known reactions where the number of re- acting molecules is much greater than three. It would appear, how- ever, that these complicated reactions take place in stages, so that each is really an action composed of successive simple reactions. To take a simple instance, hydrogen arsenide is decomposed by heat into hydrogen and arsenic vapour. From the vapour density of arsenic it is known that the arsenic molecule contains four atoms under the conditions of experiment. We therefore write the equation for the decomposition as follows : 4AsH 3 = As 4 + 6H 2 . xxiii KATE OF CHEMICAL TRANSFORMATION 259 A study of the rate of the reaction, however, shows that it is not quadrimolecular, as the equation would lead us to suppose, but that it gives numbers agreeing with the unimolecular formula. This is evident from the following table, which gives the values of the expression Jc= - log , characteristic of unimolecular reactions, for t CL X different times : t k t k 3 hours 0*0908 4 0-0905 5 i 0-0908 6 hours 0-0905 7 ,, 0-0906 8 '. 0-0906 The action whose rate we measure, therefore, is a unimolecular action, most probably AsH 4 = As + 4H, which is then followed by the actions 4As = As 4 2H = H 2 . Now, in order that the course of the total action may appear as a unimolecular action, it is a necessary assumption that the rate of the first action given above is much smaller than the rate of the succeeding actions. This is so because, in the first place, it is the slowest of a series of actions which will principally determine the rate from the initial to the final stage, and in the second place, if the total rate is to coincide with the rate of this slowest action, the rate of the others must be incomparably greater than this. An analogy may serve to 'make the point clear. The time occupied in the transmission of a telegraphic message depends both on the rate of transmission along the conducting wire, and on the rate of the messenger who delivers the telegram; but it is obviously this last, slower, rate that is of really practical importance in determining the total time of trans- mission, and, indeed, the speed of electricity may be neglected alto- gether in the calculation. When we measure the rate of a complicated action, then, we are in general measuring an average rate of a series of reactions which may be progressing successively or simultaneously, and it is the rate of the slowest of these actions which plays the principal part in determining the total rate. Should the other actions proceed at an immeasurably faster rate than the slowest action, the whole action will appear to go at a rate and be of a type regulated by this action alone. Hence it is that we so frequently find complex actions pro- ceeding in such a way as to suggest that they are much simpler in type than the total action really is. The saponification of an ethereal salt of a bibasic acid affords a 260 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. good instance of the progress of a reaction in stages. For example, the saponification of diethyl succinate by caustic soda has been shown to proceed, not according to the equation C 2 H 4 (COOC 2 H 5 ) 2 + 2NaOH = C 2 H 4 (COONa) 2 + 2C 2 H 5 OH, but according to the equations C 2 H 4 (COOC 2 H 5 ) 2 + NaOH = C 2 H 4 (COOC 2 H 5 )(COONa) + C 2 H 5 OH, C 2 H 4 (COOC 2 H 5 )(COONa) + NaOH = C 2 H 4 (COONa) 2 + C 2 H 5 OH. On the last assumption, Guldberg and Waage's principle leads to a certain expression for the velocities of the total action involving the two velocity constants of the single actions, and the experimental rate has been found to agree with the theoretical requirements. The fact that the action does in reality proceed in two stages may be easily demonstrated by treating ethyl succinate in alcoholic solution with half the calculated quantity of caustic potash necessary for complete saponification. Instead of half the ethereal salt being completely saponi- fied, and half being left untouched, about three-fourths of the original quantity is, under favourable conditions, converted into the potassium ethyl salt C 2 H 4 (COOEt)(COOK), the product of the first stage, one- eighth remaining unattacked, and one -eighth being converted into the dipotassium salt. In general, then, it may be accepted as a fact that in actions which are expressed by comparatively complicated chemical equations, in- volving the interaction of a large number of molecules, we are dealing with a series of simpler actions, not more than three molecules being involved in each of these. The velocity of a given action is usually greatly affected by change of temperature and change of the medium in which the action occurs. Almost invariably a rise of temperature is accompanied by a large increase in the rate of the reaction, the speed being very frequently doubled for a rise of five or ten degrees starting at the ordin- ary temperature. It is somewhat difficult to account for this very high temperature coefficient. On any of the usual hypotheses regarding the rate of molecular motion and its variation with the temperature, it is impossible to assume that the speed of the molecules increases so greatly that they encounter each other twice as often when the temperature rises from 15 to 20. It has been suggested that only a certain small proportion of the total number of molecules are active at any one temperature, and that this number increases rapidly as the temperature rises. On this supposition ions might be supposed to be almost all in the active state, at least so far as double decompositions amongst acids, salts, and bases are concerned, for these actions progress so rapidly that their speed has never been measured. An action which lends some support to this supposition is that of very xxiii EATE OF CHEMICAL TRANSFORMATION 261 dilute acids on zinc, which progresses at a rate which is almost independent of the temperature. This action is, in terms of the dissociation theory, Zn + 2H' = Zn" + H 2 The solid zinc can scarcely have its active part greatly affected by change of temperature (since at most only the superficial part of it can take part in the action), and the hydrogen ions must be considered nearly all active at the ordinary temperature, as a rise of tempera- ture does not increase the rate of the reaction. Very slight changes in the nature of the medium also greatly affect the speed of a reaction. Thus if we remove 1 per cent of the water in which the conversion of ammonium cyanate into urea is taking place, and bring the solution up to its original volume by adding acetone, which takes no part in the reaction, the rate of the conversion increases by nearly 50 per cent. In ethyl alcohol the rate of transformation of the cyanate into urea is thirty times as great as it is in pure water, other conditions remaining the same, notwithstanding the fact that the number of ions in the alcoholic solution is much smaller than the number in pure water of the same concentration (cp. p. 223). The following table contains the coefficients of velocity observed for the bimolecular reaction the actual in different solvents. To avoid unnecessary ciphers, numbers have been multiplied by 1000. Solvent. k Hexane . 0-180 Heptane . 0-235 Xylene 2-87 Benzene . 5-84 Ethyl acetate 22-3 Ethyl ether 0757 Methyl alcohol 51-6 Ethyl alcohol 36-6 Allyl alcohol 43-3 Benzyl alcohol 133 Acetone . 60-8 The great range of the speed of one action is very well shown by this table. We cannot imagine that the reacting molecules meet each other in benzyl alcohol 700 times as often in a given time as they do in hexane. Either we must assume that there is a greater proportion of active molecules in the former solvent than in the latter, or that the addition takes place at a larger proportion of the encounters of the reacting molecules, a supposition which is practically identical with the first. We very frequently find that below a certain temperature chemical action apparently will not occur, while above that temperature the 262 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. action takes place freely. Thus we speak of the temperature of ignition of a mixture of gases, meaning usually thereby the lowest temperature which, when given to one part of the mixture, will produce chemical action in the whole mass. For example, we say that the temperature of ignition of a mixture of air and saturated carbon bisulphide vapour is about 160, because a glass rod heated to that temperature and applied to any part of the mixture will inflame the whole. If we inquire more closely into such actions, however, we generally find that the action occurs at temperatures below the ignition temperature, but that it will not then propagate itself under ordinary circumstances. The experimental proof of this is given by maintaining the whole mixture at a temperature somewhat below the ignition point, and noting after a time if the action has progressed. The reason for the non- propagation at lower temperatures is that the action then progresses only slowly. The heat evolution consequent on the chemical change is therefore spread over such an interval of time that the temperature of the mixture remains below the ignition temperature, the action becoming slower and slower if no external heat is supplied, eventually to cease. At the ignition temperature the action takes place at such a rate that the heat evolution is sufficiently rapid to keep the temperature of the gaseous mixture up to the ignition point, and even to raise it still higher, with the result that the action proceeds at an ever-increasing rate. We therefore see that the so-called temperature of ignition depends on the generally - observed rapid increase in the rate of chemical action with rise of temperature, and may vary with the original temperature of the whole mixture. The same thing holds good, and may be followed more easily, with certain solids. Solid ammonium cyanate, for instance, may be kept for many months at the ordinary temperature without undergoing any notable transformation into urea, according to the equation NH 4 CNO = CO(NH 2 ) 2 . At 60 the action is fairly rapid, if the temperature is kept up externally, but the heat evolution is still too slow to enable the action to go on at an increasing rate. If the external temperature is 80, however, the action proceeds swiftly, and the rapid heat evolution raises the temperature to such an extent that the whole passes almost instantaneously into fused urea, the melting point of which is 132. If the heat evolution is comparatively small, as it is with ammonium cyanate, the increasing rapidity of the reaction may be observed through a considerable range of temperature. If, on the other hand, the heat evolution is great, as it is with most explosives, the action when it takes place perceptibly is usually propagated at once, owing to the rapid rise in temperature of the particles in the immediate neighbourhood of the reacting particles. The rate of propagation of explosion in solid explosives when fired is very great, rising in some xxm EATE OF CHEMICAL TRANSFORMATION 263 instances to about five miles per second. The rate of propagation of the explosion wave in gaseous mixtures, such as that of oxygen and hydrogen, is somewhat less than this, averaging about a mile and a half per second, and being therefore of the same dimensions as the average rectilinear velocity of the gaseous molecules at the temperature of the explosion (cp. p. 87). Substances which react vigorously at the ordinary temperature usually lose their chemical activity entirely when cooled to the tem- perature of boiling liquid air. This we must attribute to the lowering of the rate of chemical action by fall of temperature, the speed being so greatly diminished that the action does not perceptibly occur at all in any moderate length of time. That there is not an entire cessation of action is probable, since in certain cases we can follow the gradual slack- ening of the action to extinction as the temperature falls. Thus sodium and alcohol, which react briskly at the ordinary temperature, with evolu- tion of hydrogen, become less and less active as the temperature is lowered, until finally the hydrogen formed by the action is so small in quantity as to escape observation. If we reflect that we often see the reaction velocity halved by a fall of 5 in temperature, we can conceive that a fall of 100 might by successive halving reduce the reaction velocity to a millionth of its original value. We may some- times find it convenient, in accordance with this, to look upon chemical inactivity as being not absolute inactivity, but rather the progress of a chemical action at a rate too slow for measurement, or even detection. We have now to consider briefly the points of resemblance between physical transformation and chemical transformation. In the change from solid to liquid, and vice versa, there is (for a given pressure) a definite temperature of transformation (p. 60). Above this tempera- ture the solid passes into the liquid; below this temperature the liquid passes into the solid. So it is for the transformation of one crystalline modification into another, as in the example of the two modifications of sulphur. Above the inversion temperature, one modification is stable; below the inversion temperature, the other. If a liquid is brought to a temperature slightly below its inversion temperature (the freezing point) and brought into contact with the more stable crystalline phase, it will assume the crystalline state, the rate of crystallisation being for small temperature differences proportionate to the degree of overcooling. A maximum rate, however, is attained when the degree of overcooling reaches a certain value, and below the temperature corresponding to this, the rate of crystallisation rapidly falls with further diminution of temperature (cp. p. 62). The same thing has been observed with the reciprocal transformation of crystalline modifications : at first the rate of transformation increases as the temperature falls below the tem- perature of inversion, afterwards however to diminish rapidly as the fall of temperature proceeds. No doubt monoclinic sulphur, cooled to a temperature considerably below zero, would exhibit little tendency to 264 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. pass into the more stable rhombic form, for even at ordinary tempera- tures the process takes a comparatively long time. We have already seen that an overcooled " glass " may remain for a very long time in contact with the more stable crystalline modification if the overcooling is sufficiently great, without crystallisation progressing at a sensible rate. If we heat the " glass," however, to a higher temperature, the transformation proceeds with ever-increasing rapidity until a tempera- ture a few degrees below the inversion point is reached. This is evidently comparable to the chemical transformation of solid ammonium cyanate into urea. Ammonium cyanate is the less stable form, and although it may be kept for a very long time at in contact with urea without undergoing appreciable transformation, the transition goes on at an increasing rate as the temperature is raised. Here the inversion point is not known, but it must at least be considerably over 80. The reverse transformation of urea into ammonium cyanate has only as yet been investigated in aqueous solution. When cyanic acid vapour condenses below 150, it forms cyamelide ; when it condenses above 150, it forms cyanuric acid; the chemical action being in both cases one of polymerisation, as cyanic acid has the formula CNOH, cyanuric acid the formula (CNOH) 3 , and cyamelide a still more complicated formula, expressed generally by (CNOH) n . This would indicate that cyanuric acid had the inversion point 150, cyamelide being the stable form below that temperature, and cyanuric acid the stable form above that temperature. The conversion of cyamelide into cyanuric acid above 150 has actually been observed, but the reverse action has not hitherto been noticed, probably on account of its slowness. If we construct a pressure- temperature diagram for the three phases, we shall find it exactly analogous to the diagram for the three physical states of aggregation of water (Fig. 9, p. 64). From the foregoing instances it is evident that considerable analogy exists between the effect of temperature on chemical and on physical change, especially when the chemical change is reversible : and it may often be found expedient to look upon apparently irreversible chemical changes as being in reality reversible but with a temperature of inver- sion so high that the reverse action has never been realised. It will be noted that in all of the above instances there can be no coexistence in equilibrium of two systems which are mutually insoluble and capable of reciprocal transformation, except at the inversion point. Above the inversion point one of the systems is stable, below the in- version point the other is stable ; at the inversion temperature itself, both systems are equally stable. This rule holds good both for chemi- cal and physical changes. If, on the other hand, the chemical systems are mutually soluble, there can be equilibrium at any temperature for which they only form one phase, the proportions of each system present changing in this case with the temperature. The fused mixture of xxiii EATE OF CHEMICAL TRANSFORMATION 265 ammonium thiocyanate and urea forms an example of this one-phase equilibrium of two reciprocally transformable systems. If we fuse thiourea, a certain proportion of it passes into ammonium thiocyanate with a measurable velocity, and we should expect that a different pro- portion would be transformed according to the temperature at which the system was maintained. Another example of one-phase equilibrium of reciprocally transformable systems is to be found in the solutions of dynamic isomerides (p. 253), or in the substances themselves if they be liquid. The actual transformation in this case also has been noted, and its velocity measured. Traces of water vapour have been found to play a very important part in determining the occurrence, or at least the rate, of many chemical actions. Thus ammonia and hydrochloric acid, when both in the gaseous state, unite readily under ordinary circumstances to form ammonium chloride. If precautions are taken, however, to have the gases absolutely dry, they may be mixed without any union taking place. On the other hand, ammonium chloride, when vaporised, dis- sociates to a very great extent into ammonia and hydrochloric acid, as is rendered evident by the vapour density being only about half the normal value calculated from the molecular formula NH 4 C1 by the help of Avogadro's principle. If the ammonium chloride is perfectly dry, however, the vapour density is normal, thus showing that no dissocia- tion has taken place. The reversible action NH 3 + HC1 : NH 4 C1 is therefore apparently dependent on the presence of traces of water for its occurrence either in the direct or the reverse sense. No satis- factory explanation of the action of the moisture has yet been given. It has not indeed been clearly established whether the action is alto- gether inhibited by the absence of water vapour or whether it still goes on, but at a greatly diminished rate. In the latter case the action of the water might be comparable to the action of acids in accelerating the inversion of cane sugar, the hydrolysis of ethereal salts, and the like. The following papers dealing with rate of chemical