摘要:

292 GENERAL DISCUSSION AUTHOR Alfrey, G. F., 218. Andon, R. J. L., 168. Atack, Douglas, 210. Barker, J. A., 111, 142, 188, 270, 279. Baughan, E. C., 283. Bell, G. M., 113, 2%. Bellemans, A., 80, 117, 275. Brown, I., 142. Brown, J. A., 254. Clark, A. M., 202. Strickland-Constable, R. F., 288. Copp, J. L., 174, 265, 268, 270, 272. Cox, J. D., 168. Din, F., 202. Eley, D. D., 262. Everett, D. H., 114, 117, 126, 174, 252, Ewald, A. H., 238. Fock, W., 188. Francis, P. G., 279. Fuchs, O., 290. Glew, D. N., 150, 260, 264, 267, 275. Glueckauf, E., 257, 28 1. Guggenheim, E. A., 24, 66, 108, 109, 1 1 1 , 121, 271. Guggenheimer, K. M., 121. Haase, R., 270. Hart, K. R., 130. Herington, E. F. G., 168, 255, 256, 273. Hildebrand, J. H., 9, 254,264,275,285,290, Moelwyn-Hughes, E. A. 150, 261, 264.Jepson, W. B., 238. Wynne-Jones, W. F. K., 161. 258, 260, 267, 268, 270, 272. 291. INDEX * Kirkwood, John G., 28. Klinkenberg, A., 276, 290. Koefoed, J., 207, 258, 282. Korvezee, A. E., 255, 274, 276. Lambert, J. D., 226. Lewin, J., 195. Longuet-Higgins, H. C., 73, 112. Martin, A. R., 265. Sarolea-Mathot, L., 259. Mathot, V. D. J., 93, 109, 114, 118. Mitchell, A. G. 161. Pople, J. A., 35, 111, 257. Prigogine,T., 80,93, 110, 112, 116, 117, 120, Rice, 0. K., 110, 210, 276, 286, 287. Robin S., 233. Rowlinson, J. S., 52, 108, 112, 238, 258, Rushbrooke, G. S., 57, 115, 258. Salzburg, Zevi W., 28. Schneider, W. G., 11 1, 218. Scoins, H. I., 57. Scott, Robert L., 44, 11 3, 118, 282. Smith, F., 142. Staveley, L. A. K., 130. Timmermans, J., 195, 278. Tompa, H., 258, 283.Trappeniers, N., 93. Tupman, W. I., 130. Ubbelohde, A. R., 124. Vodar, B., 233. Waals, J. H. van der, 113, 261. Wakefield, A. J., 57. Webster, T. J., 243. 286, 288. 290. * The references in heavy type indicate papers submitted for discussion.292 GENERAL DISCUSSION AUTHOR Alfrey, G. F., 218. Andon, R. J. L., 168. Atack, Douglas, 210. Barker, J. A., 111, 142, 188, 270, 279. Baughan, E. C., 283. Bell, G. M., 113, 2%. Bellemans, A., 80, 117, 275. Brown, I., 142. Brown, J. A., 254. Clark, A. M., 202. Strickland-Constable, R. F., 288. Copp, J. L., 174, 265, 268, 270, 272. Cox, J. D., 168. Din, F., 202. Eley, D. D., 262. Everett, D. H., 114, 117, 126, 174, 252, Ewald, A. H., 238. Fock, W., 188. Francis, P. G., 279. Fuchs, O., 290. Glew, D. N., 150, 260, 264, 267, 275. Glueckauf, E., 257, 28 1.Guggenheim, E. A., 24, 66, 108, 109, 1 1 1 , 121, 271. Guggenheimer, K. M., 121. Haase, R., 270. Hart, K. R., 130. Herington, E. F. G., 168, 255, 256, 273. Hildebrand, J. H., 9, 254,264,275,285,290, Moelwyn-Hughes, E. A. 150, 261, 264. Jepson, W. B., 238. Wynne-Jones, W. F. K., 161. 258, 260, 267, 268, 270, 272. 291. INDEX * Kirkwood, John G., 28. Klinkenberg, A., 276, 290. Koefoed, J., 207, 258, 282. Korvezee, A. E., 255, 274, 276. Lambert, J. D., 226. Lewin, J., 195. Longuet-Higgins, H. C., 73, 112. Martin, A. R., 265. Sarolea-Mathot, L., 259. Mathot, V. D. J., 93, 109, 114, 118. Mitchell, A. G. 161. Pople, J. A., 35, 111, 257. Prigogine,T., 80,93, 110, 112, 116, 117, 120, Rice, 0. K., 110, 210, 276, 286, 287. Robin S., 233. Rowlinson, J. S., 52, 108, 112, 238, 258, Rushbrooke, G. S., 57, 115, 258. Salzburg, Zevi W., 28. Schneider, W. G., 11 1, 218. Scoins, H. I., 57. Scott, Robert L., 44, 11 3, 118, 282. Smith, F., 142. Staveley, L. A. K., 130. Timmermans, J., 195, 278. Tompa, H., 258, 283. Trappeniers, N., 93. Tupman, W. I., 130. Ubbelohde, A. R., 124. Vodar, B., 233. Waals, J. H. van der, 113, 261. Wakefield, A. J., 57. Webster, T. J., 243. 286, 288. 290. * The references in heavy type indicate papers submitted for discussion.

ISSN:0366-9033    

DOI:10.1039/DF95315FX001

出版商:RSC    

年代:1953

数据来源: RSC

摘要:

292 GENERAL DISCUSSION AUTHOR Alfrey, G. F., 218. Andon, R. J. L., 168. Atack, Douglas, 210. Barker, J. A., 111, 142, 188, 270, 279. Baughan, E. C., 283. Bell, G. M., 113, 2%. Bellemans, A., 80, 117, 275. Brown, I., 142. Brown, J. A., 254. Clark, A. M., 202. Strickland-Constable, R. F., 288. Copp, J. L., 174, 265, 268, 270, 272. Cox, J. D., 168. Din, F., 202. Eley, D. D., 262. Everett, D. H., 114, 117, 126, 174, 252, Ewald, A. H., 238. Fock, W., 188. Francis, P. G., 279. Fuchs, O., 290. Glew, D. N., 150, 260, 264, 267, 275. Glueckauf, E., 257, 28 1. Guggenheim, E. A., 24, 66, 108, 109, 1 1 1 , 121, 271. Guggenheimer, K. M., 121. Haase, R., 270. Hart, K. R., 130. Herington, E. F. G., 168, 255, 256, 273. Hildebrand, J. H., 9, 254,264,275,285,290, Moelwyn-Hughes, E. A. 150, 261, 264.Jepson, W. B., 238. Wynne-Jones, W. F. K., 161. 258, 260, 267, 268, 270, 272. 291. INDEX * Kirkwood, John G., 28. Klinkenberg, A., 276, 290. Koefoed, J., 207, 258, 282. Korvezee, A. E., 255, 274, 276. Lambert, J. D., 226. Lewin, J., 195. Longuet-Higgins, H. C., 73, 112. Martin, A. R., 265. Sarolea-Mathot, L., 259. Mathot, V. D. J., 93, 109, 114, 118. Mitchell, A. G. 161. Pople, J. A., 35, 111, 257. Prigogine,T., 80,93, 110, 112, 116, 117, 120, Rice, 0. K., 110, 210, 276, 286, 287. Robin S., 233. Rowlinson, J. S., 52, 108, 112, 238, 258, Rushbrooke, G. S., 57, 115, 258. Salzburg, Zevi W., 28. Schneider, W. G., 11 1, 218. Scoins, H. I., 57. Scott, Robert L., 44, 11 3, 118, 282. Smith, F., 142. Staveley, L. A. K., 130. Timmermans, J., 195, 278. Tompa, H., 258, 283.Trappeniers, N., 93. Tupman, W. I., 130. Ubbelohde, A. R., 124. Vodar, B., 233. Waals, J. H. van der, 113, 261. Wakefield, A. J., 57. Webster, T. J., 243. 286, 288. 290. * The references in heavy type indicate papers submitted for discussion.292 GENERAL DISCUSSION AUTHOR Alfrey, G. F., 218. Andon, R. J. L., 168. Atack, Douglas, 210. Barker, J. A., 111, 142, 188, 270, 279. Baughan, E. C., 283. Bell, G. M., 113, 2%. Bellemans, A., 80, 117, 275. Brown, I., 142. Brown, J. A., 254. Clark, A. M., 202. Strickland-Constable, R. F., 288. Copp, J. L., 174, 265, 268, 270, 272. Cox, J. D., 168. Din, F., 202. Eley, D. D., 262. Everett, D. H., 114, 117, 126, 174, 252, Ewald, A. H., 238. Fock, W., 188. Francis, P. G., 279. Fuchs, O., 290. Glew, D. N., 150, 260, 264, 267, 275. Glueckauf, E., 257, 28 1.Guggenheim, E. A., 24, 66, 108, 109, 1 1 1 , 121, 271. Guggenheimer, K. M., 121. Haase, R., 270. Hart, K. R., 130. Herington, E. F. G., 168, 255, 256, 273. Hildebrand, J. H., 9, 254,264,275,285,290, Moelwyn-Hughes, E. A. 150, 261, 264. Jepson, W. B., 238. Wynne-Jones, W. F. K., 161. 258, 260, 267, 268, 270, 272. 291. INDEX * Kirkwood, John G., 28. Klinkenberg, A., 276, 290. Koefoed, J., 207, 258, 282. Korvezee, A. E., 255, 274, 276. Lambert, J. D., 226. Lewin, J., 195. Longuet-Higgins, H. C., 73, 112. Martin, A. R., 265. Sarolea-Mathot, L., 259. Mathot, V. D. J., 93, 109, 114, 118. Mitchell, A. G. 161. Pople, J. A., 35, 111, 257. Prigogine,T., 80,93, 110, 112, 116, 117, 120, Rice, 0. K., 110, 210, 276, 286, 287. Robin S., 233. Rowlinson, J. S., 52, 108, 112, 238, 258, Rushbrooke, G. S., 57, 115, 258. Salzburg, Zevi W., 28. Schneider, W. G., 11 1, 218. Scoins, H. I., 57. Scott, Robert L., 44, 11 3, 118, 282. Smith, F., 142. Staveley, L. A. K., 130. Timmermans, J., 195, 278. Tompa, H., 258, 283. Trappeniers, N., 93. Tupman, W. I., 130. Ubbelohde, A. R., 124. Vodar, B., 233. Waals, J. H. van der, 113, 261. Wakefield, A. J., 57. Webster, T. J., 243. 286, 288. 290. * The references in heavy type indicate papers submitted for discussion.

ISSN:0366-9033    

DOI:10.1039/DF95315BX003

出版商:RSC    

年代:1953

数据来源: RSC

摘要:

AND SEVENTH SPIERS MEMORIAL LECTURE BY JOEL H. HILDEBRAND Received 16th April, 1953 The invitation to deliver the Spiers Memorial Lecture before this Society upon so notable an occasion is an honour which I prize as highly as any that has come my way. And it is more than an honour. The topic chosen for the symposium to follow has been the major interest of my lifetime. It has been a long life, thanks to my wise choice of ancestors, and I have seen the gradual development of our knowledge of liquids and solutions from its elementary status a half-century ago to the relatively advanced stage testified by the contributions to this programme. I was strongly impressed by this process of growth while working with my co-author, Dr. Scott, upon the third edition of Solubility of Non-electrolytes.1 The first edition, published in 1924, I could write largely out of my head.The rela- tions set forth were mainly qualitative. The second edition, in 1936, recorded many notable advances, as reflected also in the meeting of this Society in Edinburgh in the same year. Writing the third edition proved a monumental task, even for two men, in view of the numerous contributions of the able investigators who have been attracted to the field. Inasmuch as the demands of a fourth edition would exceed my probable span of years, as well as my competence to keep up with the intense activity in the field, I am taking advantage of this occasion to give my farewell general appraisal of the status of theory. The approach to the subject which is congenial to my nature is predominantly inductive.It is the one aptly described by a former colleague when he said, " One of the main functions of a university is to examine the discrepancies between actual phenomena and the currently accepted theories about them ". Faraday expressed essentially the same thought when he wrote to Tyndall, " The more we can enlarge the number of anomalous facts and consequences the better it will be for the subject, for they can only remain anomalies to us while we continue in error ". My purpose in this lecture is to call attention to a number of experimental facts that are frequently not taken into account in the models set up by various investigators, including myself. As I have often remarked, it is our good fortune that some of our formulae apply as well as they do in cases that depart consider- ably from the simplified assumptions of the models.But it is well to know what we are doing, and not be so lost in admiration for our handiwork in building imposing mathematical structures that we fail to appreciate the limits of their applicability. My chief ambition has been to give reasonably approximate answers to the practical questions involving solubility that constantly confront workers in every field of chemistry. These questions nearly always involve the behaviour of molecules having features not considered in the models. I present what follows with no sense of authority. I am the very opposite of the old maid, defined as one who knows all the answers but has never been asked any of the questions.I have been asked many of the questions, but I do not know all the answers. In what follows there will be more questions than answers. The phenomena under discussion are various and complicated, and no one of us is in a position to tell the whole truth about any aspect of it. If I inject only half- truths into the discussions, I do so because, as Stephen Leacock once wrote, " in 910 SEVENTH SPIERS MEMORIAL LECTURE an argument, a half-truth is often like a half-brick, it carries better ". I under- stand that I am immune to at least vocal criticism at this time, but I desire no immunity during the subsequent sessions. I shall feel more complimented by criticism than by silence. I wish to begin with certain facts and opinions bearing upon the structure and solutions is a very attractive one.It extends to the liquid state the methods already familiar for calculating certain properties of solids. It provides a frame of reference for counting the number of possible configurations in liquid solu- tions and permits one to apply to them the same type of calculation of energy and entropy of mixing as is applied to mixed-crystals. However, the model is subject to several disadvantages. I mention, first, the fact that few molecular species are either so nearly equal in size as to permit replacement in a " cell " or in a fixed lattice, or of such relative dimensions as to permit them to occupy respectively single and multiple sites in a fixed lattice. Let me emphasise the spread in the molal volumes of liquids by listing a number of values.Of liquids and SOlUtiOnS. THE QUASI-CRYSTALLINE OR LATTICE MODEL of liquids TA~LE ~.-MOLAL VOLUMES OF LIQUIDS cm.3 at 25" €3 r2 51 C6H6 89 C(CH3)4 122 cs2 61 cc14 97 n-C6H14 132 p4 70 Tic14 1 1 1 n-C8Hls 164 CHC13 81 SnC14 118 JZ-C~FI~ 227 Nature avoids this difficulty in solid solutions, because two substances do not normally form mixed-crystals unless their pure crystals have nearly identical lattices, but not so with liquids. One effect of such inequalities is seen in the unsymmetrical shape of the excess functions and of liquid-liquid solubility curves when plotted against mole fraction. This is strikingly evident in the liquid-liquid solubility curves for perfluoroheptane, in fig. 1, determined for me by Fisher and Benesi.2 The dotted lines mark the critical compositions calculated by the formula +l/q5z = (V2/V#, derived from the solubility equation in which both the heat and the entropy are expressed in terms of volume fractions 4 and molal volumes K The agreement with the maxima of the curves is excellent. The case of stannic iodide and dicetyl3 is especially striking.Their molal volumes at the critical temperature are very different, 151 cm 3 and 615 cm3 respectively, in the ratio of 1/4. The formula gives 91 mole %, which agrees with experiment within the limit of its accuracy, as seen in fig. 2. Our monograph 1 givcs many other examples of the superiority of volume fractions over mole fractions whenever, as is usually the case, the constituents have significantly different molal volumes. Our recent investigations of the supercooling and crystallization velocity of phosphorus 4 seem to me to indicate that the structure of the liquid is far from that of a solid somewhat " mussed up ", or containing a few holes.Although the P4 molecule is tetrahedral, and doubtless possesses rotational energy-it melts with exactly R entropy units, and has a transition at - 79", so that one would expect it to form its cubic crystals quite easily, it can be kept in the liquid state at room temperatures, some 25" below its melting point, for weeks on end, and we have cooled isolated liquid droplets to below - 70" without their becoming glassy. It is hard to see how microcrystalline aggregates, if present, could fail to grow and the mass solidify when supercooled so far.When crystallization does set in it proceeds at lower temperatures with almost explosive velocity ; we measured a linear velocity of crystallization of over 2 m/sec at 22". A similar inference can be drawn from recent work by Thomas and Staveley.5 As a further check on the degree of similarity of solid to liquid structure, D. W. Frage, in our laboratory, under the direction of my colleague, Dr. Alder,A B FIG. 3. A, X-ray picture of solid gallium at 22" C with an 8-h exposure. B, X-ray picture of liquid gallium at 30.5" C with a 5-h exposure. [To f u c e p . 11JOEL H . HILDEBRAND 11 has photographed the X-ray scattering of gallium, as liquid just above its melting point and as solid powder just below. Fig. 3 reproduces these photographs 200 190 f80 J70- J60- 150- I40 N O side by side.The considerable difference between them indicates that the structure of the liquid is far from a quasi-crystal with a few more holes. These photo- graphs are being analyzed in order to obtain and compare the two distribution functions. Rowlinson and Curtiss 6 have recently published a critical analysis of four modi- fications of lattice theory of the liquid state, showing their degree of divergence from experiment (see also Peek and Hill 7). Krigbaum and Flory,* discussing the statistical mechanics of dilute ternary solutions of two polymers, stated that “use of equations based on the lattice model to evaluate polymer-polymer inter- action parameters from dilute solution data is shown to bc valid only in special instances ”. In short, I agree with Fiirthg who wrote in an excellent review of the struc- ture of liquids, “Curiously enough, the ‘ fashion ’ to regard the liquid as an im- perfect solid, which was particularly advo- cated by Frenkel, has not remained long in favour with the physicists and is now giving way l o a ‘ new look ’ which, as so often is the case in fashion, is in fact a very ‘old look’ indeed, namely, to regard ‘fluids’, that is gases and liquids, as essentially the sainc thing (Kirkwood, Born and Grcen, etc.).I I I I - Stannic iodide -Dice fyl solution - y u’ +$ $ - c?! $ - - Mole fraction of stannic iodjde I I I I A 0 0.2 0.4 0.6 0.8 0 FIG. l.-Xz = Mole fraction of second component FIG. 2. “ Whatever the reasons for this change of opinion are, it must be admitted that there is no conclusive experimental evidence for the fundamental assumption12 SEVENTH SPIERS MEMORIAL LECTURE of a ‘ solid-like ’ structure of liquids, namely, the assumption that the arrangement of the near neighbours of an arbitrary particle in a liquid is identical with that in the corresponding crystal.As early as 1927 it had been pointed out by Zernicke and Prins (31) that thc observed X-ray patterns of liquids could be explained by the mere assumption of the existence of a ‘ radial distribution function ’.” It was for this reason that Wood, and I,lO in 1933, turned to the radial dis- tribution function in order to derive expressions for the potential energy of a liquid and of a regular solution. In my opinion, this function provides a far more realistic description of liquid structure than does any lattice model.Its form is reasonably well known from X-ray scattering, and it is with good approxi- mation a function of volumc only. My former collaborator, J. A. Campbel1,ll showed that its form with xenon is the same at - 110” and 1 atm as at - 90” and 130 atm, when it is compressed to the same volume ; and furthermore, that the curves for xcnon and argon, at practically corresponding states, are almost exactly superimposable (cf. fig. 4). It seems evident that it would be very useful to have tables of the normalized distribution function for a “perfect ” liquid at I I I 2 I C 5 1.0 I .5 2.0 FIG. 4. various degrees of relative expansion. In my Edinburgh address,12 in 1936, I had this to say : ‘‘ The problem of deriving mathematically the probability function as used by Prins, Zernike, Debye and Menke, for the structure of a pure liquid is one from which all mathematicians to whom I have broached the subject have so far retreated.I mention it, however, because it seems to be thc kind of problem that ought to be soluble. The problem presented by solutions of molecules of different sizes and shapes is, of course, vastly more difficult. Most of us would tackle the statistical problems that could be presented by arrangements of checkers on an ordinary checkerboard with some hope for success, but we would shy off in dismay from a game such as might be played in Wonderland with pieces of varying shapes and sizes on a board whose squares would change size to ac- commodate the pieces.It would seem, however, that some future Lewis Carroll should arise mathematician enough to train an Alice to play such a game.” I subsequently mentioned to Kirkwood the desirability of deriving the radial distribution function, saying that he should do it because it was beyond my powers but well within his. The event has shown that I was right. He and his co-workers have published a notable series of studies of this problem, culminating in a recent remarkable one by Kirkwood, Lewinson and Alder.13JOEL H . HILDEBRAND 13 The other problem, that of the checkerboard, also was solved independently by Flory and by Huggins for long chain molecules by imagining their segments strung on multiple sites of a lattice with the solvent molecules occupying single sites.The relation they derived is not dependent, however, upon the assumption of a lattice, because I found 14 that essentially the same relation could be obtained by considering the increase in entropy occurring when N 1 and N 2 moles of the pure liquid components are mixed, their free volumes, V1f and Vzf, both increasing to the free volume in the mixture, Vf. The increase in entropy can then be written Vf ASM = R N1 In - -I- N 2 In - ( Vlf If the expansion on mixing is neglected, and if the free volumes are assumed to be proportional to the molal volumes, this reduces to the familiar Flory-Huggins equation, where the 4’s denote volume fractions. Huggins 15 also showed that this equation could be derived without reference to a lattice. The convenience of being able to compute entropy by counting configurations upon a lattice or in assumed cells should not blind us to the fact that such a frame of reference is quite artificial for a liquid. Counting is strictly in order only when there are natural discrete positions, as in a crystal, or, discrete quantum states.I have more to say upon the subject of VOLUME CHANGES ON MIXING LIQUIDS. In my original definition 16 of a “ regular solution ” as one in which thermal agitation suffices, despite unequal intermolecular forces, to give maximum random- ness, and therefore ideal entropy of mixing, I added the condition that “ We may expect, further, that a small correction should be added to take care of the change in entropy accompanying changes in volume, given by ( ~ S / ~ V ) T , or, we may state our principle in the following form.A regular solution is one involving no entropy change when a small amount of one of its components is transferred to it from an ideal solution of the same composition, the total volume remaining unchanged.” Scatchard 17 was the first to calculate the magnitude of the difference between the entropies of mixing at constant pressure and at constant volume, and he and his collaborators measured the expansion on mixing and determined accurately the difference between the two entropies. Scatchard, Wood and Mochel18 found, for example, that the maximum expansion for carbon tetrachloride + cyclohexane mixtures is 0.175 %, and the difference between the entropy of mixing at constant pressure and at constant volume is 0.03 cal/mole deg.Wood 19 has continued to study this problem. These calculated values of entropy and energy combine to give free energy correctly, as pointed out by Scatchard 20 in 1949. It is only when they are compared separately that the discrepancy becomes serious. May I emphasize the situation with respect to free energy by giving here again the success with which the equations reproduce21 the measured solubilities of iodine in a series of solvents covering a 420-fold range, as shown by the good agreement of the calculated values of 8 2 , the “ solubility parameter ” of iodine. ASM = - R(N1 In + d- N2 In 421, TABLE 2.-IODINE SOLUTIONS AT 25°C 61 82 molal mole % volume 12 solvent n-C7F16 227.0 0.0185 5.7 14.2 Sic14 115.3 0.499 7.6 13.9 cc14 97.1 1.147 8.6 14.2 Tic14 110.5 2.15 9.0 14-1 CS? 60.6 5-46 9.9 14.1 CHBr3 87.8 6.16 10.5 14.114 SEVENTH SPIERS MEMORIAL LECTURE Recently Scott and 122 have returned to this question, examining solutiolls where the discrepancy between - R In x2 and the actual entropy of solution is very large.The straight lines resulting when the logarithm of the mole fraction of a solid solute is plotted, not, in the customary way, against the reciprocal of the absolute temperature, but against its logarithm, permit an accurate determi11- ation of the entropy of solution from solid to saturated solution. Subtracting the entropy of fusion, we obtain figures for the entropy of transfer from pure liquid to solution and compare it with the ideal entropy of transfer, or, alternatively, the Flory-Huggins value, -R [ In $1 - $2 ( 1 - - ;)I, The excess of the actual entropy of solution of iodine over that calculated for mixing at constant volume was 3.7 ca1,’mole deg.in solution in CCl4 and 7.1 cal/mole deg. in C7F16. In both cases, however, these large differences were in satisfactory agreement with the calculated entropy of expansion upon mixing. I wish here to rake the question of the proper definition of the term “ REGULAR SOLUTION”. In 8 Lote in Mature,z3 in 1951, I called attention to the fact that this term, which I proposed in 1927 and further defined in 1929, has come into general use, but in such a variety of senses as can lead only to confusion and misunderstanding. This symposium, devoted to the theory of solutions, seems an ideal occasion for a frank discussion of the concept in an effort to arrive at a consensus.In 1923 I wrote,16 in part, “ Now suppose that the component X2 is transferred from an ideal solution to any regular solution in which it has the same mole fraction. From our picture of a regular solution as one in which the distribution and orientations are random, just as in the ideal sclution, we may conclude that the probability of X2 is the same in the two solutions and, therefore, that the difference in entropy is zero. We cannot expect this conclusion to hold unless the random distribution of the molecules persists.” I then added the statemcnt quoted earlier concerning entropy of expansion. When I wrote the foregoing, maximum randomness and ideal entropy of mixing appeared to be synonymous.The role of unequal molecular volumes seemed inconsequential because Z knew of various solutions that approached closely to ideal behaviour despite consider- able differences in molal volume between the pure components. It was not till high polymer solutions came under scrutiny that tte role of unequal volumes was truly appreciated. Similarly, the entropy of expansion received little atten- tion until Scatchard studied it quantitatively. As a matter of fact, its neglect made very little difference to solubility relations, for the reason stated above. The solubiiity relations of iodine, for example, can be satisfactorily correlated without reference to the entropy of expansion, But the discrepancy between the actual partial molal entropy of solution and the ideal, for iodine in carbon tetrachloride and in perfluoroheptane, satisfactorily accounted for by the entropy of expansion, recalled to me the necessity of retaining the stipulation that the mixing must take place at constant volume if one is to expect ideal entropy.But the fact that maximum randomness cannot be expected to yield ideal, i.e. Raoult’s law, entropy of mixing unless the components have molecules of equal size raises the question whether it would be better to define a regular soIution as (i) one formed from its component liquids at constant volume with ideal entropy, or (ii) one in which thermal agitation is sufficient to give practically complete randomness. The former has the advantage that it can be expressed in a simple mathematical form, with a simple relation to activity coefficient, but it is far more restricted than the latter in that it applies strictly only to the comparatively few solutions whose components have practically equal volumes.One might, of course, try to avoid tbis difficulty by adopting the Flory-Huggins equation as the definition of regular entropy of mixing, but that equation is frankly one of several formulations of the entropy of the process. I have expressed it in terms of free volumes and Scatchard has suggested the use of surface areas. May I therefore amplify upon my note to Nature.JOEL H . HILDEBRAND 15 I have for some time wavered between the two alternatives, but I am now inclined to favour the second, for the following reasons. (i) It includes solutions whose components have unequal molal volumes but which give, nevertheless, solubility curves for a solid component that belong to a mathematically regular family.This is the fact that led originally to the designation “ regular solution ”, and it affords a simple, practical, graphic method of distinguishing a class of solu- tions that offer prospect for common theoretical treatment simpler than the solu- tions whose curves have divergent slopes. (ii) It excludes solutions in which there are specific, “ chemical ”, or hydrogen bonding, or orientational effects. (iii) It is not tied to any particular artificial, probably oversimplified model. (iv) It leaves open for further study all questions regarding the best mathematical formulation of the several thermodynamic quantities involved. It is a concept of permanent significance rather than a formula sure to be superceded.The fact that unequal molecular fields, even where none but pure dispersion forces are involved, always tend to prevent complete randomness is no more serious than the fact that there are no “ strictly ” ideal solutions. The “ strictly ideal ” and the “ strictly regular ” are both concepts that invite profitable study of the variety of factors causing actual solutions to deviate from them. May I point out that the term “ strictly regular solution ” is now being used by several British scientists in a sense very different from the forcgoing, the adjective “strictly” not merely modifying but rather profoundly altering the original idea of a “ regular solution ”.I make no plea for orthodoxy, and would gladly agree to a re-definition o€ the term if that would clarify our intercommunication. But a Tower of Babel is not a symbol of communication. I trust there will be an exchange of opinion on this matter. Another question having to do with structure is that of CLUSTERING in a solu- tion of two molecular species of unequal intermolecular attractive potentials. The distribution approaches the complete randomness postulated for a regular solution with rising temperature, also with increasing dilution of either com- ponent ; in the former, because of increasing thermal agitation, and in the latter because of the difficulty of forming clusters between molecules that rarely collide. It i$ in the critical region that clustering is most pronounced and the question is, how rapidly does it diminish upon departing from the critical point, either in temperature or composition, and how does this affect the form of the liquid-liquid solubility? My colleague, Jura and I, with the aid of re3earch assistants, have been studying these questions.The basic difficulty with any equation of state in the critical region is that two distinct phases do not merge abruptly into a single, homogeneous phase upon heating through the critical “ point ”, but into a region that only gradually loses its heterogeneity, and where a special treatment is indicated, as recognized by Mayer 26 and co-workers. The criteria mainly used for defining a critical point have been (i) the disappearance of the interfacial surface and (ii) zero values for the first and second differentials of pressure with respect to volume, for the liquid- vapour equilibrium of a single component, or of the differentials of activity with respect to composition for a liquid-liquid system of two components.But addi- tional significant thermodynamic data are available, and we have been studying the changes in heat capacity and volume occurring when mixtures of the critical composition arc go through the critical region. Fig. 5 shows the A-type curve obtained for the molal heat capacity of a critical mixture of perfluoroheptane and isu-octane (2 : 2 : 4-trimethylpentane) in an exploratory investigation of only moderate accuracy.2* Fig. 6 shows a careful investigation of the change in volume occurring when the same mixture is very slowly cooled.The investigation is being carried further to complete the thermodynamic record. I quote from a brief, preliminary paper : 27 “ In order to have limited miscibility, the intermolecular potentials within the two pure liquids must be very different. The potential between the unlike molecules in the solution is approximately the geometric mean of the like potentials,16 SEVENTH SPIERS MEMORIAL LECTURE which is less than the arithmetic mean. Therefore, as mutual solubility increases with increasing thermal motion, 1-1 and 2-2 neighbours are replaced by 1-2 neigh- bours, with a decrease in the total intermolecular potential energy of each phase, causing expansion and absorption of heat. The connection between these two quantities is well recognized in the theory of regular solutions.1 Both increase more and more rapidly as the top of the liquid-liquid solubility curve is approached, but still continue, although at'a greatly reduced rate, after the disappearance of FIG.5.-Critical mixture of C7F16 and iso-C8Hl8. 192.40 192.39 192.38 $ 19237 192.36 5 192.35 > 192.34 w -I 0 FIGS. B.-Molal volumes of mixture of C7F16 with hdsH18 in critical region. the meniscus, because the mixing is still far from complete, as shown by the turbidity that persists in gradually decreasing amount as the temperature is further raised. " Conversely, on lowering the temperature of a mixture of the critical com- position from well above the critical region, there occur, first, aggregations of like molecules of continually increasing size, as revealed in the beautiful work of Zimm.26 These remain suspended by Brownian movement until they become sufficiently large for the gravitational field to separate them into an upper and a lower phase.It seems obvious that the precise temperature at which this occursJOEL H . HILDEBRAND 17 is a function of the density difference and the strength of the field, and might be measurably raised in a centrifugal field. We are investigating this possibility. The role of gravitation in the critical region of a one-component system has recently been strikingly shown by Murray and Mason,27 and by Weinberger and Schneider.28 Furthermore, there must be a sort of Brownian movement perpendicular to the plane of the interface just below the temperature at which it appears.The microscopic area of the interface must be greater than the macroscopic." Our attention has recently been called to a paper by Semenchenko and Skripov,29 published in the J. Physic. Chem., U.S.S.R., reporting measurements of the heat capacity of critical mixtures of nitrobenzene + hexane, and water + triphenyl- amine. A-Type curves were obtained, but the scatter of the points was very wide, probably as the result of using very small quantities of liquid in order to avoid stirring. As a corollary to the use of a FLUOROCARBON in the foregoing section, I wish to emphasize the service this class of compounds can render in putting strains upon our theories. Their intermolecular forces are so much smaller than those of other substances liquid at ordinary temperatures as to afford solutions deviating enormously from Raoult's law.Refore they became available there were known only a few pairs of non-polar liquids sufficiently different to yield two liquid phases. But the fluorocarbons have solubility parameters so low as to give critical unmixing with other non-polar liquids at convenient temperatures. Even heptane and perfluoroheptane give two liquid phases below 50". I take the liberty here to recall CERTAIN OBSERVATIONS UPON TETRAHALIDES and their mixtures that J. M. Carter and I made many years ag0,30 which I think are still significant. These substances were selected because of their molecular symmetry. We measured (3P/3T)v. We found it to be constant over the range studied, and hence were able to determine it with considerable precision.From this one can calculate (3E/3Y), by the thermodynamic equation The relation of these terms to those of the van der Waals equation is apparent on writing the latter in the form a/ V2 = RT( V - b) - P. TABLE 3.-vALUES FOR CARBON TETRACHLORIDE to at 1 atm GV'lbVv, atmldeg. molal vol., cm3 W h 4 b V) atm 1.2 T, 20.40 11.40 96.54 31.18 20.63 1 1 *40 96.57 31.23 25.84 1 1 *02 97.19 31.18 31.21 10.72 97.84 3 1 -23 37.09 10.36 98.56 31.22 Table 3 gives values for carbon tetrachloride which show the remarkable con- stancy of V2(3E/3V),, corresponding to a van der Wads a. We found the con- stancy to be equally good for 6 tetrahalides and 8 of their binary mixtures. More- over, the a-values for the mixtures agree within approximately l % with values calculated by aid of the Berthelot relation, a12 = a,,k&, as illustrated by the following values.31 31'21 46.86 (obs.) 46.46 (calc.).CCl4 SnBr4 64-79 I wish to comment upon the question of ATTRACTIVE FORCES BETWEEN POLY- ATOMIC MOLECULES. At Edinburgh in 1936 I expressed the following opinion : " In most of the simplified. treatments heretofore used for dealing with inter- molecular potentials, the centre of the molecule is taken as the measure of inter- molecular distance and corresponding potential. The question suggests itself18 SEVENTH SPIERS MEMORIAL LECTURE whether it would not be better to consider the atoms as the attracting centres. The range of attraction is so short, falling off as it evidently does with the sixth power of the distance, that the adjacent atoms contribute most of the inter- molecular potential.The tetrahalides, again, can furnish excellent material for analysis on this basis.” London,32 in 1942, published an important paper, “ On Centres of van der Waals Attraction ”, in which he reviewed progress that had been made in follow- ing up a suggestion fist made by Meyer and Otterbein 33 that the polarizability of a compound be regarded as built up from a system of polarization ellipsoids which are co-ordinated not to the single atoms but rather to the bonds of the compounds. One simple piece of evidence for a significant difference between the shapes of the intermolecular potential curves of monatomic and polyatomic molecules is the failure of the latter to follow the requirements of the theory of corresponding states.Pitzer 34 showed that what he called “ perfect liquids ”, all possessing, among other criteria, potential curves of the same shape, should have equal entropies of vaporization at temperatures where the ratios of vapour to liquid volume are the same. This holds well for the Group 0 liquids, but we found upon comparing closely similar pairs of polyatomic molecules-one, chlorine and hexachloroethane, another, ethane and di-isopropy1,-that the entropy of vapor- ization of each pair agreed most closely when the vaporization occurs not at equal ratios of vapour to liquid volume, or of temperature to critical temperature, but at equal vapour volumes : the so-called “ Hildebrand Rule ”. My collabor- ator, Gilman, and I,35 interpreted this to mean that the intermolecular attraction is peripheral rather than central.The matter is sufficiently significant, I think, to justify repeating here a sample of the evidence. The entropy of vaporization of ethane at 148” K is 25.8 cal/mole deg. That of di-isopropyl at the same vapour volume is 25.5 cal/mole deg., whereas at the same ratio of vapour to liquid it is 28.3 and at the same reduced temperature it is 31.8. MacCarmack and Schneider 36 have determined the second virial coefficient for CC14 and SF6 and concluded that the evidence is against central forces for these molecules. And quite recently Thomaes,37 of the notable Brussels group, has made a calculation of the intermolecular potential between these molecules in terms of localized centres of London dispersion, local dipole-dipole, dipole- quadrupole, and repuslive forces. While much more complicated than a simple Lennard-Jones potential, this is unquestionably far more realistic.REPULSIVE POTENTIALS BETWEEN POLYATOMIC MOLECULES also merit comments. One of the most serviceable formulations used in current theories of solution is the Lenr,ard-Jones formula for intermolecular potential, E = j/rn - k/r6. Although an exponential expression for repulsion has been suggested, the possible theoretical advantage has not been sufficient to offset the mathematical complica- tions of a repulsive term different in form from the attractive term. Moreover, most properties of liquids are not very sensitive to the form of the repulsive term, because ordinary liquids are sufficiently expanded to minimize the effect of differences in the repulsive term. The mathematical simplicity achieved by setting n = 12 has favoured this particular assumption, but it is certainly an over-simplification.Atoms with different electron structures are not equally soft. My colleague, Jura, has calculated the parameters of the Lennard-Jones equation for neon, argon and krypton from their heats of sublimation, and informs me that the best fit is given by n = 8 for neon, 10 for argon, and 13 for xenon. For mercury, Hildebrand, Wakeham and Boyd 38 found n == 9, and this has recently been confirmed by Moelwyn-Hughes.39 Kerr and Lund 40 found n = 8.3 for mercury and n = 10 for argon, agreeing with Jura.JOEL H .HILDEBRAND 19 It is hardly necessary to do more than mention the fact that the repulsive forces between polyatomic molecules, like their attractive forces, do not necessarily vary with distance in the same way, because they involve bond-bending and perhaps form-fitting. We know a great deal about the force constants of certain molecules, especially the tetrahalides, and it should be possible to translate this into repulsive potentials. Still more significant would be the study of yet more complex molecules of maximum symmetry, such as 2 : 2 : 3 : 3-tetramethyl butane (hexamcthyl ethane), or perfluorocyclohexane. It seems highly desirable to learn more about THE BEHAVIOUR OF LARGE POLY- ATOMIC MOLECULES WHICH APPROACH AS NEARLY AS POSSIBLE TO SPHERICAL SYM- METRY.Questions involving their attractive and repulsive forces and their con- formity to the Flory-Huggins formula for the entropy of mixing are particularly important. The question raised by Scatchard, for example, whether this entropy should be referred to ratios of molal volumes or of surfaces, cannot be answered by studying linear polymers, where surface and volume are proportional. A practical difficulty confronting anyone wishing to study these matters is the short liquid range of substances whose molecules are spherically symmetrical, because they are able to pick up rotational energy at a lower transition point which postpones melting to a higher temperature. Perfluorocyclohexane, for example, would be suitablc if it had any liquid range whatever at atmospheric pressure.We have turned to 2 : 2 : 3 : 3-tetramcthyl butane, " hexamethyl ethane ", which I shall refer to as HME. It melts at 100.63" and boils at 106.3O.41 Its cntropy of fusion is very low,42 4.82 cal/mole deg. (That of normal octane is 22.27 cal/mole deg.43) There is a transition at 152.5" K where it doubtless begins to acquire rotational energy ; its heat capacity increases from 34 to 70 cal/mole from there to its melting point. It seems, therefore, to furnish a reasonably good model of a quasi-spherical, polyatomic molecule. Rotariu and I have made the follow- ing two comparisons between it and n-octane. Its density in the liquid range is greater than that of n-octane, which would not be the case if the molecules of these substances approximated to hard spheres and cylinders respectively.We thought it would be interesting to compare their molal volumes as solids, there- fore we measured their densities by weighing solid samples of each immersed in methyl alcohol at - 75" C and in nitrogen and hydrogen at their boiling points. We also weighed HME under water at 1.4" C. The results, together with data from the literature, are represented in fig. 7 as molal volumes against absolute temperature. The other property measured was the partial molal volume of methane in each of these octanes in the liquid state at 102.2". The experiment was a difficult one, and the probable error was about 5 cm3/mole. The molal volume of methane was thc same in the two liquids within the limit of error, 67 cm3. GjaldbaekM found the following partial molal volumes V for methane at 25" C : solvent C7F16 C6H14 cs2 V 68.4 60.0 56.1 It sec ms evident that there are not in HME intersphcrical spaces into which methane could enter without very much enlargement, such as would be available to cherries in a box of oranges.I wish to mention briefly a piece of work being carried out by my associates, Haycock and Alder, designed to test the relative merits of two models for liquid diffusion, the one, according to which a molecule diffuses, as in a crystal, by moving into an adjacent hole, the other according to which two molecules may squeeze past each other if they possess sufficient kinetic energy in opposite directions to overcome the repulsive barriers. I realize that diffusion is not an equilibrium property but our experiments are designed to throw light upon liquid structure, which is such a property.20 SEVENTH SPIERS MEMORIAL LECTURE We have measured the diffusion of iodine in carbon tetrachloride through a temperature range of 40" and at 1, 65 and 200 atm.The full lines in fig. 8 repre- sent the temperature coefficients of diffusion at the three pressures, and the dotted I90 I80 17c I60 I50 I40 I 3 0 120 0 /ILiquid i / I I w Liquid I Y xamet hyl ethane - 100 200 300 400 500 FIG. 7. 9 5.89 I X 7 I ATMOSPHERES 200\ 65\ i\a 3. I 3.2 3.3 3.4 3.5 FIG. 8. lines the same at constant volume, where the number of "holes", if be assumed really to exist, must remain essentially constant. The such can value of (3 In Dl3T)v at 1-atm and 25' taken from the slope of the appropriate isochore is 0.0063.The coefficient of kinetic energy at 25" is, of course, 0.00335. It seemsJOEL H . HILDEBRAND 21 evident that the increase in molecular agitation also provides more frequently barriers low enough to permit interchange. We postpone any effort at quanti- tative theoretical treatment till we secure similar measurements on self-diffusion of carbon tetrachloride. The possibility of ACID-BASE INTERACTIONS should not be overlooked. For a long time I was puzzled by the fact that iodine solutions in benzene are not violet in colour, despite the non-polar nature of both substances. The mystery dis- appeared with the discovery of the strong ultra-violet absorption of this solution and the convincing evidence of the presence of a 1/1 chemical solvation.45 A symposium on generalized acids and bases, held in connection with the meetings of the American Chemical Society in Septembcr, 1950, brought to light a con- siderable number of interactions of this sort, and research in this field has been very active ever since.Mulliken46 and his group have done notable work on the spectroscopy of this kind of intcraction and collected and classified a large number of specific reactions. It has become evident that many more systems than we have been aware of involve specific, non-additive, chemical interaction. Weiss's 47 discovery of the relation between the strength of the interaction and the strengths of the electron donors and acceptors, as revealed respectively by their ionization potentials and electron affnities, provides a basis for including such systems into solution theory.Benesi and I pointed out, for example, thst the exceptional, non-regular solvent powers of benzene and mesitylene for iodine corresponds well to the strength of acid-base interaction as measured by their ultra-violet absorption. Electron donors include a variety of familiar liquids, such as ethers, olefins, aromatics, ethyl iodide, and even cyclopropane. Ac- ceptors include the halogens, nitro-compounds, sulphur dioxide, carbon disulphide, and many others. One should not use components capable of donor-acceptor interaction, at least without investigating the ultra-violet absorption, to test a model based upon norr-selective, physical, intermolecular potentials. FREE VOLUME is an important concept in theories of liquids and solutions, but it has been defined in different ways which yield different numerical values for the same substance.Scott and I discussed them at some length in our monograph 38 and Bondi and I have recently been considering the matter further. We suggest that several " free volumes " be distinguished as follows. (1) Empty volume. The total volume less the sum of the volumes of the individual molecules calculated from their actual dimensions as determined by X-ray diffraction methods or gas kinetic collision cross-sections. (2) Expansion volume. The difference between the molal volume at a tem- perature t and the volume extrapolated to 0" K. This would be the van der Waals V - h if the molecules were incompressible.( 3 ) FIuctuation volume. The volume through which the centre of a molecule moves in its " cage ". In a paper which Bondi is preparing for publication he shows that these defini- tions, apparently in conflict, are actually related in reasonably quantitative agree- ment with each other and with the experimental facts appropriate to each. In conclusion, although I have referred to a number of respects in which the simple models we set up for dealing with liquids and solutions would have to be extended to take account of the various complicating factors, I do not advocate trying to include them all in a mathematical formulation of vast dimensions. A theory passes through an optimum in practicality as it takes cognizance of more and more complications. I quote again from the review by Fiirth : " Now, if full advantage is to be taken of these rigorous theories for the pre- cise theoretical treatment of a particular liquid, the potential function would first have to be known accurately, which is never the case.Thus, in spite of the very great theoretical progress made, when it comes to practical applications so many simplifications and approximations have to be used that the intricate22 SEVENTH SPIERS MEMORIAL LECTURE apparatus of the theory is largely wasted. Therefore, the development of more primitive theories of the liquid state should not be neglected, starting with ap- proximations right from the beginning by replacing the actual structure of the system by some simple kind of “ model ” structure, so that the treatment becomes mathematically more tractable.A number of such simplified theories exist already and are more or less successful within their limited spheres.” The physical chemist who wishes to get answers to questions about solubility is in a position similar to a man who wants to know just how to soft-boil an egg to suit his exacting taste. The complete procedure would be to study the kinetics of every reaction involved and the heat conductivities of the several parts ; combine all results into an equation which could doubtless be applied only by the aid of an electronic computer. The other, cruder way, is to boil eggs at a series of systematically chosen times and temperatures. I do not presume to say which method is the more scientific, but you can easily see from the contcnt of this address how I prefer to operate in attacking “ the equilibrium properties of solutions of nonelectrolytes ”.I like the practical philosophy of scientific research descri bed by Gilbert Lewis in these words : “The scientist is a practical man and his are practical aims. He does not seek the ultimate but the proximate. He does not speak of the last analysis but rather of the next approximation. His are not those beautiful structures so delicately designed that a single flaw may cause the collapse of the whole. The scientist builds slowly and with a gross but solid kind of masonry. If dissatisfied with any of his work, even if it be near the very foundations, he can replace that part without damage to the remainder. On the whole, he is satisfied with his work, for while science may never be wholly right it certainly is never wholly wrong ; and it seems to be improving from decade to decade.” 1 Hildebrand and Scott, Solubility of Non-electrolytes (Reinhold Pub. Corp., New 2 Hildebrand, Fisher and Benesi, J .Amer. Chem. Soc., 1950, 72, 4348. 3 Dice and Hildebrand, J . Amer. Chem. SOC., 1935, 57, 866. 4 Hildebrand and Rotariu, J. Amer. Chem. SOC., 1951, 73, 2524; Hildebrand, Powell 5 Thomas and Staveley, J. Chem. Suc., 1952,4568. 6 Rowlinson and Curtiss, J. Chem. Physics, 1951, 19, 1519. 7 Peek and Hill, J . Chem. Physics, 1950, 18, 1252. 8 Krigbaum and Flory, J. Chem. Physics, 1952,20, 873. 9 Furth, Sci. Prog., 1949, 146, 202, 10 Hildebrand and Wood, J . Chem. Physics, 1933, 1, 817. 11 Campbell and Hildebrand, J. Chem. Physics, 1943, 11, 334. 12 Hildebrand, Trans. Faraday Soc., 1937, 33, 189. 13 Kirkwood, Lewinson and Alder, J. Chem. Physics, 1952, 20, 929. 14 Hildebrand, J. Chem. Physics, 1947, 15, 225. 15 Huggins, J. Physic. Chem., 1948, 52, 248. 16 Hildebrand, J. Amer. Chem. Soc., 1929, 51, 66. 17 Scatchard, Trans. Faraday SOC., 1937, 33, 160. 18 Scatchard, Wood and MocheI, J . Amer. Chem. SOC., 1939, 61, 3206. 19 Wood, J. Chem. Physics, 1947, 15, 358. 20 Scatchard, Chem. Rm., 1949,44,7. 21 Hildebrand, Chem. Rev., 1949, 44, 37. 22 Hildebrand and Scott, J. Chem. Physics, 1952, 20, 1520. 23 Hildebrand, Nature, 1951, 168, 868. 24 Mayer, J. Chem. Physics, 1937, 5, 67 ; Mayer and Harrison, J. Chem. Physics, 25 Jura, Fraga, Maki and Hildebrand, Proc. Nat. Acad. Sci., 1953, 39, 19. 26 Zimm, J. Physic. Chem., 1950, 54, 1306. 27 Murray and Mason, Can. J. Chem., 1952, 30, 550. 28 Weinberger and Schneider, Can. J. Chem., 1952, 30, 422, 847. 29 Semenchenko and Skripov, J . Physic. Chem., U.S.S.R., 1951, 25, 362. York, 3rd edition, 1950). and Gilman, J. Amer. Chem. Soc., 1951, 73, 2525. 1938, 6, 87, 101 ; McMillan and Mayer, J. Chem. Physics, 1945, 13, 276.JOEL H . HILDEBRAND 23 30 Hildebrand and Carter, J. Amer. Chem. SOC., 1932, 54, 3592. 31 ref. 1, p. 131. 32 London, J. Physic. Chem., 1942, 46, 305. 33 Meyer and Otterbein, Physik. Z., 1931, 32, 290 ; 1934, 35, 249. 34 Pitzer, J. Chem. Physics, 1939, 7, 583. 35 Hildebrand and Gilnian, J. Chem. Physics, 1947, 15, 229. 36 MacCormack and Schneider, J. Chem. Physics, 1951, 19, 849. 37 Thomaes, J. Chim. Phys., 1952, 49, 323. 38 Hildebrand, Wakcham and Boyd, J. Chem. Physics, 1939, 7, 1094. 39 Moelwyn-Hughes, J. Physic. Chem., 1951, 55, 1246. 40 Kerr and Lund, J. Chem. Physics, 1951, 19, 50. 41 Seyer, Bennett and Williams, J. Amer. Chem. SOC., 1949, 71, 3447. 42 Scott, Douslin, Cross, Oliver and Huffman, J. Amer. Chem. SOC., 1952, 74, 883. 43 Parks and Todd, Ind. Eng. Chem., 1929, 24, 1235. 44 Gjaldback, J. Amer. Chem. SOC., 1950, 72, 1077. 45 Benesi and Hildebrand, J. Amer. Chem. SOC., 1949, 71, 2703. 46 Mulliken, J. Chem. Physics, 1951, 19, 514; J. Amer. Chem. SOC., 1952, 74, 811; J . Physic. Chem., 1952, 56, 801 ; see also McConnell, Ham and Platt, J. Cliem. Physics, 1953, 21, 66. 47 Weiss, J. Chem. SOC., 1942, 245 ; 1943, 462. 48 ref. 1, pp. 72 ff.

