Statistical Associating Fluid Theory Coupled with Restricted Primitive

Journal of Chemical & Engineering Data 0 (proofing), ... SAFT2: 1. Single Salt + Water Solutions. Sugata P. Tan, Xiaoyan Ji, Hertanto Adidharma, and M...
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Ind. Eng. Chem. Res. 2005, 44, 7584-7590

Statistical Associating Fluid Theory Coupled with Restricted Primitive Model to Represent Aqueous Strong Electrolytes: Multiple-Salt Solutions Xiaoyan Ji, Sugata P. Tan, Hertanto Adidharma,* and Maciej Radosz Department of Chemical and Petroleum Engineering, University of Wyoming, Laramie, Wyoming 82071-3295

Statistical associating fluid theory coupled with the restricted primitive model (SAFT1-RPM), previously proposed for representing single-salt solutions in water, is extended to multiple-salt solutions using a mixing rule for the hydrated diameter. A binary adjustable parameter in this mixing rule, for a pair of salts, is obtained from experimental osmotic coefficients of the corresponding ternary system. The ternary systems considered contain water and all pairs of NaCl, KCl, NaBr, and KBr. LiCl + NaCl and LiCl + KCl pairs are also correlated. The adjustable parameters are used to predict the density of the ternary systems with or without common ions, the solubility of two ternary systems, and the osmotic coefficient of a quaternary NaCl-KClLiCl-H2O solution. Introduction Aqueous electrolyte solutions, especially aqueous multiple-salt solutions, are encountered in wastewater treatment, extraction, seawater desalinization, and distillation. In enhanced oil recovery by waterflooding, the composition of the connate and invading brines could have a major influence on wettability and, consequently, improve the oil recovery at reservoir temperature.1,2 In enhanced oil recovery by CO2-flooding, the presence of salts in water reduces the solubility of CO2 in water.3 As indicated in a simulation study, the oil recovery increases with the salinity of the brines.4 Because of the important role of electrolyte solutions in industrial processes and oil recovery, the interest in the modeling of the thermodynamic properties of these solutions, including activity coefficients, osmotic coefficients, density, and saturated pressure, has grown over the past decade. The models proposed by Pitzer,5 Chen and Evans,6 and Clegg and Pitzer7 have been widely used. These models, based on the excess Gibbs energy (GE), are not directly applicable to density calculation. To be able to calculate density, models based on an equation of state (EOS) are better choices. Recently, Myers and Sandler8 included the Born charging and discharging contributions and the mean spherical approximation (MSA) in the Peng-Robinson EOS to describe the activity coefficients, osmotic coefficients, and density of binary systems and the osmotic coefficients of two ternary systems. In this work, the binary system means water with a single salt, the ternary system means water with two salts, and so on. The EOS proposed by Harvey and Prausnitz9 is based on perturbation theory, also coupled with Born and MSA terms. In their approach, the adjustable salt/solvent parameter was correlated from osmotic-coefficient data, and the gas solubility in aqueous salt solutions was then predicted. Jin and Donohue10,11 and Liu et al.12 used perturbation theory coupled with MSA to correlate the mean activity coefficients of salts in water and to predict the density of such binary systems and the activity * Corresponding author. Tel.: (307) 766-2909. Fax: (307) 766-6777. E-mail: [email protected].

coefficients and salt solubilities in ternary systems. On the basis of statistical associating fluid theory (SAFT), other models have been proposed13-16 to represent the properties of binary systems. However, those models have not yet been elaborately extended to aqueous multiple-salt solutions. We proposed the SAFT1 EOS coupled with MSA, specifically with the restricted primitive model (RPM), referred to as SAFT1-RPM,17 which was found to represent the mean activity coefficients, osmotic coefficients, vapor pressures, and densities of binary aqueous solutions of alkali halide salts. The goal of this work is to extend this SAFT1-RPM to multiple-salt solutions using the same alkali halide salts as examples. SAFT1-RPM The EOS used is SAFT118 coupled with RPM,19 which is defined in terms of the dimensionless residual Helmholtz energy,

a˜ res ) a˜ seg + a˜ assoc + a˜ ion

(1)

where the superscripts refer to terms accounting for the residual, segment, association, and ionic interactions, respectively. The first two terms on the right side of eq 1 belong to the original SAFT1 EOS described in detail elsewhere.18 The chain term is absent because the water molecules and ions are assumed to be spherical segments. The third term (a˜ ion) is the RPM contribution to the residual Helmholtz energy,19