action may be con- sulted : HARCOURTand ESSON (Philosophical Transactions, 1866, p. 193, and 1867, p. 117) : Interaction of Permanganate with Oxalic Acid, and of Hydrogen Peroxide with Hydriodic Acid. R. WARDER (American Chemical Journal, vol. iii. No. 5) : Saponifica- tion of Ethyl Acetate. A. A. NOTES and G. J. COTTLE (ibid. vol. xxi. p. 250) : Reduction of Silver Acetate by Sodium Formate. J. WALKER and others (Journal of the Chemical Society, Ixvii. p. 489 ; Ixix. p. 195 ; Ixxi. p. 489): Transformation of Ammonium Cyanate into Urea. CHAPTER XXIV RELATIVE STRENGTHS OF ACIDS AND BASES IT is customary and correct to speak of sulphuric acid as a strong acid, and of acetic acid as a weak acid, and the statement is the outcome of our general experience of the chemical behaviour of these substances. In comparing two such acids there is no difficulty ; they are so different in their properties that no one could mistake their relative strengths. But if we compare two acids which are closer together in the scale of strength, say hydrochloric and nitric acids, it is impossible from 01 general chemical experience alone to say which is the stronger, am we must resort to a more exact definition of strength and to moi accurate experiment. A method which, when properly applied, leads to useful and consistent results, is that of the displacement of one acid from its salts by another. If an equivalent of sulphuric acid be added to a given quantity of an acetate in solution, the change in properties of the solution is sufficient to indicate that practically the whole of the sulphuric acid has been neutralised, and the corresponding quantity of acetic acid liberated. There is therefore no doubt that the sulphuric acid is much stronger than the acetic acid, being capable of turning the latter out of its salts. But the method must be applied with caution, or it leads to contradictory and inconsistent results. If, for example, we take sodium silicate in aqueous solution and add hydro- chloric acid, sodium chloride will be formed and silicic acid liberated, but yet at a very high temperature, as in the process of glazing earthen- ware, silicic anhydride in presence of water vapour is capable of decomposing sodium chloride with expulsion of hydrochloric acid. Without any further principle than that of displacement to guide us, the first experiment would show that hydrochloric is stronger than silicic acid, and the second that silicic acid is stronger than hydro- chloric acid. Again, if we pour aqueous acetic acid on sodium carbonate, there is immediate effervescence due to the expulsion of carbonic acid. Yet, if we pass a stream of carbonic acid into a saturated aqueous solution of sodium acetate, a precipitate of sodium CH.XXIV RELATIVE STRENGTHS OF ACIDS AND BASES 267 hydrogen carbonate soon makes its appearance, the acetic acid having been expelled from its salt by carbonic acid. Again, a solution of potassium acetate in nearly absolute alcohol is decomposed by carbonic acid to a great extent with precipitation of carbonate. From these experiments it is impossible to say whether acetic or carbonic acid is the stronger, since the expulsion takes place in different senses accord- ing to the conditions of experiment. There is, of course, no doubt among chemists that hydrochloric acid is much stronger than silicic acid, and that acetic acid is much stronger than carbonic acid. We must | therefore inquire more closely into the experiments which seem to j; point to the contrary conclusions. In the first place, it is obvious from the experiments themselves that we are dealing with actions which can, according to circumstances, take place in either sense, i.e. with balanced actions. Carbonic acid, or its anhydride and water, can always displace a little acetic acid from its salts, although it is a much weaker acid. If all the substances remain within the sphere of the reaction this displacement will not go far, as the reverse reaction will set in and soon establish equilibrium. The action expressed by the equation CH 3 . COOK + C0 2 + H 2 = KHC0 3 + CH 3 . COOH would therefore speedily come to an end if the products of the action were to remain and accumulate in the system. But if one of the ; products, say potassium hydrogen carbonate, is insoluble, or nearly so, j its active mass cannot increase beyond a certain small amount, however ; much of it may be formed, for it falls out of solution, i.e. the true : sphere of action, as soon as it is produced. Whilst, therefore, in aqueous ; solution, in which all the substances remain dissolved, the carbonic jj: acid succeeds in displacing only a small proportion of the acetic l| acid, in alcoholic solution it displaces a much larger amount, owing to tj the insolubility of one of the products of reaction. The comparatively ! great displacement in saturated sodium acetate solution is referable to the same cause, sodium hydrogen carbonate being very slightly soluble I' in a saturated solution of sodium acetate. In the case of the action of silica on a chloride, we have again one of the substances removed from the sphere of action as it is produced, viz. hydrochloric acid, which at the high temperature of the experi- ment escapes as vapour. In solution, only an infinitesimally small proportion of hydrochloric acid would be displaced by silicic acid, owing to the reverse reaction which would at once set in, but at the high temperature no reverse action is possible at all, since one of the reacting substances leaves the sphere of action as soon as it is formed. Similarly, we are not in a position to judge of the relative strengths of hydrochloric and hydrosulphuric acids by experiments with sulphides insoluble in water. Sulphuretted hydrogen will at once expel hydro- 268 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. chloric acid from copper chloride, even if excess of hydrochloric acid is present in solution. Yet sulphuretted hydrogen is a very feeble acid compared to hydrochloric acid. The displacement is due to the insolubility of the copper sulphide, which is removed from the sphere of action as it is produced. That such experiments lead to no definite conclusion may be seen by taking the same acids with another base. Though sulphuretted hydrogen is passed through a solution of ferrous chloride until the solution is saturated with it, a scarcely percep- tible precipitate of ferrous sulphide will be formed, and if a little hydrochloric acid is added to the solution from the beginning, no precipitate will be formed at all. Here, then, the same two acids exhibit totally different behaviour relatively to each other, according to the nature of the base for which they are competing. Now, in the above instances the acids have been selected of as widely different natures as possible, so that the fallacy of the reasoning based on the experiments mentioned is obvious. But similar fallacious reasoning is prevalent, and passes without detection, when experiments are discussed regarding acids where common chemical experience supplies no answer as to their relative strengths. Thus it is almost invariably a settled conviction in the minds of students that sulphuric acid is a stronger acid than hydrochloric acid because it expels the latter from its salts. No doubt this is the fact if we evaporate a solution of the chloride and sulphuric acid to dryness, or nearly so. But, of course, in this case the hydrochloric acid is expelled as vapour, and cannot therefore participate in the reverse reaction. The hydrochloric acid is expelled, not because it is a feebler acid than sulphuric acid, but because it is more volatile. These examples suffice to show that expulsion of one acid from its salts by another cannot be used as a proof that the latter is the stronger acid, unless the two acids are competing under circumstances equally favourable to both. The proper conditions are secured when the reacting systems form only one phase, namely, that of a solution. As soon as one of the components of the reacting systems is removed as a gas or as an insoluble solid or liquid, the system which does not contain that substance as one of its components is unduly favoured at the expense of the other system. If we are to compare the strengths of hydrochloric and sulphuric acids, then, we shall do best to take a soluble sulphate and add to it an equivalent of hydrochloric acid, the base being so chosen that the chloride formed is also soluble. Since all salts which have potash or soda as base are soluble, a potassium or sodium salt is generally selected, and the competing acid added to its aqueous solution. Thus equivalent solutions of hydrochloric acid and sodium sulphate may be mixed, and the composition of the resulting solution investigated, in order to ascertain in what proportion the base distributes itself between the two acids, the assumption being that the stronger acid takes the greater xxiv KELATIVE STRENGTHS OF ACIDS AND BASES 269 share of the base. In general, it is necessary to use a physical method for determining the composition of the solution, since the application of a chemical method would disturb the equilibrium. That an equilibrium is actually being dealt with is ascertairiable from the I fact that the solution has exactly the same properties in all respects, | whether the base was originally combined with the sulphuric acid or I with the hydrochloric acid, as will be presently shown in a numerical i example. The two methods which have been most extensively applied I are the thermochemical method of Thomsen and the volume method ! of Ostwald. When a gram molecule of sulphuric acid in fairly dilute solution > (about one-fourth molecular normal) is neutralised by an equivalent amount of caustic soda of similar dilution, a production of 313*8 I centuple calories is observed. The same amount of soda neutralised by hydrochloric acid is attended by a heat evolution of 274'8 K. Now if \ the addition of hydrochloric acid to a solution of sodium sulphate pro- duced no effect chemically, we should also expect no thermal effect. If, \: on the other hand, all the sulphuric acid were expelled from combination i with the soda, we should expect an absorption of 313'8 - 274'8 K = 39 K. Now an actual heat absorption of 3 3 '6 K per gram i molecule was observed by Thomsen. This would indicate that the greater proportion of the base was taken by the hydrochloric acid, an equivalent quantity of sulphuric acid being expelled from its combina- tion with the soda. If we assumed that the heat absorption were j directly proportional to the amount of chemical action, the proportion of sulphate converted into chloride would be 33'6^-39 = 0-86. This ;; proportion, however, is not the correct one, for the sulphuric acid liberated reacts with the normal sodium sulphate remaining, to produce ! i a certain quantity of sodium hydrogen sulphate, the formation of which is attended by absorption of heat, so that the total heat absorp- tion observed is too great. Special experiments show that the correc- tion to be applied in order to eliminate the effect of this action is 7 '6 K, the heat absorption due to the displacement of sulphuric by hydro- chloric acid thus being 33*6 - 7*6 = 26 K. The proportion of sulphuric acid expelled is therefore 26-7-39, or two-thirds. When, therefore, equivalent quantities of sulphuric and hydrochloric acids compete for a quantity of base sufficient to neutralise only one of them, the hydro- chloric acid takes two-thirds of the base and the sulphuric acid one- third. The reverse experiment of adding sulphuric acid to a solution of sodium chloride showed that the final distribution of the base between the acids was the same as above so far as could be judged from the heat effect. Since the hydrochloric acid always takes the larger share of the base, we conclude that it is the stronger acid, at least in aqueous solution. Thomsen, by working in this way, compiled a table of the avidi- ties of different acids, from which it is possible to tell at once how a 270 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. base will distribute itself between any two of them if all three sub- stances are present in equivalent proportions. The avidities of some of the commoner acids are given below : Acid. Nitric Avidity. 100 100 Sulphuric . Oxalic Orthoph osphoric Monochloracetic Tartaric . Acetic 49 24 13 9 5 3 In order to find the distribution ratio from this table, we proceed as follows. Let the acids be sulphuric and chloracetic, the avidities being 49 and 9 respectively. If the base and these acids are present in equivalent proportions, the base will share itself between the acids in the ratio of their avidities, i.e. the sulphuric acid will take -f--f and the chloracetic acid T 9 . Ostwald's volume method is based on similar principles. Instead of heat changes, the changes of volume accompanying chemical reactions are measured. The substances used by him were contained in aqueous solutions of such a strength that a kilogram of solution contained 1 gram equivalent of acid, salt, or base. The specific volumes of these solutions were carefully determined so that the change of volume pro- duced by chemical action might be ascertained. Thus the volume of a kilogram of potassium hydroxide solution was found to be 950 '6 6 8 cc., and of a nitric acid solution 96 6 '6 2 3 cc. If, on mixing these solutions, no change of volume occurred, the total volume would be 1917'291 cc. But the volume actually found on mixing the solutions was 1937'338 cc. The neutralisation of the acid and base is thus accompanied by an expansion of 20 '04 7 cc. Similarly, changes of volume accompany other chemical reactions, and the extent to which a given action has occurred can be measured by the volume change. A solution of copper nitrate had a volume equal to 3847 '4 cc., and an equivalent solution of copper sulphate 3 8 40 '3 cc. Solutions of nitric and sulphuric acids had the volumes 1933'2 and 1936'8 respectively. If no action occurred on mixing the copper sulphate solution with the nitric acid solution, the total volume would be 3840'3 + 1933-2 = 5773-5 ; if complete trans- formation into copper nitrate and sulphuric acid took place, the total volume would be 3847-4 + 1936-8 = 5784-2. The actual volume found by mixing the copper nitrate and sulphuric acid solutions was 5 780 '8, and by mixing the copper sulphate and nitric acid solutions 5 781 '3. These two volumes are practically identical, and we may take as their mean 5781'0. We have therefore the numbers All Copper Sulphate. Actual. No Copper Sulphate. 5773-5 5781-0 5784'2 Difference 7'5 3 '2 xxiv RELATIVE STRENGTHS OF ACIDS AND BASES 271 The actual equilibrium is evidently nearer the system containing no copper as sulphate than the system containing all the copper as sulphate, and if we assume direct proportionality, the base is shared by the nitric and sulphuric acids in the ratio of 7*5 to 3*2, or nitric acid takes 70 per cent of the base, leaving the sulphuric acid 30 per cent. This result is not quite accurate, as allowance has to be made for the slight volume changes consequent on the action of the respective acids on their neutral salts. When this correction is applied, it appears that I the nitric acid takes 60 per cent of the base and the sulphuric acid 40 I per cent. A table of avidities can be constructed for the different acids from ' similar data, and a comparison with Thomsen's avidities derived from r thermochemical experiments shows that the two methods yield results in harmony with each other, at least so far as relative order of the I acids is concerned. The actual avidity numbers differ considerably in many instances, but it has to be borne in mind that the thermochemi- cal measurements are on the whole much less accurate than the volume < measurements, and the numbers derived from them consequently less trustworthy. In special cases the distribution of a base between two acids may be studied by making use of other physical properties than those j already mentioned. For example, measurements of the refractive in- I dex of solutions often lead to satisfactory results, and also measure- r: ments of the rotatory power when optically active substances are in j question. The principle involved is identical with that just described, : any differences being merely differences in detail. A method which differs in principle from the distribution of a base I between two competing acids, and may also be applied to the deter- mination of the relative strengths of acids, is to ascertain the accelerat- ing influence exerted by different acids on a given chemical action. For example, the inversion of cane sugar has long been known to take place much more rapidly in presence of an admittedly strong acid like sulphuric or hydrochloric acid, than it does in presence of an equiva- lent quantity of an admittedly weak acid like acetic acid. The strong mineral acids have thus a greater accelerating effect than the weak organic acids, and it is natural to infer that an exact determination of i the specific accelerating powers of different acids might lead to a ?S knowledge of their relative strengths. There is no obvious connection ;' between this method and the preceding method of relative displace- I ment, but a connection exists, as will be shown later, and the results \ obtained are in general quite in harmony with each other. The method of sugar inversion as practised by Ostwald was performed in the following manner. Normal solutions of the various acids were mixed with an equal volume of 25 per cent sugar solution and placed in a thermostat whose temperature remained constant at 25. The rotation of each solution was taken from time to time, and 272 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. a velocity constant calculated according to the formula given on p. 255. The order of these velocity constants is the measure of the accelerating powers of the acids, and presumably a measure of their relative strengths. Another action well adapted to investigating the accelerating power of acids is the catalysis of methyl acetate (cp. p. 256). Ostwald mixed 1 cc. of normal acid with 1 cc. of methyl acetate, and diluted the mixture to 15 cc. This solution was then placed in a thermostat at 26, and its composition ascertained at appropriate intervals in the manner already indicated. A calculation of the velocity constant by the usual formula for unimolecular reaction gave the required measure of the accelerating power. A comparison of the results obtained by the different methods is given in the following table, the value for hydrochloric acid being made in each case equal to 100, in order to assist the comparison : Velocity Constants. Avidity. Sugar Inversion. Catalysis of Acetate. Hydrochloric 100 100 100 Nitric 100 100 91 '5 Sulphuric 49 53 547 Oxalic 24 18-6 17'4 Orthophosphoric 13 6 '2 Monochloracetic 9 4 '8 4 '3 Tartaric 5 ... 