Global Fluid Phase Equilibria and Critical Phenomena of Selected

1350

J. Phys. Chem. B 2006, 110, 1350-1362

Global Fluid Phase Equilibria and Critical Phenomena of Selected Mixtures Using the Crossover Soft-SAFT Equation Fe` lix Llovell and Lourdes F. Vega* Institut de Cie` ncia de Materials de Barcelona, (ICMAB-CSIC), Consejo Superior de InVestigaciones Cientı´ficas, Campus de la U.A.B., Bellaterra, 08193 Barcelona, Spain ReceiVed: September 12, 2005; In Final Form: NoVember 19, 2005

We present here the extension of the crossover soft-statistical associating fluid theory (soft-SAFT) equation of state to mixtures, as well as some illustrative applications of the methodology to mixtures of particular scientific and technological interest. The procedure is based on White’s work (White, J. A. Fluid Phase Equilib. 1992, 75, 53) from the renormalization group theory, as for the pure fluids, with the isomorphism assumption applied to the mixtures. The equation is applied to three groups of mixtures: selected mixtures of n-alkanes, the CO2/n-alkane homologous series, and the CO2/1-alkanol homologous series. The crossover equation is first applied to the pure components of the mixtures, CO2 and the 1-alkanol family, while an available correlation is used for the molecular parameters of the n-alkane series (Llovell et al. J. Chem. Phys 2004, 121, 10715). A set of transferable molecular parameters is provided for the 1-alkanols series; these are accurate for the whole range of thermodynamic conditions. The crossover soft-SAFT equation is able to accurately describe these compounds near to and far from the critical point. The theory is then used to represent the phase behavior and the critical phenomena of the selected mixtures. We use binary interaction parameters ξ and η for dissimilar mixtures. These parameters are fitted at some particular conditions (one subcritical temperature or binary critical data) and used to predict the behavior of the mixture at different conditions (other subcritical conditions and/or critical conditions). The equation is able to capture the continuous change in the critical behavior of the CO2/n-alkane and the CO2/1-alkanol homologous series as the chain length of the second compound increases. Excellent agreement with experimental data is obtained, even in the most nonideal cases. The new equation is proved to be a powerful tool to study the global phase behavior of complex systems, as well as other thermodynamic properties of very challenging mixtures.

1. Introduction Most industrial processes require a detailed knowledge of the thermodynamic properties, including phase behavior and transport properties of their working fluids. Although the preferred method for obtaining these data would be the experimental one, there are several difficulties associated with it, mainly because of the great amount of data required to have a reliable database for multicomponent mixtures over a wide range of thermodynamic conditions. Theoretical approaches can be used as an alternative in this case. However, the intrinsic nonideal behavior of these mixtures and the limited range of available experimental data pose a challenge to any theoretical method aimed at quantitative predictions of thermodynamic properties for these complex fluids. One of the most challenging situations appears when the process works near the critical region. Because of their classical formulation, most equations of state are unable to accurately describe the density (and concentration) fluctuations appearing as the critical region is approached. Hence, the accurate prediction of phase equilibria and critical behavior of mixtures is a problem of scientific and technical relevance. Several efforts have been made in recent years toward the development of a global equations of state (EoS) with a crossover treatment that is able to describe with equal accuracy the regions far from and close to the critical point. The procedure followed to develop these crossover equations consists of linking * Corresponding author: [email protected].

an equation which incorporates the fluctuation-induced scaled thermodynamic behavior of fluids asymptotically close to the critical point, with an accurate equation which accounts for the classical behavior of the thermodynamic properties sufficiently far away from the critical point, where the effect of fluctuations becomes negligible.1 The efforts toward this direction can be summarized in three different approaches: a renormalized Landau expansion, followed by Senger and coauthors;1-4 numerical approaches based on the hierarchical theory, by Reatto and co-workers;5-7 and the phase-space cell approximation method of White and collaborators.8-11 An excellent review on the subject was provided by Anisimov and Senger,2 and a brief summary of the different approaches has been detailed in our previous work13 and also in the work of Sun et al.14 Of particular interest to the present work, also to be mentioned for historical reasons, is the work done by Kiselev and coauthors on developing crossover equations of state for engineering purposes. With the help of the crossover theory, Kiselev15 formally separated the Helmholtz free energy, calculated from a classical EoS, into two parts similar to those used in the crossover-Landau method.1-4 He then applied the crossover function to the critical part of the free energy, obtaining accurate results far from and close to the critical point. The main limitation of the equation comes from the phenomenological nature of the crossover theory used, which requires several (empirical) adjustable parameters to fit experimental data. Kiselev and Ely16 were the first ones who applied a crossover

10.1021/jp0551465 CCC: $33.50 © 2006 American Chemical Society Published on Web 12/23/2005

Global Fluid Phase Equilibria of Selected Mixtures treatment, in the spirit of Kiselev,15 to a SAFT-type equation, the statistical associating fluid theory (SAFT) equation of Huang and Radosz (SAFT-HR).17,18 Although this was a step forward on developing global equations of state from molecular-based theories, and theoretical predictions of the near-critical region were improved with this approach, the underlying limitations of the SAFT-HR equation showed up at lower temperatures. Later, McCabe and Kiselev19,20 applied Kiselev’s formalism to a more accurate SAFT-type equation (the SAFT-VR equation21), developing the SAFT-VRX equation of state. SAFT-VRX provides excellent predictions for several pure fluids and binary mixtures of n-alkanes and CO2/n-alkanes.14 A different approach on developing crossover molecularbased equations of state was taken from Prausnitz and coauthors,22-26 following White’s work. On the basis of Wilson and co-workers’ phase-cell space approximation,27,28 White derived a recursive procedure for pure fluids by using a modified free energy for the nonuniform fluid;8-12 the nonuniformity takes into account fluctuations in density appearing as the critical region is approached. White recursion theory provides coarsegraining to obtain an effective Hamiltonian in the renormalization group theory (RG). Then, one step of RG correction (i.e., integrating the short wavelength fluctuation) is applied to the effective Hamiltonian. By repeating the RG corrections, longer and longer wavelength fluctuations are included. Prausnitz and collaborators22-26 have used this approach for pure fluids, also extending it to mixtures, with the needed assumptions, as will be explained later. The procedure has been successfully applied to several pure fluids and binary mixtures of n-alkanes. In a recent work,13 we have extended the soft-SAFT equation27-32 to the critical region by combining it with a global renormalization term, in a similar way as done by Prausnitz and collaborators. The so-called crossover soft-SAFT equation is based on a set of recursive equations that let us describe the properties of Lennard-Jones (LJ) chain fluids near to and far from the critical region with a unique set of parameters. The accuracy of the equation was first checked by comparison with simulation data of pure LJ chains, finding excellent agreement in the critical region and reducing to the original equation far away from the critical point. The equation was also applied to the n-alkanes family, modeled as LJ chains, providing a new reliable correlation of molecular parameters, including the crossover parameters. The equation predicted, equally well, the regions far from and close to the critical point. Predictions of supercritical isotherms and isobars near the critical point were in excellent agreement with experimental data, improving the predictions of the original soft-SAFT equation. The equation was also able to provide the correct values of the universal critical exponents. As mentioned, most situations of practical importance in chemical engineering require the accurate predictions of multicomponent systems; however, one of the main problems of the RG theory is that it cannot be easily extended to mixtures. Some authors have proposed to calculate the fluctuations in density of each component in an independent way.22,25 The limitation of this approach is that it cannot capture the real nature of the system, because no coupling between the fluctuations of all the compounds is considered. A more practical solution can be adopted from the proposal of Fisher.33 This author assumed that the thermodynamic potential of a mixture has the same universal form as the thermodynamic potential of the onecomponent fluid in the case that an appropriate isomorphic variable is chosen. This is the so-called isomorphism assumption, which makes the RG theory suitable for mixtures. Since