3x2 + 6x + 2 - 2(1 + 2x)3/2 a˜ ion ) 12πFNAd3

(2)

where F is the molar density, NA is the Avogadro number, d is the hydrated diameter, and x is the dimensionless quantity defined by

x ) κd

(3)

where κ is the Debye inverse screening length given by

κ2 )

4π wkT

∑j

Fn,jqj2 )

10.1021/ie050488i CCC: $30.25 © 2005 American Chemical Society Published on Web 08/19/2005

4πe2 wkT

∑j Fn,jzj2

(4)

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where w is the dielectric constant of water, k is the Boltzmann constant, T is the absolute temperature, Fn,j is the number density of ion j, qj is the charge of ion j () zje), zj is the valence of the charged ion j, e is the charge of an electron, and the summation is over all ions in the mixture. In SAFT1-RPM, the salt-specific hydrated diameter d in the ionic term is an adjustable parameter. The other parameters for short-range interactions are SAFT1 parameters,18 i.e., the segment volume (v), the segment energy (u/k), and the reduced well range of the squarewell potential (λ), which belong to the individual ions. The segment number is set equal to one for all species.17 SAFT1-RPM treats a salt as one component containing two different segments representing cation and anion,17 so that the components in a binary system, for example, are water and salt, not the ions. To extend the RPM term to multiple-salt solutions, the hydrated diameter d is estimated as follows,

d)

∑j ∑i x′ix′jdij

(5)

where the summation is over all salts in the mixture, x′ is the salt mole fraction on a solvent-free basis, and

dii ) di dij ) dji )

di + dj (1 - lij) 2

(xdi + xdj)2 di + dj

ion

v25, cc

u25/k, K

λ

salt

d25, Å

Li+ Na+ K+ ClBr-

1.8515 1.2797 7.6715 0.7797 3.4268

3560.686 3349.798 580.4178 413.9908 400.3327

2.0 1.7 1.2 1.8 1.2

LiCl NaCl KCl NaBr KBr

5.7958 5.7765 3.6611 4.5054 4.2321

Table 2. Correlation of lij in Equation 7 system A

B

experiment C

I, mol/kg

1 NaCl KCl

H2O 3.3-3.7 0.5-4.8 2 NaCl NaBr H2O 2.8-4.4 3 NaCl KBr H2O 2.9-4.6 4 NaBr KBr H2O 1.0-4.5 5 NaCl LiCl H2O 2.0-6.0 0.5-6.0 6 KCl NaBr H2O 1.8-4.3 7 KCl LiCl H2O 2.0-5.0 8 KCl KBr H2O 2.0-4.4

correlation

yBa

ref.b

0-1 0.09-0.88 0-1 0-1 0-1 0.33-0.66 0.09-0.88 0-1 0.33-0.71 0-1

a b a a a c d a c a

lij

ARDc, %

0.6279

0.9969

0.2138 0.8822 0.6133 -0.5652

0.2393 0.3381 0.6509 0.5573

0.5567 0.9206 0.1814

0.6353 0.9647 0.2582

a y is the mole fraction of salt B on a solvent-free basis [) m / B B (mA + mB)]. b References: a, Covington et al.;23 b, Robinson;24 c, Robinson and Lim;25 and d, Robinson et al.26 c ARD is the average relative deviation: ARD(%) ) ∑N i ((|φw,exp - φw,cal|)/φw,exp)(100/N) where N is the number of data points.