2'3 Acetic 3 0'4 0'35 It is at once evident that the order in which the acids follow each other is the same in all cases, and in especial it will be seen that the numbers expressing the accelerating powers of the acids are closely similar, although the accelerating influence was exerted on entirely different chemical actions. The avidity numbers differ considerably from the others, but the general parallelism of the results cannot be denied, and we are therefore justified in adopting the acceleration method as a means of measuring the relative strengths of acids, although its theoretical justification is not immediately obvious. It was pointed out by Arrhenius that if we arrange the acids in the order of their relative strengths, they are also arranged in the order of the electrical conductivities of their equivalent solutions. This may be seen in the following table, the first column of which contains the mean value of the velocity constants of sugar inversion and catalysis of methyl acetate, and the second that of the electric conductivities of equivalent solutions, all values being referred to that for hydrochloric acid as 100 : Velocity Constants. Electric Conductivity. Hydrochloric 100 100 Nitric 96 99'6 Sulphuric 54 65'1 Oxalic 18 19-7 Orthophosphoric 6*2 7 '3 Monochoracetic 4 '5 4 '9 Tartaric 2'3 2'3 Acetic 0'4 0'4 cxiv EELATIVE STRENGTHS OF ACIDS AND BASES 273 ?he parallelism is here unmistakable, the numerical values in the two olumns being often practically identical. At first sight it appears a matter of difficulty to associate the lectric conductivity of an acid with its strength, i.e. its chemical ctivity in so far as it behaves as an acid ; but the dissociation ypothesis of Arrhenius furnishes the clue to the nature of the con- ection. All acids in aqueous solution possess certain properties eculiar to themselves which we class together as acid properties, 'hus they neutralise bases, change the colour of certain indicators, are our to the taste, and so on. We are therefore disposed to attribute them the possession of some common constituent which shall account these common properties. On asking what aqueous solutions of he various acids have in common, we find for answer "hydrogen ions" f we adopt the hypothesis of electrolytic dissociation. Let us suppose he peculiar properties of acids to be due to hydrogen ions. How in hat case are we to explain the different strengths of the acids 1 Evidently on the assumption that different acids in equivalent solution r ield different amounts of these ions. The acid which in a, normal olution produces more hydrogen ions will be the more powerful acid, ' o that on this hypothesis the degree of dissociation of an acid urnishes a measure of its strength. But if we compare equivalent olutions of different acids under the same conditions, the electrical onductivity is closely proportional to the degree of dissociation of the dissolved substance. This arises from the fact that the speed of the .lydrogen ion is much greater than the speed of any negative ion with vhich it may be associated. The conductivity, then, of any acid solu- |don is due principally to the hydrogen ions it contains, so that if we lompare the conductivities of solutions of different acids, the values re obtain are nearly proportional to the relative amounts of hydrogen ons in the solutions, and thus to the relative strengths of the acids. Jince it is a very easy matter to measure the conductivity of solutions, ihis method of determining the relative strengths of acids has practi- ^ally superseded the other methods, especially in the case of the weaker >rganic acids. For them it is possible to calculate a dissociation con- ; -tant according to the formula given on p. 224, and this constant is r ery generally accepted as a measure of their strength, for which eason it is sometimes spoken of as the affinity constant of the acids. It will be remembered that the theoretical dissociation formula only .pplies to half-electrolytes, the strong and highly dissociated mineral ,cids giving only constants with the empirically modified formulae of iludolphi and van t' Hoff. These empirical constants might be used as * affinity constants " for the strong acids, since they, like the true disso- iation constants, give a measure of the relative dissociations of different icids independent of the dilution, but as their significance is doubtful, md the degree of constancy attained is not after all very great for ; : If now we mix these isohydric solutions, the volume becomes v + v', and the number of hydrogen ions m + m'. For the acid HA under the new conditions we have now the equilibrium equation (m + m'}m ^ ' (1 -m)(v + v) Dividing this equation by the first, we obtain (m + m')v m + m' v + v (v + v')m ~ ' m v m v' whence = - , m v m m' or = v v Now is the concentration of hydrogen ions in the acid HA, and v v is the concentration of hydrogen ions in the isohydric solution of the acid HA', and these two concentrations prove to be equal. We may say, therefore, that solutions of electrolytes containing a common ion are isohydric when the concentration of the common ion in the different solutions is the same. It is often convenient for purposes of calculation to imagine an actual mixed solution to be split up into its component isohydric solutions, which may then be ideally mixed at any time without any change in the dissociation of the electrolytes occurring. For example, a solution containing equivalent quantities of hydrogen acetate and sodium acetate may be imagined to exist in a rectangular vessel with a movable vertical partition through which water can freely pass, but not the dissolved substances. Let the hydrogen acetate be on one side of the partition and the sodium acetate on the other. The partition is now to be moved until the concentration of the common ion, acetion, is the same on both sides. Since the sodium acetate is highly dissociated, it must receive most of the water in order that the concentration of the acetion may be as small as that derived from the slightly dissociated hydrogen acetate. The position of the partition for isohydry must therefore be as shown in Fig. 42, which represents a horizontal section of the vessel, the liquid rising to the same level on both sides of the diaphragm. It must be borne in mind that con- 288 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. HI a Ac HAc centrating the solution of hydrogen acetate does not involve similar concentration of the hydrogen or acetate ions, for as the dilution diminishes, the degree of dissocia- tion, and therefore the proportion of ions, diminishes also, although at a smaller rate. Beginners are apt to reason that if the solution of one electrolyte is ten times more dissociated than an equival- Fia 42 - ent solution of another, it is only necessary to concentrate the second solution to a tenth of its volume in order that the degrees of dissociation of the two solutions may become equal. In view of the diminution of the degree of dissocia- tion as the dilution diminishes, a much greater degree of concentra- tion is necessary. If we consider the mixing of two salts which have no common ion, say NaCl and KBr, the problem becomes much more complicated. When the two salts are mixed, we have not only the substances origin- ally in solution, i.e. the undissociated salts NaCl and KBr, and their ions, Na, K, Cl, and Br, but also the new undissociated substances NaBr and KC1. Let there be prepared isohydric solutions of the different salts, NaCl being made isohydric with NaBr, by getting the sodium ions of the same concentration in both solutions ; KC1 may then be made isohydric with NaCl by making the chloride ions of the same concentrations in the two solutions. KBr may finally be made isohydric with KC1 by equalising the concentration of the K ions. Any two of these solutions then which possess a common ion may be mixed in any proportions without change in the dissociation. If we wish to mix all four, we must take volumes of the solutions such that the products of the volumes of reciprocal pairs are equal. If a, b, c, d be the volumes of the iso- hydric solutions of NaCl, NaBr, KC1, and KBr respectively, such that ad = be, then the solutions may be mixed in these proportions without change in the dissociation. Using a dia- grammatic representation similar to that adopted for mixtures of pairs of electrolytes with a common ion, we get the diagram Fig. 43, which fulfills the desired condition. The proof of the condition may be given on the supposition that 6 a NaBr NaCl d c KBr KCI FIG. 43. xxv EQUILIBRIUM BETWEEN ELECTROLYTES 289 the substances obey Ostwald's dilution law. Let a, b, c, d be the volumes of the isohydric solutions which when mixed will produce no change in the dissociation. Before mixing, the equilibrium of the sodium chloride will, assuming its quantity to be unity, be represented by the formula m m a a m 2 L I - m (1 -m)a where m is the degree of dissociation. Let the solutions be now mixed. The quantity of the sodium ion has now increased in the ratio of a + b to a, since b volumes of NaBr have been added to the original a volumes of NaCl, and the concentration of the sodium ions is the same in both solutions. But the volume in which this quantity is contained is now a + b + c + d, so that the active mass of the sodium ion is now a + b I m(a + b) . .. . . m x x = ; \ ' - Similarly the quantity of a a + b + c + d a(a + b + c + d) the chloride ion increases in the ratio of a + c to a, and its active mass m (a + c) becomes - . The undissociated proportion of sodium a(a + b + c + d) chloride remains the same as before, viz. 1 - m. We have therefore for the new equilibrium the equation m(a + b) m(a + c) a(a + b + c + d)' a(a + b + c + d) l-m ' a + b + c + d m 2 whence, since k is also equal to -, r- 5 (1 -m)a m 2 (a + b)(a + c) m 2 ( 1 - m)a\a + b + c + d)~ (I - m)a ' (a + b)(a + c) _ a(a + b + c + d)~ a? + ab + ac + be = a 2 + ab + ac + ad, be = ad, which was to be proved. According to Guldberg and Waage's Law, we should have for equilibrium in the balanced action the expression [NaCl] x [KBr] u 290 INTRODUCTION TO PHYSICAL CHEMISTEY CHAP. where the formulae in square brackets represent the active masses of the respective substances. This equilibrium formula takes no account of electrolytic dissociation of the various salts, and is only valid under certain conditions of dissociation. The correct formula is [diss. NaCl] x [diss. KBr] = [diss. NaBr] x [diss. KC1] ' as may be deduced from the above relation ad - be. Since all the solutions to which these letters refer are isohydric in pairs, i.e. have the same concentration of ions, the volumes a, b, c, d are proportional to the quantities of the ions in the various solutions, i.e. to the dissociated quantities of the salts, and not to their total quantities. When all , the substances are highly dissociated, Guldberg and Waage's Law leads to very nearly the same result as when the dissociation is con- sidered, and the same holds true when two of the four substances are highly disso- ciated. When, however, one or three of the substances are highly dissociated, there is usually a great discrepancy no. 44. between the two modes of calculating the equilibrium, Guldberg and Waage's Law being no longer even approximately true, except in special circumstances. As an example of the application of the theory of isohydric solu- tions as applied to the equilibrium of four dissociated substances, we may take the distribution of a base between two acids obeying Ostwald's dilution law and find the relation of the distribution ratio to the ratio of the dissociation constants of the acids. Let, as before (p. 275), one gram molecular weight of each of the substances HA, HA', and NaOH be dissolved in a certain volume of water, and let the solution thus obtained be ideally split up into isohydric solutions of the same ionic concentration i. We thus get the diagram Fig. 44. The volumes are again represented by a, &, c, d, and the following table gives the data necessary for the calculation, if x is the amount of HA neutralised by the soda : b NaA x a HA r-x d c NaA' HA' l-X X cxv EQUILIBRIUM BETWEEN ELECTROLYTES 291 HA. NaA. HA'. NaA'. Total quantity 1 -x x x I -x Dissociated quantity ia ib ic id ia ib ic id Degree of dissociation (m) = I -x x x I -x abed Dilution (v) 1 X X X I -X the concentration of the ions is the same in all the solutions, the lissociated quantities are equal to the volumes of the solutions multi- )lied by the common ionic concentration, i. The degree of dissociation, n, is the ratio of the dissociated to the total quantity, and the volume livided by the quantity contained in it gives the volume which contains init quantity measured in gram molecules, i.e. the dilution v. The icids by supposition obey the theoretical dilution law m 2 (1 m)v I8y a simplification which has already been adopted when the degree |D dissociation is small, we may neglect m in comparison with 1, and Iwrite the dilution formula = k Now, substituting the above values Df m and v for the acid HA, we obtain 2 1- x) - = k, l-x and similarly for the acid HA', we obtain = K. X Division then gives ax Now for equilibrium we have ad = be, or a/c = b/d, whence As before, we may assume that the two sodium salts are dissociated 292 INTRODUCTION TO PHYSICAL CHEMISTEY CHAP.J i to an equal extent, so that in their case the ratio of the dissociated quantities is the ratio of the total quantities, i.e. ib x id I -x We thus obtain finally the relation x 2 _k x /fc (T^~&' )0r r^~ v &" that is, the ratio of distribution of the base between the two acids is- equal to the ratio of the square roots of the dissociation constants of the acids, a result already obtained on p. 275 under the same assump- tions. The student who desires to familiarise himself with the equilibrium electrolytes in solution is advised to study the subject from the point of view of isohydric solutions, in particular when dealing with two electrolytes containing a common ion, or with double decompositions between electrolytes. In this last case the important fact to bear in \ mind is that the product of the dissociated quantities on one side of the equation is equal to the product of the dissociated quantities on| the other. If, for example, we are dealing with the double decom- position CH 3 . COONa + HC1 ^ CH 3 . COOH + NaCl, 7H.J 9712 ^3 ^4 and the quantities of these substances, when equilibrium has been) attained, are m v m 2 , m 3 , m 4 , with the degrees of dissociation d v d y d^ d \ in the mixed solution, we have always the relation This relation we can combine with our knowledge of the general nature of the dissociation of the various substances, and its variation with dilution, to ascertain the actual character of the equilibrium. Thus Arrhenius, to whom the theory is due, has shown that the avidities of two monobasic acids at a given dilution are approximately proportional to the degrees of dissociation which they would have if each were dissolved separately in the given volume of solvent. This we showed above to be the case for two weak acids, but it is equally true if both acids are strong, or if one is strong and the other weak. In the case of dibasic acids, like sulphuric acid, the theory cannot easily be applied owing to the excessively complicated nature of the equilibrium caused by the presence of acid salts (cp. p. 285). It is easy, too, to prove from the theory that the degree of dissociation of a weak acid in presence of one of its salts is nearly ixxv EQUILIBRIUM BETWEEN ELECTROLYTES 293 nversely proportional to the quantity of salt present. If the weak icid should be in presence of several strongly dissociated electrolytes, t can also be shown that its degree of dissociation will be the same as f the dissociated parts of these electrolytes were the dissociated parts j)f a salt of the acid itself. We have now to examine the nature of the equilibrium between hich determines the solubility is here supposed to be that between le solid silver bromate and the undissociated silver bromate in the )lution. The concentration of this last will remain constant if the .jmperature remains at 24*5 and the solvent remains water. The ddition of a small quantity of a perfectly neutral substance, such as .cohol, sugar, and the like, does not appreciably affect the solubility ; any substance in water, since the nature of the solvent practically ins the same. Besides the undissociated silver bromate in the tion, we have silver ions and bromate ions. We can increase the mcentration of silver ions by adding a soluble silver salt, and we in increase the concentration of bromate ions by adding a soluble iromate. Suppose that we add such a quantity of silver nitrate as to 294 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP double the number of silver ions after equilibrium has been attained The concentration of the undissociated silver bromate will, by hypothesis remain the same as before. But the dissociation equilibrium of silve] bromate requires that the product of the concentrations of the ion; will be equal to a constant into the concentration of the undissociatec salt, i.e. will remain constant. If, therefore, the concentration of th( silver ions is doubled, the concentration of the bromate ions must b< halved, in order that the product of the two may have the same valu< as before. Bromate ions can only fall out of solution along with ar equivalent quantity of some positive ion, and since the only kind o positive ion in the solution is the silver ion, bromate of silver must b< precipitated in order to re-establish equilibrium. The effect, then, o adding silver nitrate to the silver bromate solution is to diminish th< solubility of the silver bromate, and that in a degree depending 01 the amount of silver nitrate added. The addition of a soluble bromati acts in precisely the same way. The number of bromate ions a equilibrium is increased, and the number of silver ions must b< proportionately diminished in order to secure the constancy of th< product of the bromate and silver ions. The following numerical example will afford an insight into th< mode of calculation. As has already been stated, the concentration o a saturated silver bromate solution at 24*5 is 0*0081 normal. If w< assume the salt to be entirely dissociated, the product of the ions is 0-0081 x 0-0081 =0-0000656. Now a quantity of silver nitrate is added, which, when dissolved ii the same water as contains the silver bromate would make the solutioi 0*0085 normal with respect to silver nitrate. Again we assume tha the silver nitrate is entirely dissociated. Suppose that the concentra tion of the silver bromate now remaining in the solution is x, a smalle quantity than before. The concentration of silver ions will then b' 0'0085 + x, and the concentration of bromate ions will be x. We hav< therefore the product of these concentrations equal to the former pro duct, i.e. (0-0085 +z)z = 0-0000656, whence We should consequently expect the addition of silver nitrate to reduc< the strength of the saturated solution of silver bromate from 0*008 '. to 0-0049. An actual determination showed that the solubility wa: reduced to 0*0051, which is in fair agreement with the theoretica result. It must be noted, however, that the theoretical result wa deduced on the erroneous assumption that the degree of dissociatioi of the various substances remained the same throughout the experi ments. This is, of course, not the case, as the degree of dissociatioi xxv EQUILIBRIUM BETWEEN ELECTROLYTES 295 at the dilutions considered is not equal to unity, and is diminished on the addition of the silver nitrate. It is easy, however, to take account of the change in the degree of dissociation of the silver salts by making use of conductivity determinations, although the formula for equilibrium then becomes somewhat complicated. Making the necessary corrections in the above case, the theoretical number comes out equal to 0*00506, which is very nearly the value observed for the solubility. Since silver nitrate and sodium bromate have practically the same effect so far as dissociation is concerned, we should expect