CURRENT THEORIES OF SOLUTIONS OF By N. G. PARSONAGE and L. A. K. STAVELEY (THE INORGANIC CHEMISTRY LABORATORY OXFORD UNIVERSITY) THE purpose of this Review is to consider some of the recent theoretical and experimental studies of the connection between the thermodynamic properties of a mixture of two liquids and the intermolecular forces existing in it. For many years the usual point of departure in discussing the thermodynamic properties of a liquid mixture has been Raoult’s law. If the vapours are assumed for simplicity to behave as perfect gases then the mixture obeys Raoult’s law if the partial vapour pressure of each component is proportional to its mole fraction x. Thus for component 1 NON-ELECTROLYTES p = p,ox . . . . . . . . . (1) where p 1 is its partial vapour pressure and pl0 the vapour pressure of the pure component 1.Such a solution is described as ideal or perfect. Alter- natively conformity with Raoult’s law requires that the chemical potential p of any component in the liquid should have the dependence on its mole fraction x which is given for component 1 by the equation . . . . . . . p = plo + RTln x1 (2) where pl0 is the chemical potential of the pure liquid component 1 at the temperature Tand at a prescribed pressure. If it is remembered that the chemical potential p,’ of the same component in the vapour phase (in which it is assumed to behave as a perfect gas) is given by . . . . . . . p,’ = pl’O + RTlnp (3) and that for equilibrium . . . . . . . . . p = p1‘ (4) then the connection between equations (1) and (2) is apparent. It follows from equation (2) that if x1 moles of a liquid 1 and x ( = 1 - x1 ) moles of a liquid 2 are mixed at a constant temperature Tand at a constant pressure to form one mole of mixture the increase AGi in the Gibbs free energy G accompanying the formation of this ideal mixture is dGi = x,(plo + RT In x,) + x2(p20 + RTln x2) - xlplo - X ~ U ~ By applying (5) to the general thermodynamic relation .. . . . . . . . . . . = RT(x In x1 + x In x,) ( 5 ) 306 PARSONAGE AND STAVELEY SOLUTIONS OF NON-ELECTROLYTES 307 (which is one form of the Gibbs-Helmholtz equation) it follows that for an ideal solution the heat content. change AH accompanying the mixing process must be zero. Likewise from the relation . . . . . . . . A Y = [ a ( A G ) / a p ] ~ (7) the volume change d Y for the mixing process must also be zero.Finally since AH = 0 from the equation dG = d H - TAS and from equation (9 the entropy of mixing ASi for an ideal solution is given by ASi = -R(x,lnx + x,lnx,) . . . . . (8) In reality it is almost unknown for two liquids to mix without showing detectable departures from the criteria given above for an ideal solution. (Even mixtures of isotopes may not necessarily form an ideal so1ution.l) Systems which were once cited as examples of ideal solutions such as benzene-ethylene dichloride have been found on a more careful experi- mental examination to show measurable departures from perfect be- haviour. In other words if two liquids are mixed at constant pressure and temperature there is an evolution or absorption of heat and a volume change and the free energy and the entropy of mixing are not correctly expressed by equations ( 5 ) and (8).It has become customary to represent quantitatively the departure of a system from ideal behaviour by means of the so-called excess functions first introduced by Scatchard,2 and now usually denoted by a superscript E attached to the appropriate thermo- dynamic symbol. Thus if dG is the actual increase in Gibbs free energy on formation of one mole of mixture the excess Gibbs free energy C E is given by . . . . . . . . G E = A G - AGi (9) The heat absorbed on formation of one mole of mixture is HE and YE is the corresponding increase in volume where . . . . YE= v - x l v l o - x x Y ~ o (10) Y10 and V20 being respectively the molar volumes of the pFre components 1 and 2 at the prescribed constant temperature and pressure and V the actual volume of one mole of mixture.Alternatively a non-ideal solution may be defined as one for which the chemical potentials of the components are no longer given exactly by equations such as (2) but by such equations modified by the introduction of the activity coefficient y. Thus for component 1 in a non-ideal mixture p1 = plo + R T l n x y . . . . . . (11) It then follows from equations (9 (9) and (1 1) that Prigoghe “The Molecular Theory of Solutions” North HoIland Publ. Co. Amsterdam 1957. * Scatchard Chem. Rev. 1931,28,321. 308 QUARTERLY REVIEWS G E = RT(x,ln y1 + x21ny2) . . . (12) The experimental determination of G E is therefore in effect a matter of determining the activity coefficients of the two components in the mixture and different methods can be employed to do this.One commonly used is to find the composition of vapour in equilibrium with mixtures of known composition i.e. to establish both the dew-point and boiling-point curves of the liquid-vapour equilibrium diagram. However for the binary mixtures of condensed gases with which we shall be largely concerned it is more convenient to measure only the total vapour pressure as a function of the composition of the liquid mixture. Provided that information is available to allow for the imperfection of the vapour phase it is then possible to evaluate G E if some suitable type of relation between the activity coefficients and the liquid composition is assumed. Here use is made of expressions originally due to Margules. In pure liquid 1 y1 is unity and In y = 0.As the component 2 is added and x 2 increases from zero In y1 departs from 0. It is reasonable therefore to express the dependence of In y1 on x2 in the form In y1 = Alx2 -t B1xz2 + C,x,3 + . . . and similarly to write for ln y 2 In y z = A,x + B2x12 + Czxl3 + . . . (14) In y1 and In y 2 are however related by the Gibbs-Duhem equation x,dln yl + x,dln y2 = 0 . . . . . . (1 5 ) and if (13) and (14) are substituted in (15) and the differentiation per- formed (remembering that x1 + x 2 = 1 and dx = - dx2) it is then found that Al = A 2 = 0 and that certain relationships must exist between the coefficients of the remaining terms. In particular if the terms in xZ2 and x12 [which are now the first terms in (13) and (14)] are considered to be adequate to represent the dependence of y1 and y 2 on composition then B and B2 have to be equal (= B say) and in this