(6) (7)

where lij is an adjustable parameter corresponding to the interaction between two salts i and j. As noted from eqs 5-7, if lij ) 0, the hydrated diameter of a salt mixture becomes a simple average of the diameters of the individual salts, i.e., ∑ix′idi. Thus, the role of lij in this case is to correct this simple average diameter. A positive lij, for example, means that the hydrated diameter of a mixture is smaller than the simple average diameter. Consequently, if the magnitude of lij is large enough, the hydrated diameter of a salt mixture could be larger (for lij < 0) or smaller (for lij > 0) than the hydrated diameters of both salts that constitute the mixture. It can be shown that the upper bound of lij is given by

lij,max )

Table 1. SAFT1-RPM Parameters for the Aqueous Electrolyte Solutions

(8)

for which the hydrated diameter of the salt mixture becomes zero at some salt composition and, thus, unphysical. For the systems studied in this work, since the hydrated diameters of the individual salts do not differ much,17 lij,max ≈ 2. Mixing rules for the other parameters (SAFT1 parameters) describing short-range interactions are the same as those for SAFT1,18 and no other binary interactions are needed. As shown in our previous work,17 the parameters v, u/k, and d are temperature dependent. However, in this study, we present our model at 25 °C, the temperature at which plenty of experimental data exist. An extension to other temperatures is straightforward. The parameters at 25 °C are then given subscripts, i.e., v25, u25/k, and d25, as defined in our previous work.17 The dielectric constant of pure water is also temperature dependent

Figure 1. Comparison of calculated activity coefficients (γ) with the experimental data for LiCl-H2O: ], Hamer and Wu;20 s, calculated.

and assumed to be density independent.17 At 25 °C, the dielectric constant used is 78.42. Results and Discussion Parameter Fitting. The hydrated diameters of LiCl, NaCl, KCl, NaBr, and KBr and the corresponding ion parameters are summarized in Table 1. The parameters listed in Table 1 are all taken from our previous work,17 except the SAFT1 parameters (ion parameters) for Li+ and the hydrated diameter of LiCl, which are simultaneously fitted in this work to the mean activity coefficient of LiCl in water20 and the density of LiCl solution21 at 25 °C up to 5.0 molality of LiCl. The average relative deviation (ARD), defined under Table 2, for the mean activity coefficient is 0.66% and for the density is 0.07%. The results are shown in Figures 1 and 2. Using the same set of parameters without further readjustment, the osmotic coefficient and the vapor pressure of LiCl solutions are predicted and found to

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Figure 2. Comparison of calculated density (F) with the experimental data for LiCl-H2O: ], Zaytsev and Aseyev;21 s, calculated.

Figure 5. Comparison of calculated osmotic coefficients (φw) with the experimental data for NaCl (A)-KCl (B)-H2O: ], Robinson;24 +, Covington et al.;23 s, calculated.

Figure 3. Comparison of predicted osmotic coefficients (φw) with the experimental data for LiCl-H2O: ], Hamer and Wu;20 s, predicted.

Figure 6. Comparison of calculated osmotic coefficients (φw) with the experimental data for NaCl (A)-NaBr (B)-H2O (S1) and KCl (A)-KBr (B)-H2O (S2): ], Covington et al.23 (S1); 4, Covington et al.23 (S2); s, calculated for S1 system; - - - - - -, calculated for S2 system.

Figure 4. Comparison of predicted vapor pressure (Psat) with the experimental data for LiCl-H2O: ], Zaytsev and Aseyev;21 s, predicted.

agree with the experimental data20,21 as shown in Figures 3 and 4. The ARD for the predicted osmotic coefficient is 0.48%.

For the record, λ of Li+ in Table 1 is a bit outside of the λ range. In the evaluation of the SAFT1 segment term in eq 1,18 the effective reduced density of hard spheres was approximated using a cubic polynomial in reduced density, the coefficients of which were derived for a square-well fluid having λ in the range of 1.11.8.22 However, since ions only have dispersion interaction with water molecules, not among themselves, the effective value of λ for Li is the combined value with that of water, i.e., (2.0+1.5423)/2 ) 1.7712, which is still in the valid range. The SAFT1 parameters of water can be found in our previous work,17 where a water molecule is modeled as a single spherical segment with four association sites. For the ternary systems in this work, we just need to fit the lij parameters, one for each pair of salts. Instead of fitting lij to mean activity coefficients, we fit lij to experimental osmotic coefficients because they are the only accurate experimental data available for the eight ternary systems used in this study: NaCl + KCl + H2O, NaCl + NaBr + H2O, NaCl + KBr + H2O, KCl + NaBr + H2O, KCl + KBr + H2O, NaBr + KBr + H2O, LiCl + NaCl + H2O, and LiCl + KCl + H2O. The osmotic

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Figure 7. Comparison of calculated osmotic coefficients (φw) with the experimental data for NaCl (A)-KBr (B)-H2O: ], Covington et al.;23 s, calculated.

