Cubic Equation of State and Local Composition Mixing Rules

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Ind. Eng. Chem. Res. 2001, 40, 2544-2549

Cubic Equation of State and Local Composition Mixing Rules: Correlations and Predictions. Application to the Solubility of Solids in Supercritical Solvents E. Ruckenstein* and I. Shulgin† Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260

A family of new mixing rules for the cubic equation of state through a synthesis between the classical van der Waals mixing rule and the local composition concept is proposed. The binary interaction parameter in the van der Waals mixing rule, which is devoid of physical meaning, is thus replaced with a more physically meaningful parameter. The new mixing rules were used to correlate the dependence on pressure of the solubilities of a number of solid substances, including penicillins, in supercritical fluids. Because the new mixing rules contain physically meaningful parameters, dependent on the energies of the binary intermolecular interactions, the calculations of those energies can allow one to predict the solubilities of solids in supercritical fluids. Such predictions were made for the solubility of solid CCl4 in the supercritical CF4. The required binary intermolecular energies were computed with the help of quantum mechanical ab initio calculations, and a good agreement between the predicted and experimental solubilities was obtained. Introduction

P)

In a number of areas of modeling phase equilibria, the cubic equation of state (EOS) provided equal or even better results than the traditional approach based on the activity coefficient concept. In fact, for certain types of phase equilibria, the EOS is the only method that provided acceptable results. The solubility of solids in a supercritical fluid (SCF) constitutes such a case. For the solubility of a solid in a SCF [SCF (1) + solid solute (2)], one can write the well-known relation1

y2 )

[

]

P02 (P - P02)V02 exp Pφ2 RT

(1)

where P is the pressure, T is the temperature in K, R is the universal gas constant, φ2 is the fugacity coefficient of the solute in the binary mixture, y2 is the mole fraction at saturation of the solute, and P02 and V02 are the saturation vapor pressure and molar volume of the solid solute, respectively. Equation 1 shows that the solubility of a solid in SCF depends among others on the fugacity coefficient φ2, and as is well-known, this coefficient is responsible for the unusually large values of the solubility. These solubilities are much larger than those in ideal gases, and enhancement factors of 104-108 are not uncommon.2 They are, however, still relatively small and usually do not exceed several mole percent. The fugacity coefficient can be calculated using a suitable EOS. The Soave-Redlich-Kwong3 EOS (SRK EOS) will be employed in this paper. Starting from the SRK EOS * Correspondence author. E-mail: [email protected]. Fax: (716) 645-3822. † E-mail address: [email protected].

a(T) RT V - b V(V + b)

(2)

the following expression for the fugacity coefficient was obtained:4

RT ln φi ) -RT ln

]

[

V-b RT a + + V V - b b(V + b)

(

)

2 a V + b ∂(nb) 1 V + b ∂(n a) ln ln - RT ln z V ∂ni nb V ∂ni b2 (3)

In eqs 2 and 3, V is the molar volume, z is the compressibility factor, n is the total number of moles in the system, and ni is the number of moles of component i. Equations 2 and 3 show that the fugacity coefficient φ2 at a given pressure and temperature can be calculated if the parameters a and b and their derivatives with respect to the number of moles of solute are known. While near the critical point the fluctuations are important and an EOS involving them should be used,1 we neglect for the time being their effect. The mixture parameters a and b can be expressed in terms of those for the pure components aii and bii, using a variety of mixing rules starting from those of van der Waals4 to the modern ones.5,6 For the solubility of a solid in a SCF, the van der Waals mixing rules are most often used. They have the form

a)

∑i ∑j yiyjaij

(4)

b)

∑i ∑j yiyjbij

(5)

and

where yi is the mole fraction of component i.

