Model Calculations of Thermodynamic Mixture Properties from Direct Correlation Integrals

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Zeitschrift fiir Physikalische Chemie Neue Folge, Bd. 166, S. 63-69 (1990) Vol.

1 Zeitschrift fiir Physikalische Chemie Neue Folge, Bd. 166, S (1990) Vol. 166, Part 1, Pages 63-69, 1999 Model Calculations of Thermodynamic Mixture Properties from Direct Correlation Integrals By Esam Z. Hamad and G.Ali Mansoori University of Illinois at Chicago (M/C 063) Chicago, Illinois , USA Direct correlation function / Mixture / Thermodynamics / Statistical mechanics/ Liquid state Abstract Exact relations among mixture direct correlation integrals in terms of concentration variables are presented. Different closure (approximation) expressions for cross-direct correlation integrals are suggested. These closures are joined with the exact relations among direct correlation integrals to derive equations for thermodynamic mixture proper ties on the basis of the hard-sphere and Lennard-Jones models. Zwischen direkten Korrelationsintegralen von Komponenten in Mischungen werden auf Grund von Konzentrationsvariablen exakte Beziehungen vorgestellt. Fiir Kreuzkorrelations-Integrale werden verschiedene Niiherungsausdriicke angegeben. Diese Niiherungen werden mit den exakten Beziehungen aus den direkten Korrelationsintegralen kombiniert, um Gleichungen fiir thermodynamische Eigenschaften der Mischungen auf Grund des Hart-Kugel-Modells und des Lennard-Jones-Modells abzuleiten. 1. Introduction The knowledge about fluctuation integrals and their relations in mixtures is important in the development of mixture theories of complex molecules [1-4]. In a recent study [4] relations among fluctuation integrals of mixtures in the grand canonical ensemble were derived. These relations were combined with a closure relation to get the three integrals in a binary mfxture. In the present report the relations among the direct correlation integrals in terms of the concentration variables; are presented and combined with different closures (approximation) expressions for cross-direct correlation integrals to calculate the chemical potentials for binary mixtures of model fluids. addresses: EZH (esam.hamad@aramco.com); GAM (mansoori@uic.edu) 63

2 2. Theory Consider a multicomponent mixture in thermodynamic equilibrium which is described by the following set of independent variables {T, Qi, {!2,..., en}, where Tis the Kelvin temperature, Q; = N;/V, N; is the number of molecules i in the system, Vis the system volume, and n is the number of components in the mixture. In the grand canonical ensemble the following expression for 8 µ ;/8 Qi is derived [4-6]: [oµi/oeih. e 1 = kt{i5u/q;-c; i } where Q J stands for the set Qi, Q 2,, Qn excluding Q j, and c;j(r) is the direct correlation function of species i andj. Exact relations can be derived among the direct correlation integrals, C i i, by equating the mixed second derivatives of the chemical potentials [4, 6]. For a binary mixture with T, Qi, e2 independent variables one has: and E.Z. Hamad and G.A. Mansoori 8C 12 /8Q1 = 8C 11/8e2 8C 12/8Q 2 = 8C 22/8Q 1. (5) The above two equations between C 11, C 22, and C 12 are exact. However, to solve for C 11, C 22, and C 12 one needs an additional relation (closure) among C 11, C 22. In what follows a number of closure (approximation) expressions for C 12 in terms of C 11 and C 22 are considered and the direct correlation integrals are solved for the hard-sphere and the Lennard-Jones model fluids. The closure expressions which are presented in this article are the simplest that one can suggest. The simplicity of the suggested closures makes it easier to use them in getting the C ii's. The first closure is suggested to be based on a relation among the contact values of the direct correlation functions in a mixture of onedimensional hard rods. It can be shown that the contact values are related by [7] where a ii is the distance of closest approach between centers of particles i and}, and For the direct correlation integrals of a mixture of hard-spheres a relation similar to Eq. (6) may be assumed 64 '(1) (2) (4) (6) (7) (8)