ISSN:0366-9033    

DOI:10.1039/DF9531500009

出版商:RSC    

年代:1953

数据来源: RSC

摘要:

GENERAL THEORY INTRODUCTION TO GENERAL THEORY BY E. A. GUGGENHEIM Chemistry Dept., Reading University It is just thirty years, to within a few months, since Joel Hildebrand finished writing the first edition of Solubility. The second edition appcared shortly before the Faraday Society Discussion on Liquids and Solutions at Edinburgh in 1936. The third edition written in collaboration with Scott was published most conveniently in 1950 so that everyone here has had ample time to read it. Having read all three editions of Solubility, and not having read any other monograph covering the field of solutions, I may claim to be a student of Hildebrand although I have never worked in his laboratory. It is difficult nowadays to realize the extremely primitive state of the subject at the time when Hildebrand entered the field.There were two contending parties, the one claiming that all deviations from ideality were due to chemical association, the other claiming that all devi- ations from ideality were explicable by van der Waals’ equation. Both parties had more exuberance than common sense. Hildebrand soon realized that both views were absurdly extreme and yet that each contained a germ of truth. I think that Hildebrand’s greatest contribution to the subject is his emphasis on the need to classify solutions bcfore one can usefully discuss the detailed behaviour of any particular solution. If the classification in the third edition of Solubility is not quite the same as in the first edition, that shows that the subject is a live one. If, moreover, my choice of a classification at any time is not completely in phase with Hildebrand’s choice that is circumstantial evidence that we are both alivc and awake.It seems to me natural and healthy that classification and definitions should change as a subject develops. Considcr the word “polarity”. In the first edition of Solubility no better measure of polarity was available than the dielectric constant. In the second edition polarity was naturally measured by the electric dipole moment p. Nowadays, as is clear from the recent work of Rowlinson, a still better measure is the ratio p2J~03, where E is a characteristic interaction energy and (T a characteristic length (collision diameter). As another example take the conception of “ perfect gas ”. The original definition was presumably empirical : a perfect gas was one whose pressure-volume-temperature behaviour closely resembled that of air.Later the definition became thermodynamic: a perfect gas is a substance for which at a given temperature the total energy U and the product pV are independent of the density. Nowadays many would prefer a molecular definition : a perfect gas is a system of molecules between which the interaction is negligible. These three definitions are closely related, but are by no means equivalent. We now know that for air p V is not quite independent of pressure. We also know that even in the absence of molecular interactions pV will not be independent of density at the lowest temperatures (Bosc-Einstein condensation). Similarly there are three alternative definitions of an ideal mixture.Empirically, an ideal mixture is one with propcrties similar to those of a mixture of isotopes. Thermodynamically, an ideal mixture is one with a molar free energy of mixing RT{(I - x) In (1 - x) + xlnx}, 24E. A . GUGGENHEIM 25 where x denotes mole fraction. Molecularly an ideal mixture consists of molecules having the same size and having interaction energies related by ~ E A ~ = EAA + EBB. Whichever definition we choose, it is evident that a precisely ideal mixture will rarely, if ever, be met in practice. An ideal mixture is rather a conception with which a real mixture may be usefully compared and contrasted. This being so, my preference is for a molecular classification, rather than an empirical or a thermodynamic one.This does not imply that other people in other contexts should not use one of the other alternatives. With this preamble I propose to outline a classification of solutions which in the present state of our knowledge seems to me useful. The first main classification is between those mixtures in which molecular orientation is unimportant and those in which it is all-important. The former class consists of mixtures of non-polar or slightly polar molecules. The latter consists of mixtures containing strongly polar molecules, acid and base molecules, and above all molecules forming hydrogen bonds. There has been significant work in the last few years on mixtures of the second class (Prigogine 1950, Tompa 1951, Barker 1952). This progress bccame possible only after obtaining a clear idea of the much simpler behaviour of mixtures of the former class, which I shall now coiisider in greater detail.For the sake of brevity I shall confine myself to binary mixtures. I shall begin by considering mixtures of molecules of sufficiently nearly the same size that one may usefully speak of two molecules of different kinds inter- changing positions. By far the most important of the quantities determining the properties of the mixture is the molecular interchange energy w. I assume that this parameter is familiar to all here and I deliberately postpone consider- ation of how best to define it. The simplest case occurs when w is zero. The molar frce energy of mixing is then purely configurational and is given by (1) where x denotes mole fraction.Such a mixture is called " ideal ". The thermo- dynamics of these ideal mixtures were expounded almost simultaneously by Bronsted (1908) and by G. N. Lewis (1908). When w is not zero the molar free energy of mixing can to a useful degree of approximation be written as AmF = RT((1 - x) In (1 - x) -t- x In x}, A m F - RT((1 - x) In (I - x) 4- x In x } 4- x(1 - x)Nw. (2) This formula has been proposed and used especially by Porter (1920), van Laar and Lorenz (1925), Heitler (1926), Hildebrand (1929). The striking feature of this forniula is the additivity between the interaction terms in w and the con- figurational terms for an ideal mixture (w = 0). This additivity implies completely random arrangement of the molecules, or rather negligible deviations from randomness.More accurate formulae which take account of the deviations from randomness have been derived by Rushbrooke (1938) and by Kirkwood (1938). These deviations can be expressed as a power series in w/kT. If we include only the leading terms, we have instead of (2) A,F == RT((1 - x) In (1 - x ) + x In x } 1- x (1 - x) Nw wliere z denotes the average number of closest neighbours of a given molecule (co-ordination number). The reason why (2) is such a useful approximation to (3) is that i n the power series the determinative quantity is not w/kT but 2wlzkT. Note that w is an energy per molecule, while 2142 is an energy per pair of neigh- bows. Since w/kT cannot exceed 2 without separation into two phases, thc first correction term cannot exceed 1/22.In most practical applications it is much smaller. The solutions which I have been discussing are related to those which26 GENERAL THEORY Hildebrand defined as “ regular ” and they have been distinguished by the name “ strictly regular ” when the distinction between the molecular and the empirical definitions seemed important. I hold no strong view on whether the name “ strictly regular ” is appropriate. No one has yet proposed an alternative name. From formula (2) we derive immediately for the entropy and energy of mixing (4) dw A d =z - R((1 - x) In (I - x) + x In x} - x(1 - x)N -- dT7 Until five ycars ago it was assumed, tacitly or explicitly, that w was independent of the temperature. There is in fact no ground either theoretical or experimental for such an assumption.Quite recently the dependence of w on temperature has been studied theoretically by Rowlinson (1952) and by Scott (this Discussion). Perhaps the simplest way to see why w depends on the temperature is to recapitulate the argument leading to tcmperature-independence and then point out where it is wrong. The partition function of a single liquid or a mixture may be regarded as the product of a “configurational” factor with each molecule fixed at an equilibrium position, and an “ acoustic ” factor taking account of the motion of the molecules about their equilibrium positions. To obtain a temperature- independent w one has to assume that each molecule contributes to the acoustic partition function a factor independent of its neighbours. It is obviously an improvement to assume that each pair of neighbours contributes a -factor, say qM or qAB or q B B according to the nature of the two molecules of the pair.This leads to an interchange energy w of the form where wo is the interchange potential energy of molecules in their equilibrium position and z is the co-ordination number. It is evident that w is a free energy (not a total energy) of interchange. The temperature dependence of each q will be more or less complicated. If, for example, in order to simplify as much as possible we assume classical simple harmonic vibrations each q would have the form k T i z I n q = 31n-, hv (7) where v is a characteristic frequency. Formula (6) would then become Even if this simplification were justifiable the v’s would still depend on the tem- perature through the thermal expansion, so that w would still not be linear in T.The sug- gestion made at the Edinburgh discussion in 1936 that these might obey different formulae from mixtures of molecules of equal size was met with a scepticism which is now amusing. We have advanced a long way since then mainly thanks to the importance attached to solutions of macromolecules. Thc most important parameter is again an interchange energy w which must now have a more com- plicated definition owing to the different sizes of the molecules. I shall not go into details of definition. Roughly speaking 2wlz is the interchange energy per contact between a pair of segments of different molecules. 1 begin by considering the simplest case of w = 0 (athermal mixtures).Then in place of formula (1) we have Flory’s (1941) formula I turn now to mixtures of molecules of significantly different size. A,F = RT((1 - x) In (1 - +) + x In 41, (9)E. A . GUGGENHEIM 27 where 4 denotes volume fraction. Incidentally this notation, used by Hildebrand and Scott, is now generally recommended. The use by some authors of the same letter v, even in different founts, for volume fraction and for molar volume is thc worst possible notation. Formula (9) is only an approximation. More accuratc formulae obtained by Chang (1939), Huggins (1942) and Miller (1943) involving the co-ordination number z all reduce to (9) when z-1 is made to tend to zero and show that (9) is in fact a useful approximation. When w is not zero, formula (2) has to be replaced by formula A m F = XT((1 - x ) In (1 - 4) -1- x In +} 4- +’(I - +‘)Nw, wherc 4’ is a quantity related to d) but more complicated; often for the sake of simplicity I$’ is replaced by 4 (Flory 1941).1 need not here go into details about 4’. The important feature of formula (10) is the additivity between the interaction terms proportional to w and the configurational terms corresponding to w = 0. This additivity, as in formula (2), implies that deviations from randomness are negligible. More accurate formulae taking proper account of the deviation from randomness have been derived by Orr (1944) and by Tompa (1949). Owing to the fact that the deviations from randomness are determined by the ratio 2w/zkT rather than by w/kT it turns out that (10) is a useful approximation to the more accurate formulae. From (10) we obtain directly x)In(l - 4) -t- x ~ n + > - +’(1 - and there is no reason to suppose that dw/dTis negligible. I need hardly mention that all equilibrium properties are derivable from the free energy, including the condition for separation into two phases and that for critical mixing. 1 will only mention the change of volume on mixing, which is fully discussed by Scott in his contribution to this Discussion. I think that the most important features of the formulae outlined above are the following. It is generally possible to a useful degree of approximation to treat the configurational (athermal) terms in the free energy and the interaction terms separately and additively. It is, however, wrong to associate the former with the entropy, and the latter with the total energy (or the enthalpy). I believe that these features will persist in the most useful future contributions to the subject of solutions.* “Detailed references to the work quoted will be found in Hildebrand and Scott (1950), Solubility of Non-electrolytes (Reinhold), or Guggenheim (1952), Mixtures ( Clarendon Press, Oxford).

ISSN:0366-9033    

DOI:10.1039/DF9531500024

出版商:RSC    

年代:1953

数据来源: RSC

摘要:

THE STATISTICAL MECHANICAL THEORY OF MOLECULAR DISTRIBUTION FUNCTIONS IN LIQUIDS * BY JOHN G. KIRKWOOD AND ZEVI W. SALSBURG Sterling Chemistry Laboratory, Yale University, New Haven, Connecticut Received 18th December, 1952 The thermodynamic functions of liquids and liquid solutions are related to the potential of intermolecular force according to statistical mechanics by the Gibbs theory of the canonical ensemble. They may be formally expressed as integrals of the probability distribution functions g(n)/v" for all subsets, n, of molecules in a canonical ensemble of average molecular density w-1, When the intermolecular force can be represented as the sum of the forces between molecular pairs, a knowledge of the pair functions g:? at all densities-for the several types of pairs EP composing a fluid mixture suffices to determine the thermodynamic functions in terms of the potential of intermolecular force.The pair functions .gLy are known in the theory of liquids as the radial distribution functions. Systems of integro-differential equations for the distribution functions g ( 4 have been formulated in equivalent forms by Yvon, Kirkwood, Mayer and Born and Green. These systems of equations havc been approximately solved by closure with the superposition approximation in triplet space, with significant results. At low densities they may also be solved by Mayer clustcr expansions in powers of the average densities. A new system of integral equations for the distribution functions g(n) is presented, which is based upon partial cluster expansions of the probability density in the canonical ensemble.For molecules possessing a rigid impenetrable core of radius b and an attractive force of finite radius a, the new cluster expansions are of finite degree v, equal to the number of spheres of radius b which can be close packed in a sphere of radius a, and thus converge for all densities. All thermodynamic functions arc expressed in terms of the small finite set of distribution functions g(*), . . . g(9. Phase transitions are related to the eigen-values of the kernels of the system of integral equations. The statistical-mechanical theory of liquids and liquid solutions provides relations between the thermodynamic functions of the system and the potentials of intermolecular force, which involve distribution functions for sets of n molecules, for example, singlets, pairs, triplets, which form subsets of the total number of molecules of which the system is composed. In the theories of Kirkwood,l Born and Green,2 Yvon,3 and Mayer,4 systems of integro-differential equations for the distribution functions have been formulated in several equivalent forms.These systems have the common feature that they involve derivatives of the distribution functions with respect to suitable parameters upon which the potential of intermolecular force is assumed to depend. Integration with respect to these parameters offers difficulties, which have allowed it to be performed only after approximations, such as the superposition assumption 1 in the space of molecular triplets, haw been introduced into the exact systems of equations.We shall undertake here the derivation of a new system of integral equations for the distribution functions by a method which does not depend upon the device * This work was carried out with support from the Ofice of Naval Research under contract with Yale University. Contribution No. 1136 from the Sterling Chemistry Laboratory, Yale University, New Haven, Conn. 28JOHN G. KIRKWOOD AND ZEVI W. SALSBURG 29 of differentiation with respect to parameters in the potential of intermolecular force. We are thus led directly to a system of integral equations rather than to a system of integro-differential equations. The method is based upon a partial cluster expansion of the Mayer type of factors in the canonical distribution function in configuration space, depending upon the co-ordinates of a single molecule.* For an intermolecular force of finite range, the partial cluster ex- pansions are found to be polynominals in the average density of small finite degree, which therefore never simulate the behaviour of divergent series, as do complete cluster expansions of the partition function and distribution functions.As a by-product of the theory, we obtain an expression for the chemical potential, which had previously been derived by Mayer by means of a less direct and more complicated rnethod.4 In the present investigation, we shall limit our attention to the derivation of the equations for a system of one component and to a brief sketch of their significance in the theory of phase transitions.Extension to multi-component systems is obvious and offers no difficulty other than that of more complicated notation. We shall postpone for later treatment a discussion of the solution of the equations by approximate methods as well as the formulation of a more detailed and exact theory of phase transitions based upon them. The density V N ( ~ ) (1, . . . n) of ordercd sets of n molecules, forming subsets of a closed system of N molecules, occupying a volume v, is defined by the relation, * 11 * I n - l + --f at a point r l , . . . r, in the 3n dimensional configuration space of the set n, when the entire set of N molecules is located at a point R1, . . ., RN in the complete 3N dimensional phase space, where the functions 6(Rii - Y i ) are three-dimensional Dirac delta functions. The average density P N ( ~ ) of ordered sets n, in a petit canonical ensemble of closed systems of N molecules, is then given by 3 3 3 + /3 = l/kT, h A=- (2nmkT)a’ where AN is the Helmholtz free energy, m the molecular mass, h is Planck’s con- stant, k is Boltzmann’s constant, and T the thermodynamic temperature. We shall assume that the potential of intermolecular force, VN, has the form N VN = 1 V(ik), i< k=l * As Mayer and Montroll4 have shown, there exist many alternative cluster expansions of the distribution functions.Their complete cluster expansions diverge in condensed phases. They also present a set of integral equations for the distribution functions based upon cluster expansions involving the potential energy of all members of sets of n mole- cules.Although our method does not differ from theirs in essential details, we believe that the new set of integral equations presented here will prove more useful than their more complicated set of equations.30 MOLECULAR DISTRIBUTION FUNCTIONS I N LIQUIDS - - + + where V(kZ) depends only on the relative co-ordinates, Rk - R[ of the molecular pail kl and I is any specified molecule of the set. For a system of one component, all of the terms of eqn. (l), defining V N ( ~ ) make identical contributions to the integral of eqn. (2), and there are N!/(N - n)! such terms. Therefore we have, where an arbitrarily selected subset of n molecules, identified as I, 2, . . ., n are located at the points ri, . . ., rn, designated for brevity as 1, 2, .. ., n, and the integration is extended over the phase space of the residual set of N - n molecules, identified as n + 1, . . ., N. We now consider the open system composed of arbitrary numbers N of molecules of the given type in the fixed volume v. The average density (1, . . ., n) of ordered sets n at the point r l , . . ., v,, is given by the theory of the grand canonical ensemble in the form, p ( ~ , . . ., n) = z P N W , . . ., n) exp {/XQ + N , - A N ] ) , 9- -pv, 3 4 -> -+ 00 ( 5 ) N = O where p is the chemical potential per molecule, p is the pressure, and it is to be understood that p ~ ( " ) is equal to zero, if N < n. From eqn. (4) and (5), we then obtain We now select any one of the molecules of the set n, say moIecule 1, and write, exp (- ~ V N ) = exp (- / 3 J " ( l ) - F J " - I), N exp (- P J G W = ~ X P (- P Y , ( ~ ) ) (1 +flu), a = n + l f l u -- exp { - P W 4 ) - 1, n We now introduce eqn.(7) into eqn. (4) and (6) and expand the cluster product (1 i-fi,) to obtain partial integrands of products, n fro, containing factors a - n f l a - n - f - 1 flu for s distinct molecules of the residual set of N - n molecules. Since there are exactly (N - n)![s! ( N - n - s)!] identical terms in each integral of eqn. (6), arising from cluster products involving s molecules of the residual set, integration over the co-ordinates of the set of N - n - s molecules not involved in the cluster product leads to a factor [(N -n)! Is! ]h3(N-1) exp (- P A N - 1 ) l ) in each integrand. FinaJly, after interchanging the order of summation over the cluster number s and the number of molecules N in each example of the grand ensemble, we obtain with the use of eqn.(3, N sJOHN G. KIRKWOOD AND ZEVI W. SALSBURG n i - s n + s u - n + l u = n + l K,(I ; n + 1, . . ., n + s) = n fi, = (exp( - B W ~ ) ) - 11, p' = p - k T log h3, 31 n = 1,2, . . ., 00. Eqn. (8) constitute a determinate system of integral equations for the average number densities, under the conditions that p(n) (1, . . ., n) is a symmetric function of the co-ordinates, 1, . . ., n, and that p(4 = 0 as n -+ 00 in the finite volume v. Intermolecular repulsion at short range insures the validity of the latter conditions for systems of actual molecules, as will presently be shown. From eqn. (4) and (3, we remark that where { N ! / ( N - n)!}av is the average number of ordered sets n in the volume v.It is frequently convenient to work with correlation functions, g(") (1, . . ., n), instead of with the average number densities p(n) (1, . . . n), defined as follows : where volume per molecule. form, is the average number of molecules in the volume v , and u/N is the average For the correlation functions, g(n), eqn. (8) assume the where p is the excess chemical potential over that of an ideal gas of the same average molecular density. Eqn. (11) are again subject to the conditions that g ( n ) (1, . . ., n) be symmetric in the co-ordinates and that g(n) = 0 as n = 00. The correlation functions evidently possess the following integrals, In particular, by eqn. (12) the integral, v-1 g(1) (l)dv, is equal to unity.We r therefore obtain the following expression fog the excess chemical potential, pe, by integration of both sides of the first of eqn. (1 1) over the volume of the system, (13) Eqn. (13) is identical with the expression for the fugacity obtained by Mayer by another method.4 The correlation functions g ( n ) are related to the potential of average W(n), acting on a set of n molecules fixed at positions 1, . . ., n, in the following manner, g(n)(l, . . ., n) == exp {- ,6W(n)(l, . . ., n)], exp (-- /lw(n)> = exp + ~p - AN]) exp (-- @ W N ( ~ ) ) , (14) N a n32 MOLECULAR DISTRIBUTION FUNCTIONS I N LIQUIDS where WjJn) is the potential of average force acting on the set n in a closed system of fixed number of molecules N.Eqn. (1 1) may be transformed into an equivalent system of equations for the potentials of average force W(n), with the aid of eqn. (14), if desired. The next question to be considered is the convergence of the cluster sums in eqn. (8), (11) and (13). In order to settle this question, we make the significant observation that if the potential of intermolecular force V(R), is idealized as that appropriate to two molecules with rigid impenetrable cores of radius 6, exerting on each other a total force of finite range a, the cluster sums on the right-hand sides of eqn. (8), (11) and (13) are finite sums, each containing at most v terms, where Y + 1 is the maximum number of spheres of radius b, which can be packed into a sphere of radius a. The breaking-off of the sums with a small finite number of terms arises from the fact that the kernels Ks(l ; 2, .. . s + 1) vanish with an intermolecular force of finite range a, unless all of the set of s molecules lie with their centres in a sphere of radius a, centred in molecule 1, since otherwise at least one of the factors fla vanishes. On the other hand, repulsion between the rigid cores causes all g(s) to vanish, except, on a region of zero measure, for configurations with s centres in the sphere of radius a around 1, if s is equal to or greater than the maximum number of spheres of radius b, which can be packed into a larger sphere of radius a. Thus when s exceeds v, the integrands vanish because there is no overlap in the regions in which Ks and g(s) are non-vanishing.For example, for rigid spheres, exerting no attraction on each other, the range a is equal to 26, and v is equal to 12. For short-range intermolecular attractive forces of the Van der Waals type, V(R) may be adequately approximated for all practical purposes by that of a force of finite range a, if a is selected as a finite multiple of the core radius 6, say of the order of magnitude of 1 0 . For real inter- molecular repulsion, the core radius need only be selected as slightly less than the first zero of a potential of the Lennard-Jones type such that V(b)/kTis, say, of the order of magnitude of 10, to avoid serious distortion of the physical situation at the temperature of interest. For the idealized type of intermolecular force which has been described, we may write eqn.(1 1) and (13) in the form, g(n)(l, . . ., n) = exp ( P [ p - Vn(l)J} n - 1 , 2 , . . . , N O , ( 1 5 4 f J - n + l and the system of equations evidently terminates with n = No, where NO + 1 is the maximum number of spheres of radius 6, which can be packed into the volume w of the system, since repulsion between the rigid cores causes all g(n) to vanish for n > ND. The cluster sums are finite polynominals in @/v, at most of degree v, and thus their convergence is assured at all densities, Also, since v is in general a number of small order of magnitude, not only do the cluster sums always con- verge, but they never simulate the behaviour of divergent series, as in complete Mayer cluster developments. The expression for the chemical potential may be simplified in the following manner. For crystalline phases, g(l)(l) is a periodic function of the co-ordinates of the point 1, with the period of the space lattice characteristic of the crystalJOHN G .KIRKWOOD AND ZEVI W. SALSBURG 33 structure, and g(s) (2, . . ., s + 1) may be regarded as a periodic function of 2 and a function of co-ordinates of 3, . . ., s 1- 1, relative to 2. The integrals over 2, . . ., s + 1 on the right side of eqn. (156) are then periodic functions of 1, and we may write, s+ 1 exp (-&+ 1 + il m(-$ 1 % rs". I . syK,(I ; 2, . . ., s+ l)g(S) (2, . . .,s+l) n dvadvl, S = a = 2 (1 6) where the integral over 1 extends over an elementary cell of the lattice, of volume A. For fluid phases, gas or liquid, the partial integrals over 2, .. ., s + 1 are in- dependent of the position of 1, and eqn. (16), reduces to s+l exp(-&e)=l+ 5 .(;)'r 1 N . . . r K S ( l ; 2, . . ., s-f- l)g(4(2, . . .) s +I) n dv,. s= 1 a=2 (17) The existence of crystalline phases of course depends upon the existence of non- trivial periodic solutions g(l)(l) of the system of equations, eqn. (151). The system of integral equations, eqn. (15) may be written in the following equivalent form, g(n)( 1, . . ., n) = exp (/3& - Vn(l)])g(n -1) (2, . . ., n) + rK(n)(l, 2, . . ., n + l)g(n) (2, . . ., n + l)dvn + 1 Sg(n+s)(2, . . ., n + s + 1) g(nl(2, . . ., n + 1) ' phs)(2, . . ., n 4- l/n + 2, . . ., n f- s + 1) = where the kernels K(4 depend upon the average densities p(n9 S) of ordered sets s when n molecules are located at points 2, .. ., n -I- 1, which in turn depend upon the correlation functions g ( n + s), for s = 1, . . ., v - 1. It is instructive to examine the first two members of eqn. (18). g(l)(l) = exp ( P p ) + K(1)(1, 2)g(1)(2)da2 s"34 From where MOLECULAR DISTRIBUTION FUNCTIONS I N LIQUIDS eqn. (12), we observe that, the last relation, eqn. (20), follows from the theory of density fluctuations. A formal theory of first order phase tlansitions, based upon the same principles as Mayer's eigenvalue theory,4 is contained in eqn. (19) and (20). Eqn. (19a) is an inhomogeneous linear integral equation of the Fredholm type. When unity is an eigenvalue of the kernal K(1) (1, 2), no solution exists and the density fluctu- ation integral, eqn. (20) becomes infinite.This condition is presumed to define states of coexistence of a fluid phase and a crystalline phase in the pressure-tem- perature plane, separating a crystalline domain for which non-trivial periodic solutions g(1) (1) of eqn. (19a) exist from fluid domains in which unity is the only solution of eqn. (19a). In the fluid domains, eqn. (196) reduces to v - 1 1 s+3 (1 + 1 (s+ l)! s = 1 where g(2) (1,2) is the radial distribution function of the theory of liquids and depends only on R12, the scalar distance between the points 1 and 2. When unity is an eigenvalue of K(2) (1, 2, 3), eqn. (21) possesses no solution and again the fluctuation integral diverges. This condition is presumed to define states of coexistence of the liquid and vapour phases in the pressure-temperature plane, separating a liquid domain from a vapour domain. The existence of a critical point means tbat there is a temperature above which K(2) (1, 2, 3 ) doesmot possess the eigenvalue unity at any density. Below the triple point, K(1) (1, 2) reaches the eigenvalue unity before K(2) (1, 2, 3) does so, and a direct transition from vapour to crystal occurs. Of course the calculation of the transition points and the determination of the eigenvalues of K(1) (1, 2) and K(2) (1, 2, 3) demands the explicit solution of the complete system of equations, eqn. (15) or eqn. (18). The theory of phase transitions, based upon the new system of integral equations, which has been roughly sketched here mainly in the form of surmises, will be developed in more exact form in a later investigation. Also, the use of the equations, with the aid of suitable approximations to terminate the system, for the calculation of the radial distribution function in the liquid state will be the subject of a later investigation. 1 Kirkwood, J. Chem. Physics, 1935, 3, 300. 2 Born and Green, Proc. Roy. SOC. A , 1946, 188, 10. 3 Yvon, Actualitis Scientifiques et Industrielles (Hermann et Cie, Paris (1935), p. 203. 4 Mayer, J . Chem. Physics, 1947, 15, 187. 5 Mayer and Montroll, J. Chem. Physics, 1941, 9, 2.

ISSN:0366-9033    

DOI:10.1039/DF9531500028

出版商:RSC    

年代:1953

数据来源: RSC

摘要:

THE STATISTICAL MECHANICS OF SYSTEMS WITH NON-CENTRAL FORCE FIELDS BY J. A. POPLE Department of Theoretical Chemistry, University of Cambridge Received 2nd February, 1953 A general method for investigating the thermodynamic effects of the dependence of the intermolecular potential on orientation is described. A systematic expansion of the angular dependent part of the intermolecular energy forms the basis of an approximate method of evaluating the partition function which is then used to discuss the thermo- dynamic properties of liquids and liquid mixtures. Although complete expressions for the intermolecular energies are not available, it is possible to draw certain general con- clusions applicable to all types of directional forces. A simple relation is found con- necting the additional cohesive energy due to directional forces with the consequent loss of entropy due to restricted rotation.A similar treatment of liquid mixtures leads to a relation between the heat, free energy and entropy of mixing which differs significantly from corresponding theories based on central forces. The theory is used to interpret some experimental data on mixtures in terms of intermolecular forces. 1. INTRoDuCTIoN.-MoSt theoretical work on the relation between intermolec- ular forces and the thermodynamic properties of liquids and liquid mixtures has been limited to potential fields which are independent of the orientation of the particles. This condition, however, is only strictly satisfied by monoatomic substances. For a great many molecular substances directional intermolecular forces are likely to be important and will have a significant effect on the thermo- dynamic properties of liquids both because of the additional cohesive energy and because of the loss of entropy associated with hindrance to free rotation.Although many of the observed properties of liquids have been attributed to directional forces in a qualitative manner,1 there has been little in the way of general quantitative theory. The development of a general theory of systems with non-central force fields can be divided into two parts. First the many types of directional interaction that may occur have to be classified within a general mathematical framework and then approximate methods of evaluating the partition function have to be devised.This paper summarizes some of the results of a method developed by the author 2 with particular reference to its application to the properties of liquid mixtures. In order to simplify the presentation only axially symmetric molecules are considered in detail. This restriction is not altogether necessary as other systems can be treated by similar methods with very similar results. In the next section a general method of separating the intermolecular field into a central-force part and directional terms of various angular symmetries is described. Simple models such as dipole-dipole forces to represent interactions between polar molecules correspond to particular terms of this expansion. The additional free energy due to the directional part of the field can then be estimated by a perturbation method, provided that the additional field is not too large. The method is applic- able at any density and enables approximate theories of monatomic systems to be extended so as to apply to more realistic intermolecular fields.3536 NON-CENTRAL FORCE FIELDS In the two final sections an approximate version of the theory, based on a lattice distribution, is used to discuss the thermodynamic properties of liquids and liquid mixtures. to discuss the thermodynamic properties of assemblies of axially symmetric molecules in a comparative manner, it is necessary to use a systematic expansion for the intermolecular field so that various types of angular dependence can be distinguished. To specify the general configuration of two axially symmetric molecules, five co-ordinates are needed, the distance between centres and two angular co-ordinates for each molecular axis.These angular co-ordinates will be taken as spherical polar angles as in fig. 1. 2. THE INTERMOLECULAR ENERGY OF AXIALLY SYMMETRIC MOLECULES.-In order FIG. 1 .-Angular co-ordinates of interacting molecules, The intermolecular energy ust of a molecule 1 of species s and another molecule 2 of species t can be written in the form where <$'2:*)(r) is a function of r only (zero if I rn I is greater than I1 or 12) and Si,,, are surface harmonics defined by P,"(x) being an associated Legendre function Since So0 = 1, (2.1) reduces to its first term <?:"' (r) if there are only central forces. The other terms represent the various types of directional field that can occur.This expansion can be applied to any intermolecular field, but it is particularly useful since many simple models of directional interaction lead to simple expres- sions for the functions ((s:112:") (r). Further it will be shown in the next section that (2.1) is a very convenient basis for a general method of evaluating the free energy. The expansion (2.1) is particularly useful if the intermolecular forces are well represented by the interaction of permanent electrostatic moments. Attempts have been made, for example, to discuss the interaction of simple polar molecules in terms of a central force field representing the dispersion and repulsion forces, together with the interaction of point dipoles situated at the molecular centres.3-5 If the moments of the dipoles are ps and pt, this directional field can be written in the form (2.1) where When considering short-range interaction, however, the point dipole model may be inadequate because the dimensions of the molecules are not small compared with the intermolecular distance.It then becomes necessary to include interactionsJ . A . POPLE 37 due to higher order moments. If the molecules are represented by dipole moments ps, pt and quadrupole moments 0,, 01 the complete directional field is given by (2.4) together with 3. STATISTICAL MECHANICS OF ASSEMBLIES OF AXIALLY SYMMETRIC MOLECULES.- To illustrate the way in which the contributions of the directional forces to the thermodynamic functions can be estimated by statistical mechanics, we shall limit ourselves to pure substances in this section.The extension to mixtures will be discussed in the last section. The Helmholtz free energy of an assembly of N identical molecules occupying a volume Y at temperature T is given by where $(T) is a molecular partition function (independent of V ) allowing for the kinetic and vibrational energies, U is the total potential energy for any configura- tion of particles and dw have been written for integration over the positional and angular co-ordinates of all molecules. If U is written as the sum of contributions from all pairs of molecules dv and J S and u(i,j) is expanded in the form (2.1), we have a complete expression for the free energy F and consequently for all other thermodynamic functions in terms of the cfunctions of (2.1).In practice, the complete integration of (3.1) is very difficult, particularly be- cause of the dependence of U on orientation. The quantity in which we are interested, however, is the extra free energy that arises from the orientational forces, that is the contribution of all terms but the first in the expansion (2.1). This can be estimated by treating these orientational components in the inter- molecular energy as small quantities compared with the central force energy <(OO:O) (r) and retaining only the leading term in the expansion. If the orienta- tional forces are not small, this procedure is on1.y approximate, but in any case it should lead to results of the correct order of magnitude. The mathematical details of the method are given elsewhere2 and only the results will be quoted here.These are simplest if the only orientational terms in (2.1) are those with both 11 and 12 greater than zero. This applies to many types of field including those described in the last section and we shall limit the discus- sion to such systems. It then follows that the total free energy can be written as the sum of two contributions (3.3) where F(O)(T, V) if the free energy of a similar assembly of molecules interacting according to the central force term <(O0:O) (r) only, and F(or)(T, V ) is the additional free energy due to the orientational forces. F(or)(T, V) is given explicitly by F(T, V) = F(O)(T, V ) -l- F ( q T , V),38 NON-CENTRAL FORCE FIELDS where nS0) (r)dr is the probability of a particle in the central force system being in a volume element du at r given that there is one at the origin.Once the pair distribution function nSo) for the central force system is known, therefore, the orientational free energy FW) can be calculated immediately. The theory of central force assemblies has not yet reached a stage where the pair distribution function can be calculated accurately from the intermolecular field. but eqn. (3.4) does enable approximate theories of monatomic systems to be extended to systems where there are orientational forces. It has the further advantages of not being restricted to any particular statistical model and of being applicable at any density. high densities characteristic of the liquid phase well below the critical temperature, the pair distribution function n$*) (r) for the central force system is dominated by considerations of packing.The simplest approximation to use in (3.4) is to assume that ni0) (r) can be replaced by the corresponding function for a close- packed lattice, the dimensions of which are chosen to give the correct total volume. This means that only certain discrete values of the intermolecular distance are allowed instead of a continuous distribution, so that the integral in (3.4) has to be replaced by a sum over lattice sites. If the sites of the lattice are Ri, a typical site being chosen as origin, the expression for the orientational free energy becomes 4. THERMODYNAMIC EFFECTS OF DIRECTIONAL FORCES IN PURE LIQumS.-At If only nearest neighbour interactions are taken into account this can be written (4.2) where z is the lattice co-ordination number (the number of nearest neighbours of a given site) and a is the nearest neighbour distance.Eqn. (4.1) and (4.2) lead to simple expressions for the extra contribution to the thermodynamic functions if the intermolecular field can be represented by dipole or quadrupole interactions as discussed in section 2. Using a face-centred cubic lattice for which z = 12 and a3 = 1/2 V/N (4.3) the contribution of dipole-dipole interactions is found to be (4.4) from (4.2). If more distant interactions are taken into account (4.1) must be used and (4.4) is multiplied by a factor 1-2045.7 Corresponding expressions in- volving quadrupole moments are easily derived using (2.5).Even if simple expressions for the directional field are not available, it is possible to draw certain conclusions about the other thermodynamic functions from (4.2). The orientational free energy F@r) leads 10 the following additional contributions to the entropy, internal energy and heat capacity at constant volume F(or)/NIE T = - (Np21 Vk T)2 The extra entropy SW), which is always negative, arises because, in order to take advantage of the additional energy of certain orientations, the molecules have to restrict their freedom of rotation. The heat capacity C v ) which, accord- ing to (4.5) is twice (- SW) is due to the loosening-up of the rotationaldegrees of freedom with rising temperature. It should be possible to determine whether directional forces are important in a liquid by examining its thermodynamic properties.The most marked effect will be probably on the heat capacity. A direct test of this sort can only be appliedJ . A . POPLE 39 to substances for which there are no other complicating factors such as internal rotation about single bonds. Accurate molar heat capacities at constant volume are not available for many liquids but the additional term given by (4.5) should also contribute to the molar heat capacity at constant pressure. As only the configurational heat capacity due to intermolecular forces is required, the cor- responding heat capacities of a perfect gas (at constant volume) must be sub- tracted from the observed Cp. The gas heat capacities can be estimated theoretic- ally from the observed vibrational frequencies. The values of ((Cp)liq - (CV),,,} for some simple liquids near their normal boiling points are given in table 1.TABLE 1 .-CONFIGURATIONAL HEAT CAPACITIES OF SOME LIQUIDS AT CONSTANT PRESSURE (cal mole-1 deg-1) A 87 7.1 7 c 1 2 240 9.9 13 Kr 1 20 7.8 8 H2S 210 10.3 14 Xe 165 7.7 9 CH3C1 249 11.1 15 N2 78 8.7 10 NH3 240 11.9 '6 cs2 3 10 9.0 11 C6HG 353 14.0 17 HCl 185 9.2 12 Cc4 330 14.1 18 substance T C K ) {(Cp)Iiq - (Cv)gas} substance T("K) {(Cphiq - (Ce)gas) As expected, the configurational heat capacities of molecular liquids are larger than those of the monatomic substances. Further, most of the values increase in a reasonable order, those for symmetrical non-polar molecules being rather less than those for simple polar molecules.The heat capacity of liquid carbon tetrachloride is surprisingly high and is rather difficult to reconcile with Hilde- brand's suggestion 19 that almost free rotation occurs. If the extra heat capacities shown in table 1 are enturely due to the increasing importance of directional forces, there will be corresponding extra entropies according to (4.5). For benzene and carbon tetrachloride the negative entropy due to hindered rotation would correspond to a value of (- TS) as large as 1 kcal mole-1. 5. THERMODYNAMIC EFFECTS OF DIRECTIONAL FORCES IN LIQUID MIXTURES.- The theory applied to pure liquids in the last two sections can be generalized to liquid mixtures and can be used to discuss the effects of directional forces on the thermodynamic functions of mixing.Classical statistical mechanics leads to a complete expression for the free energy of a multicomponent system in terms of the intermolecular energies ust for all pairs of components s and t . Each usr can be expanded in the general manner (2.1), so that it is separated into a spherically symmetric part and various directional terms. The general method is to suppose that all the intermolecular energies ust differ only slightly from the corresponding energy uoo for some reference substance, urn being a central force energy and consequently a function of r alone. If all the u,t were equal to uoo, the mixture would be ideal and there would be no excess thermodynamic functions. The aim of the present method is to express the excess functions in terms of the differences between the intermolecular fields usr and urn, these differences being assumed small.The difference (ust - uoo), which is to be treated as a perturbation, can be writ ten I- us,(rlh+l, 02, +2) - Moo (r) = We are again considering only directional terms for which both I1 and i2 are greater than zero. The two parts of (5.1) representing central force and directional force differences respectively lead to distinct contributions to the free energy. If only leading terms are retained these two contributions are additive and can40 NON-CENTRAL FORCE FIELDS be considered separately. The excess Helmholtz free energy of mixing at constant volume, therefore, can be written (5.2) where the two contributions arise from the first and second parts of (5.1). For binary mixtures the explicit expressions for these two quantities are~2 (A*F), y = (A*F(cenr))~, y + (A*F(or))r, y where xA, x'B are the mole fractions of the components and ni0) (r) is the pair distribution function for the reference liquid.This paper is mainly concerned with the excess free energy ( A * F ( ~ ) T , v but we shall first gi\e a short discussion of theories of the central force term ( A * F ( c e n t ) ) ~ , v which has been evaluated by a variety of methods. If the continuous distribution function .Io) is repIaced by a lattice distribution as in the previous section and if interactions between non-neighbouring sites are neglected, (5.3) becomes (A*F(cent))~, = NwxAxB, (5.5) . where a being the distance between neighbouring sites and z the co-ordination number of the lattice.This is the zeroth approximation in the theory of strictly regular solutions.20 One important consequence of (5.5) is that (A*F(cenO)l, v is inde- pendent of temperature so that there i s no entropy of mixing at constant volume. A lattice model will only give an excess entropy if higher order calculations are carried out and then (A*S), v is smaIl and always negative. An improvement that has recently been introduced 2 1 s 22 is to treat the particles as moving in cells rather than restrict them to lattice sites. This improved model does lead to an excess entropy of mixing because the vibrational motions of particles in their cells may differ in the mixture and pure liquids. Quantitative estimates of this entropy vary according to the type of cell field used, but the more realistic fields lead to a value of A*S which is of the same sign as A*F.The relative contributions of the excess entropy predicted by the various theories are shown in table 2, together with some experimental data on equimolar non-polar mixtures. It is important that the theoretical predictions should be compared with measurements of mixing functions at constant volume, for a considerable part of the measured excess entropy of mixing a t constant pressure can be attributed directly to the volume change. The figures of table 2 show that, even after the entropy due to the volume change has been allowed for, the remaining excess entropy is considerably larger than that predicted by any of the models of the reference liquid. The cell model of Lennard-Jones and Devonshire, in particular, has proved very successful in other respects, so that we have some justification for concluding that the observed entropies cannot be explained in terms of central forces, however good the statistical model.Longuet-Higgins 30 has shown how, under certain circumstances, it is possible to express ( A * F ( c e n t ) ) ~ , y and other thermodynamic functions in terms of experi- mentally measurable properties of the reference liquid, thereby eliminating appeal to any statistical model. In particular it is found that (5.7) (A*G), P : (A*S)T, P = RT - Qo : dQo/dT - R,J . A. POPLE 41 TABLE 2.-EXCESS FREE ENERGIES AND ENTROPIES OF MIXING FOR EQUIMOLAR MIXTURES (A*+, y T(A*S)=, p T(A*SIT, v T(A*s)T. V (cal mole-1) (cal mole-1) (cal mole-1) (A*F)T, v - - 0.00 central forces, lattice theory - central forces, harmonic oscillator - - 0.08 cell theory 21 - central forces, smoothed potential - - 0.00 cell theory 21 - central forces, Lennard-Jones and - - 0.10 Devonshire cell theory 22 - benzene + cyclohexane (298" K) 74.4 23 119.1" 66*2* 0.89 benzene + carbon tetrachloride 19.5 24 11.2" 10*5* 0-54 cyclohexane + carbon tetrachloride 16.7 25 21.3" 8-2* 0.49 (298" K) (298" K) * calculated using heats of mixing given by Scatchard et a1.26 and volume changes measured by Wood et a1.27-29 where Qo is the latent heat of conversion of a mole of reference liquid to its vapour at the same temperature and zero pressure. If either of the components is used as reference liquid and the quantities on the right-hand side of (5.7) are obtained experimentally, this equation predicts a larger excess entropy than any of the com- pletely theoretical treatments quoted in table 1 and consequently rather better agreement with experiment. It should be noted, however, that this only means that the observed properties of mixtures can be interpreted in terms of central forces if the properties of the pure liquids are also interpretable in this way.In fact dQo/dT is closely related to the heat capacity at constant pressure and we have seen in the previous section that this shows a systematic variation which is possibly due to directional forces. If we now consider the expression (5.4) for the excess free energy due to differ- ences in directional forces, it is immediately clear that (A*F(o~))T, v is more tem- perature dependent than (A*I;(cen'))~, v because of the extra factor T-1.If the simple lattice distribution function is used for nSo), (A*F(orl)~, v is given by or, if only interactions between nearest neighbours are considered, a being directly related to the total available volume. sign as the excess free energy and which is such that Both these expressions lead to an excess entropy of mixing which has the same (5.10) If more accurate statistical models were used allowing for the temperature de- pendence of n$'), these ratios would no doubt be modified, but as these refinements only make small differences to the central force theory, (5.10) is probably sub- s tan t ially correct. The most important qualitative conclusion to be drawn is that the excess entropy of mixingdue to directional force differences is approximatelyequal to the excess free (A*F(or))r, v = T(A*SW)T, v = &(A*E(~~))T, v.42 NON-CENTRAL FORCE FIELDS energy divided by the temperature. This is more in accord with the experimental data of table 2.The observed values of the ratio T(A*S)r, v/(A*F)r, v lie between 0.5 and 1.0 as would be expected if both effects are operative. The entropy change due to directional forces is to be interpreted physically in terms of hindrance to free molecular rotation. If the directional forces between molecules of different species are stronger than those in the pure components, for example, there will be less random orientation in the mixture and consequently less entropy.This has often been suggested as the qualitative explanation of observed excess entropies, but simple quantitative expressions for the effect in terms of intermolecular forces have not previously been put forward. In the remainder of this section, we shall examine the expressions (5.8) or (5.9) in some particular cases to see whether observed data can be interpreted in terms of reasonable intermolecuIar fields. (a) Benzene + cyclohexane.-This system has been extensively studied 239 26927 and is found to show positive deviations from Raoult's law together with a con- siderable excess entropy of mixing (table 2). Scatchard, Wood and Mochel23 suggested that the excess entropy might be due to incomplete randomness of orientation in either of the pure liquids.This might be expected on the grounds that the plane or puckered hexagons fit together better among themselves than with each other. Neither'molecule is truly axially symmetric, but we may consider them to be approximately so about axes perpendicular to the hexagons. Suitable approximate forms for the intermolecular fields are where &(x) is i-9-i and < r' ') (r) are negative so that face-to-face configurations are preferred. If it is now assumed that the central forces are all the same, that the directional forces only operate between like particles and that the additional intermolecular energy of the face-to-face configurations in both pure liquids is - q at the nearest neighbour separation a so that (5.12) (22: 0) (22 : 0) (a) = [BB ('1 = - $q, then the expression (5.9) for the excess free energy becomes (A*F)r, v/NkT = O*O2~(q/kT)2 XAXB.(5.13) Taking z = 8 and using the observed value of (A*F)r, y (table 2), we find that q/kT = 1.77 corresponding to q= 1050 cal mole-1. According to (5.10), (A*S)=, y will be equal to (A*&, v/T in approximate agreement with observation. These figures are, of course, derived from special assumptions and may not represent the actual interactions at all well, but the calculation does show how the quali- tative suggestion of Scatchard, Wood and Mochel can be put in approximate quantitative form. It should be mentioned that in a later paper Scatchard, Wood and MocheI25 withdrew their explanation on the grounds that the excess entropy of C6H12+CgHg is considerably greater than the sum of excess entropies for the mixtures C6H6 f CCl4 and C(jH12 -I- Cc14.Equality is ody to be expected, however, if there are no orientation effects either in pure CCk or in the other two mixtures. As has been pointed out in the previous section, it is by no means certain that CC14 acts as a spherically symmetric molecule. (b) SimpZe polar rnoZecules.-If the directiona 1 forces between simple polar molecules can be represented by the interactions of dipoles and quadrupoles, then eqn. (5.8) or (5.9) lead to simple expressions for (A*F(or))r, v. As in the corresponding treatment of pure liquids, we shall use a face-centred cubic lattice. If onIy the dipole moments /LA and pB are important, we find, using (2.4), (5.14)J. A . POPLE 43 the corresponding energy and entropy of mixing being given by (5.10).If the quadrupole moments @A and OB also have to be taken into account, (5.14) has to be replaced by (A*F(o~))T, v/NkT = { 1 *2045[ZV(& - &)I VkTl2 + 2.540 Ng/3(pi - pi)(@: - Oi)/V8/3k*T2 $- 2*716[N5/3(@: - O ; ) / V 5 l 3 k a 2 ) X ~ X ~ . (5.15) Eqn. (5.14) can be tested by calculating the free energy of mixing from observed vapour phase dipole moments. The results for two simple mixtures are compared with the observed values of (A*Q=, p 31 (which should be approximately equal to ( A * F ) , V) in table 3. (V is taken as the mean of the molar volumes of the pure components.) TABLE 3 .-EXCESS FREE ENERGY FOR POLAR-NON-POLAR MIXTURES (EQUJMOLAR COMPOSITION) CHCl3 -I- CS2 1 -05 73.4 9 76 (CH3)2CO -1 CS2 2.74 68 460 267 The large difference between the two calculated values is due to the fact that the expression (5.14) varies as the fourth power of p.It would appear reasonable to attribute the mixing properties of acetone + carbon disulphide mainly to the dipolar interaction, but other types of intermolecular force must play a significant part in the system chloroform + carbon disulphide. 1 Pitzer, J. Chem. Physics, 1939, 7, 583. 2 Pople (to be published). 3 Stockmayer, J. Chem. Physics, 1941, 9, 398. 4 Rowlinson, Trans. Faraday SOC., 1949, 45, 984. 5 Pople, Proc. Roy. SOC. A, 1952, 215, 67. 6 Lennard-Joncs and Inghani, Proc. Roy. SOC. A , 1925, 107, 636. 7 Frank and Clusius, 2. physik. Chem. B, 1939,42, 395. 8 Clusius, 2. physik. Chem. B, 1936, 31,459. 9 Clusius and Riccoboni, 2. yhysik. Chem. B, 1937, 38, 81. 10 Clayton and Giauque, J. Amer. Chem. SOC., 1933, 55, 4875. 11 Brown and Manov, J. Amer. Chem. SOC., 1937,59,500. 12 Giauque and Wiebe, J. Amer. Chem. SOC., 1928, 50, 101. 13 Giauque and Powell, J. Amer. Chem. SOC., 1939, 61, 1970. 14 Giauque and Blue, J. Amer. Chem. SOC., 1936, 58, 831. 15 Messerly and Aston, J. Amer. Chem. SOC., 1940, 62, 886. 16 Overstreet and Giaque, J. Amer. Chem. SOC., 1937, 59, 254. 17 Oliver, Eaton and Huffman, J. Amer. Chem. SOC., 1948,70, 1502. 18 Lord and Blanchard, J. Chem. Physics, 1936,4, 707. 19 Hildebrand, J. Chem. Physics, 1947, 17, 727. 20 Guggenheim, Mixtures (Oxford University Press, 1952). 21 Prigogine and Mathot, J. Chem. Physics, 1952, 20, 49. 22 Pople, Trans. Furaday SOC. (in press). 23 Scatchard, Wood and Mochel, J. Physic. Chem., 1939, 43, 124. 24 Scatchard, Wood and Mochel, J. Amer. Chem. SOC., 1940, 62, 713. 25 Scatchard, Wood and Mochel, J. Amer. Chem. SOC., 1939, 61, 3206. 26 Scatchard, Ticknor, Goates and McCartney, J. Amer. Chem. SOC., 1952, 74, 3721. 27 Wood and Austin, J. Amer. Chem. SOC., 1945, 67, 480. 28 Wood and Brusie, J. Amer. Chern. SOC., 1943, 65, 1891. 29 Wood and Gray, J. Amer. Chem. SOC., 1952, 74, 3729. 30 Longuet-Higgins, Proc. Roy. SOC. A, 195 1 , 205, 247. 31 Hirschberg, Bull. SOC. chim. Belg., 1932, 41, 163.

ISSN:0366-9033    

DOI:10.1039/DF9531500035

出版商:RSC    

年代:1953

数据来源: RSC

摘要:

THEORETICAL MODELS AND REAL SOLUTIONS BY ROBERT L. Sco-m Department of Chemistry, University of California, Los Angeles, California, USA. Received 22nd January, 1953 Various theoretical modeIs of solutions (Hildebrand’s regular solution, Guggenheim’s strictly regular solution, Longuet-Higgins’ conformal solution, and Prigogine and Mathot’s cell model) are discussed and compared. To the first approximation, limited to first order terms in (E2 - El)/E, all are equivalent. Volume changes on mixing at constant pressure are emphasized, particularly for dispersion-force solutions, where Prigogine and Mathot predict contraction on mixing. INTRODUCTION.-ReCmt years have seen increasing interest in the development of a comprehensive theory of non-electrolyte solutions. Several of the theories which have been advanced account satisfactorily (in a qualitative or semi- quantitative way) for some of the properties of many solutions.A completely adequate treatment, however, must interpret, in terms of the same consistent picture of interactions between molecules, all the properties of solutions of non- polar and polar substances alike-heats of mixing, entropies, volume changes, solubilities, critical phenomena, surface tensions, etc. Considering the not-very-satisfactory status of theories of liquids, it is at first surprising that so much progress has been made in our understanding of liquid solutions. The reason for this paradox must lie in the fact that the solution properties which interest us are usually differences or ratios referred to the pure liquid as standard state.If we assume that the factors for which we cannot satisfactorily account are the same in the solution as in the pure substances, these cancel out and can be ignored. That this view must be, to a first approximation, a correct one, is indicated by the considerable success of even very artificial models of solutions (e.g., the quasi-lattice model which regards the liquid as a crystalline solid and the liquid solution as a solid solution). Much of the advantage of this situation depends, however, upon our using the experimental properties of the pure components as parameters in otherwise theoretical equations. While a completely theoretical approach is more im- pressive and ultimately more desirable, the semi-empirical approach is as yet more useful.No one can yet expect an accurate prediction of the vapour pressure of a solution from a theory which uses theoretical vapour pressures for the pure components. FOUR DIFFERENT MODELS.-AII but completely empirical equations are based upon certain theoretical assumptions which can usually be related to a physical model. The success of the equations in fitting experimental data is by no means convincing evidence for the correctness of the model, for several different models may lead to the same result. Nor is failure to fit experiment always grounds for rejecting a model ; the difficulty may lie in mathematical approximations. All models so far proposed are much oversimplified and correspond even approximately to few, if any, real systems. Much can be learned, however, from a consideration and comparison of these models, and we can frequently estimate where each might be most successful and where it might be expected to fail.44ROBERT L . SCOTT 45 In this paper, four different approaches will be compared : I. The first “ model ”, if it may be called that, is Hildebrand’s “ regular solution ”. Initially suggested by the “ regular ” family of solubility curves for various solid non-electrolytes in a series of non-polar solvents,l this was later defined 2 as a solution in which the distribution and orientations of the various species are random, leading to an ideal entropy of mixing : (1) where N is the total number of molecules (Nl + Nz), and x1 and x2 are the mole fractions of each component. For such a system, deviations from ideal behaviour must be due to the heat (or energy) of mixing.If the molecules are distributed randomly, and there is no change of volume on mixing, AEMis not difficult to calculate ; 39 4 for components of equal molar volumes, one obtains 5 ASM = Nk(x1 In x1 + x2 In x2), where A12 is a constant characteristic of the system. If the molar volumes differ significantly, a less exact treatment 3 9 4 suggests substitution of volume fractions for mole fractions, a procedure justified empirically by much experimental evidence. As Hildebrand has recently re-emphasized,a the basic feature of the regular solution is the ideal entropy of mixing, not any detailed specification of the physical situation. As such, a regular solution is more nearly a semi-empirical concept than a physical model ; in fact, it cannot be derived exactly from any self-consistent model (only an ideal solution can have exactly ideal entropy).It is, however, the zeroth approximation obtained from the statistical formulations of a number of different physical models. Moreover, the free energy equation obtained by combining eqn. (1) and (2) has a remarkably wide range of approximate applica- bility, far more than could be expected from any of the special physical models. 11. The first reasonably rigorous statistical formulation was the “quasi-chemical” treatment of “ strictly regular ” solutions developed by Guggenheim 7 and Rushbrooke.8 Their model is a quasi-lattice of co-ordination number z in which each molecule occupies a single lattice site, and so is more nearly appropriate to solid solutions than to liquid solutions.The major accomplishment of this treatment has been the evaluation of the preferential ordering induced by differ- ences in the magnitudes of the various intermolecular interactions. The effect of this ordering upon the thermodynamic properties while small is not insignificant. Moreover, very recently, Barker9 and Tompa 10 have extended the method to solutions of polar molecules where ordered orientations seem to be very important indeed. The thermodynamic functions of the strictly regular solution are all expressible in terms of a single parameter, an “ interchange energy ” w, which is the decrease in potential energy on exchange of a single pair of molecules between the pure liquid components.In terms of the pair energies €11, €12, and €22 w = Qz(2q2 - €11 - €22). (3) When ordering effects are ignored, the strictly regular solution reduces to the regular solution with A12 = w. The strictly regular solution has some serious limitations in addition to the artificiality of the lattice model itself (which, because of the cancellations suggested above, is probably less serious than frequently supposed). In order to fit into the regular lattice, molecules of the different species must be of nearly the same size, although the method has been extended to include molecules occupying a series of lattice sites.11 Moreover, the strictly regular solution like Hildebrand’s regular solution assumes no change of volume on mixing, although such a change can be allowed for indirectly as will be seen later.Once the model is accepted,46 THEORETICAL MODELS AND REAL SOLUTIONS however, the application to it of the quasi-chemical method is a very nearly exact one.12 111. Recently Longuet-Higgins 13 has extended the Pitzer 14 treatment of corresponding states for perfect liquids to yield a theory of “conformal soh- tions ”. All the thermodynamic functions depend on two parameters : &, the molar energy (essentially - dZ?:v) of a reference substance (usually one of the pure components), and a differen_ce function dl2 = 2f12 - fil - f22. The f’s are ratios of two energies (f11 = &,lEo) measured in corresponding states of temperature and pressure, and in principle are equal to the ratios of critical temperatures.If the f ’ s were so evaluated, however, we would lose part of the advantage of dealing with solutions; consequently it seems better to regard them as empirical. The equations are good only to the first order in (f- 1) ; because most liquids are not perfect liquids and obey the laws of corresponding states only roughly at best, any attempt to extend the theory of conformal solutions to higher orders of (f- 1) would probably be useless.” IV. Very recently Prigogine and Mathot 15 have used the “ cell-method ” to develop from a smoothed-potential free-volume model a detailed theory of solutions. All the relevant thermodynamic functions can be obtained from a set of equations in which the only adjustable parameters are three energies 1111, A12, n22, which measure intermolecular attractions.(In the equilibrium con- figuration at zero temperature and pressure, the potential energy per molecule is - 0.7178A ; were only nearest neighbour interactions considered, it would be - A/2.) Like the strictly regular solution this is a lattice model-with all the limitations which that implies-and like the conformal solution, it follows a law of corre- sponding states. While the behaviour of this solution under varying conditions of temperature, pressure, etc., can be calculated in any desired detail, it is not certain that conclusions drawn from terms of second order in kT/A are relevant to real solutions. A GENERAL TREATMENT OF DEvrATIoNs.-Since it is always necessary to make restrictive assumptions in order to make the mathematical treatment tractable, it is important to be able to assess the influence of these restrictions on the final result.16 As a zeroth approximation, we start with a fairly restrictive model in which a particular property of the system, 8, is assumed to have the fixed value 80.On this basis we calculate certain thermodynamic functions. However, in the real solution, 4 actually has the value s’ rather than t o . We can relate a thermodynamic function X to its model value by developing a Taylor series expansion around 5’. If there is more than one such parameter, a multi-variable expansion can be used : Assuming that terms higher than first order in the difference functions can be neglected, eqn. (4) suggests that corrections of different kinds can be separately added on to a crude zeroth approximation, say that of the regular solution. This is important, since one kind of approach may be best for evaluating one type of correction and be very poor for another ; eqn.(4) permits us to be eclectic. *Brown and Longuet-Higgins (Proc. Roy. SOC. A , 1951, 209, 416) have derived equations for a second order theory of conformal solution, but conclude that some of the parameters cannot be directly related to the thermodynamic properties of the reference species and cannot be evaluated without appealing to some sort of special model for the reference assembly.ROBERT L. SCOTT 47 If the thermodynamic function Xis the Helmholtz free energy F (for isothermal constant volume systems) or the Gibbs free energy G (for isothermal constans pressure systems), a special simplification may be available.If the parametert 6 and q are ones which are arbitrarily fixed by theoretical assumptions, but are not in fact macroscopic variables of the system which can be fixed in the laboratory, the system will of course choose those values which minimize the free energy. In this case, the leading terms on the right-hand side of eqn. (4) vanish. If the parameter 4 is not self-adjusting, but can be fixed by the experimenter, 3F/3[ need not vanish. In a most important case of this kind, however, where is volume, (3F/3 V ) T = - P and since we normally work near zero pressure, the leading term again vanishes. This accounts for the strange fact that the free energy calculated from a grossly oversimplified model frequently agrees well with experimental results, even though a more careful examination reveals that other thermodynamic properties (e.g.heat, entropy, etc.) agree not nearly so well. The disappearance of this first order correction term in the free energy can mask the unsatisfactory nature of the model; on the other hand, it permits very crude models to yield useful free energy equations. VOLUME EFFECTS-A consideration of the change of volume on mixing and of the effect of the volume expansion with increasing temperature can be most helpful in illuminating the relationships between these various approaches. In what follows we assume a random distribution ; the strictly regular solution then reduces to the zeroth approximation regular solution ; presumably correction factors for ordering effects can later be applied by using eqn.(4). Both the regular solution and the strictly regular solution assume zero volume change, but we can use eqn. (4) to assess the effect of the actual change in volume on mixing at constant pressure.17 Disregarding correction terms in powers of AVM higher than the first (except in eqn. (5) where the first power has vanished) the constant (zero) pressure functions (subscript p ) are related to the constant volume functions (subscript v ) by the equations (5) 1 AF,M = AG,M = AFvM - - (A VM)2 AFvM, 2P v TOC A E ~ -- AH,M = AE,M + -(AvM), B (7) where cc and /3 are the coefficients of thermal expansion and of isothermal com- pressi bility, respectively. a ASpM = ASvM -1- -(A vM), P In all these models (except for order-disorder effccts), the entropy change at consrant volume has the ideal value (eqn.(l)), and must therefore be pressure independent. Consequently where Eu is the energy of the unmixed components. The internal pressure can always be written as (3E/3 V)T = - n(E/V), where n = - VTa/PE. For many normal liquids n = 1, but keeping the more general relation,* and assuming n the same for mixed and unmixed, we obtain A V M AEvM * If n is assumed to be unity, one obtains the equation : AVM = FAEVM. __ = - aT- V E *48 THEORETICAL MODELS AND REAL SOLUTIONS The assumption about constant n can be shown to be approximately valid provided that AEvM/E is not so small that it is of the order of magnitude of (E2 - E1)2/E2; it is that small for dispersion forces, a situation which leads to complications which we discuss later.Substitution of eqn. (10) back into eqn. (6) and (7) leads to the relation A H ~ M - AE,M = T(ASpM - ASvM) = .- V(ciT)2AEvMf/3E. ( 1 1 ) EQUIVALENCE OF THE EQUATIONS.-we can now compare the equations of the different models (RS for regular solution, LH for Longuet-Higgins conformal solution, PM for Prigogine and Mathot cell model). Terms of order higher than the first-beyond the limit of validity of conformal solution theory-are omitted.* (The superscript E symbolizes excess over the ideal value, and ‘2 order to express everything in molecular quantities, Longuet-Higgins’ molar EO and To are reduced to molecular €0 and VO.) Volume change on mixing (constant zero pressure) : = 1*9296kT($ - 2 - ”> = 1*9296kTA(l/A>.(12PM) A22 Excess free energy of mixing (constant zero pressure or constant volume) : Heat of mixing (constant zero pressure) : = - 0.7178M 4- 12*733(kT)2A (1/A). (1 3RS) ( 1 3LH) (13PM) ( 1 4RS) ( 1 4LH) (14PM) * The approximate equations given by Prigogine and Mathot (their numbers 57-5-10 and 6.9-6.12) are not self-consistent (for exampIe, - dAGf/dT i AS:). The reason for this lies in their peculiar expansion of their function a (not the coefficient of thermal expansion) around the value a = 0.95 at kT/A = 0.05412; when they drop all terms involving powers of [(kT/A) - 0.054121 higher than the first, some terms in the difference functions depending upon TO and T1 are lost as well as, intentionally, the T2 terms. We have expanded the exact equation for their a around kT/A = 0, obtaining a = 1.19145 - 5.0190 (kT/A) + 17.427 (kT/A)2 and from this derived a different set of approximate equations.ROBERT L.SCOTT 49 Excess entropy of mixing (constant zero pressure) : - _ - (3dl2XlX2 (15RS) (1 5LH) = 25.466 (kT)zA( 1 /A>. (1 5PM) We shall now show that these three sets of equations are to this approximation First it is clear that the energy constants are all very similar: entirely equivalent. w = @El2 - E l l - €221, cod12 = ~o(2fi2 - f i l - f 2 d , (1 6RS) (1 6LH) and since A for a random mixture is merely x12A11 1- 2~1x21112 + x22A22, M = A - xlA11 - x~A22 = (2A12 - 1111 - A22)~1~2. (16PM) If the conformal solution reference point is chosen properly, so that etc., and since to this approximation E w €0, a w CCO, etc., it follows that 1 f- 1 I < 1, fll = E l l / € = M A , Moreover, since dEo/dT is really 1/N times the temperature derivative of the configurational energy, (1/N)(3Ec/3T)p taken at P = 0, we can evaluate it from the thermodynamic relation, This, for (bEc/3T)v = 0, which is true for all these models, reduces at zero pressure to (3Ec/3T)p = TVa2//? or dco- 1 dw T& dT d12dT- /3 Eqn.(17) and (19), when substituted into eqn. (12)-(15) complete the proof of equivalence of the regular solution and conformal solution equations. Guggenheim 18 has suggested that w be regarded as a temperature-dependent local free energy ; we see here that a term dw/dT allows for the expansion of the lattice with temperature and automatically converts the simple constant volume equations into constant (zero) pressure equations.It remains to relate the Prigogine and Mathot equations to the others. From their basic equations, we can derive the following, in which €00 denotes the energy at zero temperature. (19) - - __--- - - €00 = - 0*7178A, (20) a = 2*1064k/A, /3 = 0,17420 v/A, = - 0.7178A 4- 12.733 (kT)2/A. The factor A(l/A> must also be evaluated :50 THEORETICAL MODELS AND REAL SOLUTIONS To this approximation, no distinction is made between the average value of l/A for the unmixed components (x1/A11 + x2/A22) and the reciprocal of the average value of A, [l/(xlAll + x2A22)]. This distinction becomes all important when dealing with dispersion force solutions. With this approximate equation, we can rewrite eqn.(13PM) as A F f N I_- - - 0.7178M - 12.733 (kT)2A(l/A> = - 0.7178M + 12.733 (kT)2M/A2 (25) This is of course the same as the regular solution and conformal solution equations (13RS and 13LH) ; the equivalence of the other functions can be demonstrated as well. It also can be shown that for the constant volume process, the excess entropy of mixing (AS,E) is zero for all three models, leaving the energy of mixing (AE,M) equal to the excess free energy (AF,E = AGf). Of course the three approaches will yield the same values for the thermo- dynamic functions only if their parameters are evaluated in equivalent ways. In regular and conformal solution theory, dw/dT, a, /3, etc., are evaluated empirically from experiment, while Prigogine and Mathot calculate them from theoretical relations involving the A’s.THE DISPERSION FORCE SOLUTION.-when the intermolecular attractive forces are believed to be of the London dispersion force type, we usually assume the interaction energy between unlike pairs to be the geometric mean of those between like pairs (26) These lead to the relations (27RS) (27LH) (27PM) = [- 0.7178A + 12.733 (kT)2/A](M/A) = ~ d 1 2 ~ 1 ~ 2 . - €12 = - 1W62214 ; f 1 2 = ( f d 2 2 ) f . ; 4 2 = (AllA22)*. w = +HI’ - ] € 1 2 1 V , 4 2 = ( f l l ’ - f 2 2 9 2 , AA = - (&* - (12292X1X2, but A(l/A) w - 3(A11* - A 2 2 + ) 2 ~ 1 ~ 2 / 1 2 = 3M/A2. (28) Note that eqn. (28) gives A( 1 / A ) an opposite sign and three times the magnitude of that predicted by eqn. (24), In this case the difference between the average (l/A) for the unmixed components and the reciprocal l/(A>, ignored hitherto, is all important, being equal to xlx2(All - A22)2/A3 rn 4x1x2(Ail* - (122WA.When eqn. (28) is used to evaluate the thermodynamic functions of the previous section, we find a positive heat of mixing associated with a negative volume change and a negative excess entropy, whereas the regular solution and conformal solution theories predict positive signs for all three. Further analysis leads to the following observations : (i) The conformal solution treatment is admittedly valid only to the first order in (f- 1) ; in the dispersion force case, the first non-zero terms involve (f- 1)2. Thus to the first approximation, as far as we can legitimately go, the conformal dispersion force solution is ideal.(ii) The extended regular solution treatment of volume change (eqn. (9), (10)) was not derived from any specific model, but rather depended upon the relative constancy of the ratio It = - (3 In E/3 In V)T. The variation of n with Vinvolves the second and higher powers of (f- 1) ; so, unless the first order terms vanish (as here), the derivation is defensible. (iii) In the cell model of Prigogine and Mathot, n is a rapidly varying function of kT/A, and it is impossible for the two pure components to have different E’s at a particular temperature without at the same time having different n’s. It is this feature, coupled with the lattice requirement that the different molecules in the mixture occupy cells all of one size, which produces the unexpected negative signs.ROBERT L .SCOTT 51 How far this model reproduces the behaviour of real solutions is very uncertain. Experimental observations on real non-polar liquids of widely varying energies, sizes, and shapes give n-values 19 ranging from 0.9 to 1-1. Moreover mixtures of these substances almost invariably show positive deviations from Raoult’s law, positive heats of mixing and positive volume changes3 Recently Hildebrand and Scott 21 showed that the entropies of solution of iodine in carbon tetrachloride and perfluoroheptane (C7F16) exceed the ideal by an amount in good agreement with values estimated by means of eqn. (7) and (10). Admittedly these systems do not conform to the requirements of the cell model, viz., spherical molecules of essentially the same size.Prigogine and Mathot 15 report volume contractions when carbon tetrachloride and neopentane are mixed. It is always possible that “ perfect liquids ” of the same molar volumes are unique in this respect, while “ imperfect ” liquids, for reasons as yet unexplained fit the cruder theory better.* The question of volume changes in dispersion force solutions is clearly one which requires much more theoretical study and much careful experimentation on well-chosen systems. Because it is a second order effect unmasked by any first order term, it should be an excellent place to test the “ fit ” of new theories of solutions. * One is reminded of the comparison of the Hildebrand rule with the laws of corre- sponding states ; cf. Hildebrand and Gilman, J . Chem. Physics, 1947, 15, 229. 1 Hildebrand and Jenks, J . Amer. Chem. SOC., 1920, 42, 2180; 1921, 43, 2172. 2 Hildebrand, Proc. Nat. Acaci. Sci., 1927, 13, 267 ; J . Amer. Chem. Soc., 1929, 3 Scatchard, Chem. Rev., 193 1,8,321 ; Kemisk Maanedsblaci(Cupenlzagen), 1932,13,77. 4 Hildebrand and Wood, J . Chem. Physics, 1933, 1, 817. 5 for details of these equations, and of various applications of regular solution theory, see Hildebrand and Scott, Solubility of Non-electrolytes (New York, Reinhold, 1950), 3rd ed. 51, 66. 6 Hildebrand, Nature, 1951, 168, 868. TGuggenheim, Proc. Roy. SOC. A, 1935, 148, 304. For a thorough discussion of strictly regular solutions, see the recent book by Guggenheim, Mixtures (Oxford University Press, 1952). 8 Rushbrooke, Proc. Roy. Soc. A, 1938, 166, 296. 9 Barker, J . Chem. Physics, 1952, 20, 1526. 10 Tompa, J . Chem. Physics, 1953, 21, 250. 11 Guggenheim, Proc. Roy. SOC. A , 1944, 183, 203, 213. 12 Bethe and Kirkwood, J. Chem. Physics, 1939, 7, 578. 13 Longuet-Higgins, Proc. Roy. SOC. A, 1951, 205,247. 14 Pitzer, J . Chem. Physics, 1939, 7, 583. 15 Prigogine and Mathot, J . Chem. Physics, 1952, 20, 49. 16 The method described here was first presented in Hildebrand and Scott,S pp. 135-136. 17 The equations here derived are equivalent to those first derived by Scatchard, 18 Guggenheim, Trans. Faraday Suc., 1948, 41, 1007. 19 ref. (5), p. 97. 20 ref. (5), p. 142. 21 Hildebrand and Scott, J . Chem. Physics, 1952, 20, 1520. Tratts. Faraday SOC., 1937, 33, 160 ; see also ref. (5), chap. 8, pp. 136-143.

ISSN:0366-9033    

DOI:10.1039/DF9531500044

出版商:RSC    

年代:1953

数据来源: RSC

摘要:

LATTICE THEORIES OF LIQUIDS AND SOLUTIONS AT LOW TEMPERATURES BY J. S. ROWLINSON Department of Chemistry, The University of Manchester Received 7th January, 1953 Several modifications of the lattice theory of solutions are tested by comparing the corresponding theories of pure liquids with experimental results. The quantities tested are the configuration energy (RTniinus the latent heat of evaporation), its rate of change with temperature, and the coefficient of thermal expansion of the liquid. Longuet- Higgins has shown that, with certain restrictions, these are related to the excess free energy, entropy and volume of a solution. To obtain good agreement for all these quantities, it is shown that a lattice theory of liquids must take into account the potential of all the neighbours of a given molecule, not only the nearest, and must not approximate this potential by replacing it with a smoothed square-well potential, or with the potential of a simpIe harmonic oscillator.Refined lattice theories, which allow for unoccupied sites, give the same results as simple theories at temperatures below the normal boiling point. -- 1. INTRODUCTION.-L~~~~C~ theories of liquids and of liquid solutions have been developed almost independently. As it is hardly desirable that the dis- tinction should be maintained, these theories have recently been extended in such a way that one set of assumptions may be made the basis both of a theory of pure non-polar liquids and of a theory of the excess thermodynamic properties of a mixture of such liquids.1.2~ 3 The results obtained apply only to liquids which obey the principle of corresponding states and which have similar molar volumes.In this paper the predictions of these theories are compared with experimental results. The quantities chosen for comparison are the configuration energy (RT minus the latent heat of evaporation), its temperature coefficient, and the coefficient of thermal expansion. Longuet-Higgins 4 has shown that these quantities are related to the excess free energy, entropy and volume of a binary solution by the equations, Y (1.1) where El and V1 are the configuration energy and volume of component 1, where XI and x2 are mole fractions, and where €11, €12 and €22 are the minimum energies of interaction of isolated pairs of molecules. These relations are correct only for the first-order deviations from unity of the ratios of €12 and €22 to €11.They were derived without reference to any model of the liquid state and must, there- fore, hold for any consistent theory based upon a model. This must be true whether or not the second-order deviations of the energy ratios from unity may be neglected. These relations provide the most convenient means of testing theories of solutions, as accurate experimental measurements of the properties of pure liquids are more plentiful than those of the excess properties of solutions. Moreover, as the lattice theory of liquids 5 has been developed further than that of solutions, this comparison shows which of these developments are worth incorporating into a theory of solutions.Lattice theories of liquids are of three kinds. In the first, the motion of a molecule near its site is governed by the smoothed potential of its interactions 52 - A* V - ~ 1 ~ 2 ( 2 ~ 1 2 - €11 - €22) - A*G - -A*S ----= 4 ( S E l / W P m W T ) P €1 1J. S. ROWLINSON 53 with its neighbours. This smoothing is usually the replacing of the actual inter- actions by a square-well potential centred on the lattice site, whose depth and diameter may be adjusted empirically. In the second kind of theory, the potential is supposed to be that which the neighbouring molecules (either all neighbours or sometimes only the nearest neighbours) would exert if they remained at rest upon their sites. This is the theory of Lennard-Jones and Devonshire.6 In the third kind, it is supposed that some of the sites may be unoccupied.The number of such sites is found by minimizing the free energy. Theories of solutions have been developed which correspond to the first kind of these theories of liquids, the smoothed potentia1,l and to the second kind at very low temperatures, when the motion of the molecule becomes that of a simple harmonic oscillator.2B 7 No theory has been put forward which corresponds to the third kind, the " hole " theories of liquids. However, the equations of these theories may be readily generalized to include two components, if these are assumed to be distributed at random upon the occupied sites. This would give a consistent theory of solutions. In order to see whether this development is worth while, the number of unoccupied sites may be calculated for a single component from eqn.(70) and (71) of a previous paper.5 This shows that, at very low temperatures, the average number of un- occupied sites (12 0) around one occupied one is given by (1 -2) At higher temperatures the equation for W is more complicated, but even at the normal boiling point ii is only of the order of 10-4. The correction to the theory of Lennard-Jones and Devonshire for the number of unoccupied sites is therefore negligible for all normal liquid solutions. 2. THE LATTICE THEORIES OF LrQurDs.-The three equations of state to be dis- cussed are : (1) the empirical smoothed potential model of Prigogine and Mathot,l (2) the theory of Lennard-Jones and Devonshire including interactions only (3) the same theory modified to include interactions with second and third In each case the intermolecular potential is assumed to be W =T exp (- 6ql/kT).with nearest neighbours, and neighbours.8 U(r) == 44(o/r)12 - (o/r)6]. (2.1) The three equations of state are then (I) pV/RT = (1 - 2-* q-*)-1 + (2417) [1.0110 4-4 - 1.2045 @I, (2.2) (2) PV/RT= 1 + (2417) [q-4(1 + 6) - 4-2(1 f ?)I, (2.3) ( 3 ) pV/RT = 1 + (2417) [1.0110 4-4 (1 + E) - 1.2045 q-2(1 -t- H)]. (2.4) The configuration energies are (1) E = 6N~[l.0110 4-4 - 2.4090 4-21, (2.5) (2) E = 6N~[q-4(1 + 6) - 2q-2(1 + 7)1, (2.6) (3) E = 6N~C1-01 lOq-4(1 + E) - 2.4090 (1 + H)], (2.7) where q is the reduced cell-size, or volume per molecule in units of 03, where T = kT/E, and where (,q, 8, H are defined by The symbols g , gi and gm are related to the partition function of a molecule within its cell and the derivative of this partition function with respect to volume.They54 LATTICE THEORIES AT LOW TEMPERATURES have been defined elsewhere.1p5s 6, 7 The symbols G, GL and GM are similar functions modified to allow for the effect of the outer neighbours.8 (It is in- consistent to we the simple partition functions, g, gl and gm, with the coefficients of 1.0110 and 1.2045, as has sometimes been done.) These equations are all derived from classical statistical mechanics. They therefore show no zero-point configuration energy and give a non-zero coefficient of thermal expansion at absolute zero. They show only one condensed phase and so make no distinction between liquids and solids.Eqn. (2.2)-(2.7) may be solved to give the volume and con- figuration energy of the liquid under its normal vapour pressure by putting (pV/RT) = 0. This approximation is quite adequate, for when the vapour pressure is one atmosphere (PV/RT) is less than 10-3 for all normal liquids, and decreases rapidly at lower temperatures. The integrals g , gr, gm, G, GL, GM have not hitherto been computed in this region of q and T. They may be found at very low temperatures by the approximation of the simple harmonic os- cillator.19 2 ~ 5 9 7 As this approximation is very poor at temperatures correspond- ing to the liquid state, new values of g, (, and r ) have been computed and are shown in table 1. TABLE 1 q = 1.00 q = 1.10 9 - 1.15 - 7 g x 104 e 91 g x 104 6 91 g x 104 E 91 0.3 0.76 0.127 0.0280 1.65 0.208 0.0453 - - - 0.4 1-14 0.165 0.0362 2.41 0.266 0.0576 3.58 0.340 0.0738 0.5 1.55 0.203 0.0440 3.22 0.319 0.0693 4.66 0.409 0.0890 0.6 1-98 0.237 0.0512 4.06 0.374 0.0811 - a The integral g is the most troublesome as it varies as yi at the origin and has very sharp maximum, still at a low value of y, where y is the variable of the integral. The values in table 1 were found by- Gregory’s method of numerical integration using the two ranges of y of 0.000 (0.00025), 0.002 (0.001) 0.035, and differences up to the fifth.All the integrands are negligible at values of y above 0.035. 3. REsuLTs.-Eqn. (2.2) can be solved to give q as a function Of T, by expanding the equation about any value of q. Eqn. (2.3) and (2.4) can be solved for the simple harmonic oscillator to give : from (2.3), = 1 f 0,19797, (3.1) from (2.4), 0.9162 (1 -J- 0.1254~).(3.2) Eqn. (2.3) can also be solved “ exactly ”, that is, with the values of 4 and 7 given by table 1. This givcs values of 4 which can be represented, to within 1 part in 10,000, UP to T = 0.5 by 1 4- 0.1979~ + 0*301072.S. (3.3) No exact valucs were computed for (2.4) because of the very great labour needed to compute B and H. The configuration energies can be found by inserting these values of q in (2.5)-(2-7), and the coeficients of thermal expansion by differ- entiating q with respect to temperature. The experimental values are taken from the tables of Landoft-Bornstein (5th ed.) and from a recent compilation by Timmermans.9 The configuration energies are shown at the normal boiling-point, except for benzene where its variation between the melting point and boiling point is shown.Values of E and u for the simple molecules are derived from thc second virial cocfficient or viscosity.8 Those for the other molecules are derived from the critical constants by using The results of these calculations are shown in fig. 1 and 2. kT, = 1.286, (3.4) ?‘c = 1.46 ( 3 ~ l V ~ 3 ) . (3.5)J . S. ROWLINSON 55 - 0.5 z 0.3 0-6 0.9 I I I I I I I I L FIG. 1 .-Graph of the configuration energy E against the reduced temperature T. Experi- mental values at the boiling point, argon (I), nitrogen (2), oxygen (3), carbon monoxide (4), methanc (9, n-butane (6), benzene (7), carbon tetrachloride (8). Theoretical lines : A, smoothed potential (2-5), B, nearest neighbours, simple harmonic approximation, C, nearest neighbours, exact value (2.6), D, all neighbours, simple harmonic approximation. 6 D 0.3 0-6 0-3 I I I L I I I I L FIG.2. Graph of the reduced coefficient of thermal expansion ( a ~ / k ) against the reduced temperature, where o! is the coefficient of thermal expansion. Experimental values, neon (l), argon (2), nitrogen (3), oxygen (4), chlorine (9, n-hexane (6), benzene (7); theoretical lines as for fig. 1.56 LATTICE THEORIES A T LOW TEMPERATURES Insofar as E and (T have any significance for these molecules,1O these equations probably give the best values. Slight errors in assigning these parameters do not affect the conclusions which may be drawn from these graphs. These conclusions are : (1) Cunfiguration energy.-The experimental values fall into two groups, a lower one of the simple liquids A, N2, 0 2 and CO and a higher one of the more complex liquids n-C4Hio, C6H6 and CC14.The former, to which these theories should apply most closely, fall between the curves which include all neighbours (A and D) and those which neglect all neighbours except the nearest (B and C). This is to be expected, for the configuration energies lie below the upper curves by, approximately, the latent heat of fusion. However, the upper curves would be quite satisfactory for solutions in benzene, etc., presumably because of a cancellation of errors. (2) The change of the configuration energy with temperature.-This is shown only for benzene. Curve C has a slope of the right order of magnitude, but it is clear that both the approximation of the smoothed potential and that of simple harmonic motion give too small a slope, i.e., too small an excess entropy.Curve A, only, gives a vanishing excess entropy at zero temperature. Because of this peculiarity, Prigogine and Mathot find the excess entropy proportional to the excess volume. The other approximations and, more convincingly, eqn. (1.1) show that there is unlikely to be any such simple relation between these excess quantities, (3) The coeficient of thermal expansion.-Here again the smoothed potential model (curve A) and the simple harmonic approximations (curves B and D) are unsatisfactory. Curve C, for the exact nearest neighbour model, appears to give good agreement with the experimental curves. It is probable that the " exact " curve which would correspond to curve D would also be satisfactory. So it may be concluded that if the laborious integrations to give 8 and H, (2.9), were performed, then (2.4) and (2.7) might be an adequate basis for a theory of solutions. The more simple approximations which have so far been used are not satisfactory. I wish to thank the Imperial Chemical Industries Ltd. for the award of a Fellowship. 1 Prigogine and Mathot, J. Chem. Physics, 1952, 20, 49. 2 Rowlinson, Proc. Roy. SOC. A , 1952, 214, 192. 3 Salsburg and Kirkwood, J. Chem. Physics, 1952, 20, 1538. 4 Longuet-Higgins, Proc. Roy. SOC. A , 1951, 205, 247. 5 Rowlinson and Curtiss, J. Chem. Physics, 1951, 19, 1519. 6 Lennard-Jones and Devonshire, Proc. Roy. SOC. A , 1937, 106,463. 7 Prigogine and Garikian, Physica, 1950, 16, 239. 8 Wentorf, Buehler, Hirschfelder and Curtiss, J. Chem. Physics, 1950, 18, 1484. 9 Timmermans, Physico-Chemical Constants of Pure Organic Compounds, (Elsevier, 10 Rowlinson and Townley, Trans. Faraday SOC., 1953, 49, 20. 1950).