event the excess free energy G E will b'e derivable from (1 2) by the simple expression G E T B R T x ~ x ~ = BRTxl(1 - XJ .. . . . - (16) A plot of G E against the mole fraction of either component will then be a symmetrical parabola with a maximum at xl = x 2 = +. This is often nearly true of many systems in which the departures from ideality are not too pronounced not only for GE but also for H E and Y E . If necessary allowance for asymmetry may be made by introducing into equation (16) terms in ( x - x,) ( x - x~)~ etc. e.g. G E == RTx~x,[B + P ( x ~ - ~3 + Q(x1 - ~3'1 . . . (17) where the coefficients P and Q can be related to the coefficients in equations PARSONAGE AND STAVELEY SOLUTIONS OF NON-ELECTROLYTES 309 (13) and (14).Although in explaining the criteria of an ideal solution we assumed for simplicity that the vapour phase was a perfect gas in practice it is usually important to allow for the imperfection of this phase. This requires a knowledge of the second virial coefficients B,, B22 B, at the temperature in question where B, and B, refer to the pure gases while B, concerns the interaction between a molecule of component 1 and one of component 2. Information about the virial coefficients of binary mixtures of vapours is still meagre. The excess volume V E can be determined from the density of mixtures of known composition or by direct measurement. Since G E and the heat of mixing HE are related by an equation similar to (6) HE can in principle be found from the temperature variation of GE(or of the activity coefficients).But since the temperature range over which the measurements can be made is usually quite limited it is better to determine HE by direct calori- metry. In the recent work on condensed gas mixtures this has required the development of calorimeters for measuring heats of mixing at low tem- perature~.~.~ It should be noted that strictly the values of the excess functions should relate to mixing carried out not only at constant temperature but also at constant pressure. If the vapour pressures of the pure components 1 and 2 are say -$ atm. and -1 atm. respectively and if the mixtures are formed and studied at the vapour pressure of the mixture itself then of course the pressure on the system is not constant throughout but varies from -t to -1 atm.Variations as small as this however have a negligible effect on the excess functions. We shall see that the theoretical expressions for these functions are usually formulated so as to be valid at p = 0 and values calculated from them can be compared without serious risk of error with the experimental values obtained under a pressure of an atmosphere or so. But non-constancy of the pressure cannot always be overlooked. Thus the extreme molecular simplicity of a mixture of monatomic sub- stances such as argon and krypton makes such a system a favourable one with which to test the theories we shall describe. But at the lowest tem- perature at which this system could be studied in the liquid state over the whole range of composition namely the triple-point of krypton the vapour pressure of krypton is -0-7 atm.while that of argon is -9 atm. Pressures such as these would not only make the experiments more difficult but in addition considerable pressure changes would accompany the mixing the effect of which could not be ignored. In the last decade several valuable books have been written about solutions. Two of outstanding importance are “The Solubility of Non- electrolytes” by Hildebrand and S ~ o t t ~ in which a wide variety of solubility J Pool and Staveley Trans. Faraday SOC. 1957 53 1186. 4 Jeener Rev. Sci. Instr. 1957 28 263. 6 Hildebrand and Scott “Solubility of Non-electrolytes” Reinhold New York 1930. 3 10 QUARTERLY REVIEWS data is correlated with notable success and “Mixtures” by GuggenheimYB which includes a particularly useful survey of the lattice-model approach to the problem of liquid mixtures.Our concern here however is to review that recent work which has attempted to account quantitatively for excess thermodynamic functions in the light of our knowledge of intermolecular energies. In this work Prigogine and his school and Kirkwood’ have played leading parts. Prigogine has co-ordinated his contributions in his book “The Molecular Theory of Solutions”,l and we acknowledge our indebtedness to this in the preparation of this Review. We must now briefly consider the intermolecular energy E of a pair of uncharged molecules as a function of their separation r both when the molecules are identical and when they are different. The general cause of attraction between molecules is the so-called dispersion effect which may be pictured as arising from the oscillation in phase of the electronic systems of the two molecules.This as London showed gives an energy term which for not too small values of r is proportional to r6. The repulsion between the molecules which becomes dominant at sufficiently close approach and which is due to the overlapping of the electronic clouds gives rise to an energy term which according to quantum-mechanical calculations may be approximately represented by an expression of the form (polynomial in r ) x e-br. This cumbersome expression is never in fact used but as the exponential term is the more important one the simplified expression Ae-*r is sometimes employed. But even this proves to be rather unwieldy and more frequently the energy of repulsion is represented by the purely empirical expression d/rn; n must then be found empirically and as there often proves to be some latitude in the acceptable values of n which usually fall between 9 and 12 the most common choice is to take n = 12.for algebraic convenience. This gives the following expression for E (the Lennard-Jones potential) A plot of E against r gives a curve like that in Fig. 1. Equation (18) can be re-written in terms of E* and r* where E* is the value of the minimum potential energy and r* the distance at which this minimum occurs giving and also in terms of E* and o where o is the intermolecular separation at which E = 0 when Guggenheim “Mixtures” Clarendon Press Oxford 1952. ‘I Salsburg and Kirkwood J. Chem. Phys. 1952,20 1538; 1953,21,2169. PARSONAGE AND STAVELEY SOLUTIONS OF NON-ELECTROLYTES 3 1 I FIG.1. Intermolecular energy ( E ) as a function of intermolecular separation (r) showing significance oj’quantities