3 In Eq. (8) C 12 is the geometric mean of C 11 and C 22. Postulating that C 12 is some kind of a mean of C 11 and C 22 the following two closures are suggested C 12 = (C 11 + C 22 )/2 (9) C12 = 1XC 11 + PC22, (10) Eq. (9) is the arithmetic mean, which, as will be shown latter, implies that the mixture is an ideal solution. As a result this closure is rejected. Eq. (10), which has been used before [4, 6], is a weighted arithmetic mean. However, before it can be used here, the parameters IX and /3 have to be known. One way of obtaining the parameters is to utilize the following limiting condition limit Cii = -2Bii( T) a O (11) where B;/T) is the second virial coefficient of the pair i and j. Using this limit in Eq. (10) and choosing p = (1/4) IX (the reason for this choice of /J will be clear later), gives the following: 21X = B12/B11 ± [(B12/B11) 2 -B22/B11] 1 12 (12) where only positive values of IX will be used in the calculations. The validity of closures (8) and (10) can be tested by comparing the direct correlation integrals calculated by the present technique with known values. The hard-sphere model has been studied extensively and the direct correlation integrals are available through the Percus-Yevick theory [7-9] which gives quite accurate expressions for the mixture direct correlation functions [8]. The results of comparisons for an equimolar mixture at u 1 if u 22 = 1.5 are presented in Table 1, which shows clearly that both closures are accurate enough. However, the geometric mean is more accurate at high densities and the weighted arithmetic mean is more accurate at low densities. Good agreements are also observed at other conditions. Table 1. Values of C 12 /C 11 for an equimolar hard-sphere mixture (a 11 /a 22 = 1.5) calculated based on different closures and compared with the Percus-Yevick theory. (n/6)qa223 P-Y (C22/C11) 1 12 % dev. C22 % dev. rx+{j Cu For the weighted arithmetic mean closure rx{j = 1/4. At a 11 /a 22 = 1.5 the negative sign in Eq. (12) gives a =

4 3. Calculations There are three relations (two exact and one closure) to solve for the three direct correlation integrals in a binary mixture. Let us start by using the geometric mean closure. Substituting Eq. (8) in Eq. (4), and using Eqs. (1) and ( 5) one can get: (0µ1,/0Q2) 2 (0 2 µ1,/oqd - 2(aµ1,/0Q1) (0µ1,/0Q2) (0 2 µ1,/0q10q2) + (0µ1,/0Q1) 2 ca 2 µ1,/0qd = o (13) where the subscript r stands for a residual property. This nonlinear partial differential equation can be solved analytically utilizing its symmetry. First one writes the differential equation in the form where (oµ1,/oq1) or/oq2 - (0µ1,/0Q2) or/oq1 = 0 R = (oµ1,/0q2)/(0µ1,/0q1)- Integrating Eq. (14) gives or E.Z. Hamad and G.A. Mansoori R = (0µ1,/0Q2)/(0µ1,/0Q1) = h(µ1,) (0µ1,/0Q2) = h(µ1,) (0µ1,/0Q1) (14) where h is an unknown function to be determined from the pure component properties. Integrating Eq. (15) gives (15) µ1, = F[Q2h(µ1,) + Qi] (16) where F is also an unknown function which can be determined from the condition µ1r(q2 = 0) = µ1,.p = ktf1(q) (17) where subscript (p) indicates a pure component property. Eq. (17) then implies that F(t) = kt f1(t), and Eq. (16) now becomes µ1,/kt = f1[q2h(µ1,) + Qi]. (18) To determine the function h one needs first to find the chemical potential of component 2. Using the following thermodynamic relation 0µ1,/0Q2 = 0µ2,/0Q1 (19) one gets (0µ2,/0Q1) = h(µ1,) (oµ1,/oq1) = 0H/0Q1 (20) where His defined by: H(µ1,) respect to Qi gives µ2, = H(µ1,) + K(Q1)- Jh(µ1,)dµ 1,. Integrating Eq. (20) with (21) 66

5 The function K((}1) can be shown to be zero from Eq. (8) and the fact that µ 1r and µ2r vanish at zero total density. Eq. (21) then becomes µ2r = H(µ1r). (22) At (} 1 = 0 the chemical potential, µ2r ((h = -0) = µ2r,p = kt fi((}), is now used to determine the function H(µ1r)- From Eqs. (18)-(22) one can show that h = dh/dt = Ji 1 (t)/fi 1 (t) where f;- 1 is the inverse function, such that f;(f; 1 (t)) thermodynamic relation Q oµir,p/8(! = ap; r,p/8(2, (23) t. Using the where Pir[Q] is the residual pressure of pure component i, the following relation for His finally obtained P2rff2 1 (!)] = P1r['1 1 (H)]. (25) Equations (18) and (25) give the chemical potential according to the geometric mean closure. The weighted arithmetic mean closure gives a simpler relation. When Eq. (10) is combined with the exact relations, Eqs. (4) and (5), and Eq. (1) is used the following linear partial differential equation is obtained: r:t.8 2 µ1r/o(}'f_-8 2 µ1r/8(}18q2 + /38 2 µ1r/8(} = 0. (26) The solution of this equation depends on the quantity (1-41:t./3). When this quantity is positive the differential equation is hyperbolic and the solution is obtained in terms of a power series expansion [6]. When 1-41:t./3 = 0, Eq. (26) is parabolic with the following solution µlr = F(21:t.(}1 + (}2) + (}2 H(21:t.(}1 + (}2) (27) where F and Hare undetermined functions. The choice of /3 = (1/4)1:t. in Eq. (10) has then the advantage of giving a closed-form solution. As in the case of geometric mean closure, the pure component chemical potentials are used to determine the unknown functions F and H. The chemical potential of component 1 then takes the form µlr = a 1r,p(Q1 + 21:t.(!2) + kt(x1 + 21:t.x2)- 1 {x1z1r,p(q1 + 21:t.(!2) + X2Z2r,p((}1 /2r:t. + (}2)} where a;r,p and Zir,p are the residual molar Helmholtz free energy and the residual compressibility factor of pure component i, respectively. The mixture compressibility factor takes the form: z = X1Z1,p ((}1 + 21:t.(}2) + X2Z2,p ((}1/21:t. + (!2)- (29) The above expression reduces to the ideal mixture case for 21:t. = 1 and this shows that the arithmetic mean closure, Eq. (9) implies an ideal mixture. (24) (28) 67