ISSN:0366-9033    

DOI:10.1039/DF9531500052

出版商:RSC    

年代:1953

数据来源: RSC

摘要:

THE VAPOUR PRESSURES OF ATHERMAL MIXTURES BY G. S. RUSHBROOKE, H. I. Scorns AND A. J. WAKEFIELD* King’s College, Newcastle-upon-Tyne, Durham University Received 9th February, 1953 On the basis of the quasicrystalline model of athermal liquid mixtures of molecules of two different sizes, expressions are derived for the partial vapour pressures of the two components as power series in the volume concentration 8 of the larger species. The coefficients of successive powers of 8 are shown to be given by direct analogues of the functions ,& occurring in Mayer’s theory of imperfect gases. The simplest approximation to Pk leads to the Flory vapour-pressure equations and the next most simple to those of Miller and Guggenheim. For monomer-dimer solutions sufficient /3’s have been calculated for closer estimates of the true predictions of the model to be obtained.For monomer-trimer and monomer- tctramer solutions the partial vapour pressures are found to depend, though apparently only slightly, on whether the larger molecules are rigid or flexible. 1. INTRoDuCTIoN.-~t is the purpose of this paper to demonstrate how the mathematical framework of Mayer’s theory of imperfect gases 1 may be used to obtain information about the partial vapour pressures of solutions. To some extent the work is a generalization of a paper by Fuchs 2 on the familiar regular solution problem involving systems of equal size with preferential energy inter- actions: but in the present instance we are concerned with athermal mixtures of molecules of two different sizes. The configurations of the assembly are enumerated against a lattice background, as first proposed by Fowler : 3 and we confine attention to mixtures of monomer + r-mer type.An excellent survey of existing work on this problem has been given recently by Guggenheim : 4 and we believe that the present work throws useful new light on the status of current formulae for the partial vapour pressures of such solutions. It is, of course, possible easily to extend the formulism to include interaction energies: but the labour involved in subsequent calculations is then very con- siderably increased. 2. GENERAL THEoRY.-we consider an assembly of N1 monomers and Nr r-mers, occupying N(= N1 + rNr) lattice sites. Restricting attention to athermal mixtures, all complexions of these systems have the same energy.Therefore the restricted grand partition function, for a total of N sites, may be written where XI and Ar are parameters related to the chemical potentials of monomers and r-mers, respectively ; and x = &/A;. Here g ( N ; N,) is the number of ways of arranging Nr r-mers, non-overlapping, on the given N sites. We may rewrite (1) as GN = Ar x N r C 1, (2) Nr W r ) * Present address: Elliott Brothers (London) Ltd., Research Laboratories, Elstree way, Borehamwood, Hertfordshire. 5758 VAPOUR PRESSURES OF ATHERMAL MIXTURES where (Nr) r-mer systems. This is equivalent to denotes a summation over all configurations of N , non-overlapping where now the Nr r-mers are distinguished by labels 1, 2, . . . j , . . . i . . .Nr, S denotes a summation over all conceivable arrangements on the N lattice sites (overlappings being permissible) and (4) 1 hj = 0, if systems i and j do not overlap ; Aj = - 1, if systems i and j do overlap. Now to (3) all the analysis of Mayer's work on the phase integral of an im- perfect gas is at once applicable : the only difference being that integration through- out a volume Y is here replaced by summation over conceivable arrangements on the N lattice sites. Therefore with where Zl denotes that in every term the I systems, 1, 2, . . . I, are mutually connected by f-factors (i.e. form a cluster) and S (1) (2) . . . ( I ) implies that we sum over all conceivable arrangements of these I labelled systems on the lattice. For large N, br is independent of N.It is an immediate consequence of the properties of grand partition functions that, at equilibrium, Writing rNr/N = 0 and 61 = w, (7) becomes Nr = N 1 Zb~x'. - e = I($))(l rw of which the solution, for x in powers of 0, is known to be where (7) For the moment the /?s are to be regarded simply as defined, in terms of the b's, by eqn. (10). Also, from the properties of the grand partition function for a macroscopic condensed phase whence, by (8) and (9), and, from the definition of x,G . S . RUSHBROOKE, H. I . SCOINS AND A . J. WAKEFIELD 59 If now we may treat the vapours as perfect gases, eqn. (12) and (13) may be written in terms of the partial vapour pressures of monomers and r-mers respectively. They become k k> 1 where p1O and Pro are the partial vapour pressures of pure monomer and pure r-mer liquids respectively, and $? is the number of ways of arranging Nr r-mers, non-overlapping, on rN, lattice sites, i.e.#+ is a pure number. Eqn. (14) and (15) form the basis of the present theory. In general, as we have said, the /3’s are defined in terms of the b’s from the eqn. (10). But for rigid r-mers, we can easily prove the analogue of Mayer’s beautiful combinatorial theorem, namely that where 222 implies that thef-factors are such that the k -t- 1 systems, 1, 2, . . . k + 1, form an irreducible cluster, in the sense of imperfect gas theory. This, however, is not true for flexible systems: in performing calculations for flexible r-mers we must first evaluate the b’s and thence deduce, from (lo), the values of the The parameter 8 is, of course, the volume fraction of r-mers in the solution.PS. When 6 is very small, (14) and (15) give which express Raoult’s and Henry’s laws, respectively. Our purpose, however, is to evaluate the later terms in these expansions as far as we are able. The evalu- ation of the P ’ s is a straightforward, though tedious (and, eventually, prohibitively laborious) arithmetical problem. Without going into the details of the calcula- tions, we shall next present the results which we have obtained. We shall then discuss such conclusions as may be drawn from them. 3. THE COEFFICIENTS &--To save space, we shall tabulate simply the co- efficients /?k; the br’s, which do not enter into eqn. (14) and (15), can be worked out from formula (10) if required.(Unlike the P’s, which are all negative, the b’s alternate in sign.) All the results of table 1 pertain to a simple cubic lattice (co-ordination number 6). Those of table 2 all refer to dimers (r = 2). It should be noted that in every case k& i s tabulated, rather than Pk, since this is usually an integer. TABLE RESULTS OBTAINED FOR A SIMPLE CUBIC LAT~ICE r-mer species u k = 1 2 3 4 5 k P k z - 1 1 -71 -419 -2511 -15031 kyk== 1 1 13 81 52 1 kyk 4 8 rigid linear r-mers dimers (r = 2) rigid linear trimers (r=3) 3 k,Bk= --23 -23s flexible trimers rigid linear tetramers - 23487 25 flexible n-tetramers 75 kPk= - -60 VAPOUR PRESSURES OF ATHERMAL MIXTURES For rigid species the tables show, besides the quantities &, other numbers yk, the significance of which will be discussed in the next section.TABLE 2.-RESuLTS OBTAINED FOR DIMERS (Y = 2). THE SECOND COLUMN GIVES THE CO-ORDINATION NUMBER 2 OF THE LATTICE AND THE VALUE OF bl (= w ) lattice Z,W k = 1 2 3 4 5 6 simple cubic 6 kpk= -11 -71 -419 -2511 -15031 - 3 kyk== 1 1 13 81 521 - diamond 4 kpk" -7 -31 -127 -511 -2017 -7939 2 kyk= 1 1 I 1 31 253 plane square 4 kpk= -7 -31 -121 -471 -1867 -7435 2 kyk= 1 1 7 41 181 757 body-centred cubic 8 kPk= -15 -127 -987 -7951 --63885 - 4 kYk= 1 1 37 241 1651 - With the exception of the values of P k (and yk) for k = 6 for the plane square lattice, k = 5 for the simple cubic lattice, and all the values pertaining to the body- centred cubic lattice, the results of tables 1 and 2 have been obtained by two independent methods (and workers).Most of the coefficients were, in effect, first calculated by one of us (A. J. W.) using an entirely different, and heuristic, approach which starts by evaluating the right-hand side of (1) explicitly for small values of Nr and then formally taking the Nth root by the binomial theorem (cp. Wakefield 5 ) . The approach through irreducible cluster sums, on which we have based this paper is, of course, mathematically more satisfying. In the direct evaluation of the Ps, when k = 5 or 6, we have been much helped by the classification of irreducible cluster diagrams given in Riddell's thesis.6 * We are indebted to Dr. D. H. Trevena for help in checking some of these calculations. 4. THE FLORY .APPRoxmiAnoN.-Inspection of the early entries in tables 1 and 2 suggests the approximate formula, for rigid linear r-mers, /?k =2 - r(rw)k/k (16) Deferring until 5 6 the mathematical interpretation of eqn. (16), we shall next investigate its consequences.Substituting from (16) into (14) and (15), we find, after some simple algebra, r - 1 -- '1 - (1 - e) exp (TB). P1" and Putting 8 = 1 in (18) gives whence (20) - _ Pr -- e exp [ - (r - i)(i - 811. PP Eqn. (17) and (20) are the well-known Flory formulae for the partial vapour pressures of monomer-r-mer solutions (see Guggenheim,4 12.02. 10). *It should perhaps be mentioned that some of Riddell's diagrams have the wrong weighting factors attached to them.G. S. RUSHBROOKE, H. I . SCOINS AND A. J. WAKEFIELD 61 5. THE MILLER-GUGGENHEIM APPROXIMATION.-Eqn. (1 6), Of Course, provides only quite a crude approximation to the quantities &.If we write then the y's have the value given in tables 1 and 2. They are always small in com- parison with the corresponding Ps, but for the higher /3's can amount to some 10 % of the whole value. For rigid linear r-mers (the only species with which, at the moment, we are concerned), and for any lattice on which these can be placed on consecutive ad- jacent lattice points, we find in particular (see tables 1 and 2) that yl = (r - 1)2 and y2 = i ( r - l)3, suggesting the general formula Again deferring till the next section the mathematical interpretation of this ap- proximation, we may proceed to investigate its consequences. Substituting from (22) into (14) and (15), straightforward algebra now leads to the formulae and Putting 9 = 1 in (24) whence 1 1 rw rm-r+l 9 Eqn.(23) and (26) are the familiar Miller-Guggenheim formulae for the partial pressures of athermal monomer-r-mer solutions (see Guggenheim,4 10 .09 . 4 and 10.09 . 5 ) . 6. INTERPRETATION OF APPROXIMATE FORMULAE FOR /3k.-Eqn. (16) and (22) above provide two successive approximations to the coefficients /3k. So far they have been obtained only by inspection of tables 1 and 2, and we have seen that they lead to the Flory and Miller-Guggenheim formulae respectively. We must now give these equations a more mathematical interpretation. In evaluating P k numerically we have to deal with two different geometrical concepts. Firstly, the set of irreducibIe cluster diagrams connecting k + 1 points and, secondly, configurations of k + 1 mutually overlapping r-mers on a lattice (we are now concerned only with rigid linear r-mers).Each irreducible cluster diagram has, of course, an appropriate weight factor. And any particular configuration of mutually overlapping r-mers will contribute to S,t only if there are sufficient mutual overlappings: in general we shall get a contribution from some of the irreducible cluster diagrams but not from others. In the special cases, however, in which all the r-mers have at least one point in common, every irreducible cluster diagram contributes to p k and it is not difficult to prove that, for a particular geometrical arrangement of the r-mers on the lattice, the contribu- tion to pk (from thef-factors and the weights of the cluster diagrams) is simply62 VAPOUR PRESSURES OF ATHERMAL MIXTURES - l / k .To obtain Pk itself, this has to be multiplied by the number of ways of arranging k + 1 r-mers on the lattice, when the position of one of them is per- manently fixed, and all the r-mers have at least one point in common. At first sight this number is r ( r o ) k : for we may choose any of the r sites of the fixed r-mer as the common site, and then put each of the other r-mers down on the lattice, through this site, in rw ways. We thus obtain eqn. (18). But further consideration reveals that we have then miscounted the geometrical arrangements in which the r-mers have more than one site in common. And the accurate formula is not r(uo)k but r(rw)k - (r - 1)k-1-1; which leads to eqn.(22). We have thus proved that the Flory and Miller-Guggenheim equations rest on the assumption that the only contributions to the irreducible cluster sums, pk, come from geometrical configurations of overlapping r-mers having at least one point in common. The Miller-Guggenheim approximation evaluates this contribution correctly, while the Flory approximation miscounts, i.e. overweights, the configurations in which the overlapping r-mers have two or more points in common. In general, however, there are other arrangements of k + 1 mutually over- lapping r-mers which contribute to the coefficients Pk, i.e. arrangements in which the r-mers do not all have a point in common. If we denote the contribution from these arrangements by 8k, then, accurately Table 3 gives the numerical values of these corrections 8 k for dimers on various lattices.We find that in every case a closed circuit of mutually overlapping dimers is involved : which is why, for example, 83 and 84 vanish for the diamond lattice (on which a closed polygon has at least six sides) but not for the plane square lattice, on which a closed circuit of four mutually overlapping dimers can be drawn. TABLE 3.-DATA FOR DIMERS ON VARIOUS LATTICES lattice k= 1 2 3 4 6k= 0 0 4 20 diamond 8k= 0 0 0 0 simple cubic plane square 6k= 0 0 2 10 body centred sk= 0 0 12 60 It seems also worthy of notice that 8k itself, rather than k&, 7. THE VAPOUR PRESSURE cuRvEs.-It remains to illustrate integer. 5 6 104 - 6 42 36 126 330 - is invariably an graphically the effect of these further contributions 8 k on the vapour pressure curves for dimer + monomer mixtures.At first sight one is tempted to employ the formulae and where z, = 2w, is the co-ordination number of the lattice concerned : equations which result from extracting the Miller-Guggenheim contribution to all the coefficients &. But although eqn. (28) is entirely satisfactory for computational purposes, this is not true of eqn. (29) when we know only a finite number of the 6’s. This is because, for a finite number of 8’s, eqn. (29) does not lead toG . S. RUSHBROOKE, H . I . SCOINS A N D A . J . WAKEFIELD 63 Raoult's law in the neighbourhood of 8 = 1. The difficulty is avoided, and Raoult's law automatically satisfied near 8 = 1, if in place of (28) and (29) we use the equations (30) In ( P ~ / P ~ O ) = 48 + In (1 - 8) + @Q(@ - 4 Q(@dS, s: and k > l k Numerically, we find that eqn.(28) and (30) are entirely equivalent. But eqn. (31), unlike (29), ensures that Raoult's law is satisfied near 8 = 1 even when we FIG. l.--A(~q/p1O) against mole fraction, x. (1) on the Flory approximation ; (2) from Miller-Guggenheim equation ; (3) from the present work. cut off the series involved in Q(@ after a finite number of terms : it thus makes the fullest possible use of our statistical information. Fig. 1 and 2 give the results of calculations based on (30) and (31) for a simple cubic lattice (z = 6). 4 2 is determined by the requirement that when 8 = 1, p2 =pz0. For this quantity, sometimes called the molecular mobility, we estimate the value 2.44 : the Miller-Guggenheim equations give 2.41.The figures show, respectively, the deviations A(p1/plo) and A(p2/p2O), of the quantities p1/'1" and p2/p2' from the Raoult's law values, 1 - x and x, plotted against the mole fraction of dimers x = 8/(2 - 8). In each case we give the deviations predicted by the Flory and Miller-Guggenheim formulae as well as those found by the64 VAPOUR PRESSURES OF ATHERMAL MIXTURES present treatment. Assuming, as we believe, that the results of the present calculations are substantially correct (i.e. do justice to the predictions of the under- lying model) we conclude that the Flory formulae exaggerate the deviations from Raoult's law by about 50 %, while the Miller-Guggenheim equations exaggerate the true deviations by 5 or 10 %.8. FURTHER CONCLUSIONS.-T~US far, the present work may reasonably be held to substantiate, rather than throw serious doubt upon, the Miller-Guggenheim formulae. It remains, however, to draw attention to three cases in which we may expect these equations to be more seriously in error. Firstly, for close-packed lattices, even three dimers can form a closed ring of mutually overlapping systems. This means that the Miller-Guggenheim 1 0.2 0.4 0.6 0.8 X FIG. 2.-A(p~/p2') against mole fraction, x. (1) on the Flory approximation ; (2) from Miller-Guggenheim equation ; (3) from the present work. partial vapour pressure curves will deviate more quickly from the true ones even though the overall deviations may still not be large. Secondly, for non-athermal mixtures it is clear from preliminary calculations made by one of us (A.J. W.) that, for much the same reason as for close-packed lattices, the inclusion of interaction energies between unlike systems leads to partial vapour pressure expressions which differ from the appropriate Miller- Guggenheim formulae (expanded in powers of 6' or x ) at lower powers of the concentration than in the case of athermal mixtures. Thirdly, the Miller-Guggenheim equations lead to the same expressions for the partial vapour pressures of monomer-r-mer solutions whether the r-mers are rigid or flexible. And it is already clear from our present calculations that, on the basis of the present lattice model, this is not strictly true. From the data of table 1 above, the following expressions for the partial vapour pressures of monomer-trimer mixtures are easily deduced :G .S . RUSHBROOKE, H . I . SCOINS AND A . J . WAKEFIELD RIGID LINEAR TRIMERS : 65 8 2 8 3 $ = 1 - 3 (!) - 30 (4) - 136 (G) - . . . \ P1 p3 -- + 14 (gy + 112 (g)3 + . . . e +3P3O - 9 (33) FLEXIBLE TRIMERS : - - P I - 1 - 3 (i) - 29-76 (:)2 - 151.1328 (g)3 - P1 O + 117.6992 In each case we are concerned with a simple cubic lattice, and 8, as always, denotes the volume fraction of r-mers. The corresponding equations for tetramers are : RIGID LINEAR TETRAMERS : - _ - 44P4O 12 (35) FLEXIBLE TETRAMERS : (The cocfficients are given in full because we are dealing with exact numbers.) We do not know sufficient terms in these expansions to draw quantitative conclusions, but it is already clear that it makes a difference whether or not the r-mers are rigid, and it seems probable that this difference will increase in impor- tance with the length of r-mer concerned. Moreover, these differences are not due solely to the possibility of flexible r-mers “ crossing ” themselves : indeed, for flexible trimers no such possibility can arise. They are due to possible contributions to the P’s (which may be thought of as excluded volumes, corrections to excluded volumes, and so on) from rings of overlapping systems : and more such rings are geometricaIIy possible when the r-mers are flexible than when they are not. We expect the differences to be rather more pronounced for close- packed than for open lattices. Further work is in progress on these problems. 1 Mayer and Mayer, Statistical Mechanics (John Wiley and Sons, New York, 1940). 2 Fuchs, Proc. Roy. SOC. A, 1942, 179, 340. 3 FowIer and Rushbrooke, Trans. Faraday SOC., 1937,33, 1272, 4 Guggenheim, Mixtures (Clarendon Press, Oxford, 1952). 5 Wakefield, Proc. Camb. Phil. Soc., 1951, 47, 419. 6 Riddell, Thesis (University of Michigan, 1951). C