c* r* and u. It will be seen from the last two equations that for all substances for which the Lennard-Jones potential is an adequate representation of the inter- molecular potential energy this energy is a universal function of the inter- molecular separation the only quantities characteristic of each substance being the scale factors E* and r* (or E* and 0). This will later be seen to have very important consequences owing to its close connection with the theorem of corresponding states. In a mixture of components 1 and 2 we are concerned not only with the intermolecular potentials 1-1 and 2-2 but also with those for the unlike molecules 1 and 2.This is a crucial matter in the theory of solutions and here unfortunately we are on less certain ground. It is usually assumed that if the Lennard-Jones potential is valid for the interaction 1-1 and 2-2 it is also valid for the interaction 1-2 with values of c12* and r12* given by the following so-called combination rules . . . . . . . . . . E12* = (Ell* x E2X*)* (21) r12* = + r,,*) (22) . . . . . . . . . . The justification for these rules is empirical rather than theoretical. London’s theory of dispersion forces related the coefficient c of equation (18) to the polarisability of the molecules and to the characteristic fre- quencies appearing in the dispersion equation and showed that the relation between the coefficients c12 cll and c22 is in general c12<(cll x c2& the equality only holding as a limiting case.Nothing whatever can be predicted about the relationship between the corresponding coefficient d for the repulsive energy term since this is quite empirical. Prigogine,l and Rowlin- son and Townley,* have reviewed the evidence in support of the rules (21) * Rowlinson and Townley Trans. Faraday SOC. 1953 49 20. 312 QUARTERLY REVIEWS and (22). This evidence is not yet as comprehensive as one could wish and a further drawback is that most of it is derived from experimental studies of gas mixtures for which a property such as the second virial coefficient depends upon the three coefficients Bll B22 and BI2. B12 is therefore only part of the quantity which is experimentally determined and this clearly affects the accuracy with which E ~ ~ * and r12* can be evaluated.The chief tests to which the validity of relations (21) and (22) has been submitted are the following (1) If equations (19) and (20) are valid for the pure com- ponents 1 and 2 then these substances conform to the law of corresponding states and (r11*)3 and (r22*)3 will be proportional to the critical volumes Ylc and V2c. Similarly ell* and E ~ ~ * will be proportional to the critical temperatures T, and T2c. For such substances the reduced virial coefficient B/Vc will be a universal function of T/Tc. If equations (19) and (20) are valid for the 1-2 interaction B12/(r12*)3 should be a universal function of T/e12*. Introducing the combination rules (21) and (22) this means that if &2/[&(V1,* + V2,Q)I3 is plotted against T’,(Tlc x T2,)g the points should fall on the curve obtained by plotting B/V against T/T for the pure substance^.^ (2) Values of B12 have been obtained by measuring the volume change on mixing of the two gases and compared with the values calculated by using (21) and (22).1° (3) Similar use has been made of experimental data on transport properties such as viscosity thermal diffusion and diffusion.(A study of interdiffusion coefficients has the advantage that they depend only upon the 1-2 interaction in the gas mixture.) As a result of his survey of the available evidence Prigogine concludes that the combination rules (21) and (22) “seem to be a fair approximation for non-polar molecules”. It should be noted however that if one of the molecules is polar we cannot strictly expect (21) to hold good.To test theories of solution it is possible to select for experimental study a number of binary systems where the molecules are sufficiently small and also non- polar. But even a diatomic molecule will have a quadrupole moment. Thus in the system argon-nitrogen in addition to the dispersion forces there will be quadrupole-quadrupole forces between the nitrogen molecules giving rise to a potential energy term which is proportional to r-l0 (and also quadrupole-induced dipole forces giving an energy term dependent on r8 though this is probably less important). Here too equation (21) cannot be strictly applicable. It is now realised that quadrupole forces can have a more important influence on the physical properties of substances with simple non-polar molecules than was at one time supposed.ll The Lennard-Jones potential (I 8) (or indeed any other tractable func- tion in which the intermolecular energy is expressed in terms of the distance r between the molecular centres) can only be expected to apply to small Guggenheim and McGlashan Proc.Roy. SOC. 1951 A 206,448. lo Michels and Boerboom BuZZ. Suc. chim. belges 1953 62 119. l1 Buckingham Quart. Rev. 1959 13 183. PARSONAGE AND STAVELEY SOLUTIONS OF NON-ELECTROLYTES 3 13 molecules. Indeed even for such simple molecules as carbon tetrachloride and sulphur hexafluoride it has been suggested that a better agreement with the thermodynamic properties of the vapours would be obtained by considering the centres of force to be situated on the periphery of the molecules. A fair test of theories based on the Lencard-Jones potential should therefore be made on binary mixtures of small preferably non- polar molecules that is molecules of substances which will probably be gaseous at ordinary temperatures.This is the reason for the recent experi- mental studies of the thermodynamic properties of binary mixtures of condensed gases. The theories which we shall consider employ the methods of statistical mechanics. If on the basis of a suitable model it is possible to evaluate the partition function Q the thermodynamic properties of the system may be calculated from general relations such as the following F = Helmholtz free energy = - k Tln Q s = - (g)v = LIn Q + kT ( a - ;)e)v . . . . . . . (23) (24) . . . . . . . . . . . . . . (25) The partition function Q is in general given by the expression .. . . . . . . . . . Q = Cexp - E,/kT (27) 9. where E is the energy of the rth level available to the molecules. In the theories we shall consider it is supposed that Q can be written as Qint x Qtr where Qtr is the translational partition