THE JOURNAL OF CHEMICAL PHYSICS VOLUME 51, NUMBER 11 1 DECEMBER 1969

Variational Approach to the Equilibrium Thermodynamic Properties of Simple Liquids. I*

G.Ali MANSOORi(1) AND Frank B. CANFIELD (2)

School of Chemical Engineering and Materials Science, University of Oklahoma, Norman, Oklahoma 73069

ABSTRACTA variational technique which is based on two different inequalities for the Helmholtz free energies is used to calculate the equilibrium thermodynamic properties of simple fluids. A system with hard-sphere potential function is used as the reference system. Helmholtz free energy of the original system is cal­culated by variation around the Helmholtz free energy of the reference system, and the other thermo- dynamic properties are calculated from free energy. By choosing a hard-sphere reference system, it is possible to calculate the equilibrium thermodynamic properties of fluids from very low densities to densities close to solid, and from high temperatures in the gas phase to low temperatures in the liquid phase, in the ranges where experimental and machine-calculated data are available. It is shown that the present variational technique is a better approach to the prediction of the equilibrium thermodynamic properties of liquids and vapor-liquid phase transition than any other approach so far developed. While the variational calculation based on a hard-sphere reference system does not predict the liquid-solid phase transition, it is argued that this might be due to the neglect of the orientation, or ordering in the formulation of the working inequality for fluids.

I. INTRODUCTION

The idea of perturbation and variational approaches to the equation of state of fluids based upon a simple system of hard spheres was originally introduced by Zwanzig.1 It was observed that at very high tempera­tures the equation of state of a gas is effectively due to repulsive potential of the molecules, and at lower tem­peratures the attractive part comes into effect. Then if we perturb or vary the attractive contribution of the partition function around the repulsive, hard-sphere contribution, we might be able to predict the properties of lower temperature gases and liquids satisfactorily. Even though the idea was sound and the perturbation expansions were rigorous, the calculations were not as successful as expected. Smith and Alder2 showed that it was possible to evaluate theoretically the thermo­dynamic quantities for a potential slightly different from the hard-sphere potential at all densities. Their calculations lead to an expansion of the thermodynamic quantities in powers of the reciprocal temperature. The convergence of the expansion for the equation of state was such that with the first two perturbation terms they were able to approximate the compressibility, pV/NkT, for a reduced temperature, T*=kT/t, greater than 2.0 to within 0.03 unit up to almost solid densities for Lennard-Jones fluid.

One disadvantage of the above perturbation method was from the fact that the repulsive part of a real inter­molecular potential, though steep, is not infinitely steep as was assumed. This introduces complications in locating the hard-sphere cutoff, a parameter to which numerical results are extremely sensitive. To solve this problem, McQuarrie and Katz3 also expanded the

* This research was supported by the Directorate of ChemicalSciences, Air Force Office of Scientific research, Grant AF­AFOSR-1020-66.

(1). Present Address: UIC, emails: mansoori@uic.edu; gali.mansoori@gmail.com (2).Emails: canner791@yahoo.com; fcanf@rkymtnhi.com1 R. W. Zwanaig, J. Chem. Phys. 22, 1420 (1954). 2 E. B. Smith and B. J. Alder, J. Chem. Phys. 30, 1190 (1959). 3 D. A. McQuarrie and J. L. Katz, J. Chem. Phys. 44, 2393

(1966).

partition function with respect to the steepness of the repulsive part of the potential. Their equation of state reliably produced PVT data up to a reduced density, p*= prr, of 0.95 and reduced temperatures as low as T*=3 for Lennard-Jones fluid. To find whether the failure of the perturbation approach at lower tempera­tures was due to the perturbation treatment of the attractive part of the potential or due to the treatment of the finite steepness of the repulsive potential, Barker and Henderson4 applied the perturbation equation ofZwanzig to square-well fluid. In this case, the effect of the attractive forces was not complicated by the "softness" of the repulsive part of the potential, which is infinitely steep for square-well model. They were also able to find approximations for the coefficient of T*-2

in the perturbation expansion. Their results indicated that the useful convergence of the perturbation expan­sion extends to very low temperatures for the square­well potential. Later, Barker and Henderson, by using a double series perturbation expansion, extended the good convergence to the perturbation equation of state to lower temperatures for more reaslistic potentials.5-

7

They were able to extend the applicability of the per­turbation equation of state to reduced temperatures as low as 0. 7 and reduced densities close to the solid phase. They also applied their perturbation approach to systems of fluids with two- and three-body forces8

•

9

and also to quantum fluids.10

Kozak and Rice,11 by considering the hard-sphere

4 J. A. Barker, and D. Henderson, J. Chem. Phys. 47, 2856 (1967).

6 Reference 4, p. 4714. 6 J. A. Barker and D. Henderson, J. Chem. Educ. 45, 2 (1968). 7 W. R. Smith, D. Henderson, and J. A. Barker, Can. J. Phys.