10.1021/ie000955q CCC: $20.00 © 2001 American Chemical Society Published on Web 05/08/2001

Ind. Eng. Chem. Res., Vol. 40, No. 11, 2001 2545

When the parameters are applied to a binary mixture and considering that a12 ) xa11a22(1 - k12) and b12 ) (b11 + b22)/2, they become

a ) y12a11 + y22a22 + 2y1y2(a11a22)0.5(1 - k12) (6)

yii ) 1 -

yji ) (7)

where k12 is the interaction parameter. Using these mixing rules, one easily obtains

[

RT a V-b + + V V - b b(V + b) V+b 1 V+b a ln b11 - ln [2y1a11 + V b V b2

RT ln φ1 ) -RT ln

]

(

( ) ( )

yji )

)

(9)

(

)

(10)

λij - λii yi + yj exp RT

and

(

)

λij - λii RT yji ) λij - λii yi + yj exp RT yj exp -

(

with λij being the interaction parameter between molecules i and j.1,8 Vera and Panayiotou used the quasi-chemical approach and suggested the following expressions for the LCs:10,11

yj

(14)

∆ yj + yi exp yj RT

( )

where ∆ ) 2e12 ) e11 - e22 and eij is an interaction energy parameter between molecules i and j. Recently,15 a modification of the Wilson expressions for LCs was proposed by assuming that the interaction energy parameter between molecules of different types depends on the composition, and the following expressions were obtained:

yii )

yi

(13)

and

Theory and Formulas

yii )

(12)

1 + [1 - 4y1y2(1 - G12)]1/2

∆ yi exp yj RT yii ) ∆ yj + yi exp yj RT

2y2(a11a22)0.5(1 - k12)] - RT ln z (8)

1. LC Concept. According to the LC concept, the composition in the vicinity of any molecule differs from the overall composition. If a binary mixture is composed of components 1 and 2 with mole fractions y1 and y2, respectively, four LCs can be defined: the local mole fractions of components 1 and 2 near a central molecule 1 (y11 and y21) and the local mole fractions of components 1 and 2 near a central molecule 2 (y12 and y22). Numerous attempts have been made to express the LCs in terms of the bulk compositions and intermolecular interaction parameters.8-15 The idea of LC acquired acceptance starting with Wilson’s paper on phase equilibria,8 where the following expressions for the LCs were suggested:

2yj

where G12 ) exp[(11 + 22 - 212)/RT] and ij is the interaction energy parameter for the pair i and j. On the basis of a lattice model, the following expressions for LC were derived:12-14

)

Among the drawbacks of the van der Waals mixing rules, an important one concerns the interaction parameter k12. This quantity being purely empirical cannot be predicted on a physical basis. In addition, being temperature-dependent, it must be calculated for each isotherm. The aim of this paper is to suggest mixing rules based on the local composition (LC) concept1 that no longer involve empirical parameters such as k12 but parameters with a more clear physical meaning. The application of such rules to supercritical mixtures is most natural, because the near critical density fluctuations generate large local density and composition changes.7

(11)

1 + [1 - 4y1y2(1 - G12)]1/2

and

and

b ) y1b11 + y2b22

2yj

[

yi

]

(15)

]

(16)

yi(λ0ji - λ0ij) + λ0ij - λii yi + yj exp RT

and

[

]

yi(λ0ji - λ0ij) + λ0ij - λii yj exp RT yji ) 0 yi(λji - λ0ij) + λ0ij - λii yi + yj exp RT

[

where λ0ij is the interaction energy parameter between molecules i and j (i * j) when yi f 0 and λ0ji is the interaction energy parameter between molecules j and i (i * j) when yj f 0. 2. LC Mixing Rules. There were attempts16-20 to express the mixture parameters a and b in terms of the LC. However, most of the suggested mixing rules belong to the so-called density-dependent mixing rules (with the mixture parameter a being a function of the mixture density) and require information about the above density. Our considerations will be restricted to densityindependent mixing rules. The van der Waals parameters a and b are measures of the attractive energy in intermolecular interactions and size, respectively. It is, therefore, reasonable to express for a mixture these parameters in terms of LCs. In the present paper we suggest a family of LC mixing rules in which some or all of the bulk compositions in expressions (4) and (5) are replaced by LCs. Numerous expressions are possible, and calculations have been performed with many of them. We provide in the tables

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only results obtained with the following mixing rules, which appear to be more meaningful from a physical point of view: For the parameter a 2