function and Qint that associated with the other degrees of freedom such as the internal vibrations of the molecules and their rotation. It is further assumed that when a mixture is formed the internal partition functions Qint remain unchanged. (This should be noted; it implies that any intramolecular vibrations are not affected by mixing which is probably a very good approximation and that the rotational movement of the molecules is the same in the mixture as in the pure liquids. The validity of this second assumption may sometimes have to be reconsidered.) Interest therefore centres on the quantity Qt, which for a classical system may be written in the form (2rrmkT)f * Qt = [-jp--] Q,y + .I . . I (28) 3 14 QUARTERLY REVIEWS 1 - U/kT where eU= N ! - J . . . . J e dx dy dz1. . . . dx dyfi dZN (29) U is the potential energy of all the molecules in the system and the integra- tion extends over its whole volume. QU is called the configurational partition function and it is with the determination of this that theories of solutions are essentially concerned since once this has been done the configurational contributions to the thermodynamic properties of the system can be derived by using equations such as (23) to (26). Throughout this Review the related quantities are the configurational contributions to the thermodynamic functions the non- configurational parts making no contribution to the excess properties in which we are ultimately interested.The correlation of the thermodynamic properties of a fluid with the intermolecular energy parameters was first attempted for a single com- ponent by Lennard-Jones and Devonshire.12 The model for the fluid which they employed the so-called cell model has played an important part in modern statistical theories of solutions and the significance of these should be more readily perceived if we first briefly outline the Lennard-Jones and Devonshire approach to the problem of the single component fluid. It is first supposed that any one molecule moves in a cell formed by its nearest neighbours the average volume of which is V/N where Y is the total volume of the system of N molecules.The potential field in which this molecule moves is determined by its interaction with its neighbours. Strictly speaking this field varies with time so it is replaced by the average field in which the molecule would move if each of its neighbours was at rest at the centre of its own cell. (Only nearest-neighbour interactions were considered at first but the treatment can be extended to include inter- actions with more distant molecules.) On this basis and by using equation (19) for the potential energy of a single pair of molecules and assuming that the total potential energy of one molecule is the sum of its energy with respect to each of its neighbours in turn the following expression is obtained for w(r) the potential energy of the molecule when its centre is at a distance r from the centre of its cell w(0) is the energy of the molecule when at the centre of its cell (r = 0).A = z J E * ~ where z is the number of nearest neighbours and I€*] the depth of the trough in the intermolecular potential energy curve (Le. [€*I is the numerical value of the E* of Fig. 1 which is a negative quantity). V* = N ( Y * ) ~ / ~ where Y* is the intermolecular separation at the minimum le tennard-Jones and Devonshire Pmc. Roy. SOC. 1937 A 163,53; 1938 A 165,l. PARSONAGE AND STAVELEY SOLUTIONS OF NON-ELECTROLYTES 3 15 in the potential energy curve and y is a numerical factor depending on the type of molecular packing assumed. (For a face-centred cubic lattice y = 4 2 . ) y = r2/a2 where a is the average distance between nearest neighbour molecules; Z(y) and m(y) are functions given by the equations I(y) = (1 + 12y + 2 5 .2 ~ ~ ~ + 12y3 + f)(1 - y)-lo - 1 (31) m(y) = (1 +y)(l - 9 ) - 4 - 1 . . . . . (32) The configurational partition function for a single molecule referred to an energy zero with the molecule at the centre of its cell which we will denote as Qu' to distinguish it from the corresponding function QU of equations (28) and (29) which relates to all the molecules is given by the equation where U(r) = W(Y) - w(0). Changing to polar co-ordinates we have The integration in (34) is then carried out between the limits r - 0 and Y = af2 (= half the cell radius). (The justification for this arbitrary choice of the. upper limit is that small values of Y i.e. < the chief contributions to the integral come from a/2).This gives Qu' = 277a3g . . . . . . . (35) If it is supposed that the molecules are not localised but that the whole volume V is available to them then the relation between Quf and Qv is (36) Qu= ~ . . . . . . . ( QulN)N - Ntl~(0)/2kT N ! since Nw(0)/2kTis the potential energy of the whole system when all the molecules are at the centres of their cells referred to the free molecules as energy zero. Thus equations (31) (32) ( 3 9 and (36) combine to give the complete function for the fluid and its thermodynamic properties follow from relations (23) to (26). In particular it will be noticed that use of equation (26) establishes a relation between p V and T i.e. an equation of state which in principle enables the volume of the system at a given temperature and pressure to be estimated.It is in this way that the problem of VE for the liquid mixture is approached. 316 QUARTERLY REVIEWS The cell model has been extended to mixtures by Prigogine and co- w o r k e r ~ . ~ ~ In their work random mixing of the components is assumed this supposition being justified by the calculations of Rushbrooke14 and of Prigogine and Garikian,13 who have shown that at least for molecules of the same size there is only a very small effect on the thermodynamic excess functions due to the departures from random mixing which occur in real solutions. This assumption is embodied in the expressions for the mean potential fields in which a molecule of each type moves. Attention was first directed to the simplest case that where rll* = r22* for which it is reasonable to assume that all the cells are of the same size.The potential field for a molecule of component 1 in such a mixture is then obtained by replacing the parameter ell* in the Lennard-Jones and Devonshire expression for pure liquid 1 by xlell* + x2e12* the potential function for a molecule of component 2 being obtained in a similar