equivalent quantities of these two salts to diminish the solubility of silver bromate equally. Experiment shows that an amount of sodium bromate equivalent to the silver nitrate added in the above experiment diminishes the solubility to 0*0052, a value very nearly identical with the former value. When two sparingly soluble salts yielding a common ion are shaken up with the same quantity of water, each diminishes the solubility of the other in a degree which can be calculated as in the previous instance. Thus the saturated solutions of thallium chloride, T1C1, and thallium thiocyanate, T1SCN, have a concentration of 0*0161 and 0*0149 respectively in gram molecules per litre. For the constant product of ionic concentrations we have, therefore, 0*0 16 1 2 and 0*0149 2 , if each is fully dissociated into ions. If the solubility of the chloride in presence of the thiocyanate is x, and the solubility of the thiocyanate in presence of the chloride is y, these two numbers give the concentrations of the chloride and thiocyanate ions respectively, while their sum, x + y, gives the concentration of the thallium ion. We thus obtain the simultaneous equations 2(2* + */) = 0-01 61 2 , whence x = 0*0 11 8, and y = 0*0101, the numbers found by experiment being in good agreement, viz. 0*0119 and 0*0107. By taking account of the actual degree of dissociation, the harmony between the experi- mental and calculated values is even more marked. The lowering of the solubility of an electrolyte by the introduction into the solution of another electrolyte possessing a common ion with the first is a phenomenon of very general occurrence, the only exceptions being when the two substances form a double salt or act on each other chemically in the solution. Instances of the application of the theoretical results will be given in the next chapter. CHAPTER XXVI APPLICATIONS OF THE DISSOCIATION THEORY WHEN many of the ordinary chemical reactions are looked upon from the standpoint of the theory of electrolytic dissociation, they present an aspect very different from that to which we are accustomed. The neutralisation of strong acids by strong bases in dilute solution is a typical example. If the acid is hydrochloric acid, and the base sodium hydroxide, we have the equation Now on the dissociation theory all the substances concerned in this action are highly dissociated in aqueous solution, the water itself being the only exception. Writing, then, the equation for the ions, we obtain H* + Cr + Ha + OH' = Na + Cl' + HOH, or, eliminating what is common to both members of the equation, The neutralisation of a strong acid by a strong base consists then essentially in the union of hydrogen and hydroxyl ions to form water. So long as the base, acid, and salt are fully dissociated, their nature makes no difference whatever on the character of the chemical act of neutralisation. This we find to be in conformity with many experimental facts. For example, the heat of neutralisation of one equivalent of a strong acid in dilute solution by a corresponding quantity of a strong base is very nearly 137 K, as the following table shows, the base used being caustic soda : Acid. Heat of Neutralisation. Hydrochloric ....... 137 K Hydrobromic ....... 137 Hydriodic ....... 137 Chloric ........ 138 Bromic ........ 138 lodic ....... ,138 Nitric . ... 137 CH.XXVI APPLICATIONS OF DISSOCIATION THEORY 297 A similar table for the heats of neutralisation of bases by an equivalent of hydrochloric acid shows that the heat of neutralisation is independent of the base, as long as it is fully dissociated. Lithium hydroxide Sodium hydroxide Potassium hydroxide Thallium hydroxide Barium hydroxide . Strontium hydroxide Calcium hydroxide Tetramethylammonium hydroxide Heat of Neutralisation. 138 K 137 137 138 139 138 139 137 When we deal with weak acids or weak bases the heats of neutralisa- tion often diverge greatly from the mean value for the strongly dissociated substances. Thus the heats of neutralisation of some comparatively feebly dissociated acids by sodium hydroxide are given in the following table, the acids not being so feeble, however, as to suffer sensible hydrolysis in aqueous solution (cp. p. 278) : Acid. Metaphosphoric acid Hypophosphorus . Hydrofluoric acid . Acetic acid Monochloracetic acid Dichloracetic acid . Heat of Neutralisation. 143 K 151 163 134 143 148 Corresponding numbers for weak bases neutralised by hydrochloric acid are : Base. Heat of Neutralisation. Ammonium hydroxide . . . . . 122 K Methylammonium hydroxide . . . . 131 ,, Dimethylammonium hydroxide . . . 118 ,, Trimethylammonium hydroxide . . . 87 ,, The explanation of these divergences from the value for highly dis- sociated substances is simple. Ammonium hydroxide is only feebly dissociated at the dilution considered; the chemical action is not chiefly but H' + NH 4 OH = H 2 + i.e. from the "normal" heat of neutralisation must be subtracted the heat necessary to decompose NH 4 OH into the ions NH 4 ' and OH'. The heat of ionic dissociation for acids and bases is not as a rule great, so that in the majority of cases the heat of neutralisation even of weak acids and bases does not greatly diverge from the value 137 K. The divergence may be in the one direction or the other, according as heat is absorbed or developed on the ionic dissociation (cp. p. 252). When acids or bases are so weak that their salts undergo extensive 298 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. hydrolysis in aqueous solution, i.e. are partially split up into free acid and base, the heat of neutralisation is very small. This is owing to the fact that the neutralisation is incomplete, free hydrogen ions or free hydroxyl ions remaining in the solution. If we consider the displacement of a weak acid from its salts by a strong acid in the light of the dissociation hypothesis, we find that the resistance of the weak acid to dissociation is the determining circumstance in the reaction. Thus if the salt is sodium acetate and the strong acid is hydrochloric acid, the customary equation becomes Na + C 2 H 3 2 ' + H' + Cr = Na' + Cl' + HC 2 H 3 2 , or C 2 H 3 2 ' + H* = HC 2 H 3 2 . The action is essentially a union of hydrogen ions with acetions, and the nature of the salt or of the strong acid is a matter of indifference, provided that they are both almost dissociated at the dilution under consideration. Similarly, the displacement of a weak base from its salts by a highly dissociated base consists essentially in the union of a positive ion with a hydroxyl ion, thus NH; + cr + Na* + on 7 = Na + or + NH.OH ' ' The solution Of a metal in an aqueous and strongly dissociated acid is principally transference of a positive electric charge from hydrogen to the metal : thus if zinc is the metal and hydrochloric acid the highly dissociated acid, we have Zn + 2H' + 2Cr = Zn" + 2C1' + H 2 , or Zn + 2H' = Zn" + H 2 . Similarly, the displacement of bromine from a soluble bromide by chlorine is chiefly a transference of an electric charge from bromine to chlorine : C1 2 + 2K* + 2Br' = Br 2 + 2K' + 201' or Cl 2 + 2Br' = Br 2 +2Cl'. When two highly dissociated salts are brought together in solution, we have an ionic equation such as the following, if double decomposition is supposed to occur : Na' + Cr + K* + Br' = Na' + Br' + K' + Cl'. Both sides of this equation are the same, i.e. no chemical change has taken place at all. This of course only holds good as long as all the xxvi APPLICATIONS OF DISSOCIATION THEORY 299 substances remain in the solution. If the solution is evaporated, that salt which is the least soluble will in general fall out first. We obtain a similar equation for the action of a strong acid on the salt of an equally strong acid, for example H' + 01' + K* + Br' = H' + Br' + K* + Cl'. Here again both sides of the equation are the same, and thus no action has taken place. It is usual to say that in this case the base is equally divided between the two acids, but it may be seen from the above equation that the ions are free to combine in any way according to circumstances. If one of the acids is less dissociated than the other, then more of it will exist in the undissociated state, so that the other acid is said to have taken the greater share of the base. From the examples already given it is evident that the equations involving ions are usually of a much more general character than the ordinary chemical equations, and more frequently bring out the essential phenomenon common to a number of actions of the same type, as, for instance, neutralisation, and the displacement of a weak acid or base by a stronger. The special actions employed in testing for metallic and acid radicals are practically always reactions of the ions, so that our ordinary tests are tests for ions. Copper, for example, gives a black precipitate with hydrogen sulphide, but not under all conditions. As long as the copper to be tested for is in the form of a cupric ion Cu", the black precipitate is formed when sulphuretted hydrogen is introduced into the solution. But if the copper ceases to be a copper ion, and becomes part of a more complex ion, the precipitation will not take place. This is the case if we add potassium cyanide to the solution of a cupric salt until the original cyanide precipitate is dis- solved. The copper in the solution is then in the state of the complex salt, usually written 2KCN , Cu(CN) 2 . The formula of this salt should be written K 2 Cu(CN) 4 , for on solution in water it dissociates electrolyti- cally into the positive ions 2K' and the complex negative ion Cu(CN) 4 ". The copper is no longer in the form of the cupric ion, but exists merely as a part of the complex ion, and has in this state no reactions of its own. Sulphuretted hydrogen added to such a solution produces no pre- cipitate, and use is made of this fact to separate copper from cadmium. Cadmium, it is true, when its salts are treated with excess of potassium cyanide, ceases for the most part, like copper, to be a positive ion, and enters into the composition of the complex negative ion Cd(CN) 4 ", but this ion is not by any means so stable as the corresponding ion containing copper. All such compound ions have the tendency to split up into simple ions, according to equations like the following : Cd(CN)/ = Cd(CN) 2 + 2CN', Cd(CN) 2 = Cd" 300 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. and this tendency exists to different extents with different ions. With the complex ion containing cadmium it is very pronounced ; with the complex ion containing copper it is much less evident. In the solution of K 2 Cu(CN) 4 there is thus a scarcely appreciable quantity of the cupric ion Cu", while in the solution of K 2 Cd(CN) 4 the ion Cd" exists in moderate proportions. The former solution then is scarcely affected by hydrogen sulphide, while the latter is freely precipitated. A solution of potassium ferrocyanide, K 4 Fe(CN) 6 is almost entirely free from ferrous ions Fe", and exhibits none of the ordinary re- actions of ferrous salts. The commonest reaction for silver is the production of a precipi- tate of silver chloride by the addition to the silver solution of a soluble chloride. This reaction is for the silver ion Ag', and not for silver in general. If we add potassium cyanide to a silver solution until the original precipitate of silver cyanide redissolves, the addition of a soluble chloride fails to produce a further precipitate. Here again the silver has become part of a fairly stable complex negative ion, Ag(CN) 2 / , which gives off very few silver ions by dissociation, so that the ordinary reagents for the silver ions may fail to detect their presence. The same holds good for the solution of a silver salt in presence of sodium thiosulphate. When we add this salt to a solution of silver nitrate, we obtain first a white precipitate of silver thiosulphate, Ag 2 S 2 3 , which on further addition of sodium thiosulphate dissolves with formation of the double salt NaAgS 2 O 3 . This salt dissociates chiefly into the ion Na' and the complex negative ion AgS 2 3 ', so that very few silver ions are at any one time in the solution, and the ordinary reagents produce none of the characteristic silver reactions. A change in the quantity of electricity associated with a positive or negative radical is accompanied by an entire change in the pro- perties of the radical. Thus the reactions of the ferrous ion Fe" are entirely different from the reactions of the ferric ion Fe*" ; and the reactions of the ion of the permanganates MnO/ differ greatly from the reactions of the ion of the manganates Mn0 4 ". In connection with such changes in the electric charges of ions, the student will find it useful to remember that addition of a positive charge or removal of a negative charge corresponds to what is generally known as oxida- tion in solution ; and that removal of a positive charge or addition of a negative charge corresponds to reduction. Thus we are said to oxidise a ferrous salt to a ferric salt when we convert the ion Fe" into Fe'", or reduce a permanganate to a manganate when we convert the ion MnO/ into the ion MnO/. In the first instance a positive charge is removed ; in the second a negative charge. If we write the equation 2FeS0 4 + H 2 S0 4 + C1 2 = Fe 2 (S0 4 ) 3 + 2HC1, on the supposition that all the electrolytes are fully dissociated, we obtain xxvi APPLICATIONS OF DISSOCIATION THEOEY 301 2Fe" + 2H' + 3SO/ + C1 2 = 2Fe'" + 2H* + 3S0 4 " + 201' or 2Fe" + C1 2 = 2Fe" + 2C1'. The whole action, from this point of view, reduces itself to the simultaneous appearance of positive and negative charges. The ferrous ion assumes a positive charge and is "oxidised" to the ferric ion. The uncharged chlorine assumes a negative charge and is " reduced " to the chloride ion. In this instance no oxygen has been transferred, so that it is only by analogy that we can call such a process one of oxidation. It is precisely in these cases, however, that the above mode of viewing the action is sometimes of service. When actual transference of oxygen takes place, the composition alters as well as the charge of the ions, and the ionic conception of the process can only be applied to groups of substances which do not change in composition. Thus if we consider the action 10FeS0 4 + 8H 2 S0 4 + 2KMn0 4 = K 2 S0 4 + 2MnS0 4 + 5Fe 2 (S0 4 ) 3 + 8H 2 to take place at such an extreme degree of dilution that all the electro- lytes are fully dissociated, the equation becomes lOFe" + 16H' + 2K* + 18S0 4 " + 2Mn0 4 ' = lOFe'" + 2K' + 2Mn" + 18SO/ + 8H 2 0; or lOFe" + 16H' + 2Mn0 4 ' = lOFe'" + 2Mn" + 8H 2 0. Here again we have the " oxidation " of the ferrous to the ferric ion. The group 16H' + 2Mn0 4 ' has lost twelve positive and two negative charges in becoming the group 2Mn" + 8H 2 0, i.e. has lost on the whole ten positive charges, and has therefore been " reduced " by this amount. When a metal passes into solution as an ion, it gains one or more positive charges, and thus acts as a reducing agent. Thus in the action Zn + H 2 S0 4 = ZnS0 4 + H 2 ; or Zn + 2H* = Zn" + H 2 , the zinc has been " oxidised " to the state of a positively charged ion, while the hydrogen has been " reduced " from the ionic condition to the state of free hydrogen. It is scarcely customary to apply the terms oxidation and reduction to the passage of hydrogen, or a metal from the free state to the state of combination in an acid or salt, but the application is obviously justifiable. Free zinc and free hydrogen are undoubtedly reducing agents under proper conditions, while the ionic zinc or hydrogen, in hydrogen or zinc sulphates, can in no sense be looked on as reducing substances in dilute solution. Many applications are made in the laboratory and in the operations 302 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. of technical chemistry of the diminution in solubility suffered by a salt, acid, or base when there is added to the solution another electro- lyte having one ion in common with the original electrolyte (p. 293). If we wish to prepare a pure specimen of sodium chloride, we make a strong solution of the impure salt, and pass hydrochloric acid gas into it, or add to it strong hydrochloric acid solution. The bulk of the sodium chloride is precipitated, and in a higher state of purity than the salt originally dissolved. Suppose the sodium chloride solution to be saturated, or nearly saturated. The addition of the highly disso- ciated and extremely soluble hydrogen chloride greatly increases the number of chloride ions, and the number of sodium ions must con- sequently be diminished by separation of sodium chloride, in order that the product of the ionic concentrations shall be maintained nearly constant. Even if the solution is not nearly saturated to begin with, the addition of chloride ions in sufficient quantity from the hydrogen chloride may bring the ionic product up to the constant value, and thus determine the precipitation of a portion of the sodium chloride. If the above process of purification is to be effective, it is essential that the added substance should be considerably more soluble than the original substance, especially if the two are about equally dissociated, as is the case with most salts, strong acids, and strong bases. If we attempted to precipitate a saturated solution of the soluble sodium chloride by the addition of the comparatively sparingly soluble barium chloride, we should find very little sodium chloride to be thrown down. This is because the ionic solubility product of barium chloride is much smaller than the corresponding product for sodium chloride, owing to the smaller solubility and also to some extent to the smaller degree of dissociation. The addition of barium chloride contributes therefore very few chloride ions to the salt solution, and consequently the sodium ions are not greatly reduced in number by precipitation. The effect is all the smaller, because the great number of chloride ions from the sodium chloride necessitates a very small number of barium ions from the barium chloride, in order that the ionic solubility pro- duct may not be exceeded, i.e. the solubility of barium chloride in saturated salt solution is much lower than in water, so that very little of the salt passes into solution to displace the sodium chloride. The sodium salts of aromatic sulphonic acids are often obtained pure from solution by the addition of sodium chloride or caustic soda. These salts are, as a rule, much less soluble than either sodium chloride or sodium hydroxide, and are therefore thrown out of solution in great part when sodium ions from strong brine or solid caustic soda are added. Organic acids, especially those which are not highly dissociated, may often be readily precipitated from aqueous solution by the xxvi APPLICATIONS OF DISSOCIATION THEORY 303 addition of hydrogen chloride, the hydrogen ion being here the active substance. Even a comparatively soluble and highly dissociated acid like sulphocamphylic acid may be thrown out of its aqueous solution almost entirely by saturating with gaseous hydrogen chloride. The salting out of soap is one of the oldest applications of the principle under discussion. When a fat is saponified with a moderately dilute solution of caustic soda, the whole gradually passes into solution, and to produce a good soap it is necessary to separate the sodium salts of the fatty acids which constitute it from the glycerine formed during the saponification and the excess of caustic alkali that was employed. The separation can