2

a ) y1 a1 + y2 a2 + 2y11y22(a1a2)

0.5

a

b

eq 6 eq 17b eq 17b eq 17b eq 17b eq 17c

eq 7 eq 18a eq 18c eq 18a eq 18c eq 18a

(17b)

a ) y112a1 + y222a2 + 2y12y21(a1a2)0.5

(17c)

and



and for the parameter b

b ) y1b1 + y2b2

(18a)

b ) y11b1 + y22b2

(18b)

and



(18c)

where ai ) aii and bi ) bii. Many of the other combinations provided comparable results. If the LCs are expressed through eqs 11-14, then the expressions for the mixture parameters a and b will contain only one unknown parameter, G12 or ∆, instead of the interaction parameter k12. It should be noted that, in contrast to the interaction parameter k12, G12 and ∆ have a clear physical meaning connected with the intermolecular interaction energies. Furthermore, it was recently21,22 shown that the latter parameters can be calculated independently through an ab initio quantum mechanical calculation. Wilson’s expressions for the LCs (eqs 9 and 10) can also be used to generate mixing rules: the combination of eqs 9 and 10 with eqs 17a and 18b leads to twoparameter mixing rules, with parameters λ12 - λ11 and λ21 - λ22. Three-parameter mixing rules with param0 0 0 0 - λ11, λ21 - λ22, and λ21 - λ12 can be obtained eters λ12 by combining eqs 15 and 16 for the LCs and eqs 17a and 18b. This flexibility allows one to use two- or threeparameter mixing rules for systems for which the oneparameter mixing rules fail. It should be also noted that eqs 18b and 18c provide a temperature dependence for parameter b. In contrast, the conventional expressions (eq 18b), used in numerous mixing rules,6 provide no temperature dependence of parameter b, even though the direct calculation of a and b from experimental data indicated that b is slightly temperature-dependent.23 This family of LC mixing rules will now be applied to binary supercritical mixtures, but of course they can be applied to any kind of phase equilibria. Correlation of Solubility Data 1. Testing New Mixing Rules for the Supercritical Mixture CO2 + Naphthalene. The mixture CO2 + naphthalene was selected to test the new mixing rules because the solubility of naphthalene in supercritical CO2 was determined in numerous papers and reliable data at several temperatures are available. The calculations were carried out at three different temperatures (308, 318, and 328 K). The critical temperatures and pressures of naphthalene and CO2 were taken from refs 24 and 25 and the values of P02 and V02 for naphthalene

adjustable parameterb AAD,c %

LC

(17a)

a ) y112a1 + y222a2 + 2y11y22(a1a2)0.5

b ) y12b1 + y21b2

Table 1. Comparison between the LC Mixing Rules and the van der Waals Mixing Rules for the Correlation of Solubilitiesa of Naphthalene in Supercritical CO2 at Three Different Temperatures T ) 308 K eqs 13 and 14 eqs 13 and 14 eqs 11 and 12 eqs 11 and 12 eqs 11 and 12

0.09803 -243.32 -220.99 -243.63 -220.88 -236.90

11.36 10.80 10.84 11.00 10.84 11.58

T ) 318 K eqs 13 and 14 eqs 13 and 14 eqs 11 and 12 eqs 11 and 12 eqs 11 and 12

0.10047 -249.43 -224.65 -243.63 -224.51 -239.67

19.39 15.91 13.91 15.87 13.90 12.27

T ) 328 K eqs 13 and 14 eqs 13 and 14 eqs 11 and 12 eqs 11 and 12 eqs 11 and 12

0.09524 -222.87 -198.33 -222.40 -198.15 -209.60

14.56 8.03 8.29 7.99 8.30 11.28

a Experimental solubilities of naphthalene in supercritical CO 2 were taken from: Tsekhanskaya, Yu. V.; Iomtev, M. B.; Mushkina, E. V. Russ. J. Phys. Chem. 1964, 38, 1173. b Adjustable parameter k12 is dimensionless, and adjustable parameters 11 + 22 - 212 in eqs 11 and 12 and ∆ in eqs 13 and 14 are given in J/mol. n

