ELSEVIER Fluid Phase Equilibria 99 ( 1994) 153- 166

Phase equilibria and thermodynamic properties of molecular fluids from perturbation theory

II. Binary fluid mixtures

Keshawa Shukla

Department of Chemical Engineering, Rice University, Houston, TX 772.51-1892, USA

Received 26 May 1993; accepted in final form 14 March 1994

Abstract

The thermodynamic perturbation theory developed in Part I of this work is applied to describe phase equilibria and thermodynamic excess properties of binary fluid mixtures. The theory is based on the spherically symmetric reference system represented by the Lennard-Jones pair interactions, whereas the isotropic and anisotropic interactions arising from two-body electrostatic, induction, dispersion and repulsion forces, and from three-body induction and dispersion forces, are treated as perturbations. These calculations differ from the previous perturbation theories based on the equation of state for argon as the reference system in two major respects. First, the reference mixture properties are computed more accurately using a reliable form of the perturbation theory of simple fluid mixtures. Second, unlike potential parameters are estimated adequately using a recently proposed set of combination rules. Comparisons of theoretical results with experimental data for phase equilibria and thermodynamic excess properties are presented for some selected fluid mixtures, namely argon + krypton, xenon + ethylene, carbon dioxide + ethane, and xenon + hydrogen bromide. These comparisons show a very good performance of the perturbation theory, which offers a significant improvement over a similar form of the perturbation theory based on the equation of state for argon as the reference mixture.

Keywords: Theory; Statistical mechanics; Vapor-liquid equilibria; Perturbation theory

1. Introduction

In Part I of this paper (Shukla, 1994), we presented a perturbation theory of fluid mixtures composed of rigid nonspherical molecules, called PTHN. The theory is based on the spherically symmetric reference system represented by the Lennard-Jones (L-J) fluid and accounts explicitly for the important effects of isotropic and anisotropic interactions arising from two-body electrostatic, induction, dispersion and repulsion forces, as well as from three-body induction and dispersion forces. This form of the perturbation theory uses only two adjustable parameters

037%3812/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved SSDZ 0378-3812(94)02509-Y

154 K. Shukla / Fluid Phase Equilibria 99 (1994) 153-166

in order to predict phase equilibria and thermodynamic properties of a number of pure fluids consisting of spherical molecules and moderately nonspherical molecules of linear shapes.

In this paper we examine the reliability of the PTHN perturbation theory in describing phase equilibria and thermodynamic excess properties of binary molecular fluid mixtures solely from the properties of their pure components. In the present calculations, properties of the reference mixture are determined using an accurate version of the perturbation theory for simple fluid mixtures, and unlike potential parameters in the mixture are obtained adequately from a suitable set of combination rules. In fact, this theory is found to be more accurate in predicting properties of the fluid mixtures consisting of nearly spherical to moderately nonspherical molecules than a similar form of perturbation theory based on the equation of state for argon as the reference system. In particular, we apply PTHN to binary fluid mixtures composed mainly of those components which were treated in Part I of this work.

The theory has been described in greater detail in Part I of this work. Therefore, we present here only a brief account of it in Section 2. Methods of calculating properties of the reference mixture are presented in Section 3, while Section 4 deals with the determination of the like as well as the unlike potential parameters that appear in the mixture calculations. In Section 5 we present comparisons of theoretical results with experimental data for a few selected binary fluid mixtures. Finally, Section 6 follows with the conclusions of the paper.

2. Theory

As outlined in the Part I of this work (Shukla, 1994), we consider a binary fluid mixture consisting of N( =NA + NB) molecules of components A and B contained in a fixed volume V, at a constant temperature T. The total potential energy of interactions of the system is approximated as the sum of two-body and three-body interactions, which depend on both orientations and related center of mass coordinates of the molecules but are independent of rotational momenta and internal vibrational states.