be easily effected by the addition of common salt or a strong brine. The sodium salts of the higher fatty acids are comparatively slightly soluble in water, and thus the addition of a soluble salt like sodium chloride throws them almost completely out of solution as a curdy mass. If the fat is saponified by a strong solution of caustic soda, there may be enough sodium in the form of ions in this solution to throw the soap out as it is produced. If a fat is saponified with potash instead of with soda, a solution of the potassium salts of the fatty acids is obtained. This solution, if strong enough, assumes on cooling a consistency expressed by the name of " soft " soap. Soft soaps are not in general salted out as such, but are used while still mixed with glycerine and excess of alkali. If we wish to obtain the potassium salts free from these admixtures, we may do so by adding to the solution a strong solution of potassium chloride. The potassium salts of the fatty acids then separate out as a somewhat gelatinous mass, containing a considerable quantity of water and still retaining the characteristics of a soft soap. In the old process of manufacturing hard or soda soaps, the fat was saponified with potash, as that alkali was most readily obtainable from wood ashes. To the solution obtained on saponification sodium chloride was then added. The effect of this was to throw out, not a soft potash soap, but a hard soda soap, in virtue of a decomposition taking place between the sodium chloride and the potassium salts, whereby potassium chloride and the sodium salts were produced. These sodium salts were then thrown out by the excess of sodium chloride. The addition of sodium chloride to a solution of the potassium salts could not to any considerable extent throw them out as such from solution, for these substances have no common ion. If we consider that potassium ions, sodium ions, chloride ions, and the ions of the fatty acids exist simultaneously in the solution, it is evident that these free ions may combine in pairs in two ways, the sodium which was originally associated with the chlorine having now the choice of combining with the ions of the fatty acids. This actually occurs, because the ionic solubility product of the sodium salts of these acids is much less than the corresponding magnitude for the potassium 304 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. salts, the sodium salts being much less soluble. Under the given con- ditions the actual ionic product of the sodium and the negative ions exceeds the solubility value, and thus a portion of the sodium salts falls out of the solution, the potassium remaining behind mostly as potassium chloride. If we consider the saturated aqueous solution of any salt, the addition of a non-electrolyte will in general affect the solubility to a much smaller extent than the addition of an electrolyte containing one ion in common with the substance originally dissolved. There will, of course, always be some effect, for the solvent is changed by the addition of the foreign substance, and any change in the nature of the solvent has its effect on the solubility of the substance considered. If the substance added is not itself a solvent for the original substance the general effect will be slight precipitation from the saturated aqueous solution. If we add a small quantity of an electrolyte which contains no ion in common with the original electrolyte dissolved, the effect is generally to diminish the ionic dissociation product of the latter, and thus increase its solubility. This comes about because the addition of the second electrolyte produces double decomposition, so that on the whole more of the original ions go to form part of undissociated molecules. If double salts or acid salts may be formed, of course the equilibrium is thereby greatly complicated, and it is impossible to tell without further information how the addition of the new substance may affect solubilities. The addition of a salt of acetic acid to the acid itself is fre- quently carried out in the operations of analytical chemistry in order to reduce the strength of the acid, i.e. in order to reduce the concentration of hydrogen ions (p. 283). If we take a solution of ferrous sulphate and add to it a large excess of sodium acetate, it is comparatively easy to precipitate the iron as ferrous sulphide by means of sulphuretted hydrogen. If no sodium acetate is added, the precipitation does not take place, a dark coloration at most being effected. The old explanation of this difference was that by the addition of sodium acetate to the ferrous sulphate solution the acid liberated by the hydrogen sulphide according to the equation would at once act on the sodium acetate, producing sodium sulphate and acetic acid : H 2 S0 4 + 2NaC 2 H 3 2 = 2HC 2 H 3 2 + Na 2 S0 4 . The ferrous sulphide was supposed to be easily soluble in sulphuric acid, but insoluble in the acetic acid. This explanation is insufficient, however, for if we take ferrous sulphate and add to it no more sodium acetate than is necessary for double decomposition, the precipitation takes place to a slight extent only ; and if we now add acetic acid, the xxvi APPLICATIONS OF DISSOCIATION THEORY 305 precipitate originally formed dissolves up. Acetic acid, therefore, in these circumstances dissolves ferrous sulphide. If, however, we now add more sodium acetate to the same solution, the ferrous sulphide is reprecipitated. The reprecipitation is therefore due to the reduction of the degree of dissociation of the acetic acid by the addition of sodium acetate, and not to double decomposition alone, as was formerly supposed. Ammonium chloride acts in a similar way on a solution of ammonium hydroxide, greatly reducing the dissociation and therefore the strength of the base. It was stated on page 293 that the addition of any strongly dis- sociated electrolyte, say sodium chloride, would have practically the same effect on the dissociation of acetic acid as the addition of a highly dissociated acetate. Whilst this is true, it must not be supposed that the addition of sodium chloride to a solution of a ferrous salt, which has enough acetic acid in it just to prevent precipitation by sulphuretted hydrogen, will have the same ultimate effect as an equivalent quantity of sodium acetate. The degree of dissociation of the acetic acid is indeed diminished in the same ratio as before, but hydrochloric acid is simultaneously formed by double decomposition, and as this is highly dissociated, the number of hydrogen ions in the solution, after the addition of the sodium chloride, is rather greater than before, so that there is an actual increase in the total active acidity of the solution. Precipitation of the sulphide therefore does not occur. In analytical chemistry it is a common practice to wash a precipi- tate with the fluid precipitant, especially in quantitative operations. The theoretical basis of this is, of course, that the sparingly soluble precipitate has an ion in common with the precipitant, so that it is less soluble in the solution of the latter than in pure water. If other circumstances permit, therefore, it is advisable to wash with a diluted solution of the precipitant rather than with water alone. When solutions of two electrolytes are brought together, and it is theoretically possible by double decomposition to obtain a substance which is insoluble, then, in general, the double decomposition actually takes place. Thus when any sulphate is added to any barium salt, double decomposition invariably takes place, barium sulphate being deposited. From the point of view of the dissociation theory the explanation is at once apparent. The solubility product of the ions of barium sulphate is very small, and as soon, therefore, as barium ions and sulphate ions are brought together in quantity to exceed this solubility limit, barium sulphate fall out. Now all barium salts give barium ions freely in solution, and all soluble sulphates behave in like manner. Precipitation of h^iium. sulphate therefore invariably occurs, unless, indeed, the solu- ti-^ are so extremely dilute that the solubility product is not reached. Exceptions to this rule are only encountered when there is the possibility of the existence or formation of complex ions, or when an acid or a base is one of the pair of substances. If we add a solution x 306 INTRODUCTION TO PHYSICAL CHEMISTEY CHAP. of silver sodium thiosulphate to a solution of sodium chloride, there is the theoretical possibility of the action NaAgS 2 3 + NaCl = Na 2 S 2 3 + AgCl, but this action does not occur because there are so few silver ions given off by the sodium silver thiosulphate that the solubility product of silver chloride is not exceeded, practically all the silver in solution being in the form of the complex ion, AgS 2 3 '. For the same reason, where cyanides or ammonia are present, there is frequently no precipitation where such might be expected by double decomposition. If sulphuric acid itself is added to a barium salt, barium sulphate is precipitated as readily as if any other sulphate had been employed. It is different in the case of tartrates. If calcium chloride is added to sodium tartrate solution, a precipitate of calcium tartrate is formed. Should tartaric acid, however, be used instead of sodium tartrate, no precipitate is produced. The reason for this difference in behaviour is not far to seek. When sodium tartrate is employed, tartrate ions are abundantly present in the solution, the sodium salt being highly dissociated, and the solubility product of calcium tartrate is far surpassed. When hydrogen tartrate is employed, we have compara- tively few tartrate ions, for tartaric acid is not highly dissociated, and their number is still further reduced by the presence of the other highly dissociated substances (i.e. the calcium chloride, and the comparatively small amounts of hydrogen chloride and calcium tartrate formed by the double decomposition), which diminish greatly the degree of dissociation of the least dissociated substance, viz. tartaric acid. The ionic product of calcium and tartrate ions therefore falls short of the solubility product of calcium tartrate, and there is no precipitation. For the same reason, some metallic solutions are easily precipitated by an alkaline sulphide and not at all by sulphuretted hydrogen. In general, it may be said that when all the substances concerned are highly dissociated in solution, and complex ions are excluded, the precipitation occurs when there is the theoretical possibility of it ; while if one of the reacting substances is feebly dissociated, the precipitation may only take place to a limited extent or not at all. The precipitation may be almost perfect, even when a feebly dissociated substance is concerned, if the solubility product is vanishingly small, i.e. if the substance is scarcely at all soluble. This is the case, for example, with silver sulphide, so that sulphuretted hydrogen, although very sparingly dissociated, easily precipitates this substance from a solution of silver nitrate. In intimate connection with the formation of precipitates on mixing electrolytic solutions, there is the solubility of precipitates in solu- tions of electrolytes. It must be borne in mind that the so-Ci'^' 'I insoluble substances are merely sparingly soluble substances, and that the presence of electrolytes in the solvent water only alters the xxvi APPLICATIONS OF DISSOCIATION THEOKY 307 solubility by affecting the number of the ions which by their union might form the precipitate. Kohlrausch, from measurements of the electric conductivity of the saturated solutions, determined the solubility of some of the commoner "insoluble" substances, and his results are given in the following table, the solubility being expressed in parts per million, i.e. milligrams per litre, at 18 : Silver chloride Silver bromide Silver iodide . Mercurous chloride . Mercuric iodide Calcium fluoride Barium sulphate Strontium sulphate . .' Calcium sulphate Lead sulphate . Barium oxalate Strontium oxalate . Calcium oxalate Barium carbonate . Strontium carbonate Calcium carbonate . Lead carbonate Silver chromate Barium chromate Lead chromate Magnesium hydroxide Solubility. 17 0-4 O'l 3-1 0'5 14 2-6 107 2070 46 74 45 5-9 24 11 13 3 28 3'8 0-2 9 The solubility product of silver chloride in water is very small, corresponding to the very slight solubility of the salt. If we have the solid salt in presence of water, and add nitric acid to the solution, we disturb the equilibrium very little. The dissolved silver chloride is almost entirely dissociated, and the silver nitrate and hydrogen chloride, which might be formed from it by the action of the nitric acid, would be likewise almost entirely dissociated. The addition of nitric acid, therefore, does not appreciably affect the silver and chloride ions in the solution, and therefore is without influence on the solubility of the silver chloride. Consider, on the other hand, the effect of the addition of hydrochloric acid on the solubility of calcium tartrate. The calcium tartrate in the aqueous solution is highly dissociated, but the addition of hydrochloric acid at once liberates tartaric acid, which is only slightly dissociated in the presence of the highly dissociated calcium chloride, etc. The concentration of the tartrate ions is therefore much reduced, and the ionic product falls below the solubility product, i.e. the solution becomes unsaturated with respect to calcium tartrate. More of this salt, therefore, dissolves up, and the process of solution goes on until the ionic product once more reaches 1 These solubilities are only approximate, and are probably too great for the least soluble salts. Thus it has been found by another electrical method that the solubilities of silver chloride, bromide, and iodide, at 25, are respectively 1*8, 0*13, and 0*0017 nag. per litre. 308 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. the solubility product. It is evident from these considerations that a given quantity of hydrochloric acid will not dissolve an unlimited amount of calcium tartrate. If, however, we take a sufficient quantity of acid, a given amount of calcium tartrate can always be entirely dissolved. In general we may say that strong acids in aqueous solution will not appreciably dissolve salts of equally strong acids, or even of acids nearly as strong, for any double decomposition that takes place in solution does not then appreciably affect the number of the ions which regulate the solubility. On the other hand, strong acids will in general easily dissolve insoluble salts of weak acids, for then the weak acid is liberated, which being feebly dissociated, reduces the ionic product, with the result that the solid dissolves to restore the solution equilibrium. Sometimes, when the solubility product of the salt of the weak acid is excessively small, as it is in the case of silver sulphide, even an equivalent of a strong acid like nitric acid, will not dissolve up an appreciable quantity, for the solubility product is soon reached when silver nitrate and hydrogen sulphide begin to accumulate in the solution. Weak acids, as we might expect, do not dissolve the " insoluble " salts of stronger acids. Whilst calcium oxalate is freely soluble in hydrochloric acid, it is almost insoluble in acetic acid. In the first case the number of oxalate ions in the aqueous solution is diminished by the addition of the hydrochloric acid, whereas in the second it is not sensibly affected. Calcium oxalate, therefore, must dissolve in hydrochloric acid solution to restore the solubility product. When acetic acid solution is added, the ionic product scarcely departs from the solubility value, so that no calcium oxalate need pass into solution. As has already been indicated, the number of ions of a given kind in a solution may be greatly altered by the formation of complex ions. If to a saturated solution of silver chloride in contact with the solid there is added a quantity of potassium cyanide, the free cyanide ions unite with a portion of the silver ions to form complex ions AgC 2 N 2 '. The ionic product of silver and cyanide ions, therefore, falls below the equilibrium value, i.e. below the solubility product, with the result that silver chloride must dissolve up in order to restore equilibrium. If the quantity of cyanide added is small, the silver chloride need not dissolve wholly, but if a sufficient excess of cyanide is employed, the solution of the silver chloride will be complete. Should the solubility product be extremely small, as is the case with silver sulphide, potas- sium cyanide has only a slight solvent action, and the silver sulphide dissolved is easily reprecipitated on addition of potassium sulphide. The solvent action of sodium thiosulphate or ammonia on " insoluble " silver compounds is similar in origin, the increased solubility being due to the formation of complex ions with corresponding disappear- ance of silver ions. It will be noticed that the solubility of silver xxvi APPLICATIONS OF DISSOCIATION THEORY 309 chloride, bromide, and iodide in water (given in the table on p. 307), is the same as the order of their solubility in ammonia, as an applica- tion of the above theory would lead us to expect. The theory of electrolytic dissociation affords some assistance in understanding the action of the indicators used in acidimetry and alkalimetry. The indicators are themselves acid or alkaline in nature, but are necessarily very feeble compared with the acids or alkalies whose presence they indicate. Their action depends on a change of colour which they undergo on neutralisation. Phenol phthaleine, for example, is a substance of very weak acid character, being in aqueous solution almost entirely undissociated. If, however, we add a strong alkali such as sodium hydroxide to it, the sodium salt is formed and imparts an intense pink colour to the solution. All the soluble salts of phenol phthaleine are thus coloured, and the colour is of the same intensity in equivalent solutions if these solutions are very dilute, so that we are justified in concluding in terms of the dissociation hypothesis, that the colour is due to the negative ion of the phenol phthaleine, as this is the substance common to all dilute solutions of salts of phenol phthaleine. The undissociated substance has no colour. Let us consider what happens as we titrate a solution of an acid, using phenol phthaleine as an indicator. In presence of the acid, the indicator, being a very feeble acid, is even less dissociated than it would be in pure water, and consequently no colour is per- ceptible. As soon, however, as the acid originally present in the solution is neutralised and a drop of alkali in excess is added, the corresponding alkaline salt of phenol phthaleine is formed, i.e. negative ions from the phenol phthaleine are produced, and the solution at once assumes the pink tint characteristic of them. In order to have a sharp indication of the neutral point, it is necessary first that the acid to be titrated should be considerably stronger than phenol phthaleine itself, and that the alkali should be a strong alkali. If the acid to be titrated is such a feeble acid that its salts even with strong bases suffer hydrolysis in aqueous solution, it is obvious that phenol phthaleine is incapable of sharply indicating the neutral point, for long before sufficient alkali for complete neutralisation has been added, the salt formed will begin to split up into free