calc exp AAD(%) ) 100[ ∑ abs(yexp 2 - y2 )/(ny2 )], where n is the number i)1 exp of experimental points, y2 is the experimental solubility, and ycalc is the calculated solubility. 2

c

from ref 25. Table 1 compares the new mixing rules with one adjustable parameter and the van der Waals mixing rule in a wide range of pressures and at three different temperatures. One can see from Table 1 that SRK EOS with the oneparameter new mixing rules describes the solubility of naphthalene in supercritical CO2 somewhat better (at 308 K) or better (at 318 and 328 K) than the van der Waals mixing rules. It should be noted that the energy parameter exhibits a weak temperature dependence. 2. Correlation of the Solubility of Solids in Various SCF. The new LC mixing rules were also tested for the solubility of a large number of solid solutes in various SCFs. The critical temperatures and pressures of solids and SCFs were taken from refs 24 and 25. The molar volumes of the solids and their saturated vapor pressures were taken from ref 25. The saturated vapor pressure of perylene was found in ref 26. The results are compared with the van der Waals mixing rules in Table 2, which shows that they are comparable. The parameters of SRK EOS (a and b) can be expressed by combining one of eqs 17a-c with one of eqs 18a-c. Only a few combinations have been included in Table 2; the other ones have also been tested and provided comparable results. 3. Correlation of the Solubilities of Penicillins V and G in SCF CO2. As already mentioned, the LC mixing rules can contain one, two, or three adjustable parameters. This flexibility has proven to be useful in representing the solubilities of antibiotic penicillins in SCF CO2. These solubilities could not be satisfactorily correlated by the cubic EOS with the conventional mixing rules,27,28 and several empirical expressions containing up to seven parameters were employed to correlate them.28 The LC mixing rules were used by us

Ind. Eng. Chem. Res., Vol. 40, No. 11, 2001 2547 Table 2. Comparison between the LC Mixing Rules and the van der Waals Mixing Rule for the Solubilities of Solids in SCFs at Various Temperaturesa AAD (%) with different mixing rules system NP + C2H4 NP + C2H4b NP + C2H4b NP + C2H4b 2,6-D + CO2c 2,6-D + CO2c 2,7-D + CO2c 2,7-D + CO2c PR + CO2d PR + CO2d PR + CO2d PR + CO2d AT + CO2e AT + CO2e AT + CO2e AT + C2H6e AT + C2H6e AT + C2H6e PH + CO2e PH + CO2e PH + CO2e PH + C2H6e PH + C2H6e PH + C2H6e PL + CO2e PL + CO2e PL + C2H6e b

T, K

M1

M2

M3

285.15 298.15 308.15 318.15 308.2 328.2 308.2 328.2 308.2 318.2 323.2 338.2 313 323 333 313 323 333 313 323 333 313 323 333 323 333 333

23.86 25.25 23.92 29.90 21.60 33.62 6.84 12.03 10.26 8.23 6.55 7.98 14.97 13.34 5.56 10.13 7.47 8.79 4.34 4.08 8.56 5.25 2.09 3.23 10.23 15.74 19.20

23.56 24.84 23.51 29.37 21.12 7.65 6.95 9.47 10.24 8.22 6.48 7.92 15.0 13.39 5.58 10.15 7.50 8.82 5.14 4.88 7.88 5.20 2.12 3.11 10.22 15.74 19.20

23.56 24.89 23.58 29.48 21.02 8.00 7.51 10.04 10.23 8.23 6.49 7.94 15.0 13.40 5.58 10.15 7.49 8.82 5.14 4.89 7.88 5.20 2.12 3.11 10.22 15.74 19.20