ISSN:0366-9033    

DOI:10.1039/DF9531500057

出版商:RSC    

年代:1953

数据来源: RSC

摘要:

ATHERMAL MIXTURES BY E. A. GUGGENHEIM Chemistry Dept., Reading University The close formal analogy proved by McMillan and Mayer between the power series for the osmotic pressure of a solution and for the pressure of a real gas masks the fact that in the former the effective interaction energy, which has to be used for the correct evaluation of the irreducible cluster integrals Pk, includes the work required to push around the solvent molecules and fit them into the spaces between the solute molecules. The solvent may not be treated as a continuum unless the smallest dimension of the solute molecules is great compared with the size of the solvent molecules. When this is not the case, the only treatment which has yet proved successful is the replacement of the cluster integrals by cluster sums, in other words the use of the quasi-crystalline model. In the case of athermal mixtures, this becomes the treatment described by Rushbrooke, Scoins and Wakefield, to which this paper may be regarded as a supplement. In particular the laws of ideal solutions are obtained when solute and solvent molecules are of the same size.1. INTRoDucnoN.-Long ago van t'Hoff 1 showed the close analogy between the expression for the osmotic pressure of an extremely dilute solution and that for the pressure of a perfect gas. Much more recently McMillan and Mayer 2 have proved that there is an equally close formal analogy between the expression for the osmotic pressure of any solution and that for the pressure of a real gas. This brilliant paper is difficult reading for anyone who is not a highly trained mathematician.A short sketch of an elementary deduction of the above- mentioned important result will therefore perhaps not be out of place. Consider a mixture of a given total volume Vat a given temperature Tcon- taining a given number Nr of molecules of the solute species Y. Let the number of molecules of the solvent species 1 be denoted by N1. For the sake of brevity all internal degrees of freedom of both solvent and solute molecules will be ignored ; in any case they cancel out from the final formulae except as discussed in 0 2. Now construct the semi-grand partition function B for the given values of T, V, Nr and given value of the absolute activity A1 of the solvent where W denotes the configurational energy and the multiple integration extends over all conceivable configurations within the total volume V.As usual the sum may without loss of accuracy be replaced by its maximum term specified by (2) It follows from the general properties of grand partition functions that the pressure P is given by Ni = N r . Thus 2 = AlNl'J . . . J CXP {- W(NF, Nr, V)IkT}(dT)NI (dr)Nr. (3) This is the pressure of a mixture with given values of T, V, Nr and A,. If, now, the value assigned to A1 is that of the pure solvent at a low (effectively zero) 66E. A . GUGGENHEIM 67 pressure and of course at the same temperature T, then P is by definition the osmotic pressure of the solution. Now compare (3) with the formula for the pressure of a single gas of N molecules in a volume Vat a temperature T 3 3V P = kT- In 1 .. . 1 exp { - W(N, (4) The analogy between (3) and (4) is striking. Mayer 3 and his co-workers have obtained exact formulae for the virial coefficients of a gas in terms of the interaction energies. Their general formula is where /& is the k’th irreducible cluster integral. For full details and definitions the reader is referred to the admirable exposition in the book by Mayer and Mayer.3 From the analogy between (3) and (4) it follows immediately that the osmotic pressure II of the solution may be expressed in the form > t3 Q: W z -?S EPARAT 10 N i > t3 CL W Z W T ’ +SEPARATION I FIG. 1. FIG. 2. In spite of the complete formal similarity between (5) and (6), both of which are exact, there is an important concealed difference between them.The irreducible cluster integral & in the gas is completely defined in terms of the configurational energy of a set of k + 1 molecules. In the solution on the other hand if, in order to attain as complete an analogy as possible with the gas, Pk is defined in terms of the configurational energy of a set of k -1 1 solute molecules, then because Win (3) depends on Nf this so-called configurational energy must include the work required to push around the solvent molecules and fit them into the spaces between the solute molecules. This important point is clear in the original exposition by McMillan and Mayer,Z but has unfortunately not always been ade- quately appreciated and has led to false interpretations of the formulae of McMillan and Mayer.These misinterpretations are most clearly revealed by a considcr- ation of athcrmal mixtures. Moreover it is evidently iinpossible to obtain correct formulae for mixtures not necessarily athermal without obtaining correct formulae for athermal mixtures. I shall accordingly confine myself to athermal mixtures and for the sake of brevity to binary mixtures. Formula (6) has been applied to a solution of large spherical molecules by Zimm4 and by Huggins.5 lncidcntally in Huggins’ formula (47) the coefficients of ~ 3 4 and v35 are admittedly wrong.6 In the former - 40 should bc replaced by + 18.36. Both these authors assume an interaction energy between a pair of solute molecules of the form shown in fig. 1, whereas the effective interaction energy, which has to be used for the correct evaluation of the irreducible cluster integrals &, is more like that shown in fig.2. This diagram is only schematic and The simplest possible model of a solute molecule is a rigid sphere.68 ATHERMAL MIXTURES consequently over-simplified because the packing of molecules is three-dimensional, not one-dimcnsional as suggested by the diagram. No importance is attached to the precise spacing of the minima, shown as equal in the diagram. The essential point is that the effective intcraction energy has pronounced minima when the separation between two solute molecules allows an integral number of solvent molecules to fill the gap and is effectively infinite when this is not so. The sub- stitution of fig. 1 for fig. 2 means treating the solvent as a continuum, which can be justifiablc only if the solute molecules are very large compared with the solvent molecules. Both Zimm and Huggins go on to consider the coefficient 4/31 for long rod-like moleculcs, again treating the solvent as a continuum.This can be justifiable only if the width, and afortiori the length, of the rod is large compared with the solvcnt molecules. A powerful and elegant method of evaluating the coefficient for rigid molecules of any shape has been described by Isihara and Hayashida.7 Again the solvent is trcated as a continuum. Consequently the formulae must not be used unless the smallest dimension of the solute molecule is large compared with the dimensions of the solvent molecules. Flory and his co-workers 8 have given formulae for the osmotic pressures of solutions not necessarily athermal.In the particular case of athermal solutions these reduce to formula which I consider misleading, if not wrong. In the first place they treat the solvent as a continuum and then apply their formulae to solu- tions of chain polymers whose cross-sections are certainly not large compared with the solute. In the second place they assign to the third virial coefficient 2/32/3 for long chain molecules the same value as for large spherical molecules. This is undoubtedly wrong. Unlcss the smallest dimension of the solutc molecule is great compared with the dimensions of the solvent molecules, it is not permissible to substitute fig. 1 for fig. 2. The solvent may not be treated as a continuum and the only known model which retains the essential features of fig. 2 and has led to tractable formulae is the quasi-lattice model.The integrals in (I), (2) and (3) become sums and the irreduciblc cluster integrals /lk likewise become irreducible cluster sums. Thus, except for solute molecules large in all directions, the formulae of McMillan and Mayer do not provide an alternative to the quasi-lattice model, but their correct application is through the use of this model. We are thus logically driven to the approach so admirably expounded in the contribution of Rushbrooke, Scoins and Wakefield9 to this Discussion. I have independently used this approach and obtained some of, but by no means all, their results. In particular I have obtained many of the @k and yk in their tables 1 and 2 for the lower values of k, but I have not studied the higher values of k.I have reached a few conclusions which are not mentioned by Rushbrooke, Scoins and Wakefield, and I think some of these are worth recording. The remainder of this communication may be regarded as an addendum to that of Rushbrooke, Scoins and Wakefield. 2. ROTATIONAL FREEDoM.-First I must mention a small matter of notation. Since this is a question of convention, it is of no importance once it is understood, but if not properly understood it might easily lead to confusion. In dealing with gases it is usual to assume that the rotational and internal degrees of freedom of the molecules are unaffected by intermolecular interaction ; or alternatively that the intermolecular interaction is suitably averaged over all orientations of the interacting molecules. Contributions from such degrees of freedom are then omitted from the formulae.In dealing with liquid mixtures it may well be better to include in the formulae the contributions from rotational degrees of freedom, including internal rotations. For a monomer there are no such degrees of freedom. For an r-mer there is an extra orientational factor in its partition function which I have previously 10 denoted by p/u where u is the symmetry number of the r-mer. In particular for rigid straight r-mers with their two ends in- distinguishable p/a = 2/2, where z is the co-ordination number. This orienta-E . A . GUGGENHEIM 69 tional factor is denoted by Rushbrooke, Scoins and Wakefield by w.Incidentally in their table 1 the value w = 75 for flexible tetramers implies that the four seg- ments are interconnected like the carbon atoms in n-butane. If they were inter- connected like the carbon atoms in isobutane, we should have p = Z(Z - 1) (Z - 2) = 6 . 5 . 4 and (T = 6, so that w = p/u = 20. In constructing the irreducible cluster sums /3k we could sum over all relative positions of the centres of the several molecules and average over all orientations of each molecule; we should then obtain formula (6) for the osmotic pressure. Rushbrooke, Scoins and Wakefield have preferred to sum over all orientations of each molecule (except the first) as well as summing over all relative positions of the centres of the several molecules. Thus each /3k contains an extra factor o k = (p/a)k and so in formula (6) each /3k has to be replaced by &/wk. At the same time on the lattice model Nr/V becomes e/r.Thus formula (6) becomes When we use the thermodynamic relation between osmotic pressure and partial vapour pressure of solvent we recover formula (14) of Rushbrooke, Scoins and Wakefield. 3. THE QUANTITY a.-In previous work 11 in this field it has proved useful to introduce a quantity a defined as the ratio of the probability that a group of sites, congruent with the r-mer, be wholly occupied by a single r-mer to the proba- bility that the group be entirely occupied by monomers (solvent moiecules). It is then proved quite generally that AF the free energy of mixing of N1 molecules of solvent monomer and Nr molecules of solute r-mer is rclated to a by where Ns = N1 + rNr is the total number of lattice sites. immediatelv From (8) we derive l n g = '6 In a dB - P1 r 0 (9) whcre pl denotes the partial vapour pressure (strictly the fugacity) of the solvent in the solution and plo that of the pure solvent.In the limit of high dilution it is evident from the physical definition of a that a/O 4 l/rw as 8 --t 0. (10) Let us now expand thc difference betwecn In a and this limiting valuc as a power series in O/r. We accordingly write whcre the pk's are defined by this cxpansion, but will be proved identical with the quantities hitherto denoted by the same symbols. When we substitutc (1 I ) into (9) and perform the integrations, we obtain Comparison of (12) with formula (14) of Rushbrooke, Scoins and Wakefield shows that the &'s defined by the expansion (1 1) are identical with the &'s defined as irreducible cluster sums.This new alternative dcfinition of the /3k's is interesting and throws light on the Factor k/(k -I- 1) occurring in (12) and in (7).70 ATHERMAL MIXTURES The auxiliary function Q(6) introduced by Rushbrooke, Scoins and Wakefield can also be related to a. In fact if we write it can be vcrifieci that Q(6) thus defined is the same quantity as used by Rushbrooke, Scoins and Wakefield. From (13) we see that Flory's approximation consists in replacing Q(e) by zero. 4. DIMERS ON CLOSE-PACKED LAmIcE.-Rushbrooke, Scoins and Wakefield define and discuss a quantity & equal to the difference between the true value of pk and that givcn by an approximate formula which they call the Miller- Guggenheim approximation.The detailed history of this formula is recorded elsewhere.12 In the interest of historical accuracy it should be called the Huggins-Miller formula. My contribution, apart from simplifying the deriva- tion through the use of a, was to extend it to mixtures of more than two kinds of molecules ; this extension is irrelevant to the present context. Rushbrooke, Scoins and Wakefield in their table 3 give values of & for dimers on various lattices. In no case quoted is 8k other than zero for k less than 3. For dimers on a face- centred cubic lattice 1 find by direct counting 2p2 =- 295 as compared with the value - 287 obtained by substituting z = 12 into the formula 2/32 = 2.22 - 1 for an open lattice.This gives 8k = - 4. This example is particularly inter- esting because for this lattice McGlashan 13 has derived a value of a to a better approximation than thc Huggins-Miller. McGlashan's formula (8.6) is ) 1 1 1 1 - -0 -1 -02 4- -63 -t . . . . 12(1 " - el2 12 36 144 a - Comparing (14) with (13) we have 1 1 1 Q(6) = - In 1 - -0 -k -62 + - 0 3 4- . . , ( 12 36 144 with z = 12. The Huggins-Miller approximation is Q<O> = - In (1 - :) = - + - ;(;)2 - + - ;(;)3+. . . . Comparison of (15) and (16) gives & = - & . 7 - 4 = - 4 in agreement with thc value obtained by direct counting. In the light of the analysis of Rushbrooke, Scoins and Wakefield it appears that McGlashan's approximation when compared with the Huggins-Miller approximation is accurate to one more term, but no more, in the expansion in powers of 81.2.5. RIGID BENT MOLECULES.-whereas Rushbrooke, Scoins and Wakefield pay considerable attention to the difference in behaviour between rigid straight mole- cules and flexible molecules, they do not mention rigid bent molecules. It is, however, clear that their method of direct counting is applicable to rigid bent molccules and that on the whole the study of rigid bent molecules will be simpler than that of flexible molecules at least for small values of r. It is also clear that Bl/w for a flexible r-mer has a value intermediate between those for a rigid straight r-mcr and rigid bent r-mers. We notice that - ~ I / O . J is smaller for a flexible or rigid bent r-mer than for the rigid straight r-mer.All previous discussions of flexible molecules have been so completely focused on energetic effects thatE. A. GUGGENHEIM 71 no light has been thrown by any of them on the simpler geometrical effect which alone is present in athermal mixtures. The reason why - &/OJ is smaller for a flexible or bent r-mer than for a rigid straight r-mer is simple. In any chosen forbidden configuration of a pair of r-mers let us use the word “ clash ” to denote the occupation of the same site by one segment of each of the two r-mers. Now to a high degree of accuracy, all distinguishable relative configurations of two r-mers, the total number of clashes is the same whether the r-mers are straight or bent. But if they are bent some of the configurations may contain more than one clash and consequently the number of configurations containing at least one clash is less than for straight r-mers.But - Br/w is proportional to the number of these forbidden configurations and is therefore smaller for bent r-mers than for straight ones. 6. IDEAL MIXTuRES.-If in an athermal mixture the solute and solvent molecules are of the same size and shape we know that the mixture becomes ideal. If C denotes the concentration of solute in moles per unit volume the formula for the osmotic pressure may be written n = ~ T c { - In (1 - 8) }, where 8 denotes either volume fraction or mole fraction of solute since they are now the same thing. Expanding the logarithm we have Comparing (18) with (7) when Y = 1, w = 1 we see that they will agree if -Pk=k-l.Zimm 14 has stated that (17) must be derivable from (7) but “ to do so would take us too far afield”. It was not altogether clear whether this statement implied that Zimm had himself succeeded in deriving (17) directly from (7). However Zimm has subsequently stated : 15 “ The most direct connection between the two equations would be made by expanding the logarithms in power series and identifying coefficients. However, this process becomes exceedingly tedious and I do not believe anyone has ever carried it through.” Actually the process is far from tedious, but rather amusing. According to its definition 3 /3k includes a factor Ilk! and for our present purpose it is more convenient to consider the quantity k!Pk which I shall denote by /3t.Consequently we have to show that - p$ = (k - l)!, in order that the value of - pk shall be k-1. In view of formula (4) of Rushbrooke, Scoins and Wakefield, &! has a simple physical sig- nificance for monomers on a lattice. Consider the irreducible clusters formed out of k labelled molecules. Some of these clusters will contain an even number of links, some an odd number of links and each link contributes a factor - 1 to the particular term in the irreducible cluster sum. It follows that -13f-r is the excess of the number of irreducible clusters that can be formed with an odd number of links over the number that can be formed with an even number of links, the k molecules being supposed distinguishable. I am indebted to my colleague Prof. E. H. Neville for an elegant proof that this number is in fact (k - l)! I am indebted to Dr.Zimm 15 for pointing out to me that the formulae for an ideal mixture can also be obtained from the relations of Kirkwood and Buff,16 which can be proved to be mathematically equivalent to those of McMillan and Mayer. As Zimm points out, we merely require that both components have identical distribution functions. The argument is thus essentially based on symmetry. This is, however, quite different from the approach through the osmotic pressure formula of McMillan and Mayer which emphatically treats the solvent in an entirely different way from the solute. It is interesting and satisfying that this intentionally lop-sided approach can in the end lead to the symmetrical formulae of an ideal mixture. His proof is given verbatim in the appendix.72 ATHERMAL MIXTURES APPENDIX THE DISPARITY OF IRREDUCIBLE CLUSTERS BY E.H. NEVILLE A cluster of k + 1 points can be made out of a set of k points with connecting links by joining this set yk by sufficiently numerous links to a new point M. If yk is not itself a single cluster the new set cannot be irreducible, since it falls apart when M is removed. If yk is reducible there is a point Q of yk such that if Q is removed, yk falls into two or more pieces. To form an irreducible cluster, M must have a link to each of these pieces, whether or not M is joined to Q. That is, to any irreducible cluster of order k + 1 formed without the link MQ there corresponds an irreducible cluster formed by adding the one link MQ.Hence, whether yk has even or odd linkage, it gives rise to equal numbers of even and odd linkages in irreducible clusters of order k + 1. Hence the disparity at order k + 1 for irreducible clusters depends only on the disparity for irreducible clusters at order k, although reducible clusters of order k are used in the formation of irreducible clusters of order k + 1. If now y k is irreducible, the new set is irreducible if and only if it is joined to M by two or more links. That is, the number of new irreducible sets that can be formed from a particular irreducible set yk is i.e. 2k - 1 - k ; of these 2k-1 - 1 have linkage of the same parity as Yk, and 2k-1 - k have linkage of the opposite parity. It follows that the disparity at order k + 1 is derived from that at order k by multiplication by the factor k - 1, and since for 3 points there is one and only one irreducible cluster and this has an odd number of links, the disparity with k + 1 points is an excess of odd over even by (k - l)! 1 Van t’Hoff, Z. physik. Chem., 1887, 1, 481, especially p. 491. 2 McMillan and Mayer, J. Chem. Physics, 1945, 13, 276 ; see especially pp. 294, 295. 3 Mayer and Mayer, Statistical Mechanics (John Wiley, New York, 1940), chap. 13. 4 Zimm, J. Chem. Physics, 1946, 14, 164. 5 Huggins, J. Physic. Chem., 1948, 52, 248. 6 Huggins, private communication, 1953. 7 Isihara and Hayashida, J. Physic. SOC. Japan, 1951, 6, 46. 8 Flory, J. Chem. Physics, 1949, 17, 1347. Flory and Krigbaum, J. Chem. Physics, 1950,18, 1086. Fox, Flory and Bueche, J. Amer. Chem. SOC., 1951,73,285. Flory and Krigbaum, Ann. Rev. Physic. Chem., 1951, 2, 383, especially pp. 395-397. 9 Rushbrooke, Scoins and Wakefield, this Discussion. 10 Guggenheim, Proc. Roy. SOC. A, 1944,183,203. Guggenheim, Mixtures (Clarendon 11 Guggenheim and McGlashan, Proc. Roy. SOC. A, 1950, 203,435, especially p. 446. 12 Guggenheim, Mixtures (Clarendon Press, Oxford, 1952), p. 184. 13 McGlashan, Trans. Farau’ay SOC., 1951, 47, 1. 44 Zimm, J. Chem. Physics, 1946, 14, 167. 15 Zimm, private communication, 1952. 16 Kirkwood and Buff, J . Chem. Physics, 1951, 19, 774. Press, Oxford, 1952), pp. 185 and 196. Guggenheim, Mixtures (Clarendon Press, Oxford, 1952), pp, 185-1 87.

ISSN:0366-9033    

DOI:10.1039/DF9531500066

出版商:RSC    

年代:1953

数据来源: RSC