acid and free base, part of which will neutralise the phenol phthaleine, and thus produce a faint pink colour which will gradually deepen in intensity as the addition of alkali progresses. Carbolic acid, and other phenols, therefore, cannot be titrated with alkali and phenol phthaleine, on account of the hydro- lysis which their salts suffer (cp. p. 278). Poly basic acids, too, very frequently form normal salts which are partially hydrolysed in aqueous solution, and with them also no definite indication of the neutral point can be obtained. Thus the ordinary sodium phosphate, i.e. di-sodium hydrogen phosphate, although formally an acid salt, has an alkaline reaction to phenol phthaleine on account of its slight 310 INTRODUCTION TO PHYSICAL CHEMISTRY OH. xxvi hydrolysis. Phenol phthaleine roughly indicates neutrality in the case of carbonic acid when sodium hydrogen carbonate exists in the solution, and gives a strong pink colour with the normal sodium carbonate. Since carbonic acid thus behaves as an acid to phenol phthaleine, i.e. since carbonic acid is a stronger acid than phenol phthaleine, it must be excluded from the alkali with which acids are titrated. This is best done by using baryta as the alkali, and keeping it protected from the carbonic acid of the atmosphere. Any carbonic acid which may have been originally present settles down as barium carbonate, and the clear liquid is therefore free from this source of disturbance. If the base employed is not a strong base, the indication in this case also is uncertain owing to hydrolysis. Thus ammonia should never be used in titrations with phenol phthaleine as indicator, for the ammonium salt of phenol phthaleine, being the product of the union of a weak base with a very weak acid, is hydrolysed in aqueous solution, and thus the neutrality point is not sharply indicated, even though the salt of ammonia with the acid originally in the solution undergoes no hydrolysis. Phenol phthaleine then forms a good indicator when weak acids are titrated with strong bases. The application of the dissociation theory to explain the action of other indicators is not so simple as in the case of phenol phthaleine, because these indicators are usually of mixed function, being them- selves acidic or basic according to circumstances, and thus forming two series of salts, one series with strong acids, and another with strong bases. There may, therefore, be more than one kind of coloured ion in solution, besides, perhaps, coloured undissociated indicator, so that the interpretation of the actual colours of the different solutions on the dissociation theory is somewhat arbitrary. Many applications of the electrolytic dissociation theory to ordinary laboratory work, will be found in W. OSTWALD, The Scientific Foundations of Analytical Chemistry. CHAPTER XXVII THERMODYNAMICAL PROOFS THE experience of practical and scientific men alike goes to show that it is impossible to construct a perpetual motion machine. In the general acceptance of the term, a perpetual motion machine is one from which more energy can be obtained than is put into it from the outside. This may be termed a perpetual motion of the first class. But another kind of perpetual motion machine might exist, one namely, which could by a recurring series of processes continuously afford mechanical energy at the expense of the heat of surrounding bodies at the same temperature as itself. This kind of perpetual motion is also impossible, and has been termed perpetual motion of the second class. It must be clearly understood that not only have such machines never been constructed, but that no increase in our experimental skill at present conceivable could lead to their construction. If there- fore in argument we find that an imaginary series of processes would lead to a perpetual motion of either of the kinds mentioned above, we conclude that such a series of processes can have no real existence. The denial of the possibility of the existence of the two sorts of perpetual motion is contained in the two following positive and general statements, which are known as the First and Second Laws of Thermodynamics, respectively. I. The energy of an isolated system remains constant. II. The entropy of an isolated system tends to increase. With the second law in its general and formal aspect we shall have here little to do, and shall use in its stead the negative proposi- tion given above. Entropy is a function, which while theoretically of great value as indicating the direction in which chemical or other processes take place, and in fixing generally the conditions of equili- brium, is not susceptible of direct measurement, and is consequently of less obvious and immediate practical importance. The first law states the principle of the Conservation of Energy, by an isolated system being meant a system which can neither give up energy to its environment nor absorb energy from it. The sum of 312 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. the different kinds of energy in such a system is always the same, no matter what forms the energy may assume. In order to ascertain practically that the sum of the energies is constant, we must obviously have one kind of unit in which all energies may be expressed. If we work with only one kind of energy, we express its amount in the appropriate unit. Thus we express amount of heat in calories ; electrical energy in volt-coulombs, or the like ; and mechanical energy in foot-pounds or gram-centimetres. In dealing with different kinds of energy, we may take any of these units as the standard in which we express all the different varieties, for we know the factors necessary for converting one unit into any of the others. One calorie, for example, is under all circumstances equivalent to 42,000 gram- centimetres, so that if we wish to add heat energy and mechanical energy together, we must either multiply each calorie of heat energy by 42,000 to convert it into gram-centimetres, or divide the number of gram -centimetres by the same number in order to find their equivalent in calories. A few simple deductions from the First Law, involving only the consideration of thermal and mechanical energy, will now be given. When a small quantity of heat dQ is supplied to a gram-molecular weight of a perfect gas at constant volume, the heat goes entirely to raise the temperature of the gas, and we may therefore write dQ C v dT ; where C v is the heat capacity of the gram molecule at constant volume (cp. p. 34), and dT the small rise of temperature which this amount of gas experiences. If the heat is supplied to the gas under constant pressure, it not only goes to raise the temperature, but is also partially converted into work done by the gas during expansion. If p is the constant pressure, and dv the change of volume, this work is repre- sented by the product pdv. Now according to the First Law (I) if no other kind of energy is involved, and if the heat and mechanical energy are expressed in terms of the same unit. In the gas equation for the gram molecule (pv = ET), p, v, and T are variable, so that on differentiation we obtain pdv + vdp = RdT. (2) By eliminating dT from equations (1) and (2) we obtain + - or since C p = C v + H, as was shown on p. 34, Now, if the pressure, volume, and temperature of the gas be XXVII THEEMODYNAMICAL PEOOFS 313 allowed to change adiabatically, i.e. in such a way that heat neither enters nor leaves the system, we have dQ = 0, and consequently Cppdv + C v vdp = 0. Introducing into this equation the ratio of specific heats k = C p /C w we have kpdv + vdp = ; p This equation, when integrated between the values^ andj^, gives k(\og e v 1 - log e v) + logo/p, - log^y = ; whence k or =(?)' These results are of importance as they enable us to ascertain the ratio of the specific heats of gases by observations of pressures and volumes under circumstances in which heat can neither leave nor enter the gas considered. For example, during the rapid compressions and dilatations which constitute the passage of sound through a gas, there is no time for the heat change at any portion of the gas to com- municate itself by conduction to neighbouring portions, so that the process instead of being isothermal, is, so far as any given portion of the gas is concerned, an adiabatic process, the gas being cooled at each dilatation and warmed at each compression. The gas then does not in these circumstances obey Boyle's Law, which holds good only for isothermal change of pressure and volume, but the law given above, where the pressure is not inversely proportional to the volume, but to the Jc-ih power of the volume. It is owing to this circumstance that we can deduce the ratio of specific heats of a gas from the speed at which sound travels in it. We have now to find the law connecting the volume of a gas with its absolute temperature when it is heated adiabatically, i.e. by com- pression, since no heat as such must enter the system. This can easily be done by eliminating pdv from equations (1) and (2), the result being C p dT-vdp P since pv = RT. For an adiabatic compression dQ = 0, so that 314 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. k-l Integrating between T, p, and T^ p v we obtain - lo ge T) - (log^ - logtf) = 0, 1) / V \^ But since in an adiabatic process ~ = f J , we obtain finally (?)'= (Vr This result we shall find useful in calculating the maximum amount of work which can be obtained from a given quantity of heat under given conditions, a problem which we shall now proceed to solve. According to our statement of the second law, no work can be obtained from the heat contained in a number of bodies all having the common temperature of their surroundings. In order to convert heat into work, we must have temperature differences. A body in cooling through a certain range of temperature parts with a certain amount of heat, and a definite fraction of that heat may be transformed into work. In actual practice, different engines will effect the conversion of different amounts of the heat, but there is a theoretical limit which no engine, however perfect, can exceed, and it is our task to find what that limit is. The process of reasoning is essentially the same as that of Carnot, who introduced the conceptions necessary for the solution of the problem, and arrived at the desired result, although to him heat was a material substance, and not, as we now believe, a form of energy. The fundamental conception is that of a reversible cycle of opera- tions. By a cycle of operations is understood a series of processes which leaves the system considered in exactly the same state as it was initially, and the term reversible applies not merely to the direction of the mechanical operation, but to the complete physical reversibility of all the processes involved. xxvii THERMODYNAMICAL PROOFS 315 A reversible heat-engine after converting a certain amount of heat into work will return to its original state in every respect if made to act backwards step by step, so that the work obtained is entirely converted into heat by its agency. Such an engine is of the maximum possible efficiency, for if any engine were more perfect a perpetual motion could be obtained. This may be shown as follows. Let A be a reversible engine, and let B be an engine which under the same conditions can convert a larger proportion of heat into work than A. Let the two engines work between the same temperatures, and sup- pose that when a quantity of heat Q is given to A, the proportion q is converted into mechanical work. If the work corresponding to q is now done upon this engine so that the processes are reversed, the system will arrive exactly at its initial state, the quantity of heat Q - q being raised from the lower to the higher temperature. Now instead of letting the engine A act directly, take in its stead the more perfect engine B, and supply it at the higher temperature with the quantity of heat Q. A greater proportion of this than before is converted into work, say /, the quantity of heat Q - q falling to the lower tempera- ture. By using the reversible engine to perform the reverse trans- formation of work into heat, we can regain the original quantity of heat Q at the original temperature, by expending mechanical energy equivalent to q, the quantity of heat Q - q being at the same time raised to the higher temperature. In the whole series of operations, then, the heat q' - q has been taken in at the lower temperature and an equivalent amount of work has been gained. This cycle of operations can be repeated as often as we choose, so that here we should have a system capable of giving an indefinitely large amount of work at the expense of heat at the lower temperature, which might be the uniform temperature of the surroundings, i.e., we should have realised a perpetual motion of the second class. We conclude, then, that an engine more perfect than the reversible engine A cannot exist. It will be noticed that nothing is said as to the nature of the working sub- stances in the reversible or the other engine, so that the conclusion is perfectly general. We are at liberty therefore to use any kind of reversible engine in our calculations in order to ascertain the maximum quantity of heat which can be converted into work, and we shall find it convenient to take for our working substance a perfect gas, on account of the simplicity of the laws which it obeys. We must first of all investigate if the processes through which we put the working substance are really reversible. The condition Of reversibility is that the state of the system at any time does not differ sensibly from equilibrium, for then the slightest variation in the conditions will determine the occurrence of the process in the one direction or the other. If, therefore, we communicate heat to the gas, we must so arrange that the gas and the heat source have temperatures differing from each other by an infinitely small 316 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. amount. Similarly, if the gas is to part with heat, it must do so to a heat-sink with a temperature lower than its own only by an infinitesimal quantity. If the gas is to be compressed, the external pressure at any instant must be only greater by an infinitely small amount than the pressure of the gas itself; and if the gas is to be expanded, the external pressure must be less than the pressure of the gas by an infinitesimal difference. The machine of course must be absolutely frictionless, for otherwise some work would have to be expended in moving the machine, and thus converted into heat, inde- pendent of the gaseous working substance which is alone considered. From all this it is obvious that a reversible engine is an engine which can never be realised in practice. For an engine to be strictly reversible, there should be no departure from the conditions of equilibrium at any stage, in which case no process could occur at all, for the occurrence of any process naturally involves a departure from equilibrium. If the departure from the equilibrium conditions were infinitely small, the process would occupy an infinitely long time in its performance. We see then that a reversible process is an ideal, just as a perfect gas is an ideal. Neither can ever be met with in practice, but this in no way impairs the value of the theoretical con- clusions deduced by their aid. The series of operations which we shall perform on the gas will be best seen in the pressure-volume diagram, Fig. 45. We begin with the gas in the state represented in the diagram by the point 1. At the con- stant temperature T we let the gas slowly expand until its pressure and volume are indicated by the point 2. The form of the curve obtained during the expansion is the rectangular hyper- bola of gases (cp. p. 76). We next ~' v isolate the gas from the heat source of FlCL 45> constant temperature T, and let the expansion continue adiabatically until the point 3 is reached. Since the pressure varies more rapidly with the volume in an adiabatic than in an isothermal process (p. 3 1 3), the line 2, 3 will be more inclined to the volume axis, than I, 2, as is shown in the diagram. As no heat can enter the system during the adiabatic expansion, the temperature will fall, say to T'. We now bring the gas into contact with a heat reservoir at the temperature T', and compress it isothermally until a point 4 is reached, such that when the compression is continued adiabatically, the adiabatic curve will pass through the initial point I, where the process is stopped and the cycle thus completed. In this cycle a certain quantity Q of heat has been absorbed by the gas at the higher temperature T, and the quantity Q' has been xxvn THEEMODYNAMICAL PROOFS 317 given out by the gas at the lower temperature T'. At the same time a certain amount of work has on the whole been performed. The gas on expanding does work, and this work is measured by the product of each pressure into the corresponding change of volume. In the diagram, therefore, the work performed by the gas on expansion is measured by the area a, 1, 2, 3, y. When the gas was compressed, work was done upon it, and this work in the diagram appears as the area y, 3, 4, 1, a. The total work obtained then from the gas during the cycle is the difference of these areas, viz., the quadrilateral 1, 2, 3, 4. To obtain actual numerical relations we may consider a gram molecule of the gas, for which the previous equations of this chapter are valid. During the isothermal expansion 1, 2, the gas absorbed Q units of heat at the temperature T, while it expanded from the volume v l to the volume v 2 . In equation (1) p. 312. we may put dT equal to zero, since we are considering an isothermal PT process, and for p we may substitute . We thus obtain -PT dv Ml , (la) which when integrated between the limits v l and v 2 , gives 0- JO 1 log, J V 2 as the amount of heat absorbed by the gas. Similarly for the isothermal compression 3, 4 we obtain or The sign of Q' is in the first equation of this pair different from that of Q above, since in the first case the system gains heat, and in the second loses it. Division now gives (3) nog.-. 