way. Unfortunately the complicated nature of the expressions which result obscures their dependence on the mole fractions. Because of this Prigogine and Mathot have used the so-called smoothed potential model in which a square well is substituted for the full Lennard-Jones and Devonshire type of potential. In this approximation to the proper cell model the depth of the potential well is that of the minimum in the Lennard-Jones and Devonshire potential I t I I I I I I I t i I 1 I I I I I I I I I I I I I I I I I I I I FIG.2. Comparison of the Lennard-Jones and Devonshire potential (full curve) witlz so-called smoothed potential (broken line). The potential energy w(r) of the molecule as a function of its distance ( r ) from the centre of its cell. l3 Prigogine and Garikian Physica 1950 16 239; PrigDgine and Mathot J. Chem. Phys, 1952,20,49; Prigogine and Bellemans J. Cliern. Phys. 1953,21 561. l4 Rushbrooke Proc. Roy. Soc. 1938 A 166 296, PARSONAGE AND STAVELEY SOLUTIONS OF NON-ELECTROLYTES 3 17 "(O)] and the diameter of the well is the radius of the cell (a) minus the molecular diameter (a) for interactions between the molecule and the wall of the cell (Fig. 2). On the smoothed potential model the configurational partition function for a single molecule of component I (Q'u,) is given by Ha-d Q'U1 = 4 ~ 1 r2 dr = &r(a - a)3 .. . . . . . . (37) 0 referred to an energy zero with the particle at the centre of its cell. Since the potential energy of the system when each molecule is at the centre of its cell is +N[x,w,(O) + x2w2(0)] referred to the completely free molecules as energy zero (where w,(O) = the potential energy of a molecule of com- ponent 1 at the centre of its cell referred to the completely free molecule as energy zero and w2(0) = the potential energy of a molecule of component 2 at the centre of its cell referred to the completely free molecule as energy zero) the configurational partition function for the whole system is w,(O) is easily shown to be given by or if interactions with second and third shell neighbours are taken into account ~1(0) = ~ ( ~ 1 ~ 1 1 + ~ 2 ~ 1 2 ) Thus w1(O) and likewise w2(0) can be expressed in terms of a which is proportional to Vg.It can be seen therefore that (38) gives Q as a func- tion of V T and the parameters characteristic of the inter-molecular interactions. By using (23) to (26) the thermodynamic functions are ob- tained in terms of the same variables. When in addition the usual combination rules for the inter-molecular parameters (21) and (22) are assumed this theory leads to the rather surprising prediction of a positive excess free energy coupled with a negative excess volume. On previous theories a positive excess volume would be expected. The possibility of contractions on mixing in simple systems was confirmed by experiments on the mixtures CClp-C(CH3)415 and CO-CHp.16 l5 Mathot and Desmyter J.Chenz. Phys. 1953 21 782. l* Mathot Staveley Young and Parsonage Tram. Faraday Soc. 1956,52,1488. 318 QUAKTEKLY REVIEWS A qualitative explanation can be given of this contraction which is predicted and observed for mixtures of molecules of equal size whose interaction energy contains only terms due to dispersion and short-range repulsion forces. Ignoring zero-point energy the inter-molecular separation of a 1-1 pair at O'K would be rll*. However as the temperature is raised the mean inter-molecular distance increases owing to the asymmetry of the potential well about r = Y*. (This is in fact partly responsible for the thermal expansion of the liquid.) If the inter-molecular potential energy is represented by the Lennard-Jones 6-12 equation it can readily be shown that if the energy of vibration of the pair of molecules relative to each other is A then the mean distance apart of the molecules (rll) is For small values of A/cll we may use the approximate formula Remembering that for molecules of the same size rlZ* = rll* = rZ2* and putting c12* = ( E ~ ~ * .E ~ ~ * ) * it then follows that 2 ~ ~ < r ~ ~ + r22 i.e. a contraction on mixing would be expected. Indeed if the assumption 2el2* = ell* + E ~ ~ * is made r12 is even smaller. The extension of the theory to the case where rll* # r22* was made by Prigogine and Bellemans. They assumed that cells of two sizes existed in the liquid mixture cells containing molecules of species 1 being different in size from those containing molecules of species 2.The ratio of the diameters of the cells was then chosen so as to minimise the free energy of the mixture. Since the existence of cells of different sizes is really in- compatible with these cells' having the simple close-packed structures of the pure liquids the Prigogine and Bellemans treatment can only be a valid approximation if the sizes of the two species of molecule do not differ too much. Otherwise the structure of the mixture might be quite different from that of the pure liquids. One important point was however established by this work namely that a relatively large positive excess volume would result from quite a small difference in the molecular size leading us to expect that the negative excess volumes which should occur for mixtures of molecules of the same size would be only rarely observed.In the cell theories of mixtures dealt with so far discrepancies between the experimental and the theoretical values of the excess thermodynamic functions could arise either from the failure of the model to lead to the correct values of the functions for any liquid even for the pure com- ponents or from errors introduced by the assumptions made about the molecular movement and relationships within the mixture. As a result of this a number of theories have been developed based on the theorem - - - PARSONAGE AND STAVELEY SOLUI IONS OF NON-ELECTROLYTES 319 of corresponding states which are liable only to the second type of error. In these theories the properties of the mixture and of the pure componznts are expressed in terms of those of a suitable reference liquid and as a consequence the values given for the pure components are necessarily correct provided only that they obey the theorem of corresponding states.For the derivation of these theories nf mixtures the form of the theorem of corresponding states due to Pitzerl' is used. He showed that if the inter- molecular potential energy of a pair of molecules ( E ) at a distance apart (r) could be expressed in the form where cj5 is a universal function and E* and r* are two parameters character- istic of the interacting molecules (and it will be noted that the Lennard- Jones 6-12 potential conforms to this equation) then the configurational contribution to the bulk thermodynamic functions could also be expressed in terms of universal functions of W/E* and T / T * .Thus for the Helmholtz free energy we would have E = E*&r/r*) . . . . . . . . (43) (44) . . . . . where $ is a universal function. the equations We may define the reduced temperature (?) and reduced volume (;> by n. ?