J. Phys. Chem. B, Vol. 110, No. 3, 2006 1351 the density is the variable chosen to describe vapor-liquid equilibria, the global density of the system will become the new order parameter, substituting the density of one pure compound. The isomorphism approximation requires the evaluation of chemical potentials as independent variables, which is a difficult task since soft-SAFT, as most EoSs, uses mole fractions as independent variables. This problem can be overcome by choosing mole fractions as independent variables, as shown in the work of Kiselev and Friend34 and later adopted by Cai and Prausnitz26 and Sun et al.35 Although some scaling behavior would not be correctly described, this is still a good approximation, as already shown by these authors. The goal of this work is to extend the crossover soft-SAFT equation to binary mixtures using the isomorphism assumption33 and the approximation by Kiselev and Friend,34 in a similar way as done by Cai and Prausnitz.26 Then, the global behavior of several mixtures is studied to test the accuracy of the new equation for highly nonideal systems. The equation is first applied to selected mixtures of n-alkanes, the n-butane series, chosen for its regularity and also because it is a standard benchmark when developing accurate models for mixtures including the critical region.31,35 The equation is then used to study the continuity of the mixtures’ critical behavior in two different types of homologous series, the CO2/n-alkane mixtures and the CO2/1-alkanol mixtures. Galindo and Blas36 have already studied the global behavior of the CO2/n-alkane mixtures within the context of SAFT-type equations of state, by using the SAFT-VR equation.21 They provided an excellent systematic study on the behavior of these systems using the available experimental data. They were able to predict the transition from type II (according to the classification of mixtures made by Scott and Konynenburg37,38) for shorter n-alkanes to type III for longer n-alkanes, with the unlikely parameter fitted to ensure the type IV behavior of the CO2/n-tridecane mixture. The main limitation of their approach comes from the use of rescaled parameters: since they used the classical formulation of the SAFT-VR equation, different molecular parameters of the pure compounds were used for the subcritical and critical regions (for discussions on the use of rescaled parameters versus a crossover equation, see ref 13). The problem was lately overcome by Sun et al.,14,35 who used the SAFT-VRX equation19,20 (the extended SAFT-VR equation with a crossover term included) to systematically study the CO2/ n-alkane binary mixtures, obtaining excellent agreement with experimental data with a unique set of parameters. The CO2/n-alkanol mixtures represent one of the most severe tests for any theoretical (and experimental) approach, because they contain polar compounds with strong associative interactions. The mixture CO2/n-pentanol exhibits a very narrow, limited liquid-liquid immiscibility,39 belonging to type IV, which is the transition between type II (CO2/n-butanol and the previous homologous mixture) and type III (CO2/n-hexanol and the following homologous mixture). This continuous change happens in a very narrow range of the molecular weight of the second compound, contrary to what happens in the equivalent CO2/n-alkane homologous series. Therefore, the combination of the strong, nonideal interactions present in these mixtures with the narrow window in which the continuous change on the type of behavior happens makes the calculation of the global phase equilibria and critical lines of these mixtures a severe test for any theoretical approach. To the best of our knowledge there are not published studies on the applications of SAFTtype equations to these mixtures. The most extensive theoretical work related to these systems has been performed by Polishuk

1352 J. Phys. Chem. B, Vol. 110, No. 3, 2006 et al.40 They used a semipredictive approach (SPA) in which the estimation of the binary adjustable parameters for a certain member of a homologous series was obtained through the quantitative phase diagram method. The data were then used to predict other homologous mixtures. This approach was combined with three classical EoSs: the Peng-Robinson equation, the Trebble-Bishnoi-Salim equation, and a fourparameter equation (C4EOS). Among the three equations investigated, C4EOS was the most accurate one, especially for the prediction of the vapor-liquid critical lines. The equation was also capable of predicting the topology of the phase behavior, being the only equation able to qualitatively describe the transitional members of the CO2/n-alkanol homologous series, i.e., Type IV for n-pentanol and Type IIIm for the hexanol homologous. All three equations slightly underestimated the low-temperature vapor-liquid critical data. As explained by the authors, this is probably due to the inability of a cubic EoS to describe the increasing aggregation of alkanol molecules at low temperatures, which increases the experimental critical pressures. The rest of the paper is organized as follows. In Section 2, we present a brief overview of the soft-SAFT equation. In Section 3, we describe the recursive relations and the approaches made for mixtures in the crossover soft-SAFT equation. Section 4 briefly presents the methodology employed to calculate phase equilibria and critical lines of pure fluids and mixtures. Section 5 is devoted to results and discussion, and it is divided into four parts: (1) the parametrization of the 1-alkanols family and CO2, including quadrupolar interactions; (2) the study of the global phase behavior and critical lines of some mixtures of n-alkanes, to check the improvements versus the original softSAFT equation; (3) the study of some binary systems containing n-alkanes and CO2 to see the transition in the critical behavior; and (4) the study of CO2/1-alkanols homologous series to investigate the capability of the crossover equation to predict the continuous change in the critical behavior in this family as well as their phase behavior. The last section summarizes our findings, giving some concluding remarks. 2. Soft-SAFT Equation The soft-SAFT27-32 EoS belongs to a family of molecularbased equations, known as SAFT-type equations, based on Wertheim’s thermodynamics perturbation theory (TPT1).41-44 The original equation was proposed fifteen years ago,45,46 and it has been used since then by several researchers in both academic and industrial environments. SAFT has been especially successful in some engineering applications for which other classical EoSs fail. In addition, the microscopic components of the equation make it a very challenging approach from the fundamental point of view, since extensions and modifications of the equation can be systematically performed and compared to simulation data for the same underlying model. The success of the equation in its different versions is proved by the amount of published works since its development. The key of the success of SAFT-based equations is their solid statistical-mechanics basis, which allows a physical interpretation of the system. It provides a framework in which the effects of molecular shape and interactions on the thermodynamic properties can be separated and quantified. Besides, its parameters are few in number, have physical meaning, and are transferable, which makes SAFT a powerful tool for engineering predictions. The different versions of the equation, its modifications, and applications have been reviewed by Mu¨ller and Gubbins47 and also by Economou.48 SAFT equations are usually written in terms of the residual Hemholtz free energy, where each term in the equation