The perturbation theory calculations are based on the expansion of the Helmholtz free energy in terms of the Pade approximation. Expansion is carried out about the spherically symmetric reference system, in which the molecules interact with a L-J (12-6) pair potential. The Hehnholtz free energy through the third-order of perturbation expansion (Shukla et al., 1983) is given by

where

and

(1)

(2)

(3)

(4)

K. Shukla / Fluid Phase Equilibria 99 (1994) 153-166 155

where AL” is the Helmholtz free energy of the reference mixture, and A: is the first-order contribution to the free energy arising from the isotropic induction and the isotropic site-site repulsion interactions. A t”” is the first-order contribution due to the isotropic (14-8) pair interaction, and A dispobn is due to the three-body nonadditive dispersion interaction. A2 and A3 are second- and third-order perturbation terms in an expansion in powers of the anisotropic part of the potentials arising from electrostatic, dispersion, induction and overlap forces. The various terms which contribute significantly to the Helmholtz free energy are collected in Appendix A of Part I of this work. In particular, the terms arising from the electrostatic, induction, dispersion and repulsion interactions and their cross terms, which contribute significantly to the Helmholtz free energy, are taken into consideration. The three-body nonadditive short-range forces and the induction forces higher than the three-body forces have been neglected, while the three-body dispersion forces are considered up to the second-order perturbation term. Since the inclusion of the full anisotropic site-site terms did not yield the proper convergence of the perturbation series in the case of pure fluids (Lucas, 1986; Shukla, 1994), only the isotropic repulsion terms (which converge adequately for the particular molecular eccentricities) are taken into account in the present calculations. For zero eccentricity of the molecules and in the absence of long-range interactions, the free energy of Eq. (1) reduces to the free energy of the mixture of spherical molecules. Once the free energy is known, other thermodynamic properties can be determined directly using the standard relations. In particular, we are interested here in calculating phase equilibria and thermodynamic excess properties (i.e. excess Gibbs free energy, excess enthalpy and excess volume) of a few selected binary fluid mixtures consisting of spherical as well as nonspherical multipolar molecules. Eq. (1) has been referred to as PTHN. When the first three terms of Eq. (1) are replaced by the equation of state for argon, PTHN reduces to PTHO given as

A =A”‘g”“+Ay+A2 (1 -:,lAl) In PTHO, thermodynamic properties of the reference mixture are determined from the equation of state for argon (Twu et al., 1980) using the van der Waals one fluid theory (Shukla et al., 1984a, b).

Note that in the present work, PTHO for mixtures differs from the previous perturbation theory of molecular fluid mixtures based on the equation of state for argon (Shukla et al., 1984a, b) in two major ways: (1) the unlike parameters are now determined using a more accurate set of combination rules, and (2) the more reliable expressions for the site-site interactions are used. Moreover, PTHN and PTHO, as applied here to binary fluid mixtures, differ in the method of determining the properties of their reference mixtures as described below.

As shown in Appendix A of Part I, perturbation contributions to the free energy involve integrals, J$, over the pair distribution functions, gZs, and Kasy (11’1”; nn’n”) and L,,, (I; nn’) over the triplet distribution functions gXBY for the reference L-J fluid. These integrals are defined as

J$& m

s dr$.:~‘“-2’g$J(r*) (6)

0

s

m

L,,,(l; nn’) = dr:2r;E,-‘“-2’ s

*d 1

r$.&-(“‘-2) s

d(cos a,)&;(& r;r3, &)P,(cos a,) (7) 0 0 -1

156 K. Shukla 1 Fluid Phase Equilibria 99 (1994) 153-166

where

Ym(h~2Q) = ,E,, w’z”; mm’mf’)~~(o12)Y~~,(ol~)Y~~,,(W23)*

P[(o) is the Legendre function, and al is the angle between y12 and y13. Y, is a spherical harmonic with sufhx * denoting the conjugate complex value. C(ZZ’Z”; mm’m”) is a Clebsch-Gor- don coefficient as defined by Rose (1957). These integrals are evaluated using the scheme described in Section 3.