3 For the adiabatic expansion 2, 3, we have (p. 314) T \*. and for the adiabatic compression 4, 1 318 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. whence - = - or = v 2 Vl v 2 Equation (3) therefore reduces to i.e. the heat absorbed is to the heat given out as the absolute tempera- ture of the absorption is to the absolute temperature at which the heat is lost by the system. A slight alteration gives the equation in the form Q-V _ T-r Q T ' i.e. the proportion of the absorbed heat which is converted into work is equal to the temperature difference between the two isothermal operations divided by the temperature of absorption. A form which we shall find useful in subsequent calculations is T- T These results although derived from a consideration of the be- haviour of gases, are valid for all reversible cycles, and can therefore be applied in every case for which we can show all the operations in- volved to be revers- 1, ,2 ible. 4" ^3 The first applica- tion of equation (4) will be to the process of vaporisation. Let us consider a quantity of liquid under a pres- sure P which is equal to the vapour pressure of the liquid at the constant temperature Volume chosen. An infinite- FIO. 46. simal diminution of the pressure on the liquid will cause it gradually to pass into vapour if this pressure and the constant temperature are maintained. Suppose one xxvii THERMODYNAMICAL PROOFS 319 gram -molecular weight of vapour to be produced in this way, and let the isothermal process be represented in the indicator diagram, (Fig. 46), by the line 1, 2 which is parallel to the axis of volumes, the pressure being constant. Now expand the vapour adiabatically from the original pressure P to a pressure which is dP lower, the temperature at the same time falling. The pressure and volume may then be represented by the point 3. At a temperature T-dT for which the vapour pressure of the substance is P dP, compress the vapour isothermally to liquid and finish the compression adiabatically until the original pressure, temperature, and volume are regained. The work done upon the system is equal to the area of the figure 1, 2, 3, 4, which is practically the product of the line 1, 2 into the vertical distance between 1, 2 and 3, 4. Now the line 1, 2 in the diagram represents the difference in volume between the liquid and the vapour, and the distance between the two horizontal lines represents the differ- ence of vapour pressure dP due to a difference of temperature dT. For the work done then we have the product dP(V-v\ where V is the molecular volume of the vapour, and v the molecular volume of the liquid. The quantity of heat absorbed at the higher temperature T is Q, the molecular heat of vaporisation of the liquid at this temperature. The quantity which has been transformed into work is consequently dT Q~ according to equation (4). If we express the heat in mechanical units, as we can do by multiplying the heat units by .7=42,350 (see p. 6) we obtain the equation a result of considerable practical importance and of wide applicability. As the volume of a liquid is only a fraction of a per cent of the volume of vapour derived from it at ordinary pressures, it is often per- missible to write simply V instead of V-v, which brings about a numerical simplification. For the gram-molecular volume of a gas we have PV=RT, where R is equal to 2<7 very nearly (see p. 29) so 2JT that V -=-. We may therefore write equation (5) in the form dP dT ~2T 2 ( ' We may calculate from this result in the form of equation (6) the latent heat of vaporisation of benzene from the change of its vapour pres- 320 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. sure with the temperature. At 5 the vapour pressure of benzene is 34*93 mm. of mercury, or 47'50g. per sq. cm. ; at 5'58 the pressure is 36*06 mm. or 49'04 g. per sq. cm. For dP then we have T54, for dT we have 0*58, for P we have the mean value 48*27, and for T the mean value 278'3. The value of Q is therefore 2T 2 dP 2(278-3) 2 x 1-54 -= 48-27 x 0-58 The value of the heat of vaporisation actually found is 8420 cal., so that the agreement between calculation and experiment is fairly close, the difference not being greater than the error of experiment. Equation (5) not only holds good for the liquid and gaseous phases, i.e. for vaporisation, but also for the equilibrium between any other pair of phases, for example, solid and liquid, or two solid phases such as the different crystalline modifications of sulphur. In the form (8), Jq it is useful for ascertaining the effect of pressure on the temperature of equilibrium. T, v, and q may all refer either to molecular quantities or to unit weight of the substance considered, as the molecular factor cuts out in the right-hand member. If we consider the transformation brought about by supplying heat to the substance, q is a positive quantity, while / and T also are necessarily positive. Consequently dP will have the same sign as dT, or the opposite sign, according as V v is positive or negative, i.e. if the substance expands on being transformed by application of heat, dP and dT will have the same sign, whilst if it contracts dP and dT will have different signs. In the first case then the transition point will be raised by increase of pressure; in the second case it will be lowered. Rhombic sulphur on melting expands ; V v is therefore positive and the melting point of rhombic sulphur is raised by pressure. Again rhombic sulphur expands on passing into monoclinic sulphur, and consequently the transition point is raised by application of pressure. Water, on the other hand, occupies a smaller volume than the ice from which it is produced; V-v is therefore negative, and increase of pressure lowers the melting point. As a numerical example of the application of formula (8), we may calculate the effect of one atmosphere increase of pressure on the melting point of ice. A cubic centimetre of water at is obtained from 1'09 cc. of ice; the change of volume on liquefaction is therefore 0'09 cc. per gram of water. The latent heat of liquefaction per gram is 80 cal., and the temperature of liquefaction is T 273. We have, therefore, if dP = I atm. = 1033 g. per sq. cm., xxvn THERMODYNAMICAL PEOOFS 321 that is, the melting point of ice is lowered 0*0075 for each atmo- sphere increase of pressure. A corresponding diminution of pressure causes the same rise in the melting point. Thus ice which melts under atmospheric pressure at 0, melts at 0'0075 under the pressure of its own vapour, so that the triple point (p. 99) lies at this tempera- ture and not at 0. Dilute Solutions When we dissolve a substance in any liquid, the process is not, under ordinary conditions, a reversible one, for we have not in general during dissolution a state bordering on equilibrium. In certain circumstances, however, it is possible to conduct the process reversibly. If we are dealing, for example, with the solution of a gas in a non- volatile liquid, we can proceed reversibly as follows. Let the gas and liquid be taken in such proportions that the gas will just dissolve in the liquid at the pressure p and the constant temperature of experi- ment t. Suppose the liquid and gas to be contained in a cylinder with a movable gas-tight piston. At first let there be a partition separating the gas and the liquid. Without removing this partition, expand the gas by gradually diminishing the pressure in such a way that at no instant during the expansion the condition of the gas differs sensibly from equilibrium. According to Henry's Law, the quantity of gas dissolved by the liquid is proportional to the pressure of the gas. Let the expansion be continued until the pressure of the gas is so small that practically none of it dissolves in the liquid when the separating partition is removed. After removal of the partition let the pressure on the gas be increased by insensible gradations. At no time does the state of the system deviate sensibly from equilibrium, and the pressure may be gradually raised until at the pressure p all the gas has dissolved. The solution of a gas in a liquid then may be made part of a reversible cycle if the process is carried out as here indicated. The concentration of a solution of a non-volatile substance in any liquid can be changed reversibly by bringing the solution into contact with the solvent under equilibrium conditions, and then by an infinitely small alteration of the conditions determine a process of concentration or dilution. It is apparent at once that we cannot bring the solution into direct contact with the liquid solvent, either by mixing directly or allowing the dissolved substance to diffuse slowly into a fresh quantity of solvent as in the experiment described on p. 149, for no slight change in the conditions can at any stage make the action proceed in the reverse direction, i.e., make a solution separate into a more concentrated solution and the solvent. If the solvent is in the form of vapour or solid, however, the dilution or concentration may be effected reversibly. Suppose the solution to be in presence of Y 322 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. its own saturated vapour at the constant temperature of experiment. An increase of external pressure, however slight, will cause part of the vapour to condense, i.e. will dilute the solution; while a slight diminution of the external pressure will cause part of the solvent to evaporate, i.e. will concentrate the solution. Similarly, if the solution is in equilibrium with the solid solvent, a very slight rise of tempera- ture will bring about partial liquefaction of the solid solvent, and thus dilute the solution, and a correspondingly slight diminution of tempera- ture will produce partial separation of solid solvent and thus con- centrate the solution. There is still another way of bringing the solution into contact with the pure solvent under equilibrium conditions, namely through a diaphragm which is permeable to the solvent and not to the dissolved substance. If the solution is enclosed in a cylinder with a semi- permeable end and a movable piston, it will be in equilibrium with the liquid solvent through the diaphragm when there is a certain pressure on the piston, the osmotic pressure. If the pressure of the piston is increased ever so slightly, solvent flows outward through the semipermeable diaphragm and the solution becomes more concentrated : if the pressure on the piston is diminished, solvent flows inwards through the diaphragm, and the solution is thereby diluted. All these methods of changing the concentration of a solution can therefore be adopted as parts of reversible cycles of operations, and we shall see that by removing a portion of solvent from a solution by one method, and by adding it to the solution again by another method, we obtain a series of results which are of great theoretical and practical importance. In the first place we shall consider a cycle in which a gas is dis- solved in a non-volatile liquid by the reversible process given on p. 321, and the system then brought back to its original condition by means of semipermeable diaphragms. We start with a volume v of the gas under pressure^, and with a volume Fof liquid just sufficient to dissolve the gas under this pressure, and we propose to find what amount of work (positive or negative) must be done in order to bring the gas into solution reversibly at constant temperature. Puring the first stage contact between gas and liquid is prevented by a partition inserted at the surface of the liquid. If the cylinder in which the gas and liquid are contained have unit cross section, and the initial distance of the piston from the liquid surface is x , we have for this state x = v . At any stage of the expansion (x) the pressure 79 9) p is given by the equation p ^-^?, and the work done by the gas tK during the expansion is represented by the expression x /7/v, //v, rf -=p v \Qg e - V ^ VQ XXVII THERMODYNAMICAL PKOOFS 323 x being a very large multiple of v . The partition is then removed, and the pressure on the gas increased. The pressure on the piston in a given position x is less than before, for the gas which was previously confined to the space x is now partly in solution. If s denote the solubility (p. 59), the available volume is practically increased in the ratio x : x + sF, so that the pressure in position x is now given by and the work required to be done during the compression is * dx x + sF On the whole, the work done on the system during the double opera- tion is or IWo | 10g e -^- f. X+SF 1 SF ) 10g ^} The quantity within the brackets of the second expression may be seen to be zero, since x is indefinitely great, so that - = 1, and since x by supposition the quantity of liquid is just capable of dissolving the gas, whence sF=v . The conclusion, then, is that there is no gain or loss of work in dissolving the gas reversibly in the liquid. j^l I |j^ The gas may now be removed from solu- tion and restored to its original state rever- sibly by means of semipermeable membranes arranged as in Fig. 47. One membrane gg permeable to gas but not to liquid, is intro- duced at the surface of the liquid, on which the piston KK rests at the commencement of the operation. A second membrane II, permeable to liquid but not to gas, is sub- stituted as a piston for the bottom of the cylinder, and is backed upon its lower side by the pure solvent. By suitable proportional movements of the two pistons, KK being raised through the space v m while II is raised through the space F, the gas may be expelled, the pressure of the gas retaining the constant value p m and the solution which remains retaining a constant strength, and therefore a constant ' Gas Solution Soli tent FIG. 324 INTRODUCTION TO PHYSICAL CHEMISTEY CHAP. osmotic pressure P. When the expulsion is complete, i.e., when the two semipermeable membranes have come together, the work done upon the lower piston is PV, and the work done by the gas in raising the upper piston is p v . The system is now in its original state, and all the operations have been conducted reversibly. A reversible cycle has therefore been T- T performed, and the equation Q-Q' Q is applicable. The temperature has remained constant throughout, so that T - T' = 0, and therefore Q - Q' = 0, as in all reversible isothermal cycles. Since no heat has been converted into work, or vice versa, the work done on the system must on the whole be equal to the work done by the system. In the first stage, i.e. the process of solution, it has been shown that there is neither loss nor gain of work, so that for the second stage we must have If the gas which occupies the volume v at the pressure p ot is made to occupy the volume J 7 ", its pressure will assume a new value p, and according to Boyle's Law we shall have j? # =pV j combining this with the previous equation, we then obtain pV-PV and p = P. The osmotic pressure then of the gas in solution is equal to the gaseous pressure which it would exert in absence of the solvent if it occupied the same space at the same temperature, a result in harmony with the calculation from experiment made on p. 164. 1 The above result holds good only for ideal substances and for very dilute solutions, since in its deduction we have assumed that the gas is a perfect gas which obeys the laws of Boyle and Henry exactly, and that the volume of the solution is exactly the same as the volume of the solvent which it contains, an assumption which can only be justifi- ably made when the solution is extremely dilute. The solvent, too, was supposed to be non-volatile, and the reversible processes themselves are purely ideal. Notwithstanding all these assumptions the conclusion arrived at is practically important, and holds good with close approxi- mation for all dilute solutions under ordinary conditions. We shall now consider an isothermal reversible cycle performed with a solution of a non-volatile solute in a volatile solvent. Let the solution contain n gram-molecules of dissolved substance in W grams of solvent, and let the constant absolute temperature of all the processes be T. From the solution let there be removed by means of a piston and a diaphragm permeable only to the solvent, a quantity of the latter which in the solution contained one gram molecule substance dissolved in it. This quantity is W\n grams, and the solution is supposed to be present in such large proportions that the removal of this amount of 1 The proof here given is taken from an article by Lord Kayleigh, Nature, vol. lv. p. 253, 1897. xxvn THERMODYNAMICAL PROOFS 325 solvent does not sensibly affect its concentration, the osmotic pressure thus remaining constant during the operation. Since the change in volume of the solution is the gram molecule volume, the work done on the system in removing the solvent is the product of this into the osmotic pressure, and is therefore equal to ET, if the gas laws apply to dissolved substances. This quantity of liquid solvent is now con- verted into vapour reversibly by expanding at the vapour pressure of the liquid, which we shall call /. The vapour pressure of the solution is smaller than this and equal to /'. Let the vaporous solvent there- fore expand reversibly till its pressure diminishes to this value. The gaseous solvent is then in equilibrium with the solution, and may be brought into contact with it and condensed reversibly at the pressure /', so that the whole system regains its initial state. We have now to consider the work involved in the expansion and contraction. The work done by the system on expanding from liquid to vapour under the pressure /, is equal to the work done on the system in condensing the gas to liquid under the constant pressure /', being equal in each case to RT for the gram molecule, if we neglect the volume of the liquid (cp. p. 29). There remains then the work done by the gas on expanding from / to /'. For isothermal expansion we have the work /-/rti RT per gram molecule of gas (equation la, p. 317). Now dp _dv p v for a gas, since pv = const., and therefore pdv + vdp = 0. For the work done during the expansion of the gas then we have - ET-, or - RT per gram molecule if the difference between / and /' is very small. For the actual amount of solvent considered we have conse- W f f quently - -^-ET , where M is the molecular weight of the J.