Figure 9. Comparison of calculated osmotic coefficients (φw) with the experimental data for NaCl (A)-LiCl (B)-H2O: ], Robinson et al.;26 O, Robinson and Lim;25 s , calculated. Figure 8. Comparison of the calculated osmotic coefficients (φw) with the experimental data for NaBr (A)-KBr (B)-H2O: ], Covington et al.;23 s, calculated.

coefficients of these systems are those measured by the Robinson group.23-26 If there is more than one group of data for the same ternary system, all of the data are simultaneously correlated with the same weighting. The experimental osmotic coefficients of ternary systems used to fit lij are summarized in Table 2 along with the correlation results and the ARD. As noted in the table, the value of lij of each system is much less than lij,max (∼2). The calculated osmotic coefficients are found to agree with the experimental data, as shown in Figures 5-11. In all the figures in this work, yi is the mole fraction of salt i on a solvent-free basis [) mi/ m] and m is the total molality [) Σmi]; I is the ionic strength on a molality basis [) mA + mB]. Predicting the Densities. There are much fewer experimental densities reported for ternary systems than there are experimental osmotic coefficients. Among the eight ternary systems studied in this work, to the best of our knowledge, the experimental densities were reported for KCl-NaBr-H2O27 and NaCl-KCl-H2O,28 representing mixed salts without and with common ions, respectively. Using the lij values obtained from the osmotic coefficients, without any further readjustment, the densities of these two ternary systems are predicted and com-

Figure 10. Comparison of calculated osmotic coefficients (φw) with the experimental data for KCl (A)-NaBr (B)-H2O: ], Covington et al.;23 s, calculated.

pared with experimental data. For KCl-NaBr-H2O, the comparison is shown in Figure 12; the ARD is 0.18%. For NaCl-KCl-H2O, the comparison is shown in Figure 13; the ARD is 0.19%. These results indicate that no special treatment is needed for the common ions. Predicting the Osmotic Coefficients for a Quaternary System. There are very few experimental

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Ind. Eng. Chem. Res., Vol. 44, No. 19, 2005 Table 3. Experimental and Calculated Osmotic Coefficients (Ow) for NaCl-KCl-LiCl-H2O m, mol/kg NaCl

KCl

exp.29

cal.

|∆φw|a

0.9650 1.0000 1.0114 1.0321

0.9650 1.0000 1.0114 1.0321

0.9650 1.0000 1.0114 1.0321

1.0553 1.0677 1.0709 1.0802

1.0803 1.0885 1.0912 1.0961

0.0250 0.0208 0.0203 0.0159

a

Figure 11. Comparison of calculated osmotic coefficients (φw) with the experimental data for KCl (A)-LiCl (B)-H2O: ], Robinson and Lim;25 s, calculated.

φw

LiCl

|∆φw| ) |φw,exp - φw,cal|.

ficients of the corresponding three ternary systems, the osmotic coefficients of this quaternary system are predicted and compared with the experimental data in Table 3; the ARD is 1.9%, which is in line with the experimental uncertainty. Predicting the Salt Solubility. For a hydrated strong electrolyte MνCXνA‚hH2O, i.e., a solid solution of 1 mole of MνCXνA in h moles of water, the solubility equilibrium can be represented as Ksp

MνCXνA‚hH2O 798 νCMzC + νAXzA + hH2O

(9)

where MzC and XzA are the constituent cation and anion, ν is the stoichiometric coefficient, z is the valence, and subscripts A and C denote anion and cation, respectively. The equilibrium constant of eq 9, also referred to as the solubility product, can be written as

Ksp )

(νC+νA) h mνMCmνXAγ((m) aw

aMνCXνA‚hH2O

(10)

where m is the molality, γ((m) is the molality mean activity coefficient of the salt, aMνCXνA‚hH2O is the activity of the solid, and aw is the activity of water. Assuming a pure solid that has an activity of 1 yields Figure 12. Comparison of predicted density (F) with the experimental data for KCl (A)-NaBr (B)-H2O: ], Kumar;27 s, predicted.