Llovell and Vega represents different microscopic contributions to the total free energy of the fluid. For associating chain systems, the equation is written as

ares ) aref + achain + aassoc

(1)

where ares is the residual Helmholtz free energy density of the system. The superscripts ref, chain, and assoc refer to the contributions from the monomer, the formation of the chain, and the associating sites, respectively. Most SAFT equations differ in the reference term,21,27,46,49 keeping formally identical the chain and the association term, both obtained from Wertheim’s theory. Soft-SAFT uses as the reference term a LJ spherical fluid, which accounts both for the repulsive and attractive interactions of the monomers forming the chain. The free energy and derived thermodynamics of a mixture of LJ fluids are obtained in this work through the accurate equation of Johnson et al.50 The chain contribution for a LJ fluid of tangent spherical segments, obtained through Wertheim’s theory, in terms of the chain length m and the pair correlation function gLJ of LJ monomers, evaluated at the bond length σ, is

achain ) FkBT

∑i xi(1 - mi) ln gLJ

(2)

where F is the molecular density of the fluid, T is the temperature, m is the chain length, kB is the Boltzmann constant, and gLJ is the radial distribution function of a fluid of LJ spheres at density Fm ) mF. We use the function fitted to computer simulation data for gLJ, as a function of density and temperature, provided by Johnson et al.51 The association term, within the first-order Wertheim’s perturbation theory for associating fluids, is expressed as the sum of contributions of all associating sites of component i,

a

assoc

) FkBT

(

∑i ∑R xi

ln

XRi

-

)

XRi 2

+

Mi 2

(3)

with Mi being the number of associating sites of component i and XRi being the mole fraction of molecules of component i nonbonded at site R, which accounts for the contributions of all the associating sites in each species:

1

XRi ) 1 + N AF

∑j xj∑β

(4) Xβj ∆Riβj

The term ∆Riβj is related to the strength of the association bond between site R in molecule i and site β in molecule j. For the square-well bonding potential and a spherical geometry of the association sites,52,53 the simplified expression is

[ ( ) ]

∆Riβj ) 4π exp

Riβj - 1 kRiβjI T

(5)

where Riβj is the association energy and kRiβj represents the association volume for each association site and compound, while I is a dimensionless integral converted into a numerical function of the temperature and density.54 Since the Hemholtz free energy density is calculated by adding different terms, each of them should be expressed in terms of composition for mixture studies. Note that the chain and association term are already applicable to mixtures. Hence, the extension needs to be performed just for the reference term.

Global Fluid Phase Equilibria of Selected Mixtures

J. Phys. Chem. B, Vol. 110, No. 3, 2006 1353

As in our previous work,30,31 the van der Waals one-fluid theory (vdW-1f) is used to describe the monomer contribution aref. In this theory, the residual Hemholtz free energy density of the mixture is approximated by the residual Hemholtz free energy density of a pure hypothetical fluid, with parameters σm and m calculated from the following: n

σ3m )

n

mimjxixjσij3 ∑ ∑ i)1 j)1 n

n

(6)

∑ ∑mimjxixj i)1 j)1 n

σ3m )

n

mimjxixjijσ3ij ∑ ∑ i)1 j)1 n

n

(7)

∑ ∑mimjxixj i)1 j)1

the density fluctuations due to the attractive part of the potential is then divided into short-wavelength and long-wavelength contributions. It is assumed that contributions from fluctuations of wavelengths less than a certain cutoff length L can be accurately evaluated by a mean-field theory. The effect of this short wavelength can be calculated using the soft-SAFT equation, or any other mean-field theory. However, the choice of the meanfield theory is of relevance for the overall behavior of the crossover equation. The RG term corrects the approach to the critical region, but it does not improve the performance of the underlying original equation far from the critical point. The strengths and limitations of the original equation will always be there. This is why different crossover equations have been developed and continue to be under development nowadays. The contribution of the long-wavelength density fluctuations is taken into account through the phase-space cell approximation.9 In a recursive manner, the Helmholtz free energy per volume of a system at density F can be described as8-11

an(F) ) an-1(F) + dan(F)

(11)

The effective chain length of the conformal fluid is given by n

m)

ximi ∑ i)1

(8)

The above expressions involve the mole fraction xi and the chain length mi of each of the components of the mixture of chains. The crossed interaction parameters σij and ij are calculated using the generalized Lorentz-Berthelot combining rules,

σii + σij σij ) ηij 2

(9)

ij ) ξijxiijj

(10)

where the factors ηij and ξij modify the arithmetic and geometric averages between components i and j, which are the adjustable binary parameters of this equation. With the expressions in eqs 6-10, the reference term is expressed as a function of the chain molar fractions. The validity of these rules to achieve excellent results has been proved in other works. 32,55,56

where a is the Hemholtz free energy density and dan is the term where long-wavelength fluctuations are accounted for in the following way,

Ωsn(F) Fmax dan(F) ) -Kn ln l , 0 e F e 2 Ωn(F) dan(F) ) 0,

(12b)

where Ωs and Ωl represent the density fluctuations for the shortrange and the long-range attraction, respectively, and Kn is a coefficient:

Kn )

kBT

(13)

23nL3

T is the temperature, and L is the cutoff length.