\J.7l J solvent in the gaseous state. Since in the whole cycle no heat is con- verted into work or vice versa, this work done by the system must be numerically equal to the work done on the system during the osmotic expulsion of the solvent, i.e. Mn f /-/' Nn * - - which is identical with the result arrived at for very dilute solutions by the method of calculation given on p. 170. Thus by a direct thermodynamical proof we can arrive at the relation between the 326 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. osmotic pressure and the lowering of the vapour pressure of liquids by substances which are dissolved in them. If the solution considered be not very dilute, we cannot write / for . For greater concentrations we integrate this expression, and f) thus obtain log e ~. By the same process of reasoning as before, we then get / Mn This expression holds good for all concentrations of solutions for which the total volume of the liquid does not change when the quantity of solvent considered is added to or removed from the solution. If this condition is not observed the result is only approximate, for in the deduction of the relation we assumed that the work done by the expressed liquid in expanding at the constant gaseous pressure of the solvent was equal to the work done on the vapour when it was con- densed at the constant vapour pressure of the solution, an assumption which is only valid if the volume of the liquid is the same before and after the reversible mixing of the solvent with the solution. The relation between the more accurate logarithmic and the usual approximate expression may be obtained by writing the former as follows : If we expand the logarithm in this second form, the first term of the expansion is - , identical with the usual expression. Having thus obtained the formula for the lowering of the vapour pressure of a solvent, we may now proceed to deduce the formula for the corresponding rise in the boiling point. From equation (6) we obtain the expression to represent the concomitant variations of temperature and vapour pressure of a solvent. Consider now a solution containing n gram molecules of dissolved substance in W grams of solvent. Let this solution have the vapour pressure P at the temperature T + dT, T being the temperature at which the solvent has the same pressure. At the temperature T + dT the solvent will have the pressure P + dP. Now dP is the lowering of the vapour pressure of the solvent, but since XXVII THEKMODYNAMICAL PROOFS 327 dP is very small compared with P we may write instead of this the expression for the lowering. But we found above that this lower- Mn mg is equal to -== t so that W Now Q is the latent heat of a gram molecule of the solvent, and M is its molecular weight (both molecular quantities referring to the gaseous state), so that Q/M=q the latent heat of vaporisation per gram. We thus obtain W or T + dT-T=dT is the elevation of the boiling point of the solvent caused by the substance dissolved in it, and we have now obtained an expression for this in terms of the boiling point of the solvent itself, its latent heat of vaporisation and the concentration of the solution, to which the elevation is proportional. The elevation of the boiling point caused by the solution of one gram molecule of substance in 100 grams of solvent is sometimes referred to as the "molecular elevation" (p. 185). For this concentration n becomes 1 and W becomes 100, so that the molecular elevation is For a solution containing one gram molecule per 1000 grams, i.e. very nearly one gram molecule Q'0027 72 per litre for aqueous solutions, the elevation is - . Since both boiling point and heat of vaporisation vary with the pressure at which ebullition takes place, there is no definite " molecular elevation " for any one solvent unless the pressure is specified. For ordinary purposes the pressure is of course the atmo- spheric pressure, and the fluctuations to which this is subject have so little effect on the molecular elevation that it may be taken as a constant magnitude in the practical molecular weight determination by the boiling point method. In order to give an example of the agreement between the calculated molecular elevation and the same magnitude as determined experimentally, we may take the common solvent ether. The boiling point of ether is 35, and therefore T = 308. The latent heat of vaporisation per gram at this temperature is 0'02T 2 90. The expression - - has thus the value 21*1. The average molecular elevation observed in the case of nine different substances 328 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. in moderately dilute ethereal solution was found to be 21*3, the extreme values being 20'0 and 21'8. The expression for the molecular depression of the freezing point has a similar form, and may be deduced by means of a reversible cycle as follows. Let a solution containing n gram mole- cules of dissolved substance in W grams of solvent be contained in a cylinder provided with a semi -permeable end, and a movable piston. At T-dT, the freezing point of the solution, let such a quantity of the solvent freeze out as originally contained one gram W molecule of substance dissolved in it, viz., grams. The quantity of ti solution is supposed to be so great that the freezing out of this amount of solvent does not appreciably affect its concentration, so that the temperature of equilibrium between solution and solid solvent does not change during the process of freezing. The solid is now separated from the solution, and the whole system is raised to the temperature T, the melting point of the solvent, and at this temperature the solid W solvent is allowed to melt. In doing so it absorbs q calories, if q 71 represents the latent heat of fusion per gram. The fused solvent is now brought into contact with the solution through the semipermeable diaphragm under equilibrium conditions, viz., with the pressure on the solution equal to the osmotic pressure P. By raising the piston under this constant osmotic pressure, the solvent passes through the diaphragm and mixes reversibly with the solution, the concentration as before remaining unchanged. The work done on the piston is equal to the product of the constant osmotic pressure P into the volume v which contains one gram molecule of solute. But this amount of work, according to the osmotic pressure theory, is equal to RT ; or if we express the gas constant in thermal units, to 2 T. The system after mixing is finally cooled to the original temperature T dT so as to complete the cycle. By selecting the solution sufficiently dilute we may make the depression of the freezing point dT as small as we choose, and consequently the heat absorbed and evolved in warming and cooling the system through this small range of temperature may W be made negligible in comparison with the finite amount of heat q absorbed by the solvent on melting. In the reversible cycle, then, we W have the amount q absorbed at the higher temperature T, and the dT W amount of this converted into work is q. But the only work done by the system is the osmotic work, since the external work brought about by the volume-changes on freezing and melting are so small as to be negligible. We have consequently xxvii THERMODYNAMICAL PEOOFS 329 or The depression of the freezing point is thus seen to be proportional to the concentration of the solution, and if we make the concentration such that n=l and W ' - 100, that is, if we dissolve one gram molecule in 100 grams of solvent, we get the molecular depression equal to Q-Q2T 2 . The expression is exactly the same as that for the elevation of the boiling point, q referring here, however, to heat of fusion instead of to heat of vaporisation. The following table exhibits the nature of the agreement between the calculated and observed values of the molecular depression in various solvents : Solvent MoKDep. Water 18 '5 18 '4 Formic acid 28 '4 27 % 7 Acetic acid 38 '8 39 Benzene 51 49 Phenol 76 74 Nitrobenzene 69 '5 707 Ethylene dibromide 119 118 Electrical and Chemical Energy It is possible, as Helmholtz showed, to establish a relation between the energy of the chemical processes occurring in a galvanic cell and the electrical energy produced by the action. At one time it was supposed that all the energy which could be obtained under ordinary circumstances as the heat of the chemical reaction, could be converted into electrical energy and made available as an electrical current. Thus if Q were the thermal effect for one gram equivalent of the sub- stances entering into chemical action in the cell, it was assumed that an amount of electrical energy equivalent to this could be obtained from the cell for each gram-equivalent of chemical transformation. In a Daniell cell, for example, where the reacting system is Cu, CuS0 4 solution ; ZnS0 4 solution, Zn, the total chemical action is Zn + CuS0 4 = Cu-fZnS0 4 , zinc being dissolved up at one pole of the battery and copper being deposited at the other. Now the thermal effect of this reaction is obtained by subtracting the heat of formation of the zinc sulphate 330 INTRODUCTION TO PHYSICAL CHEMISTRY CHAP. from that of the copper sulphate, both numbers referring to aqueous solutions of the salts (cp. p. 121). We thus get 106,090 - 55,960 = 50,130 cal. per gram molecule, or 25,065 cal. per gram -equivalent. That is, when 3 2 '5 grams of zinc displace an equivalent amount of copper from a solution of copper sulphate, 25,065 calories are evolved. Now, if the displacement takes place indirectly in a galvanic cell with production of electric current, 96,500 coulombs of electricity will be obtained, according to Faraday's Law, for every 3 2 '5 grams of zinc dissolved. To express the numerical equivalence of thermal and electrical energy we have the equation 1 volt-coulomb = 0'24 cal. The electrical energy then equivalent to the heat of the chemical reaction is 25,065-^0-24 = 104,270 volt-coulombs. If we now divide this number by 96,500, the number of coulombs produced, we get T08 for the electromotive force of the cell expressed in volts, on the assumption that all the chemical energy of the reacting substances has been converted into electrical energy. Direct measurement of the E.M.F. of the Daniell cell gives 1'09 to 1*10 volt. The assumption then that the chemical energy is wholly converted into electrical energy is in this case very nearly true, and several other cells are known to which a similar simple calculation for their electromotive force is applicable. These cells, however, are all of a special type y and we shall therefore proceed by means of a reversible cycle of operations to deduce a formula of more general application. In the first place the cell considered must be, like Daniell's, of a completely reversible or non-polarisable type ; i.e. it must be of such a kind that if we pass a current through it in the reverse direction of the current which the cell would of itself generate, the chemical action in the cell will be exactly reversed. Thus when the Daniell cell is in action the positive current within the cell moves with the positive ions from the zinc electrode to the copper electrode. If we now by using an external electromotive force make the current pass from the copper pole to the zinc pole, the chemical action which occurs is which is the reverse of the primary action of the cell, copper being dissolved and zinc deposited. Let the cell act at the constant temperature T, and generate the quantity (7=96,500 coulombs of electricity. If the electromotive force of the cell is E, the electrical energy afforded by the cell is EC, which may or may not be equal to $, the diminution in chemical energy, which under ordinary circumstances would be the heat of the chemical action. Suppose the electrical energy produced to be less than the fall in chemical energy, then the element on working will give out Q - EC as heat at the constant temperature T. Let now the system be heated to the slightly higher temperature T + dT, and let the quantity C of electricity be sent through the cell in the reverse xxvii THERMODYNAMICAL PROOFS 331 direction, the temperature being kept constant. If the electromotive force has diminished by the amount dE owing to the change of temperature, the work done on the cell will be C(E-dE), and the amount of heat absorbed will be Q - C(E - dE) on the supposition that the heat of the reaction does not change appreciably with the temperature. The system is finally cooled to the original temperature T, so that everything regains its initial state, a reversible cycle having been performed. On the whole, the system has done the external electrical work C(E - dE} - CE= - CdE, which must be equal to the fraction dT/T of the heat given out at the lower temperature. Now the quantity of heat given out is Q - CE, so that the heat transformed into work is and we have therefore the equation or From this relation we see that in order to ascertain the electromotive force of an element from the heat of the chemical action within it, we must know the rate of change of the electromotive force with the temperature. If the electromotive force does not vary with the temperature, i.e. if dE/dT be zero, then the simple formula P Q E = G may be used. The electromotive force of a Daniell cell has a very small temperature co-efficient, so in its case the simpler formula gives a result closely approximating to the truth. The more accurate formula gives in this case a still closer approximation to the observed electromotive force, and has been experimentally verified in many other instances for which the temperature co-efficient is larger. If we write the expression in the form we see at once that the electrical energy and the chemical energy of the process are equal if dE/dT is zero ; that the electrical energy is greater than the chemical energy if the temperature coefficient is positive, i.e., if the electromotive force increases with rise of tempera- ture; and that the electrical energy is less than the chemical energy if the temperature coefficient is negative, i.e. if the electro- 332 INTRODUCTION TO PHYSICAL CHEMISTRY CH. xxvn motive force falls with rise of temperature. If a cell with a positive temperature coefficient of electromotive force is allowed to act, it will make up for the difference between the electrical and chemical energies by abstracting heat from neighbouring bodies; or, if no external heat is available, it will cool itself by working. The student is apt to imagine that this is a contradiction of the Second Law of Thermo- dynamics ; but, like the self-cooling of a freezing mixture, the process here involved is not a cycle of operations, and the system cannot regain its original state without work being done upon it. What the Second Law contradicts is the existence of a system which, working in a cycle, by repeated self-cooling, can convert into work the heat of neighbouring bodies. An excellent elementary account of the Principles of Thermodynamics is that by CAREY FOSTER in WATTS' Dictionary of Chemistry, original edition, 3rd supplement, pt. ii. pp. 1922-1951. INDEX Accelerating influence of acids, 254, 271 Acids, dissociation constants, 144, 224, 285 electric conductivity, 224 strength (avidity), 266, 269, 275, 292 Active mass, 236 of solids, 243 Additive properties, 136 Adiabatic processes, 313 Affinity constants, 273 Allotropic transformation, 101 Amorphous state, 60-62 Argon, 17, 36, 49 Association, molecular, 193 Asymmetric carbon atom, 140 Atom, 8 Atomic heat, 31 hypothesis, 8-21 refraction, 139 volume, 42, 137 weights, 12, 49 table of, 20 unit of, 13 Avidity, 269, 275, 292 Avogadro's Law, 11 Balanced Actions, 234-253 Bases, conductivity, 224 strength, 276 Blagden's Law, 63 Boiling points, 131, 198 of solutions, 80, 173 Border curve, 77 Boyle's Law, 27 deviations from, 89-91 for dissolved substances, 163 Calorie, 6 Calorimeter, 125 Capillarity, 189 Catalysis of ethereal salts, 256, 272 Circular polarisation, 139, 141, 150 Colligative properties, 147 Combining proportions, 9 Combustion, heat of, 122, 130 Complex ions, 305, 308 molecules, 193 Condensation and vaporisation, 73-83 Conductivity, molecular, 211, 220-224 Conservation of energy, 4, 119, 311 Constant boiling mixtures, 81 Constitution and physical properties, 136- 147 Continuity of gaseous and liquid states, 77, 91 Corresponding conditions, 94 Critical constants, 73 solution temperature, 76 Cryohydrates, 65 Crystalline liquids, 62, 104 Crystallisation, 61 Cycle of operations, 314 Dalton's law of partial pressures, 55, 79 Decomposition by water (hydrolysis), 247, 278-282 Degree of dissociation, 221, 231, 273 Dehydration of hydrates, 111, 244 Deliquescence, 114 Density, 3, 127 of gases, 13, 176 of solutions, 155 Depression of freezing point, 174, 186, 329 Desmotropy, 253 Diffusion of gases, 148 in liquids, 149, 165 Dilution formulae, 224, 227 Diminution of solubility, 293, 302 Dissociation constants, 144, 224, 285 electrolytic, 217-233 gaseous, 241, 265 pressure, '244, 250 Distillation of mixtures, 80 with steam, 82 Double decomposition, 305 Dulong and Petit's Law, 30 Dynamic isomerism, 253, 265 334 INTRODUCTION TO PHYSICAL CHEMISTRY Efflorescence, 113 Electrical eiiergy, 7, 329 units, 6 Electrolysis, 201-211 Electrolytes, 202 Electrolytic conductivity, 201, 207, 211. 272, 277 dissociation, 217-233, 272, 296-310 Electromotive force, 6, 330 Elements, 8 periodic classification, 44 table, 20 Elevation of boiling point, 173. 181, 327 Endothermic compounds, 123 Energy, 4 conservation of, 4, 119, 311 intrinsic, 118 Equations, chemical, 22-26 thermochemical, 120 Equilibrium, 97-116 chemical, 234-253 of electrolytes, 283-295 Equivalent weights, 9 determination of, 17 Eutectic mixtures, 71 Exothermic compounds, 123 Explosions, 262 Extraction with ether, etc., 57 Faraday's Law, 204 Freezing point, 61 depression of, 174, 186, 329 of mixtures, 70 of solutions, 63, 174 Fusion and solidification, 60-72 Gas-constant, 29 Gas-laws, 27-29 deviations from, 89-91 for solutions, 162-164 Gaseous diffusion, 148 liquefaction of, 74 Gases, solvent action of, 79 Gay-Lussac's Law of Volumes, 11 of expansion, 27 Heat, atomic, 31 mechanical equivalent of, 6 molecular, 33, 35 specific, 30-37 units, 6 of combustion, 122, 130 of formation, 121 of transformation, 117 Henry's Law, 55, 198, 321 Homologous series, 127-135 Hydrates, 51, 67, 69, 108 dehydration of, 111, 244 Hydrolysis of ethereal salts, 238, 256 of salts, 247, 278 Indicators, 309 Intrinsic energy, 118 Inversion of cane-sugar, 254, 271 Inversion points, 102 Ions, 204 migration of, 207 speed of, 212 Isohydric solutions, 286 Isoinerism, 145, 146 dynamic, 253, 265 Isothermal curves, 76, 91 Isotonic solutions, 167 Jellies, 213 Kinetic theory, 84-96 Liquefaction of gases, 74 Liquids, molecular weight, 188 associated, 53, 195, 200 crystalline, 62, 104 normal, 53, 188, 199 Lowering of vapour-pressure, 171, 325 of freezing point, 174, 328 Magnetic rotation, 141 Mass, 2 active, 236 Mechanical equivalent of heat, 6 Medium, influence on rate of chemical change, 261 Melting points, 60, 69, 72 in homologous series, 133 Metastable conditions, 70, 101, 115 Migration of ions, 207 Miscibility of liquids, 52 Mixtures, constant-boiling, 81 distillation of, 80 eutectic, 71 of liquids, 52 of gases, 55 Moduli, 155 Molecular complexity, 193-200 conductivity, .211, 220-224 depression, 187, 329 elevation, 185, 327 heat, 33/34, 128, 137 magnetic rotation. 142 refractive power, 139 rotation, 139 volume, 14, 128 weights, 14, 176-192, 193-200 Neumann's Law, 33 Neutralisation, heat of, 119, 296 Normal liquids, 53, 188, 199 Octaves, law of, 42 Ohm's law, 217 Optical activity, 139, 150 magnetic, 141 INDEX 335 Osmotic pressure, 158-168, 324 Ostwald's dilution formula, 224 Oudeman's Law, 153 Oxidation in solution, 300 Partial pressure, 55, 79, 159 Partition coefficient, 59, 197 Periodic law, 38-49 table, 44 Phases, 97 new, 114 Physical properties and chemical constitu- tion, 142-147 Precipitation, 303-304 Pressure, atmospheric, 3 osmotic, 158-168, 324 partial, 55, 79, 159 Prout's hypothesis, 21 Rate of chemical action, 254-265 of crystallisation, 61 Reactions of salts, 299 Recrystallisation, 57 Reduction in solution, 300 Refractive power, 138 Reversible cycles, 314 processes, 315, 321 Rotatory power, 139 magnetic, 142 Rudolphi's dilution formula, 227 Salting-out, 303 Salts, acid, 285 S aponification of ethereal salts, 256 Saturated solutions, 50 vapour, 75 Semipermeable membranes, 160 Solid solutions, 191 Solidification and fusion, 60-72 Solubility, 50-59 curves, 51, 107 of electrolytes, 293 of gases, 55, 59 of "insoluble" sajts, 307 of isomers, 135 of precipitates, 306 in homologous series, 134 Solutions, boiling point, 80 freezing point, 63 isotonic, 167 vapour pressure, 79 Solution-tension, 96 Solvent action of gases, 79 Specific gravity, 3, 127 heats, 30-39 refractive power, 138 magnetic rotation, 142 rotatory power, 139 volume, 3, 127 Speed of gas-molecules, 87 of ions, 212 Strength of acids and bases, 266-282 Sublimation, 78 Sugar-inversion, 254, 271 Superfusion, 61, 62 Supersaturated solutions, 50, 115 Surface tension, 188 Tautomerism, 253 Temperature, 5 influence on molecular conductivity, 222 rate of chemical change, 260 of ignition, 262 Thermochemical change, 117-126 Thermodynamics, 311-332 Transition points, 102 Transport numbers, 210 Triple point for water, 99, 321 Trouton's rule, 124 Units, 1-7 for atomic weights, 12 Valency, 9, 46 Valson's moduli, 155 Van der Waals's equation, 89-96 Van 't Hoif's factor i, 229 dilution formula, 227 Vaporisation and condensation, 73 - 83, 318 Vapour density, 176 pressure, 75, 98 of solids, 78 of solutions, 79, 171, 325 Velocity, constant, 237 of chemical action, 254-265 Volume, atomic, 42, 137 critical, 73 molecular, 14, 128 specific, 3 Water, decomposition by, 247, 278 influence of vapour, 265 Welter's rule, 122 Work done by an expanding gas, 29 Wiillner's Law, 79 THE END Printed by R. & R. CLARK, LIMITED, Edinburgh -1 AM; 1N1T.AU n^OF uR f To CENTS AUG 3O wnr 6 1936 FEB 101940 JUN 7 1941 . ,. '00 LD 2l-20m-b, o- re 16787 UNIVERSITY OF CALIFORNIA LIBRARY