Figure 13. Comparison of predicted density (F) with the experimental data for NaCl (A)-KCl (B)-H2O: ], Zhang and Han;28 s, predicted.

osmotic coefficients reported for quaternary systems. Among the salts considered in this study, only NaClKCl-LiCl-H2O29 with four data points was reported. Using the parameters obtained from the osmotic coef-

(νC+νA) h Ksp ) mνMCmνXAγ((m) aw

(11)

If the solubility product is known, the equilibrium concentration can be calculated from eq 11 as long as we can estimate the mean activity coefficient of the salt and the water activity. We take two systems as our examples, NaCl-KCl-H2O and NaBr-KBr-H2O; both salts form pure solids, which makes eq 11 applicable. For a system with a common ion, such as NaCl-KClH2O (chloride ion in common) or NaBr-KBr-H2O (bromide ion in common), the molality of the common ion in eq 11 is the total molality of the ion that comes from both salts. The mean activity coefficient of the salt involved is usually calculated considering the commonion effect. However, as demonstrated in the Appendix, when we use SAFT1-RPM, we do not need to consider such a common-ion effect on the activity coefficient of the salt. The solubility of each salt in water containing multiple salts depends not only on the salt activity coefficient but also on its solubility product. Generally, the experimental Ksp (Ksp,exp) obtained from the experimental mean activity coefficient and the water activity is more reliable than the calculated and, hence, modeldependent Ksp,cal. For NaCl-KCl-H2O, the solid phase may be NaCl, KCl, or both. The calculated solubility products are found to be close to the experimental ones, as shown in Table 4. However, we use Ksp,exp to predict the solubility

Ind. Eng. Chem. Res., Vol. 44, No. 19, 2005 7589

Figure 14. Comparison of predicted solubility (m) with the experimental data for NaCl-KCl-H2O: ], Linke and Seidell;30 s, predicted.

Figure 15. Comparison of predicted solubility (m) with the experimental data for NaBr-KBr-H2O: ], Linke and Seidell;30 s, predicted.

Table 4. Solubility Products (Ksp) for Solid NaCl and KCl in Water

Conclusions

Ksp,cal solid

Ksp,exp

this model

Pitzer model

NaCl KCl

38.05 8.003

38.66 8.270

37.67 8.704

Table 5. Estimated Solubility Product (Ksp) for Solid NaBr·2H2O and KBr in Water msat, mol/kg system

exp. data30

NaBr-H2O 9.1897 9.1934 KBr-H2O 5.6963 5.7367 5.7438

extrapolation

average

solid phase

9.1915

NaBr‚2H2O 2.001

5.7256

KBr

γ((m)

0.6424

aw

Ksp

0.5825 114.7 13.53

The SAFT1-RPM model is extended to multiple-salt solutions using a mixing rule with one adjustable binary parameter for the salt hydrated diameter. The adjustable parameter is obtained from the osmotic coefficients of ternary systems. Such binary parameters are found to predict the densities of ternary systems with and without common ions, the osmotic coefficients of a quaternary system, and the solubility of two ternary systems. Acknowledgment University of Wyoming’s Enhanced Oil Recovery Institute provided funding for this work. Appendix

of this system. The predictions, using eq 11, agree with the experimental data,30 as shown in Figure 14. For NaBr-KBr-H2O, two types of solid may exist: NaBr‚2H2O and KBr. The experimental equilibrium concentrations of NaBr and KBr in water are given in Table 5. However, the available experimental mean activity coefficient of NaBr and the water activity (the osmotic coefficient) in NaBr-H2O are only up to 9.0 mol‚kg-1, a bit less than the equilibrium concentration (9.19 mol‚kg-1). The situation is the same for KBr-H2O, as shown in Table 5; the experimental mean activity coefficient of KBr and the water activity in KBr-H2O are available only up to 5.5 mol‚kg-1. Therefore, it is impossible to directly obtain the solubility products from the experimental data. To obtain the solubility products for solid NaBr‚2H2O and KBr, the experimental mean activity coefficients and osmotic coefficients for the NaBr-H2O and KBr-H2O binaries are first extrapolated to the state of saturation. Then the Ksp’s are estimated, as shown in Table 5, and then used to predict the solubility of NaBr-KBr-H2O. Figure 15 shows the predicted and experimental solubilities.30 The model overestimates the solubility at high ionic strength because it is far beyond its correlation range. The maximum ionic strength is ∼11 mol‚kg-1 for this system, while the model was correlated up to 6 and 5.5 mol‚kg-1 for NaBr-H2O and KBr-H2O, respectively.17