46, 1725 (1968).

8

J. A. Barker, and D. Henderson, Phys. Rev. Letters 21, 134 (1968).

9 J. A. Barker, D. Henderson, and W. R. Smith, Proceedings of the Fourth Symposium on Thermodynamic Properties, ASME, p. 30, 1968.

10 S. Kim, D. Henderson, and J. A. Barker, Can. J. Phys. 47,

99 (1969). 11 J. J. Kozak and S. A. Rice, J. Chem. Phys. 48, 1226 (1968).

4958

8

DOI: 10.1063/1.1671889

4960 G.A. MANSOORI AND F.B. CANFIELD

andN

Uo= L ui;°.,>j=l

Relation (21) is based on the assumption that N;, thenumber of molecules in different spherical shells around

( 15) a central molecule in the reference system, are un-Then the basic inequalities (13) and (14) will become,respective! y,

correlated:ir"-j. (22)

N

F�Fo+ L (u;/)oi>j=l

In a manner similar to Barker and Henderson,4 if we(16) also assume the following relations between the number

of molecules in different shells:andN N

F�Fo+ L (u;/)o-(/j2/2!) ([ I: (u;/-(u;/)) ]2)0

N

+(/33/3!)([ I: (u;/-(u;/)o)]3)o, (17)•>i=l

whereU;/=u;;-U;;°.

In a fluid, the nth correlation function g<n> (r1, • • •, rn)is defined asg<n) (r1 • • •rn) = [Vn J · • • J exp(-,6U)dtn+1 • • •dtn]/ Z

(18)for n= 2, g<2>(r1, r2)=g<2>(r12) is the pair correlation, orthe radial distribution function. With this definitionof the correlation functions, inequalities ( 16) and ( 17)will be as follows:

and

-(J32/2!)w2+ (S3/3!)wa. (20)In inequality (20), w2 and ws also could be shown withrespect to the correlation functions. The resultingrelations are lengthy and involve g<2), • • ·, g<6), whichprogressively complicate the results. Zwanzig1 hasprovided the relation for w2 with respect to the correla­tion functions. In a similar manner to formulation ofw2 with respect to correlation functions, w3 could also beformulated. Since no satisfactory relations are availablefor g(3), • · ·, g<6), the method of Barker and Henderson4

will be used to approximate w2 and wa.IV. APPROXIMATIONS INTRODUCED IN THE

SECOND INEQUALITY

Barker and Henderson4 have defined two approximaterelations for w2 both of which are very close. The simplerone which is called the macroscopic compressibilityapproximation is:

w{'"'½Np[fl-1(op/iJp)0] iv [u1(r)]2g0(r)dr. (21)

(NJ\T/\\)-(N;Ni)(Nk)=O, kr"-i,j,(N;2NJ-(A?)(N;)=0, ir"-j,

(N;)2(Ni)- (N;N;)(N;)=O, ir"-j,it can be shown thatw{ ... -'½N (p//32)[ ( op/iJp )02+½P( of iJp )o(op/ap )i]

(23)(24)(2S)

Xiv [u1(r)]3g0(r)dr, (26)where p=N /V is the number density and {J1(op/op)0 is the macroscopic compressibility coefficient in thereference system.

V. VARIATIONAL TECHNIQUE

The above mentioned inequalities suggest that byvarying the properties of the reference system such thatthe right-hand side of the inequalities go to a relativeminimum, it may be possible to bring the inequalitiesto equality or, at least, closer to equality. Principles ofvariational calculus must be used to attack this prob­lem. What is needed here is the properties of the refer­ence system. In other words, we must know the Helm­holtz free energy and the radial distribution function ofthe reference system as functions of intermolecularparameters of the reference system. Also, inequalities(19) and (20) suggest that as u1z°�12, the inequalitiesgo closer and closer to equalities. Consequently, if areference system can be chosen such that its inter­molecular potential function is close to the functionalform of intermolecular potential function of theoriginal system, then the inequalities will be closer toequality, and the process of variational calculation willproduce a better approximation of the original system.