3. Properties of the reference mixture

In the present work, the integrals J,,, Kasy and L,,,, are evaluated by employing the mean-density approximation (Mansoori and Leland, 1972). In this approach, the full expression for the L-J distribution function in a mixture is replaced by that of the pure fluid evaluated at a mean density according to

g&es; T; P; x; {G/I}; {Q >I = g(r/a.s; WE,,; pai)

where

(9)

(10)

For typical ratios of the potential parameters, eAA/eBB < 3 and cAA/crBB < 1.3, this approxima- tion has been found to describe the most important integrals (Hoheisel et al., 1983) adequately. In the MDA, the integrals are reduced to the corresponding pure-fluid integrals given by

Ja, = J(P& k%,,) (11)

&y = K(pd k%Mpd k%Mp& Wed”3 (12)

L,, = [UP& kT/&Up~:, k%,W(po:, Wqdl”’ (13)

where k is the Boltzmann constant, p is the mixture number density, and X, and X, are the compositions of the species CI and p, respectively. Values of J, K, and L for the pure fluids are determined using the empirical relations described elsewhere (Nicolas et al., 1979; Shukla et al., 1983). In writing the above K and L equations, the three-body distribution functions are replaced by the corresponding pair distribution functions using the usual superposition approximation.

In order to determine the thermodynamic properties of the reference L-J mixture in PTHN, we adopt the recently developed first-order hard sphere perturbation theory, known as MPT2 (Shukla, 1991a, b). MPT2 describes accurately phase equilibria and thermodynamic properties of the model as well as the real fluid mixtures in different nonideal conditions without requiring any mixing rules. In fact, this form of the theory represents a significant improvement over the

K. Shukla I Fluid Phase Equilibria 99 (1994) 153-166 157

common one-fluid approximations. In this theory, the Hehnholtz free energy of the mixture is expressed as

ALJ=AHs+A~~+A~T* + AyPA (14)

where AHS is the free energy of the hard sphere mixture. A,,, is the first-order term in the expansion of free energy of the reference mixture and AFTA

is the first-order perturbation term due to the high temperature approximation (HTA) (Shukla, 1987). The last term in Eq. (14) arises due to the random phase approximation (RPA) (Andersen et al., 1972), and is extended to the mixture in a semi-empirical way. The method for calculating thermodynamic properties and phase equilibria of binary fluid mixtures using Eq. ( 14) has been presented before (Shukla, 1987, 1991a).

In the case of PTHO, thermodynamic properties of the reference mixtures are determined from the equation of state for argon using the well-known van der Waals one-fluid theory (Shukla et al., 1984a).

4. Potential parameters

Molecular parameters (dipole moment p, quadrupole moment Q, octopole moment R, isotropic polarizability a and anisotropic polarizability K) are taken from independent sources (Gray and Gubbins, 1984). Potential parameters (Ei and ai) are obtained from the fit of the saturated liquid density and the vapor pressure of the pure components of the mixtures. The site-site parameters (pi) are determined from the experimental values of the bond-length of a molecule. All these parameters are the same as given in Table 1 of Part I of this work. The method for determining unlike site-site parameters is also presented in Part I. The only unknown parameters needed in our mixture calculations are the unlike molecular size CAB and the energy EAB parameters of the reference mixtures. These two parameters are determined from the known values of the parameters of the pure components (CA*, cBB; EAA, EBB) using a suitable set of combination rules given below.

In this work we adopt the most reliable set of combination rules (Kohler, 1957; Smith, 1972; Kong, 1973), called Kohler- Smith-Kong (K-S-K), which has been found to describe ade- quately the various thermodynamic properties of numerous binary fluid mixtures (Shukla, 1987,

Table 1 Potential parameters used in the mixture calculations

Mixture a

Ar+Kr 1.0017 0.9859 1.0647 1.4206 1.5128 Xe + C,H, 1.0001 0.9983 1.0325 0.9658 1 .O%O CO2 + CzH, 1.0046 0.9497 1.1642 1.1789 1.5242 Xe + HBr 1.0004 0.9947 0.9540 0.9795 0.8987

a Component A is cited first for each binary.