Ωβn (F) )

3. Crossover Soft-SAFT Equation for Mixtures White’s global RG theory consists of a set of recursion relations where the contribution of longer and longer wavelength density fluctuations up to the correlation length is successively taken into account in the free energy density. In this way, properties approach the asymptotic behavior in the critical region, and they exhibit a crossover between the classical and the universal scaling behavior in the near-critical region. We follow here the implementation of White’s global RG method, as done by Prausnitz and collaborators.22,26 Since the method has been widely explained in our previous work,13 only a brief summary of the most relevant details is presented here, with emphasis to the extension to mixtures. To include the long-wavelength fluctuations into the free energy, the interaction potential is divided into a reference contribution, due mainly to the repulsive interactions, and a perturbative contribution, due mainly to the attractive interactions. The RG theory27,28 is only applied to the attractive part, since it is considered that the other term contributes mostly with density fluctuations of very short wavelengths. The effect of

Fmax e F e Fmax 2

(12a)

Gβn (F,

x) )

∫0F exp

(

)

-Gβn (F, x) dx Kn

ajβn (F + x) + ajβn (F + x) - 2ajβn (F) 2

(14)

(15)

The superindex β refers to both long-range (l) and short-range (s) attractions, respectively, and Gβ is a function that depends on the evaluation of the function aj, calculated as,

ajln(F) ) an-1(F) + R(mF)2 ajsn(F) ) an-1(F) + R(mF)2

(16)

φw2 2 L

2n+1 2

(17)

where m is the chain length (number of LJ segments forming the chain), φ is an adjustable parameter, R is the interaction volume with units of energy-volume, and w refers to the range of the attractive potential. For the LJ fluid, R and w are given by

R)

-1 2

∫σ∞ uLJ4πr2 dr ) 16πσ 9

3

(18)

1354 J. Phys. Chem. B, Vol. 110, No. 3, 2006

w2 )

-1 3!R

∫σ∞r2uLJ4πr2 dr ) 9σ7

Llovell and Vega

2

(19)

The above procedure can be interpreted as the ratio of nonmeanfield contributions to mean-field contributions as the wavelength is increased. a0 is the value obtained from the equation of state employed, and it will be corrected when the long fluctuations are taken into account through each iteration. In practice, we have observed, as some other authors did,9,22,23,57 that, after five iterations, there is not a further change in the Helmholtz free energy. Fmax is the maximum possible molecular density, and it depends on the selected model. In the case of a LJ model, we therefore set the maximum density as

Fmax )

1 mNAσ3

n

xiLi3 ∑ i)1

(21)

The crossover parameter φ is calculated in the same way as the chain length: n

φ)

xiφi ∑ i)1

4. Phase-Equilibria and Critical Lines Calculations For phase-equilibria calculations, the commonly used fugacity method is employed.59 That is, chemical, thermal, and mechanical stability are satisfied by imposing the equality of chemical potentials of each component in the coexisting phases at fixed temperature and pressure. Because SAFT is formulated as an explicit function of temperature, density, and phase compositions, the fugacity method is applied by equating chemical potentials and the pressure at a fixed temperature:

(20)

The integral in eq 14 is evaluated numerically, by the trapezoid rule. The density step has been set to obtain good accuracy, without compromising the speed of the calculations. As for the case of pure fluids, a density step F/500 has been used for mixtures. To extend White’s theory to mixtures, the isomorphism assumption33 is used. Following the RG theory, it is assumed that the order parameter plays a crucial role in the Hamiltonian. Since the density is the order parameter in our case, according to the isomorphism assumption, the order parameter to describe vapor-liquid equilibria in mixtures is the total density of the system. Therefore, the one-component density must be replaced by the total density of the components. However, according to the isomorphism assumption, the chemical potentials must be chosen as independent variables when we calculate the integration in eq 14, i.e., the integration should be performed at a fixed chemical potential. This requirement makes computation extremely difficult. Therefore, following other authors,26,35 we have decided to use the approximation of Kiselev and Friend,34 replacing constant chemical potentials by constant mole fractions in performing the integral. It is important to remark that the choice of one order parameter or another will make the equation suitable for a particular kind of mixture equilibria. For instance, the approximations made here make the equation suitable for vapor-liquid equilibria; a different order parameter will be required in the case of liquid-liquid equilibria.58 In addition to the approximation of Kiselev and Friend,34 the extension of the equation to mixtures also requires the introduction of the mixing rules to determine the crossover parameters L and φ. Because of the fact that the cutoff length L is the diameter of the three-dimensional space for characterizing density fluctuations, we have chosen the following expression for the mixture:

L3 )

parameters for mixtures presented in eqs 6-10 are used, instead of the pure-component parameters mi, σi, and i.

(22)

In addition to these crossover parameters, the soft-SAFT

PI(T,FI,xI) ) PII(T,FII,xII)

(23)

µIi (T,FI,xI) ) µIIi (T,FII,xII)

(24)

The calculation of critical properties of pure fluids and mixtures is done by numerically solving the necessary conditions,60 which involves second and third derivatives of the Gibbs free energy with respect to the molar volume (pure systems) or the composition (in mixtures):

( ) ( ) ∂2G ∂x2

∂ 3G ∂x3

)

P,T

)0

(25)

P,T

In addition to these equations, classical stability for critical points is required through the following extra condition:

( ) ∂4G ∂x4

>0

(26)

P,T

Since the soft-SAFT EoS is given in terms of the Helmholtz energy, its natural thermodynamic variables are temperature, volume, and composition. Hence, it is more convenient, as in other equations of state (van der Waals type), to express the critical conditions in terms of derivatives of the Helmholtz free energy with respect to volume and composition, at constant temperature,31

A2xA2V - A2Vx ) 0 A3x - 3AV2x

( )

( )

(27)

( )

AVx AVx 2 AVx 3 + 3A2Vx - A3V ) 0 (28) A2V A2V A2V

where the notation AnVmx ) (δn+mA/δVnδxm) is used for the derivatives of the Helmholtz free energy. 5. Results and Discussion When the original soft-SAFT equation is applied to associating chain systems, five molecular parameters are needed to describe the molecule: m, the chain length, σ, the diameter of the LJ spheres forming the chain, , the interaction energy among them, and, finally, the association volume kHB and association energy HB of the sites of the molecule. These parameters are treated as adjustable when using the equation for real fluids. The inclusion of the crossover treatment leads to two additional parameters, the cutoff length, L, related to the maximum wavelength fluctuations that are accounted for in the uncorrected free energy, and φ, the average gradient of the wavelet function.11 Moreover, when dealing with nonideal mixtures, binary parameters η and ξ are usually required. A. The 1-Alkanols Family and CO2. Following the work done by Pa`mies,61 1-alkanols are modeled as homonuclear



TABLE 1: Molecular Parameters for the 1-Alkanols (C1-C8) and CO2, Optimized Using Experimental Data from Refs 64 and 65, Using Soft-SAFT with the Renormalization Term Included M methanol ethanol 1-propanol 1-butanol 1-pentanol 1-hexanol 1-heptanol 1-octanol

CO2

1.481 1.710 1.941 2.210 2.470 2.686 2.920 3.148

σ (Å) /k (K) 3.390 3.659 3.815 3.934 4.020 4.110 4.170 4.212

227.4 240.0 249.8 266.5 279.5 291.0 299.5 306.0

/kB (K) k (Å3)