In the above case where the intermolecular forcesbetween the particles of a system consist of only two­body forces, it is assumed that the intermolecularpotential function of the original system is the Lennard­J ones 12:6 potential;

u(r) =4f[(u/r) 12- (u/r)6], (27)

where � is the depth of the minimum in the potentialcurve and u is the diameter of the particles. This choiceof the potential function is due to the abundance of theso-called semiexperimental or Monte Carlo and molec­ular dynamics data for systems with Lennard-Jonespotential function.

4962 G.A. MANSOORI AND F.B. CANFIELD

(20) becomesF Fo 48'11 f°' 96'11 [ (

op)] [

128'11 1 [ (op)2 (op)2]- < -- + - U1(s)G(s)ds- - [r1 - U2(s)G(s)ds+ - - - - _!.P (a/ap}o -

NkT - NkT T* 0 T*2 op O O T*3 /32 op O 2 op o

X j°' Ua(s)G(s)ds, (39) 0

where U2(s) and Ua(s) are inverse Laplace transforms of xu*2(x) and xu*3(x), respectively:s22 sI6 srn

U2(s) = (c-1)24 - -2(c-I)IB _ + (c-1)12 _ (40)22! 16! 10! and

s34 s'lil s22 si& Ua(s) = (c-1)36 _ -3(c-I)30 _ +3(c-I)24 __ (c-1)1s _ (41) 34! 28! 22! 16! VII. PROPERTIES OF THE REFERENCE

HARD-SPHERE SYSTEM

According to thermodynamics, the following relationexists between the pressure and the Helmholtz freeenergy of a system PV/NkT= Z= p(o/op) (F/NkT)T.N;, (42)

where Z is the compressibility. By integrating Eq. ( 42)with respect to p, one gets �=JP p-1 (

PV -1) dp- ln(pA3)-1. (43) NkT

O NkT

Then by having the compressibility of a system inhand, we can calculate F / NkT for that system. For thehard-sphere reference system in mind, there are severalrelations available for compressibility. As will be shownsubsequently19 the average Percus-Yevick equation,while simple in form, (PV/NkT)o= (l+'IJ+'IJ2-J,,3)/(1-'IJ)3, (44)

is in good agreement with the machine-calculated datafor the hard-sphere fluid. By inserting ( 44) in ( 43) wehave F0/NkT= ln(l-'11)1

'2+[3/(l-'IJ)]

+[3/4(1-'11)2]-(19/4)- ln(pA3). (45) Also from ( 44) we can show that

and i (

op)

(1-'11)4 (46) fr op o = (1+2'1J)2-1.5'1Ja(4-.,,) ,

-(1/.82) [(op/op )o2+½p(o/op) (op/op)/]

(1-.,, )7 ( 1.5.,,5-7 .5.,,4+ 207,2+5.,,-1) [(1 +2.,,)2-1.5.,,3 ( 4-.,,) ]3

(47)Compressibility coefficient of the hard-sphere reference

19 G. A. Mansoori, J. A. Provine, and F. B. Canfield, "Note on the Perturbation Equation of State of Barker and Henderson," J. Chem. Phys. (to be published).

system, fr1(op/op)0, as introduced by (46), and alsorelation ( 47) are parts of the coefficients of T*-2 andT*-3 in the inequality (39), respectively. VIII. CONSIDERATION OF THE PARAMETER c,

AS THE VARIATIONAL PARAMETER

For a single-component thermodynamic system,according to phase rule, if two of the intensive proper­ties of the system are defined, the other properties willbe defined. Of course, along the phase transition lineone intensive thermodynamic variable of the systemwill be enough to get the other variables. Then, ingeneral, if we choose the two independent variables p,the density and T, the temperature, one could writeF= F(p, T).

Now if we define the reduced quantities p*= p<r andF*= F/NkT, one could write for the first inequality(38)

F*(p*, T*) 5.Fo*(p*, T*, c)+[r1(p*, c)/T*] (48) and for the second inequality (39) F*(p*, T*) 5.Fo*(p*, T*, c)

+ I\(p*, c) + r2(p*, c) + ra(p*, c) (49) T* T*2 T*3 '

where I\, r2, and r3 are the coefficients of the inversefirst, second, and the third powers of temperature as shown in (38) and (39). The right-hand sides of in­equalities ( 48) and ( 49) are functions of p*, T*, and c, while the left-hand sides are only functions of p* and T*. This suggests that, to bring the inequalities (48)and ( 49) closer to equality, their right-hand sidesshould satisfy the following conditions, respectively:

oFo* 1 or1 &+ T*ac

= O;(50)