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

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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.

1 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 ABSTRACT A 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 thermodynamic 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 Alder 2 showed that it was possible to evaluate theoretically the thermod yn amic quantities for a potential slightly different from the hard-sphere potential at all densities. Their calculations lead to an expansion of the thermod yn amic 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 Katz 3 also expanded the * This research was supported by the Directorate of Chemical Sciences, Air Force Office of Scientific research, Grant AF AFOSR (1). Present Address: UIC, s: mansoori@uic.edu; gali.mansoori@gmail.com (2). s: canner791@yahoo.com; fcanf@rkymtnhi.com 1 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*= pr r, 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 Henderson 4 applied the perturbation equation of Zwanzig 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 forces 8 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 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). 8 9 J. A. Barker, D. Henderson, and W. R. Smith, Proceedings of the Fourth Symposium on Thermodynamic Properties, ASME, p. 30, 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). DOI: /

3 4960 G.A. MANSOORI AND F.B. CANFIELD and N Uo= L ui;.,>j=l Then the basic inequalities (13) and (14) will become, respective! y, and N N F Fo+ L (u;/)o i>j=l F Fo+ L (u;/)o-(/j 2 /2!) ([ I: (u;/-(u;/)) ] 2 )0 where N N +(/3 3 /3!)([ I: (u;/-(u;/)o)] 3 )o, (17) >i=l U;/=u;;-U;;. In a fluid, the nth correlation function g< n > (r1,, r n) is defined as g <n) (r1 rn) = [V n J J exp(-,6u)dtn+1 dtn]/ Z Relation (21) is based on the assumption that N;, the number of molecules in different spherical shells around ( 15) a central molecule in the reference system, are un- correlated: ir"-j. (22) 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: (18) for n= 2, g< 2 >(r1, r 2 )=g< 2 >(r12) is the pair correlation, or the radial distribution function. With this definition of the correlation functions, inequalities ( 16) and ( 17) will be as follows: and -(J3 2 /2!)w2+ (S 3 /3!)wa. (20) In inequality (20), w 2 and ws also could be shown with respect to the correlation functions. The resulting relations are lengthy and involve g< 2 ),, g< 6 ), which progressively complicate the results. Zwanzig 1 has provided the relation for w2 with respect to the correlation functions. In a similar manner to formulation of w2 with respect to correlation functions, w3 could also be formulated. Since no satisfactory relations are available for g(3),, g< 6 ), the method of Barker and Henderson 4 will be used to approximate w2 and wa. IV. APPROXIMATIONS INTRODUCED IN THE SECOND INEQUALITY Barker and Henderson 4 have defined two approximate relations for w2 both of which are very close. The simpler one which is called the macroscopic compressibility approximation is: w{ '" '½Np[fl- 1 (op/ijp) 0] iv [u 1 (r)] 2 g 0(r)dr. (21) (NJ\T/\\)-(N;Ni)(Nk)=O, (N; 2 NJ-(A?)(N;)=0, (N;) 2 (Ni)- (N;N;)(N;)=O, it can be shown that kr"-i,j, ir"-j, ir"-j, w{... -'½N (p//3 2 )[ ( op/ijp )0 2 +½P( of ijp )o(op/ap )i] (23) (24) (2S) Xiv [u 1 (r)] 3 g 0 (r)dr, (26) where p =N /V is the number density and {J 1 (op/op) 0 is the macroscopic compressibility coefficient in the reference system. V. VARIATIONAL TECHNIQUE The above mentioned inequalities suggest that by varying the properties of the reference system such that the right-hand side of the inequalities go to a relative minimum, it may be possible to bring the inequalities to equality or, at least, closer to equality. Principles of variational 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 of the reference system as functions of intermolecular parameters of the reference system. Also, inequalities (19) and (20) suggest that as u1z 12, the inequalities go closer and closer to equalities. Consequently, if a reference system can be chosen such that its intermolecular potential function is close to the functional form of intermolecular potential function of the original system, then the inequalities will be closer to equality, and the process of variational calculation will produce a better approximation of the original system. In the above case where the intermolecular forces between the particles of a system consist of only twobody forces, it is assumed that the intermolecular potential function of the original system is the Lennard J ones 12:6 potential; u(r) =4f[(u/r) 12 - (u/r) 6 ], (27) where is the depth of the minimum in the potential curve and u is the diameter of the particles. This choice of the potential function is due to the abundance of the so-called semiexperimental or Monte Carlo and molecular d yn amics data for systems with Lennard-Jones potential function.

5 4962 G.A. MANSOORI AND F.B. CANFIELD (20) becomes F Fo 48'11 f ' 96'11 op 128'11 1 op 2 [ ( )] [ [ ( ) NkT - NkT T* 0 T *2 op O O T* 3 2 /32 op O - < U1(s)G(s)ds- - [r 1 - U 2(s)G(s)ds _!. P (a/ap}o - ( op ) 2 ] op o X j ' U a(s)g(s)ds, (39) 0 where U 2 (s) and Ua(s) are inverse Laplace transforms of xu* 2 (x) and xu* 3 (x), respectively: and s22 s I6 s rn 22! 16! 10! U2(s) = (c-1 ) (c -I)IB _ + (c -1 )12 _ (40) s34 s'lil s22 si& U a(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 relation exists between the pressure and the Helmholtz free energy 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 =J P p-1 ( PV -1) dp- ln(pa 3 )-1. (43) NkT O NkT Then by having the compressibility of a system in hand, we can calculate F / NkT for that system. For the hard-sphere reference system in mind, there are several relations available for compressibility. As will be shown subsequently 19 the average Percus-Yevick equation, while simple in form, (PV/NkT)o = (l+'ij+'ij 2 -J,, 3 )/(1-'IJ) 3, (44) is in good agreement with the machine-calculated data for the hard-sphere fluid. By inserting ( 44) in ( 43) we have F 0 /NkT = ln(l-'11) 1 ' 2 +[3/(l-'IJ)] +[3/4(1-'11) 2 ]-(19/4)- ln(pa 3 ). (45) Also from ( 44) we can show that i ( op (1-'11) 4 ) (46) fr op o = (1+2 '1J ) '1J a(4-.,, ), and -(1/.8 2 ) [(op/op )o 2 +½p(o/op) (op/op)/] (1-.,, ) 7 ( 1.5.,, ,, , 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, fr 1 (op/op) 0, as introduced by (46), and also relation ( 47) are parts of the coefficients of T *-2 and T *-3 in the inequality (39), respectively. VIII. CONSIDERATION OF THE PARAMETER c, AS THE VARIATIONAL PARAMETER For a single-component thermod yn amic system, according to phase rule, if two of the intensive properties of the system are defined, the other properties will be defined. Of course, along the phase transition line one intensive thermod yn amic variable of the system will be enough to get the other variables. Then, in general, if we choose the two independent variables p, the density and T, the temperature, one could write F= F(p, T). Now if we define the reduced quantities p*= p<r and F*= 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 inverse first, 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 sides should satisfy the following conditions, respectively: ofo* 1 or1 & + T*ac= O; (50)

Note on the Perturbation Equation of State of Barker and Henderson*

fhe JOURNAL OF CHEMICAL PHYSICS VOLUME S1, NUMBER 12 15 DECEMBER 1969 Note on the Perturbation Equation of State of Barker Henderson* G.Ali MANSOORI, (1) Joe A. PROVINE, (2) AND Frank B. CANFIELD (3) School