= kT/E* . . . . (45) v* = ~ / r * ~ . . . . 446) where e = the volume per molecule = r3/y y being a constant character- istic of the liquid strccture. It follows from equation (44) thst F/eTis a universal function of T' and G which function we denote as (F/RT)(T,v"). The important step is now taken upon which depends the form of the expressions for the excess functions the Helmholtz frce enErgy is related to the thermodynamic proper_ties of a reference liquid (T = To w" = u",,) by a Taylor series expansion in l/Tand 5 Pitzer J. Chin. Phys. 1939 7 583.2 320 QUARTERLY REVIEWS The coefficients of the series expansion are readily expressible in terms of the thermodynamic functions of the reference substance. Thus for a given liquid the reference liquid the coefficient of the third term is {*j-)? = (,,,> E*{ a(l/T) 1 - E* a R where E is the internal energy of the reference liquid. If the two liquids are at the same temperature and occupy the same volume this is tantamount to an expansion in terms of the differences of their force parameters E* - E,* and r* - ro*. The functions S and p can be obtained by differentiation of I; with respect to T and V respectively. Of considerable interest are the functions G H and V which are the quantities handled experimentally. These quantities can be obtained provided the equation of state-is first solved.Alternatively GIRT can be expanded in terms of p” and 1/T but here the problem is more complicated than the one dealt with above because of the nature of p” [ = P ( ~ * ) ~ / E * ] . Longuet-HigginslS was the first to employ a treatment of this type. In his theory of conformal solutions he considered only terms in the expres- sions for the excess functions which were proportional to ( E ~ ~ * - ell*) and (rZ2* - rll*) where these represent the differences between the force parameters of the two pure substances. For a binary solution the results which he obtained may be written as . . . * (48) a(FtRT) a(F/RT) k W R T ) -5 E VE - S E HE (dQo/dT - R) = T(dQo/dT) - Qo - G E RT - Qo JQPo - Tao) = x1x& . . . I . . . . . . (49) where Q is the latent heat of conversion of a mole of the reference liquid at temperature T and pressure p to its vapour at T and p cco and Po are respectively the coefficient of thermal expansion and the isothermal compressibility of the reference substance and S12 is an adjustable para- meter characteristic of the mixture.The insertion of typical numerical values into the terms in the deno- minators of equation (49) shows that all of the denominators are negative except (dQ,/dT) - R which is positive. It can be seen therefore that in this form the theory could not explain the results on the CO-CH4 and CC1,-C(CH,) systems for each of which GE and YE have opposite signs. Prigogine and his co-workers19 and Scott2 have therefore considered also higher terms in the series for the excess functions which are proportional to ( E ~ ~ * - (r22* - r,l*)2 and (eZ2* - ell*) (rZ2* - rll*) if e12* and r12* are eliminated by using (21) and (22).In Prigogine’s so-called average potential model it is assumed that all cells are of the same size whether Longuet-Higgins Proc. Roy. Soc. 1951 A 205 247. lo Prigogine Bellemans and Englert-Chwoles J. Chern. Phys. 1956 24 518. 2o Scott J. Chem. Phys. 1956 25 193. PARSONAGE AND STAVELEY SOLUTIONS OF NON-ELECTROLYTES 32 1 occupied by a molecule of component 1 or by one of component 2. Hence the volume ZI to be used in evaluating the reduced volume of the liquid is just the molecular volume of the mixture. The usual assumptions of random mixing and additivity of the inter-molecular potential energies are also made which lead to the expression for the inter-molecular potential energy of a pair of molecules in the pure liquid e*((r*/r)12 - 2(r*/r)6} being replaced by + X2€22*{(-) r22* l2 - 2 (+*y] * .This may be written as ~m*{(r,*/r)12 - 2 (r,*/r)6) . . . . where { X12Ell*(rll*)6 + 2X1X2E12*(r12*)6 + -qJ2c22*(r22*)6}2 Ern* = x12Ell*(r11*)12 + ~ x ~ x ~ E ~ ~ * ( ~ ~ 2*)12 + x ~ ~ E ~ ~ * ( ~ ~ ~ * and Thus these expressions give the values of E* and r* to be used mixture. Prigogine has also carried out a more refined treatment in which the cells occupied by molecules of types 1 and 2 are no longer assumed to be of the same size. The sizes of the two types of cell cannot now be expressed in terms of the molar volume of the mixture alone an additional equation now being necessary since there are two quantities to be chosen. This extra equation is provided by requiring that the cell sizes must be chosen so as to minimise the free energy as was done in the Prigogine and Belle- mans theory.Since two types of cell are present it is necessary to choose average values of E* and r* for each type of cell. By the same process as before that is assuming additivity of the potentials and random mixing the following expressions are derived where cl* and rl* are the average parameters to be used for the cells con- taining molecules of component 1. Similar expressions are obtained for the 322 QUARTERLY REVIEWS corresponding parameters for cells containing molecules of component 2. The partial molar thermodynamic functions for component 1 can then be expressed in terms of el* rl* and the volume of a type 1 cell and the corresponding functions for component 2 can be calculated in a similar way.If the partial molar Helmholtz free energies are found to be Fl and Fn respectively then the total Helmholtz free energy of the mixture is - F=xlF1 +x2F2 . . Scott has also considered a third type of approach in which the mean distance apart of unlike molecules is allowed to adjust itself independently of the mean separations of 1-1 and 2-2 pairs. But because of the strain on the lattice brought about by the presence of in effect cells of three different