158 K. Shukla / Fluid Phase Equilibria 99 (1994) 153-166

1989, 1991b). For the mixture following the L-J (12-6) pair potential, unlike size and energy parameters can be related to those of the corresponding like parameters through

~AB=Y(~AA+ CBB)/~ (1%

EAB = ~XEAAEBB)~'~

where q and 5 are the binary size and energy interaction parameters, respectively. In the K-S-K combination rules, q and 5 are defined as

and

(16)

(17)

Here Qi is the isotropic polarizability of the component i. Note for q = 1 and 5 = 1, Eqs. ( 17) and ( 18) reduce to the well-known Lorentz-Berthelot

combination rules. The unlike potential parameters in addition to the isotropic polarizabilities used in the present

calculations are given in Table 1.

5. Results and discussion

We have examined the adequacy of PTHN by performing extensive comparisons of theoreti- .

cal predictions with experimental data for a variety of fluid mixtures. In this section we present results only for some selected binary fluid mixtures consisting of spherical and/or moderately nonspherical molecules. In order to demonstrate the superiority of PTHN over PTHO, we also present a few comparisons between the two theories.

Fig. 1 shows results for the vapor-liquid phase equilibria for the mixture argon + krypton (pressure P versus compositions of liquid X( argon) and vapor Y( argon)) at two temperatures, T = 104 K and 116 K. The agreement between theory and experiment is seen to be very good over the whole range of the compositions. Fig. 2 presents results for the excess Gibbs free energy (GE) and the excess enthalpy (HE), and Fig. 3 shows results for the excess volume ( VE) at pressure P = 0. As can be seen, for GE and VE theoretical predictions are in very good agreement with their experimental results. In contrast, for HE, PTHN is seen to disagree with experiment. Although similar results for the phase equilibria of the mixture argon + krypton could be obtained from PTHO using the van der Waals one-fluid theory, the PTHN results for the excess properties are nevertheless superior to those determined from the former. For instance, for an equimolar condition of argon + krypton at T = 116 K and P = 0 bar, the experimental values for the excess properties are GE = 82 J mol-‘, VE = -0.464 cm3 mol-’ and HE = 43 J mol-‘. The corresponding values of the excess properties in PTHN are GE = 78 J mol-‘, VE = -0.489 cm3 mol-’ and HE = 14 J mol-‘, while those in PTHO are GE = 58 J mol-‘,

K. Shukla 1 Fluid Phase Equilibria 99 (1994) 153-166 159

Fig. 1. Vapor-liquid phase equilibria for argon + krypton from theory (-) and experiment ( 0) (Davies et al., 1967).

100

20

0

0 0.2 0.4 0.6 0.6 1.0

Fig. 2. Excess Gibbs free energy and excess enthalpy for argon + krypton from theory (GE, -; HE, ---) and experiment (GE, 0; HE, 0) (Davies et al., 1967; Lewis et al., 1978) at T = 116 K and P = 0 bar.

160

-0.2

2

ii

B

3 -0.4

-0.6

Fig. 3. Excess (Davies et al.,

K. Shukla 1 Fluid Phase Equilibria 99 (1994) 153-166

0 0.2 0.4 0.6 0.6 1.0

xow

volume for argon + krypton from theory (-) and experiment (0) at T = 116 K and P = 0 bar 1967).

VE = -0.7 cm3 mol-’ and HE = 1 J mol-‘. These comparisons suggest that PTHN performs still better than PTHO, and that the excess enthalpy seems to be very sensitive to the equation of state.

0.0

-4.0

h -8.0 5 E 3 -12.0

% 4 -16.0

-20.0

-24.0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700

P (bar) P (bar)

180.0

150.0

- 120.0 "0 E 5 90.0

?c a 60.0

30.0

0.0

Fig. 4. Excess Gibbs free energy for argon + krypton from theory and experiment (Barreiros et al., 1982).

Fig. 5. Excess enthalpy for argon + krypton from theory and experiment (Barreiros et al., 1982).

K. Shukla 1 Fluid Phase Equilibria 99 (1994) 153-166 161

-2.5

T - 134.32K

9... I..., 1 .‘.““““““““~“‘~l 0 100 200 300 400 500 600 700

P (bar) So I I / I

04 x (Xe), Y (X4 0.8 1

Fig. 6. Excess volume for argon + krypton from theory and experiment (Barreiros et al., 1982).