φ

L/σ

7.70 7.00 7.30 7.65 7.83 8.00 8.10 8.25

1.390 1.300 1.320 1.350 1.370 1.380 1.390 1.395

3193 3470 3600 3600 3600 3600 3600 3600

4907 2300 2300 2300 2300 2300 2300 2300

m

σ (Å)

/k (K)

φ

L/σ

Q (C m2)

1.606

3.158

159.9

5.79

1.180

4.40 × 10-40

chainlike molecules of equal diameter σ and the same dispersive energy . The hydroxyl group in alkanols is mimicked by two square-well sites embedded off-center in one of the LJ segments, with volume kHB and association energy HB. According to our model, these five molecular parameters, plus the crossover parameters φ and L, are enough to describe all thermodynamic properties. The quadrupolar interactions present in CO2 are taken into account by an additional term in the equation. This term involves a new molecular parameter Q that represents the quadrupolar moment of the molecule.62,63 All the parameters have been calculated fitting experimental saturated liquid densities and vapor pressures for each compound. Since the length of the chain should not affect the strength of the association bonds, except for very short molecules, the parameters of the association sites are set at constant values, except for those of methanol and ethanol. The values of the fitted molecular parameters are presented in Table 1. All the parameters show physical trends (i.e., they increase as the molecular weight of the compound increases, until they reach an asymptotic value for relatively long chains, as should happen since this is a united atom approach). The exception is methanol, which is already known for its anomalous behavior. As done in previous works,13,31,32 a correlation from optimized parameters for the eight first members (excepting methanol) of the 1-alkanols series is obtained:

m ) 0.0173Mw + 0.921

(29a)

mσ3 ) 1.83Mw - 0.725

(29b)

m/kB ) 6.73Mw + 93.1

(29c)

mφ ) 0.168Mw + 4.31

(29d)

mL/σ ) 0.0262Mw + 1.03

(29e)

Units of σ and /kB are Å and K, respectively. Mw is the 1-alkanol molecular weight expressed in g/mol. The advantage of having a correlation is the ability to describe heavier 1-alkanols with the same degree of accuracy through the parameters extrapolation from these correlations. Moreover, this new set of parameters is able to describe equally well the vaporliquid equilibria diagram and the critical region of these fluids, in contrast to previous correlations which were accurate for only one of the two regions. Figure 1a shows the coexistence curve for the eight first members of the n-alkanols series. Symbols are experimental data64 and the solid line is from the crossover soft-SAFT equation.

Figure 1. (a) Temperature-density diagram for the light members of the 1-alkanols series, from methanol to 1-octanol. (b) Pressure-density diagram for light members of the 1-alkanols series, from methanol to 1-octanol. Symbols represent the experimental data taken from ref 64, and the critical points are from ref 65. The solid lines represent softSAFT + crossover predictions.

Agreement is excellent in all cases, correcting the behavior in the critical region due to the incorporation of density fluctuations into the equation. Figure 1b shows vapor pressures for the same members of the series.65 Again, excellent results are achieved. Parts a and b of Figure 2 depict the coexistence curve and the vapor pressure calculations for CO2 with and without crossover treatment. Quadrupolar interactions are included in both cases. Once more, crossover soft-SAFT quantitatively reproduces the behavior of this compound near to and far from the critical point, avoiding the overestimations that were obtained with the original soft-SAFT equation. It is important to remark that a good description could also have been obtained without considering the quadrupolar contribution. However, although it does not seem important for the pure fluid, the quadrupole moment has a remarkable effect in mixtures and it should be considered in order to have a better physical description of the system.62,63


Figure 2. (a) Temperature-density diagram for CO2. b) Pressuredensity diagram for CO2. Symbols are experimental data from ref 65, dashed lines are soft-SAFT predictions, and the solid lines are the crossover soft-SAFT estimation.

The experimental critical temperature, pressure, and density and those obtained from the original soft-SAFT and crossover soft-SAFT equations for the first eight members of the 1-alkanols series and CO2 are presented in Table 2. While the critical temperature and pressure are estimated with great accuracy by the crossover soft-SAFT equation, the critical density is, as in previous works with alkanes,13 slightly overestimated. This is the price paid by fitting the crossover parameters to the critical temperature in the temperature-density diagram, as already explained in ref 13. B. n-Alkanes Mixtures. As has been mentioned above, we intend to check the ability of the crossover soft-SAFT equation to model different kinds of binary mixtures. A first test is made here with some n-alkane binary mixtures. The molecular parameters of the n-alkanes considered have been taken from the correlation proposed in our previous work.13 Although all compounds are of the same nature, they become more dissimilar as the chain length of the second compound is increased. The binary parameters η and ξ take into account deviations between the size and energy of the segments forming the different compounds, allowing again a quantitative description of the mixture. When needed, we have obtained them by fitting to one mixture at an intermediate temperature (pressure).

Llovell and Vega Once this Pxy (Txy) diagram is optimized, the parameters are used to predict the phase equilibria of the mixture at different temperatures as well as the critical lines in a purely (semi)predictive manner. Binary parameters for these mixtures are presented in Table 3. Because n-butane and n-pentane are very similar compounds, no binary parameters are needed for this mixture. The binary parameters of the other mixtures are also very close to unity, although they deviate from unity as the two components become more dissimilar; they have been fitted just to obtain quantitative agreement with the experimental data, though a good description of the mixtures could be achieved when fixing them to unity. Figures 3 and 4 show Txy projections at two different pressures for the mixtures n-butane/n-hexane and n-butane/n-octane. Quantitative agreement is obtained in all cases; it is impossible to distinguish which one was the pressure used to fit the binary parameters and which one is the predicted one. The same binary parameters were used to predict the critical lines of the mixtures. Figure 5 shows the PT projections of the PTxy surfaces for the n-butane series, with mixtures between n-butane and n-pentane, n-hexane, n-heptane, and n-octane. In these mixtures, the two components are of similar nature and the mixture exhibits a continuous gas-liquid critical line connecting the critical points of the two pure components, corresponding to type I behavior. It is important to notice that, if we rescale the parameters of the original soft-SAFT equation to the critical point of the pure compounds,31 excellent predictions are also obtained. However, with the rescaling method, the behavior in the subcritical region is not accurately reproduced, and a different set of parameters is needed to predict the phase equilibria of the binary mixture far from the critical point. The advantage of the crossover soft-SAFT equation is the ability to have a unique set of parameters for the whole range of a fluid. C. Mixtures of CO2 with Alkanes. The next step of this study is to test the accuracy of the extended equation when dealing with mixtures of CO2/n-alkanes. It is especially interesting to check if the equation is able to reproduce the continuous change of behavior observed in these mixtures when the chain length of the second compound is increased. For this purpose, four different systems have been selected with increasing chain length: CO2/ethane, to see the effect of carbon dioxide in a short alkane, and then the three mixtures CO2/n-decane, CO2/ tridecane, and CO2/hexadecane, as representatives of the three different types of behavior experimentally observed in these mixtures. It has been shown that the CO2/decane mixture shows type II behavior. When the chain length of the second compound increases and evolves till tridecane, the type IV behavior appears and the critical line is split in two parts, with a liquid-liquidvapor region joining both. Finally, the CO2/hexadecane mixture has already evolved to type III, with a continuous critical line ending at infinite pressures. Figure 6a shows the vapor-liquid equilibria of the mixture CO2/ethane at different temperatures. Symbols represent experimental data,69 and solid lines are the equation estimations. The energy binary parameter was fitted to the intermediate subcritical temperature of 283 K (ξ ) 0.990, also given in Table 3), while the size binary parameter was kept equal to unity. Very good agreement is observed between the modeling and the experimental values at all temperatures. The azeotropes are predicted at the correct pressure and mole fraction. Figure 6b depicts the predicted critical line of this mixture, with the binary parameters obtained from the fitting at T ) 283 K. The trend of the curve is perfectly captured by the equation, providing an excellent description of the behavior of this mixture. It is