The salt activity coefficient calculated from SAFT1RPM is equivalent to the mean activity coefficient of the salt,17 and can be expressed as the ratio of the fugacity coefficient of the salt in the mixture to its value at infinite dilution:

γsalt )

φˆ salt

(A.1)

∞ φˆ salt

The fugacity coefficient of salt i is equivalent to the mean fugacity coefficient of the constituent ions: ν+(1) ν-(1) 1/(ν+(i) + ν-(i)) φˆ -(i) ] ln φˆ (i) ) ln[φˆ +(i)

(A.2)

In SAFT1-RPM, the fugacity coefficient of species i in the mixture is derived from the residual Helmholtz energy,

ln φˆ i ) a˜ +

∂a˜ ∂xi

-

∂a˜

∑i xi ∂x

i

+F

∂a˜ ∂F

(

- ln 1 + F

)

∂a˜ ∂F

(A.3)

where xi is the mole fraction of species i and the summation is over all species in the mixture. In calculating the fugacity coefficient of different species in a mixture using eq A.3, the only different term is the first derivative of the Helmholtz energy with respect to composition. Nevertheless, this first derivative is also

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the same for species with the same parameters (equivalently the same kind of species). In multiple-salt solutions with common ions, the common ions have the same transferable ion parameters in different salts and the same average hydrated diameter estimated from eq 5. For example, for salt 1 and salt 2 with a common anion, there is only one fugacity coefficient for the common anion,

ln φˆ -(1) ) ln φˆ -(2) ) ln φˆ (-)

(A.4)

The last term in eq A.4 is the logarithm of the fugacity coefficient of the common anion if the mole fractions of the common anion are lumped. Hence, the salt fugacity coefficient calculated in SAFT1-RPM, which is equivalent to eq A.2, will have the same value regardless of how the common ion is treated, separately or not. Literature Cited (1) Tang, G. Q.; Morrow, N. R. Salinity, temperature, oil composition, and oil recovery by waterflooding. SPE Res. Eng. 1997, 12, 269. (2) Morrow, N. R.; Tang, G. Q.; Valat, M.; Xie, X. Prospects of improved oil recovery related to wettability and brine composition. J. Pet. Sci. Eng. 1998, 20, 267. (3) Stalkup, F. I., Jr. Miscible Displacement; SPE Monograph Series, Vol. 8; SPE: New York, 1984; Chapter 8 (4) Enick, R. M.; Klara, S. M. Effects of CO2 Solubility in Brine on the Compositional Simulation of CO2 Floods. SPE Res. Eng. 1992, 7, 253 (5) Pitzer, K. S. Thermodynamics of Electrolytes. I. Theoretical Basis and General Equations. J. Phys. Chem. 1973, 77, 268-277. (6) Chen, C. C.; Evans, L. B. A Local Composition Model for Excess Gibbs Energy of Electrolyte System. AIChE J. 1986, 32, 444-454. (7) Clegg, S. M.; Pitzer, K. S. Thermodynamics of Multicomponent, Miscible, Ionic Solution: Gereralized Equations for Symmetrical Electrolytes. J. Phys. Chem. 1992, 96, 3513-3520. (8) Myers, J. A.; Sandler, S. I. An Equation of State for Electrolyte Solutions Covering Wide Ranges of Temperature, Pressure, and Composition. Ind. Eng. Chem. Res. 2002, 41, 32823297. (9) Harvey, A. H.; Prausnitz, J. M. Thermodynamics of HighPressure Aqueous Systems Containing Gases and Salts. AIChE J. 1989, 35, 635-644. (10) Jin, G.; Donohue, M. D. An Equation of State for Electrolyte Solutions. 1. Aqueous Systems Containing Strong Electrolytes. Ind. Eng. Chem. Res. 1988, 27, 1073-1084. (11) Jin, G.; Donohue, M. D. An Equation of State for Electrolyte Solutions. 3. Aqueous Systems Containing Multiple Salts. Ind. Eng. Chem. Res. 1991, 30, 240-248. (12) Liu, W.-B.; Li, Y.-G.; Lu, J.-F. A New Equation of State for Real Aqueous Ionic Fluid Based on Electrolyte Perturbation Theory, Mean Spherical Approximation and Statistical Associating Fluid Theory. Fluid Phase Equilib. 1999, 158-160, 595-606.