Fig. 7. Vapor-liquid phase equilibria for xenon + ethylene from theory (PTHN, -; WHO, ---) and experiment (0) (Calado, 1984) at a subcritical temperature T = 203 K.

It has been shown in the past that the pressure dependence of the excess properties is a direct measure of the effects of intermolecular forces in the mixtures (Rowlinson and Swinton, 1982). Therefore, such a comparison between theory and experiment provides a stringent test of PTHN. Figs. 4-6 show the effect of pressure on the excess properties of an argon + krypton mixture at a temperature T = 134.32 K and a composition X(argon) = 0.485. The values for GE and HE presented in Figs. 4 and 5 are the differences between their corresponding values at a finite pressure and those at zero pressure for the same temperature. They are defined as

AGE = GE(T, P) - GE(T, P = 0) (19)

AHE = HE(T, P) - HE(T, P = 0) (20)

These comparisons show that unlike PTHO, PTHN predicts AGE, AHE and VE within experimental uncertainties. The only exceptions are the high pressure experimental data for HE which seem to be of questionable accuracy, hence the difference between theory and experiment (Barreiros et al., 1982).

We have also considered the more nonideal simple fluid mixtures krypton + xenon and argon + methane. As for the argon + krypton system, the present theory describes very well the vapor-liquid phase equilibria and excess properties of these mixtures. Because of the very small octopole moment on methane, the argon + methane system behaves like a mixture of ar- gon + krypton. Therefore, those results are not presented here.

We now investigate a different class of binary mixture: xenon + ethylene. In this mixture, the first component is spherical, whereas the second component is nonspherical exhibiting a strong quadrupole moment. Calculations for phase equilibria are performed over a range of tempera- tures. Sample results for phase equilibria are shown in Figs. 7 and 8 at two temperatures; T = 203 K and 282.6 K. PTHN predictions for phase equilibria (including the azeotrope at T = 203 K and critical behavior at T = 282.6 K) are seen to be in very good agreement with

162 K. Shukla / Fluid Phase Equilibria 99 (1994) 153-166

Fig. 8. Vapor-liquid phase equilibria for xenon + ethylene from theory (-) and experiment ( 0) (Calado, 1984) at a supercritical temperature T = 282.6 K.

Fig. 9. Excess Gibbs free energy for xenon + ethylene from theory and experiment (Calado and Soares, 1977) at T = 161.38 K and P = 0 bar. Symbols as in Fig. 7.

experimental data. Figs. 9 and 10 show similar comparisons for the excess Gibbs free energy and excess volume at T = 161.38 K, P = 0 bar. In Figs. 7, 9 and 10, we have also presented the PTHO results for the phase equilibria, excess Gibbs free energy and excess volume, respectively. As is quite clear from these Figures, PTHN performs better than PTHO, particularly in the

3

i

.7@2

0

x (X4

‘r----

I 50 I I I I

o-2 0.L O-6 0.8

x (CO& Y (CW

experiment (Calado and Soares, 1977) at T = 161.38 Fig. 10. Excess volume for xenon + ethylene from theory and K and P = 0 bar. Symbols as in Fig. 7.

Fig. 11. Vapor-liquid phase equilibria for carbon dioxide +ethane from theory and experiment (Fredenslund and Mollerup, 1974; Mollerup, 1975) at T = 223 K. Symbols as in Fig. 7.

1

K. Shukla / Fluid Phase Equilibria 99 (1994) 153-166 163

I I I 1 1600 500-

I I I I

\ /

1200- \

\ /

8 geoo- / \

“X

3\

/ \

LOO- / \

/

00 o-2 0-I 0.6 0.8 x PXM x cod

Fig. 12. Excess Gibbs free energy for carbon dioxide + ethane from theory and experiment (Wallis et al., 1984) at T = 243 K and P = 0 bar. Symbols as in Fig. 7.