TABLE 2: Critical Constants for the 1-Alkanols (C1-C8) and CO2 (Experimental Data from Ref 65) Tc (K)

Pc (MPa)

Dc (mol/L)

compound

exp.

crossover soft-SAFT

soft-SAFT

exp.

crossover soft-SAFT

soft-SAFT

exp.

crossover soft-SAFT

soft-SAFT

methanol ethanol 1-propanol 1-butanol 1-pentanol 1-hexanol 1-heptanol 1-octanol CO2

512.6 513.9 536.8 563.1 588.1 611.4 633.0 655.0 304.1

513.4 513.3 536.4 563.0 587.5 611.3 633.0 653.5 303.9

541.7 540.5 560.8 588.0 616.4 642.3 668.5 695.5 316.0

8.10 6.15 5.18 4.42 3.90 3.51 3.06 2.78 7.38

8.00 6.18 5.17 4.64 4.08 3.62 3.16 2.80 7.46

10.92 8.47 7.06 5.97 5.18 4.53 4.06 3.76 9.11

8.48 5.99 4.55 3.85 3.06 2.63 2.30 2.01 10.62

9.03 5.52 4.40 3.85 3.14 2.68 2.32 2.05 11.45

7.96 5.60 4.31 3.42 2.82 2.35 2.02 1.76 9.80

TABLE 3: Binary Parameters Employed for Different Binary Mixtures between Alkanes, CO2/Alkanes, and CO2/1-Alkanols mixture

η

ξ

n-butane/n-pentane n-butane/n-hexane n-butane/n-heptane n-butane/n-octane CO2/ethane CO2/decane CO2/tridecane CO2/hexadecane CO2/1-propanol CO2/1-butanol CO2/1-pentanol CO2/1-hexanol

1.000 1.000 1.010 1.030 1.000 1.025 1.025 1.025 1.010 1.010 1.010 1.010

1.000 0.990 0.980 0.960 0.990 0.950 0.950 0.950 0.970 0.970 0.970 0.970

Figure 4. Temperature-composition diagram of the binary mixture n-butane/n-octane at 2.07 and 2.76 MPa. Symbols are experimental data from ref 68. Solid lines are crossover soft-SAFT predictions.

Figure 3. Temperature-composition diagram of the binary mixture n-butane/n-hexane at 2.58 and 3.10 MPa. Symbols are experimental data from ref 67. Solid lines are crossover soft-SAFT predictions.

important to notice that, although this mixture corresponds to the type I behavior, the shape of the critical line, with a temperature minima between the two compounds, is not straightforward to obtain with an EoS. It is assumed that the particular behavior of this mixture is due to the quadrupolar interactions of the CO2 molecule as well as the short length of the n-alkane. For instance, in some other works,36 it was necessary to use two binary parameters with values far away from unity, since quadrupolar interactions were not explicitly considered in that case. In our case, just one binary parameter, very close to unity, is enough to obtain an excellent representation of this mixture. Parts a-c of Figure 7 show the PT projections for the mixtures CO2/decane, CO2/tridecane, and CO2/hexadecane, respectively, as obtained from the crossover soft-SAFT equation.

Figure 5. PT projections of the binary mixtures of the n-butane series: n-butane/n-pentane, n-butane/n-hexane, n-butane/n-heptane, and n-butane/n-octane. Symbols are experimental data from ref 66. The solid lines are the predictions from the crossover soft-SAFT equation.

Symbols represent the experimental data, taken from refs 70, 71, and 72, respectively. As we have said, our goal is to study the evolution of the critical behavior. In this case, and because of the lack of experimental vapor-liquid equilibria data, the parameters η ) 1.025 and ξ ) 0.950 have been fitted to the critical points data of the CO2/tridecane mixture and used to


Llovell and Vega

Figure 6. (a) Pxy projections of the CO2/ethane mixtures at 263.15, 283.15, and 293.15 K. (b) PT projection of the PTx surface for the CO2/ethane mixtures. Symbols represent experimental data,69 and solid lines are the soft-SAFT predictions. The dashed line corresponds to the critical line of the mixture.

predict the behavior of the homologous series. In general, very good agreement is achieved in all cases. CO2/decane shows type II behavior, with the critical line joining the critical point of both pure compounds. CO2/tridecane, belonging to type IV behavior, becomes the most difficult mixture to predict, because it is in the intermediate changing behavior with a very narrow range before the series evolves to type III as the chain length of the second compound increases. However, crossover softSAFT is able to reproduce this shape, with the appropriate binary parameters. The three-phase line is found and the upper critical end point (UCEP) and the lower critical end point (LCEP) are compared to experimental values in Table 4. Finally, CO2/ hexadecane exhibits a clear type III behavior. The critical line goes to infinite pressures, and very good agreement is obtained with experimental data, although the border between the gasliquid and the liquid-liquid could be better predicted. Note that this could be achieved if binary parameters were optimized for this particular mixture, but in this case, the binary parameters were taken from the CO2/tridecane mixture. The purpose of this

Figure 7. PT projections of the PTx surface for several CO2/n-alkane mixtures: (a) CO2/n-decane, (b) CO2/tridecane, and (c) CO2/hexadecane. Gas-liquid and liquid-liquid critical point data are obtained from refs 70-72. The lines are crossover soft-SAFT predictions.

study is to use the same binary parameters to predict the evolution in the CO2/n-alkanes homologous series; the results indicate this achievement and the capability of the crossover



Figure 8. PT projection of the PTx surface for the CO2/1-propanol binary mixture. Symbols represent experimental data,75 and solid lines are the crossover soft-SAFT predictions.