a In Table 2 the following abbreviations were used: NP, naphthalene; 2,6-D, 2,6-dimethylnaphthalene; 2,7-D, 2,7-dimethylnaphthalene; PR, pyrene; AT, anthracene; PH, phenanthrene; PL, perylene; M1, van der Waals mixing rules (eqs 6 and 7); M2, LC mixing rule (eqs 17b and 18a with LCs given by eqs 13 and 14); M3, LC mixing rules (eqs 17b and 18a with LCs given by eqs 11 and 12). b Tsekhanskaya, Yu. V.; Iomtev, M. B.; Mushkina, E. V. Russ. J. Phys. Chem. 1964, 38, 1173. c Iwai, Y.; Mori, Y.; Hosotani, H.; et al. J. Chem. Eng. Data 1993, 38, 509. d Bartle, K. D.; Clifford, A. A.; Jafar, S. A. J. Chem. Eng. Data 1990, 35, 355. e Anitescu, G.; Tavlarides, L. L. J. Supercrit. Fluids 1997, 10, 175.

to correlate those data. The critical temperatures and pressures and the acentric factor ω of penicillins V and G estimated in refs 27 and 28 were used. The saturated pressure of penicillin V was taken from ref 27, and that of penicillin G was estimated using the Lee and Kesler correlation.29 The results of the calculations are listed in Table 3, which shows that SRK EOS with one adjustable parameter LC mixing rules provided values comparable to those obtained with the van der Waals mixing rule. However, the two- and three-parameter LC mixing rules provided improvements in the correlations of experimental solubilities. While the improvement was achieved by adding additional parameters, it should be emphasized that the six-parameter empirical equation28

provided for the solubility of penicillin G in SCF CO2 the same accuracy as the new mixing rule with only three adjustable parameters. In addition, those three parameters have clear physical meaning, and there is the possibility for their prediction on the basis of quantum mechanical ab initio calculations, as shown for a more simple case below. 4. LCs Obtained during the Calculations Listed in Table 4 for the CO2/Naphthalene Mixture. One may note that y1 is somewhat smaller than y12, hence, that there is some enrichment of the solvent around a solute molecule. Experimental30 and integral equation studies31 have shown that the local density of the solvent around a solute is higher than its bulk density. The calculation of the densities from the mole fractions involves the correlation volume (where the local density differs from the bulk one) which is not available. Consequently, either the enhancement is mainly a density effect or the results reflect a limitation of the model employed. Prediction of the Solubility of Solid Substances in SCF To our knowledge, no successful prediction of the solubilities of solid substances in SCFs has been made. Because of the physical meaning of the parameters contained in the LC mixing rules, such an attempt becomes possible. Indeed, the new parameters depend on the intermolecular energies which can be calculated independently. Quantum mechanical ab initio calculations were performed recently to calculate the interaction energies between pairs of molecules in binary systems of water and alcohols or other organic compounds.21,22 These energies were used to calculate the Wilson and UNIQUAC parameters and then to successfully predict the activity coefficients. The interaction energies were calculated as follows:21,22 (a) A cluster composed of eight molecules (four of each kind) was considered to represent a dense fluid. (b) The cluster geometry was identified by an optimization procedure involving the PM3 semiempirical method, followed by the Hartree-Fock method with a 6-31 G** basis set. (c) Interacting molecular pairs (like and unlike pairs) were selected from the above optimized cluster. (d) The interaction energy of each molecular pair was computed using the Hartree-Fock method for the separation distances and orientation obtained in the previous steps. A similar approach with the following modifications was used in the present paper: (1) The more rigorous Møller-Plesset (MP) perturbation theory32,33 was selected instead of the Hartree-Fock method. (2) Clusters of two molecules were employed for the geometry

Table 3. Solubilitity of Penicillins V and G in Supercritical CO2, Described by SRK EOS with Various Mixing Rulesa one-parameter mixing rules, AAD (%)

two-parameter mixing rules, AAD (%)

three-parameter mixing rules, AAD (%)

system

temp, K

M1

M2

M3

M4

penicillin V + CO2b penicillin V + CO2b penicillin V + CO2b penicillin G + CO2c penicillin G + CO2c penicillin G + CO2c

314.85 324.85 334.85 313.15 323.15 333.15

39.09 41.42 51.96 29.08 28.37 41.87

39.52 41.51 51.96 29.06 28.31 41.82

33.50 41.46 51.81 29.06 28.30 41.81

33.16 40.78 46.51 24.32 21.50 27.40

a M1 ) van der Waals mixing rule (eqs 6 and 7), M2 ) LC mixing rule (eqs 17b and 18a with LCs given by eqs 13 and 14), M3 ) LC mixing rule (eqs 17b and 18a with LCs given by eqs 9 and 10), and M4 ) LC mixing rule (eqs 17a and 18c with LCs given by eqs 15 and 16). b Experimental solubility data were taken from ref 27. c Experimental solubility data were taken from ref 28.