Fig. 13. Excess enthalpy for carbon dioxide + ethane from theory and experiment (Wallis et al., 1984) at T = 230 K and P = 0 bar. Symbols as in Fig. 7.

equimolar range of the compositions, demonstrating that PTHN accounts for the nonideal solution behavior in these quadrupolar mixtures very well.

In order to further test the adequacy of PTHN in describing phase equilibria and excess properties of even more nonideal fluid mixtures, we consider the mixture carbon dioxide + ethane, which is the most interesting nonideal fluid mixture investigated here. The molecules of

0 1 1

I I I I

0.I 0.8 x (Xe), Y (Xd

Fig. 14. Excess volume for carbon dioxide + ethane from theory and experiment (Wallis et al., 1984) at T = 243 K and P = 0 bar. Symbols as in Fig. 7.

Fig. 15. Vapor-liquid phase equilibria for xenon + hydrogen bromide from theory and.experiment (Calado et al., 1978) at T = 195.42 K. Symbols as in Fig. 7.

164 K. Shukla 1 Fluid Phase Equilibria 99 (1994) 153-166

Fig. 16. Excess Gibbs free energy for xenon + hydrogen bromide from theory and experiment (Calado et al., 1978) at T = 195.42 K and P = 0 bar. Symbols as in Fig. 7.

both components have strong multipole moments as well as large nonsphericity in their shapes. In addition, the potential parameters (E and o) of the pure components differ significantly (as can be seen in Table 1) where the usual van der Waals one-fluid theory generally fails, in particular, in describing the excess properties (Shukla et al., 1986; Shukla and Haile, 1987, 1988). Phase equilibriacalculations are performed at several temperatures. In all cases, temperature as well as composition dependences of the phase equilibria (including azeotropes) are explained reliably. As an example, Fig. 11 presents comparisons of PTHN predictions with experimental results for the phase equilibria at a single temperature, T = 223 K. Figs. 12- 14 show comparisons of the excess properties at P = 0. Even excess properties are explained very well. Also shown in Figs. 11-14 are results based on PTHO indicating that the van der Waals one-fluid theory is inadequate in describing the properties of the reference mixture with PTHO.

Finally, we consider a dipolar system xenon + hydrogen bromide, in which the first component is spherical, while the second one is strongly dipolar but almost spherical. As shown in Figs. 15 and 16, PTHN predictions for the phase equilibria and the excess Gibbs free energy are again in very good agreement with experimental data and are superior to those calculated with PTHO. The remaining discrepancy between PTHN and experiment in the hydrogen bromide- rich region might be attributed to the strong dipolar effect of the hydrogen bromide molecules which may not be well accounted for in the theory.

6. Conclusions

We have shown that the third-order perturbation theory presented in Part I of this work describes successfully the phase equilibria and thermodynamic excess properties of binary fluid mixtures consisting of spherical and moderately nonspherical molecules without fitting unlike parameters to the mixture data. A good comparison between theory and experiment has been achieved by computing the properties of the reference mixture using an accurate form of the perturbation theory

K. Shukla /Fluid Phase Equilibria 99 (1994) 153-166 165

of simple fluid mixtures (instead of the less reliable van der Waals one-fluid theory), and also by determining the unlike potential parameters adequately from the recently proposed set of combination rules. It is also demonstrated that FTHN represents a significant improvement over the similar form of the perturbation theory based on the equation of state for argon (PTHO), in which the properties of the reference mixture are calculated using the van der Waals one-fluid theory.

List of symbols

A

g:

J k K

L

Q r

ra T VE

Hehnholtz free energy excess Gibbs free energy excess enthalpy two-body integral Boltzmann constant three-body nonadditive integral three-body additive integral quadrupole moment intermolecular separation distance location of a site absolute temperature excess volume

Greek letters

4 isotropic polarizability E potential well-depth rc anisotropic polarizability

P dipole moment

P number density

: molecular diameter octopole moment

Abbreviations

dispObn three-body nonadditive dispersion interaction K-S-K Kohler-Smith-Kong combination rules (Eqs. 17 and 18) F’THN theory based on Eq. (1) PTHO theory based on Eq. (5)

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