6 Eq. (29) has the interesting feature that all the mixture virial coefficients, obtained by expanding the equations in powers of density, have the correct dependence on composition (quadratic for B, cubic for C and so on). 4. Results and discussion The two models developed in the previous section are used here to predict the properties of hard-spheres and Lennard-Jones mixtures. Special procedures were devised to use Eqs. (18) and (25) without having to invert the equation for the pure component chemical potential. For hard-spheres the Carnahan-Starling [10] equation of state is used to describe the behaviour of the pure fluid. The predictions of the two models are compared to the Mansoori et al. [11] hard-sphere mixture equation, which is known to reproduce the simulation data very accurately. In Table 2 the comparison is shown for the chemical potential of component 1 for a mixture with a 11 /a 22 = 1.5 and Q0' 2 = 0.25 [this density corresponds Table 2. The variation of the chemical potential of component 1 in a mixture of hardspheres (au/a22 = 1.5, ea12 = 0.25) with composition based on different closures and compared with the MCSL [11] equation of state. Xi µ1,/kt MCSL Eq. (18) % dev. Eq. (28) % dev Table 3. The chemical potential at infinite dilution of Lennard-Jones mixture. kt/e22 = 1.2, QC1 2 = 0.7). (a12/a22) 3 e12/e22 µ001,/kt MC Eq. (18) Eq. (28) Simulation data from ref. [13]. 68

7 to 0.95 of the solidification density of component (1)]. The comparison shows that both models represent well the mixture chemical potentials. Similar results are obtained for the chemical potential of component 2, and at other conditions. The hard-sphere potential has only repulsive forces. To test the present models for potentials which have attractive forces the chemical potential of a mixture of Lennard-Jones fluids at infinite dilution is considered. Pure component fluid is described by the Nicolas et al. [12] equation of state. The predictions are compared to simulation data [13] in Table 3. For molecules with the same intermolecular energies, 812/822 = 1, Eq. (28) shows better agreement with simulation data for (J12/(J22 > 1, and Eqs. (18) and (25) show better agreement for small values of (J 12 /(J 22. For molecules with different intermolecular energies, Eqs. (18) and (25) do not give real values of the chemical potential when the intermolecular energy ratio is outside the range 0.8 < 812/e 22 < 1.6. For the weighted arithmetic mean closure, when the intermolecular energy ratio is larger than about 1.2, Eq. (12) gives a complex a, and no chemical potential calculations are possible. Acknowledgement This research is supported by the Chemical Science Division of the Office of Basic Energy Sciences, the US Department of Energy Grant No. DE FG02-84ER References 1. E. Matteoli and L. Lepori, J. Chem. Phys. 80 (1984) E. Matteoli and L. Lepori, J. Chem. Thermodyn. 18 (1986) E. Matteoli and L. Lepori, J. Chem. Thermodyn. 18 (1986) E. Z. Hamad, G. A. Mansoori, E. Matteoli and L. Lepori, z. Phys. Chem. Neue Folge 162 (1989) L. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 5, Part 1 (Pergamon Press, Oxford, 1980). 6. E. Z. Hamad, Ph. D. Thesis, University of Illinois, Chicago (1988). 7. J. L. Lebowitz and D. Zomick, J. Chem. Phys. 54 (1971) J. L. Lebowitz, Phys. Rev. 133A (1964) J. K. Percus and G. J. Yevick, Phys. Rev. 110 (1958) N. F. Carnahan and K. E. Starling, J. Chem. Phys. 51 (1969) G. A. Mansoori, N. F. Carnahan, K. E. Starling and T. W. Leland, J. Chem. Phys. 54 (1971) J. J. Nicolas, K. E. Gubbins, W. B. Street and D. J. Tidesley, Mo!. Phys. 37 (1979) K. S. Shing, Ph.D. Thesis, Cornell University, Ithaca, New York (1982). 69

Theory of infinite dilution chemical potential

Theory of infinite dilution chemical potential Fluid Phase Equilibria Journal Volume 85, 141-151, 1993 141 Esam Z. Hamada and G.Ali Mansoorib, * a Department of Chemical Engineering, King Fahd University of Petroleum and Minerals, Dhahran 31261 (Saudi