sizes Scott himself considers this to be not a good representation of real liquids. He has also compared the theories previously discussed in which only one (one-liquid model) or two (two-liquid model) sizes of cell are permitted and he has come to the conclusion that the latter should be preferred since the one-liquid model overestimates the effect of difference in size of the molecules by not allowing any adjustment of the lattice which might lower the free energy.In the remainder of this Review the two-liquid approximations will be meant when reference is made to the Prigogine corresponding-states treatment or the Scott treatment. As with the cell theories the corresponding-states treatments are limited in their validity to mixtrxes the components of which have molecules of not too widely different size. Byers-Brown21 has examined very closely the errors which might be introduced into the results of a corresponding-states theory by departures from random mixing and has concluded that these errors might well be large for mixtures of molecules of appreciably different size.He has also pointed out that previous workers who have discussed this problem and reached the conclusion that non-random mixing is un- important have always considered systems of molecules of equal size whilst it is only in the case of mixtures of molecules of different sizes that non-random mixing would be expected to be important. A comparison of the thermodynamic properties of several binary mixtures of simple substances with the values predicted for these quantities on the refined version of Prigogine’s corresponding-states theory is made in the Table where the figures given are for the equimolar mixture. For some systems the values obtained from Scott’s two-liquid model are also included. Bearing in mind that the effects concerned are relatively small one may note that for the majority of systems the theoretical values are of the correct sign and order of magnitude.Particularly remarkable are the successful predictions of the volume contractions found for the systems CO-CH, 02-N2 and CCl,C(CH,), all of which approximate quite closely to the simple case of a mixture of molecules of equal size. Indeed a1 Byers-Brown Phil. Trans. 1957 250 221. PARSONAGE AND STAVELEY SOLUTIONS OF NON-ELECTROLYTES 323 CO-CHp (~cH** = 4.29 UCO* = 4-21) one of the few systems for which a complete set of experimental data is available also shows quite good agreement between theory and experiment for the other excess functions. The same is true of the system CC14-C(CH3)4. Krypton and methane molecules are also similar in size ( r ~ ~ * = 4-04 r c ~ ~ * = 4.29) and it is unfortunate therefore that only G E has been determined.It will be noticed that the theoretically predicted values are frequently very sensitive to the choice of reference substance. This would not be so if the reference substances conformed exactly to the theorem of correspmd- ing states and a sufficiently large number of terms was retained in the expansion (47). The series however is not very rapidly convergent and the number of terms which can be evaluated is severely limited by the lack of accurate data for the higher derivatives with respect to temperature and volume of the ordinary thermodynamic functions. The most striking case of the effect of the choice of reference substance on the value predicted is that of VE for the system Ar-N, where the change of reference substance leads to a change in the sign of the effect to be expected.The system Ar-CH4 appears to provide a good test of the adequacy of the theory to predict effects due to size differences of the molecules since complications due to the polar character of the components or their lack of spherical symmetry should be almost entirely absent. The comparison between theory and experiment for this system is however very disappoint- ing the experimental values being all very much smaller than those pre- dicted. Here it may well be that the difference in size of the component molecules is too large to justify the supposition that the mixtures and the pure liquids all have the same structure ( r ~ ~ * = 3.82 ~ c H * * = 4.29 r ~ r - t ~ * / r ~ ~ * = 1.12).If modifications to the structure of the mixture occur because of the difference in size of the components the free energy of the mixture and hence GE would be reduced and it is also likely that V E would become smaller. Both of these expectations are in agreement with the data. At first sight it might seem surprising that the real value of S E is less than is predicted in spite of the increase in disorder resulting from the distortions of the liquid structure. However it must be borne in mind that any con- traction in volume brought about by the distortions would lead to a large drop in the thermal entropy of the molecules and this might well be more than sufficient to counterbalance the increase in entropy previously mentioned. Appreciable effects may also arise from non-random mixing in the system.As already mentioned such effects have been carefully con- sidered by Byers-Brown. The contrast between theory and experiment for the system CO-N seems particularly significant. The close similarity b2tween these iso- electronic components (which is reflected in the small differences in many of the physical properties of the pure substances) would lead us to expect that their solutions would be almost ideal as is indeed quantitatively predicted by the theories under discussion. The actual values of G E and Y E 324 QUARTERLY REVIEWS are however much larger and are comparable in magnitude with those of many of the other systems in the Table. Admittedly carbon monoxide has a dipole moment but it is far too small to account for the discrepancy and the only feasible reason for this seems to be that carbon monoxide and nitrogen have considerably different quadrupole moments (& = 1-7 1 x e.s.u.).Calculations by one of us (N.G.P.) and Dr. A. D. Buckingham show that effects of the size observed could arise from quadrupole-quadrupole interactions. It is not surprising that the results for the Ar-0 system fit the theory well since the constituent molecules are similar in size (rg ,*/r~* = 1-01 6) and the quadrupole moment of oxygen is very small (<+@N~). But all the other systems containing diatomic molecules are more complicated in that the errors arising both from differences in size and also from neglected quadrupole effects are involved. The two causes of error seem to lead to opposing shifts in the values of the excess functions.