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Table 4. Comparison between Local and Bulk Mole Fractions in CO2 (1) + Naphthalene (2) Mixture at T ) 308.15 K Found from Solubility Dataa pressure, bar

y1

y12

60.8 79.5 80.6 92.2 106.4 152.0 192.5 243.2 293.8 334.4

0.999 0.997 0.996 0.992 0.990 0.985 0.983 0.982 0.982 0.982

0.999 0.997 0.996 0.993 0.990 0.986 0.985 0.984 0.984 0.984

a The calculations for Table 4 were carried out using the LC mixing rules 17b and 18a with the LCs given by eqs 13 and 14. The calculations for the mixtures CO2 (1) + pyrene (2) at T ) 308.15 K and C2H6 (1) + phenanthrene (2) at T ) 313.15 K indicated a similar behavior.

Figure 1. Comparison between predicted (O) and experimental35 (b) solubilities of solid CCl4 in the SCF CF4 at 249 K. Parameter a is given by eq 17a and parameter b by eq 18a. The LCs are given by the Wilson equations (eqs 9 and 10).

Table 5. Interaction Energies by Ab Initio Calculations Using the MP2 Method with a 6-31G Basis Seta system CF4 (1) + CCl4 (2)

E11 E22 E12 ∆u12 ∆u21 (kJ/mol) (kJ/mol) (kJ/mol) (J/mol) (J/mol) -1.968

-5.528

-2.944

-976

2584

a

Eij is the energy of intermolecular interactions between molecules i and j, and ∆uij ) λij - λii are the Wilson parameters in eqs 9 and 10.

optimization and calculation of the intermolecular energies. Of course, it would have been preferable to consider larger clusters. However, the computational cost of such calculations for clusters containing seven to eight molecules by the MP method would have been prohibitively high. For the sake of illustration, the relatively simple system of nonpolar and symmetrical components [CF4 (1) + CCl4 (2)] was selected. Each pair (CF4 + CCl4, CF4 + CF4, and CCl4 + CCl4) was treated using the MP2 method (with the 6-31G basis sets) available in the standard Gaussian software.34 The results of these calculations are summarized in Table 5 and used to predict the solubility of solid CCl4 in the SCF CF4. Experimental data regarding the solubility of solid CCl4 in the SCF CF4 are available in the literature.35 The required data for the pure-component properties were taken from refs 24 and 36. The solubilities of solid CCl4 in the SCF CF4 were predicted for three different temperatures. Parameter a was calculated using eq 17a and parameter b using eq 18a. The LCs were expressed through the Wilson equations (eqs 9 and 10). A comparison between the predicted and experimental solubilities is presented in Figures 1-3, which show that the suggested method provides excellent predictions regarding the pressure dependence of the solubility of solid CCl4 in the SCF CF4 at 244 and 249 K but only satisfactory agreement at 234 K. However, the authors of ref 35 pointed out that the experiments at 234 K “proved to be unexpectedly difficult” and may have been affected by the presence of a third liquid phase. Conclusion A family of mixing rules for the cubic EOS was suggested in which the empirical binary interaction parameter k12 in the van der Waals mixing rule was replaced by a physically more meaningful parameter. In the new mixing rules, some mole fractions in the expressions of parameters a and b in the van der Waals mixing rules were replaced with various expressions for



the local mole fractions. The family of the new mixing rules can contain one, two, or even three adjustable parameters. The mixing rules were applied to the correlation of the solubilities of a number of solids in SCFs. One of the advantages of the new mixing rules is their flexibility regarding the number of adjustable parameters. In particular, it was shown that the new mixing rules with two or three adjustable parameters provided better correlations of the experimental data for the solubilities of the antibiotic penicillins in SCF CO2 than the conventional mixing rules or the empirical expressions containing many more parameters. Another attractive feature of the new mixing rules is that they allow one to predict the solubilities of solids in SCFs using only data for the pure components and the intermolecular interactions. In this paper, the solubilities of solid CCl4 in the SCF CF4 were predicted for three different temperatures. The energies of the intermolecular interactions (CF4 + CCl4, CF4 + CF4, and