TABLE 4: Critical End Point (CEP) Temperature and Pressure for Type IV Binary Mixtures CO2/n-Tridecane and CO2/1-Pentanol (Experimental Data Obtained from Ref 76) CO2/tridecane mixture

Texp

TSAFT

Pexp

PSAFT

UCEP LCEP UCEP

279.0 310.8 314.0

289.0 305.6 315.2

3.933 8.114 8.716

4.408 5.751 7.625

CO2/1-pentanol mixture

Texp

TSAFT

Pexp

PSAFT

UCEP LCEP UCEP

273.5 316.0 317.0

283.4 294.7 301.8

3.413 8.752 8.950

3.981 4.689 6.325

soft-SAFT equation to predict critical behavior of this type of mixture in excellent agreement with experimental data. D. Mixtures of CO2 with Alkanols. To check the continuous change of behavior in this homologous series, we have selected the mixtures from 1-propanol till 1-hexanol to examine the critical region. According to the literature,39,40 CO2/alkanol mixtures evolve from type II to type III behavior. CO2/1propanol and CO2/1-butanol have been confirmed to show type II behavior, while the mixture with 1-hexanol is already a type III. Some discussion has been reported about the CO2/1-pentanol binary mixture: Raeissi et al.39 have identified it as a type IV, finding that this particular mixture exhibits a very narrow part of the limited liquid-liquid inmiscibility, which has been missed by other authors.73,74 Because of the narrow range in which this behavior appears, this is a very challenging mixture for any theoretical approach. We investigate here the performance of the crossover soft-SAFT equation with these mixtures, trying to find a global modeling for the homologous series, as we did for the CO2/n-alkanes homologous series. Figures 8-11 show the critical lines for the CO2/1-propanol, CO2/1-butanol, CO2/1-pentanol, and CO2/1-hexanol binary mixtures, respectively. Symbols represent the experimental data taken from refs 75-77. As previously done for the n-alkane binary mixtures, binary parameters could be fitted to experimental data from a single subcritical isotherm for each mixture and used to predict the behavior of the same mixture at other thermodynamic conditions, including the critical lines. Alternatively, as we did with the CO2/n-alkanes mixtures with heavy compounds, binary parameters can also be fitted to one selected mixture and used to predict the evolution of the critical line of the homologous series and the subcritical behavior of these

Figure 9. PT projection of the PTx surface for the CO2/1-butanol binary mixture. Symbols represent experimental data,75,76 and solid lines are the crossover soft-SAFT predictions.

Figure 10. PT projection of the PTx surface for the CO2 /1-pentanol binary mixture. Symbols represent the experimental critical end points as obtained in ref 39, while lines are crossover soft-SAFT predictions.

mixtures. Because we are interested in describing the critical line evolution in the homologous series as the length of the second compound increases, we have decided to follow this approach in order to check the capability of the equation to predict this behavior with a unique set of binary parameters for the whole series (although in some cases, this may not be the best fit to the available data). Using a value of η) 0.970 and ξ ) 1.01, an accurate description of the overall critical behavior is achieved. CO2/1-propanol and CO2/1-butanol show type II behavior, with the critical line ending at the critical point of the pure compounds. Excellent agreement between the softSAFT description and the experimental data75,76 is achieved for the two mixtures. The CO2/1-pentanol mixture falls into type IV behavior, as experimentally measured by Raeissi et al.39 and theoretically obtained by Polishuk et al.40 A short, three-phase line is found and a UCEP and an LCEP could be determined with the equation. However, it is important to remark that a slight variation of the binary parameters makes this region disappear, resulting in a type II or type III mixture. As expected, this mixture is particularly difficult to model, since it is extremely sensitive to the parameters selected. Finally, the CO2/


Figure 11. PT projection of the PTx surface for the CO2/1-hexanol binary mixture. Symbols represent experimental data,77 and solid lines are the crossover soft-SAFT predictions.

Figure 12. Pxy projections of the CO2/1-propanol mixtures at four different temperatures: 313.4 (3), 322.36 ()), 333.4 (0), and 352.83 K (O). Symbols are experimental data. Data for T ) 322 and 352 K were taken from ref 78, while data for T ) 313 and 333 K were taken from 79. Solid lines represent soft-SAFT predictions.

1-hexanol mixture shows type III behavior. Note that, although the binary parameters chosen for the whole homologous series are not the best to quantitatively describe this last data, they are still good enough to capture the overall behavior of the mixture. In particular, the gas-liquid critical line of the CO2/ 1-hexanol mixture is overpredicted, as compared to experimental data.77 We now check the capability of the two binary parameters (η ) 0.970 and ξ ) 1.01), fitted to describe the critical behavior of the mixtures, to predict the subcritical behavior of them. The four mixtures of this homologous series studied above, representing type II, type III, and type IV behavior, are also investigated here. Pxy projections of the CO2/1-propanol, CO2/ 1-butanol, CO2/1-pentanol, and CO2/1-hexanol mixtures as obtained by crossover soft-SAFT are compared to available experimental data in Figures 12-15, respectively. Experimental results have been taken from refs 78-82. It is remarkable to see the excellent agreement obtained for the CO2/1-propanol

Llovell and Vega

Figure 13. Pxy projections of the CO2/1-butanol mixtures at five different temperatures: 324.16 (3), 333.58 ()), 355.38 (0), 392.72 (4), and 426.95K (O). Symbols are experimental data from ref 80. Solid lines represent soft-SAFT predictions.

Figure 14. Pxy projections of the CO2/1-pentanol mixtures at five different temperatures: 333.08 (3), 343.69 ()), 374.93 (0), 414.23 (4), and 426.86K (O). Symbols are experimental data from ref 81. Solid lines represent soft-SAFT predictions.

and CO2/1-butanol mixtures, in quantitative agreement with experimental data. The agreement is also very good for the CO2/ 1-pentanol and CO2/1-hexanol mixtures far from the critical point, although, as expected from the PT projections (Figures 10 and 11), it deteriorates as the mixtures approach their critical points, overestimating them. It should be emphasized that a better description of each mixture could be achieved if the parameters were fitted to their particular data. However, we have decided to use a unique set of parameters to describe the behavior of the homologous series, in a transferable manner. A final remark should be made regarding the quadrupolar interactions found in the two homologous series: since in the case of CO2/1-alkanol there are strong associating forces, this effect screens the quadrupolar interactions, which are more relevant in the CO2/n-alkane mixtures where no association is present.


Figure 15. Pxy projections of the CO2/1-hexanol mixtures at four different temperatures: 353.93 (3), 397.78 ()), 403.39 (0), and 432.45K (O). Symbols are experimental data from ref 82. Solid lines represent soft-SAFT predictions.

