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. 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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.