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CCl4 + CCl4) were computed using quantum mechanical ab initio calculation. A good agreement was obtained. Acknowledgment The authors are indebted to the Center for Computational Research (CCR) of the University at Buffalo for the use of its facilities and to Dr. J. L. Tilson (CCR) for his help concerning the Gaussian software. We are also indebted to Mr. Amadeu Sum and Prof. S. I. Sandler (University of Delaware) for useful discussions regarding the ab initio calculations. Literature Cited (1) Prausnitz, J. M.; Lichtenthaler, R. N.; Gomes de Azevedo, E. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1986. (2) Eckert, C. A.; Knutson, B. L.; Debenedetti, P. G. Supercritical fluids as solvents for chemical and materials processing. Nature 1996, 383, 313-318. (3) Soave, G. Equilibrium constants from a modified RedlichKwong equation of state. Chem. Eng. Sci. 1972, 27, 1197-1203. (4) Elliot, J. R.; Lira, C. T. Introductory chemical engineering thermodynamics; Prentice-Hall PTR: Upper Saddle River, NJ, 1999. (5) Fischer, K.; Gmehling, J. Further development, status and results of the PSRK method for the prediction of vapor-liquid equilibria and gas solubilities. Fluid Phase Equilib. 1996, 121, 185-206. (6) Orbey, H.; Sandler, S. I. Modeling Vapor-Liquid Equilibria. Cubic Equation of State and Their Mixing Rules; Cambridge University Press: Cambridge, U.K., 1998. (7) Ruckenstein, E.; Shulgin, I. On density microheterogeneities in dilute supercritical solutions. J. Phys. Chem. B 2000, 104, 2540-2545. (8) Wilson, G. M. Vapor-liquid equilibrium. XI: A new expression for the excess free energy of mixing. J. Am. Chem. Soc. 1964, 86, 127-130. (9) Renon, H.; Prausnitz, J. M. Local composition in thermodynamic excess functions for liquid mixtures. AIChE J. 1968, 14, 135-144. (10) Panayiotou, C.; Vera, J. H. The quasi-chemical approach for non-randomness in liquid mixtures. Expressions for local composition with an application to polymer solution. Fluid Phase Equilib. 1980, 5, 55-80. (11) Panayiotou, C.; Vera, J. H. Local composition and local surface area fractions: a theoretical discussion. Can. J. Chem. Eng. 1981, 59, 501-505. (12) Lee, K.-H.; Sandler, S. I.; Patel, N. C. The generalized van der Waals partition function. 3. Local composition models for a mixture of equal size square-well molecules. Fluid Phase Equilib. 1986, 25, 31-49. (13) Aranovich, G. L.; Donohue, M. D. A new model for lattice systems. J. Chem. Phys. 1996, 105, 7059-7063. (14) Wu, D. W.; Cui, Y.; Donohue, M. D. Local composition models for lattice mixtures. Ind. Eng. Chem. Res. 1998, 37, 29362946. (15) Ruckenstein, E.; Shulgin, I. Modified local composition and Flory-Huggins equations for nonelectrolyte solutions. Ind. Eng. Chem. Res. 1999, 38, 4092-4099. (16) Mollerup, J. A note on excess Gibbs energy models, equations of state and the local composition concept. Fluid Phase Equilib. 1981, 7, 121-138.

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Received for review November 8, 2000 Revised manuscript received March 27, 2001 Accepted April 9, 2001 IE000955Q

Cubic Equation of State and Local Composition Mixing Rules

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