(Yet another possible complication is that the rotational partition functions of the diatomic molecules are not in fact independent of their environment.) When considering solutions containing carbon tetrafluoride it must be remembered that although it has zero dipole and quadrupole moments (on account of its symmetry) it must have large higher electric moments because of the polarity of the individual C-F bonds. It seems probable that such higher moment interactions would cause modifications to the excess properties which are of the same sign as those produced by the quadrupolar interactions already mentioned. Comparing therefore the systems CI-1,-CF and Kr-CF, the fact that in the latter G E (expt.) falls short of the theoretical figure whilst in CH,-CF the experimental value exceeds that predicted is consistent with the greater disparity in molecular size in the Kr-CF system.It would seem to be highly desirable that theories capable of dealing with orientational forces should be developed. Some work has been done Comparison of the predicted and observed values of the excess thermodynamic functions of eqzcimolar mixtures of simple substances. e.s.u. 0~~ = 1.27 x GE = Excess Gibbs free energy (cals./mole of mixture) HE = Excess heat content (cals./mole of mixture) SE = Excess entropy (cals./deg.mole of mixture) V E = Excess volume (c.c./mole of mixture) Expt. = Experimental value Theory Po() = Value predicted from Prigogine's refined corresponding states treatment using X as reference liquid Theory S = Value predicted from Scott's two-liquid model System T('K) G E HE TSE V E Theory P(C0) 32 27 -5 -1.2 Theory P(CH,) 19 16 -3 -0.1 CO-CH4 90.7" Expt.2816 263 -2 -0.3016 PARSONAGE AND STAVELEY SOLUTIONS OF NON-ELECTROLYTES 325 System T('K) Ar-CH 91 Ar-CO 83.8 Ar-0 83.8 OZ-N 83.8 A-N 83.8 CO-N 83.8 Kr-CH 115.5 CH,-CF 110 Kr-CF 117.1 CC1,- 273 Expt. Theory P(Ar) Theory P(CH4) Expt. Theory P(Ar) Theory P(C0) Expt. Theory P(Ar) Theory P(0,) Expt. Theory P(0,) Theory P(N,) Expt. Theory P(Ar) Theory P(N,) Theory S Expt. Theory P(C0) Theory P(N,) Expt. Theory P(CH,) Expt. Theory P(CH,) Theory P(CF4) Theory S Expt. Theory P(Kr) Theory P(CF4) Expt. GE 1 822 49 53 gz4 7 7 8.2,' 30 28 5*428 0-9 0.6 1429 7 8629 55 74 113 7529 115 126 7615 C(CH3)4 Theory P[C(CH3),] 84 Theory P(CC1,) 75 Ar-Kr* 83 Expt. 4532 Theory P 41 Theory S 27 Theory P 139 Kr-Xe* 105 Expt.8132 * Solid solutions. 22 Mathot personal communication. 23 Shields and Staveley unpublished work. 24 Pool Shields and Staveley unpublished work. * 5 Pool and Stavelev. Nature. 1957. 180. 11 18. HE TSE V E 2522 7 +0*18 71 22 +1.02 +090 +0.30 +0*17 1 42s 5 +0*1425 11 4 +0.13 11 4 +0.10 -0.3 126 - 0.09 -0.4 1 -0.1826 +0x2 -0.32 +O* 17 +0.13as +0.1023 +0*02 +0-88ao +3.7 7531 - 1 -0.515 71 -13 -2.2 -0.5 803 35 31 -10 30 3 17032 89 149 10 26 Pool and Stavelei; unpublished work. 27 Herrington Saville and Staveley unpublished work. 28 Herrington and Staveley unpublished work. 29 Thorp and Scott J. Phys. Chem. 1956 60 670. 30 Croll and Scott J. Phys. Chem. 1958 62 954. 31 Englert-Chwoles J . Chem. Phys. 1952 20 925; 1955 23 1168. 38 Walling and Halsey J . Phys. Chem. 1958 62 752.326 QUARTERLY REVIEWS in this field notably by P ~ p l e ~ ~ by Rowlinson and his co-worker~,~~ and by Prig0gine.l However all of these treatments consider the orientationally dependent part of the energy of interaction to be a small perturbation and from what we have seen of the results for the system CO-N2 in particular this would not appsar to be so. The development of more precise theories valid for strong orientationally dependent forces is hindered by the co- operative nature of these forces. Thm the probability of any partichlar relative orientation of two molecules is influenced by the orientations of their neighbours which in turn are influenced by the orientations of more remote molecules. Perhaps the best possibility for advance in this field lies in the work of Wood and Parker35 using high-speed electronic com- puters.So far this work has reached the stage of deriving by Monte Carlo methods values for the thmnodynamic functions of three-dimen- sional arrays of molecules interacting according to the Lennard-Jones 6-12 law of force. Systems of 108 molecules are normally studied but it has been shown that the results obtained for a 32-molecule system differ only insignificantly from those for the larger system. Thus it would appear that the sizes of the samples are quite adequate. The theoretical difficulties of treating any system containing molecules for which the fields of force are non-spherical makes particularly desirable the determination of the properties of mixtures of the inert gases. Un- fortunately measurements on the Ar-Kr system (and similarly on the Kr-Xe system) would be beset by considerable difficulties for the reasons given earlier.It is interesting therefore to have the data for the solid solutions Ar-Kr and Kr-Xe. The results of this work by Halsey and his co-workers are included in the Table. It was of course not possible to measure the heats of formation of the solid solution directly as is normally done for liquid mixtures. Instead 4 P was derived from values of G E obtained over a range of temperature [equation (6)]. Since this range was only 26" for Ar-Kr and 30" for the Kr-Xe system this method is less satisfactory than the direct method. Halsey et al. give 12 cals. mole-1 as the uncertainty in the heat of mixing. In all these experiments great care was exercised to ensure that true equilibrium was reached a waiting period of -24 hours being allowed before measurements were made. Presumably therefore the quantity most accurately known is G E . For Ar-Kr there is good agreement between the observed and the calculated values of GE but for Kr-Xe the agreement is only moderate. Finally the large differences between the observed and the calculated entropy values are a reminder that the swcessful prediction of excess entropies is an exacting test for any theory of solutions. Is Pople Proc. Roy. SOC. 1954 A 223 498. 84 Cook and Roulinson Proc. Roy. SOC. 1953 A 218 405; Rowliilson and Sutton 36 Wood and Parker J. Chew. Phys. 1957,27 720. Proc. Roy. Soc. 1955 A 229 271.