6. Conclusions We have presented here the extension to mixtures of the crossover soft-SAFT equation, which uses the RG theory to take into account the long density fluctuations present in the fluid in the near-critical region. The equations of White and coworkers8-12 could be extended to mixtures by using the isomorphism assumption33 and the approximation of Kiselev and Friend.34 The equation has been proved to be able to accurately describe the phase envelope and the critical lines of different kinds of associating and nonassociating systems, including the continuous change in the critical behavior of two homologous series as the chain length of the second compound increases: the CO2/n-alkane and the CO2/1-alkanol homologous series. The extended equation was first applied to the pure components, the 1-alkanol family and the CO2 molecule; they were parametrized using the crossover soft-SAFT equation for pure fluids, including associating and quadrupolar interactions, respectively. Very good agreement with experimental data was found in all cases, with molecular parameters showing physical trends. We were able to provide a correlation of the molecular parameters of the 1-alkanol series as a function of the molecular weight, allowing predictions for heavier members of the series without the need of experimental data for fitting. The molecular parameters of the pure compounds, together with parameters for the n-alkanes series obtained in a previous work,13 allowed us to use the extended equation for the study of several binary mixtures of interest. When the compounds forming the n-alkanes mixtures differ in chain length, binary mixture parameters were fitted to a single mixture and then used in a predictive manner for the rest of the subcritical mixtures and the critical lines of the homologous series. These parameters slightly deviate from unity as the two components become more dissimilar. For the CO2/n-alkane and the CO2/1-alkanol homologous series, the binary parameters were fitted to one selected binary mixture and used, in a predictive manner, for the rest of the homologous series. The crossover soft-SAFT equation has been able to capture the evolution from type II to type III behavior with an intermediate type IV behavior. This has been achieved with great success for both homologous

J. Phys. Chem. B, Vol. 110, No. 3, 2006 1361 series, the CO2 /n-alkane and the CO2/1-alkanol series. In general, predictions were in excellent agreement with experimental data, proving the reliability of the equation. Finally, the two unique binary parameters chosen to describe the critical behavior of the CO2/1-alkanol mixtures were used to predict the subcritical behavior of the same mixtures at different temperatures. The overall agreement of these predictions as compared to experimental data is excellent. Deviations appear for the CO2/1-pentanol and CO2/1-hexanol mixtures near their critical points, as expected from the description of the PT diagrams. The choice of a unique set of parameters to describe the critical and subcritical behavior of these mixture acts in favor of their transferability for several applications. Results obtained here confirm that the approach selected to describe the critical region is accurate to describe the vaporliquid equilibria of pure fluids and mixtures, as well as the critical lines of different types of mixtures, despite the limitations and assumptions made when developing the equation. Moreover, although this treatment can be applied to any other equation of state, the good results obtained here demonstrate the capabilities of the soft-SAFT EoS as an accurate equation to describe the global phase behavior of a compound or a mixture. Acknowledgment. We are indebted to Josep C. Pa`mies for his continuous help and contributions to this work. Several helpful discussions with Andre´s Mejıá and Felipe J. Blas are gratefully acknowledged. We are also thankful to Jan V. Senger for providing us with reprints from his work and his continuous encouragement. We thank the anonymous referees for suggesting the study of the subcritical CO2/1-alkanol mixtures and for providing references of the experimental data. This research has been possible thanks to the financial support received from the Spanish Government (projects PPQ2001-0671, CTQ2004-05985C02-01, and CTQ2005-00296/PPQ). F.L. acknowledges a predoctoral FPU grant from the Ministerio de Educacioń y Ciencia of Spain (MEC). References and Notes (1) Wyczalkowska, A. K.; Senger, J. V.; Anisimov, M. A. Physica A 2004, 334, 482. (2) Anisimov, M. A.; Senger, J. V. Equations of State for Fluids and Fluid Mixtures; Senger, J. V., Kayser, R. F., Peters, C. J., White, H. J., Eds.; Elsevier: Amsterdam, The Netherlands, 2000; Chapter 11. (3) Chen, Z. Y.; Albright, P. C.; Senger, J. V. Phys. ReV. A 1990, 41, 3161. (4) Chen, Z. Y.; Abbaci, A.; Tang, S.; Senger, J. V. Phys. ReV. A 1990, 42, 3370. (5) Parola, A.; Meroni, A.; Reatto, L. Phys. ReV. Lett. 1989, 62, 2981. (6) Parola, A.; Meroni, A.; Reatto, L. Int. J. Thermophys. 1989, 10, 345. (7) Meroni, A.; Parola, A.; Reatto, L. Phys. ReV. A 1990, 42, 6104. (8) White, J. A. Fluid Phase Equilib. 1992, 75, 53. (9) Salvino, L. W.; White, J. A. J. Chem. Phys. 1992, 96, 4559. (10) White, J. A.; Zhang, S. J. Chem. Phys. 1993, 99, 2012. (11) White, J. A.; Zhang, S. J. Chem. Phys. 1995, 103, 1922. (12) White, J. A. J. Chem. Phys. 1999, 111, 9352; 2000, 112, 3236. (13) Llovell, F.; Pa`mies, J. C.; Vega, L. F. J. Chem. Phys. 2004, 121, 10715. (14) Sun, L.; Zhao, H.; Kiselev, S. B.; McCabe, C. J. Phys. Chem. B. 2005. 109, 9047. (15) Kiselev, S. B. Fluid Phase Equilib. 1998, 147, 7. (16) Kiselev, S. B.; Ely, J. F. Ind. Eng. Chem. Res. 1999, 38, 4993. (17) Huang, S. H.; Radosz, M. Ind. Eng. Chem. Res. 1990, 29, 2284. (18) Huang, S. H.; Radosz, M. Ind. Eng. Chem. Res. 1991, 30, 1994. (19) McCabe, C.; Kiselev, S. B. Fluid Phase Equilib. 2004, 219, 3. (20) McCabe, C.; Kiselev, S. B. Ind. Eng. Chem. Res. 2004, 43, 2839. (21) Gil-Villegas, A.; Galindo, A.; Whitehead, P. J.; Mills, S. J.; Jackson, G.; Burguess, A. N. J. Chem. Phys. 1997, 106, 4168. (22) Lue, L.; Prausnitz, J. M. J. Chem. Phys. 1998, 108, 5529. (23) Jiang, J.; Prausnitz, J. M. J. Chem. Phys. 1999, 111, 5964. (24) Lue, L.; Prausnitz, J. M. AIChE J. 1998, 44, 1455. (25) Jiang, J.; Prausnitz, J. M. AIChE J. 2000, 46, 2525.

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