(13) Cameretti, L. F.; Sadowski, G.; Mollerup, J. M. Modeling of Aqueous Electrolyte Solutions with Perturbed-Chain Statistical Associated Fluid Theory. Ind. Eng. Chem. Res. 2005, 44, 33553362. (14) Galindo, A.; Gil-Villegas, A.; Jackson, G.; Burgess, A. N. SAFT-VRE: Phase Behavior of Electrolyte Solutions with the Statistical Associating Fluid Theory for Potentials of Variable Range. J. Phys. Chem. B 1999, 103, 10272-10281. (15) Gil-Villegas, A.; Galindo, A.; Jackson, G. A Statistical Associating Fluid Theory for Electrolyte Solutions. Mol. Phys. 2001, 99, 531-546. (16) Liu, Z.; Wang, W.; Li, Y. An Equation of State for Electrolyte Solutions by Combination of Low-Density Expansion of Non-Primitive Mean Spherical Approximation and Statistical Associating Fluid Theory. Fluid Phase Equilib. 2005, 227, 147156. (17) Tan, S. P.; Adidharma, H.; Radosz, M. Statistical Associating Fluid Theory Coupled with Restricted Primitive Model to Represent Aqueous Strong Electrolytes. Ind. Eng. Chem. Res. 2005, 44, 4442-4452. (18) Adidharma, H.; Radosz, M. Prototype of an Engineering Equation of State for Heterosegmented Polymers. Ind. Eng. Chem. Res. 1998, 37, 4453-4462. (19) Lee, L. L. Molecular Thermodynamics of Nonideal Fluids; Butterworth Publishers: Stoneham, MA, 1988. (20) Hamer, W. J.; Wu, Y. C. Osmotic coefficients and mean activity coefficients of uni-univalent electrolytes in water at 25 °C. J. Phys. Chem. Ref. Data 1972, 1, 1047-1100. (21) Zaytsev, I. D., Aseyev, G. G., Eds. Properties of Aqueous Solutions of Electrolytes; CRC Press: Boca Raton, FL, 1992. (22) Gil-Villegas, A.; Galindo, A.; Whitehead, P. J.; Mills, S. J.; Jackson, G. Statistical Associating Fluid Theory for Chain Molecules with Attractive Potentials of Variable Range. J. Chem. Phys. 1997, 106, 4168-4186. (23) Covington, A. K.; Lilley, T. H.; Robinson, R. A. Excess Energies of Aqueous Mixtures of Some Alkali Metal Halide Salt Pairs. J. Phys. Chem. 1968, 72, 2759-2763. (24) Robinson, R. A. Activity Coefficients of Sodium Chloride and Potassium Chloride in Mixed Aqueous Solutions at 25 °C. J. Phys. Chem. 1961, 65, 662-667. (25) Robinson, R. A.; Lim, C. K. The Osmotic Properties of Some Aqueous Salt mixtures at 25 °C. Trans. Faraday Soc. 1953, 49, 1144-1147. (26) Robinson, R. A.; Wood, R. H.; Reilly, P. J. Calculation of Excess Gibbs Energies and Activity Coefficients from Isopiestic Measurements on Mixtures of Lithium and Sodium Salts. J. Chem. Thermodyn. 1971, 3, 461-471. (27) Kumar, A. Densities and Excess Volumes of Aqueous KClNaBr up to Ionic Strength of 4 mol‚kg-1. J. Chem. Eng. Data 1989, 34, 446-447. (28) Zhang, H. L.; Han, S. J. Viscosity and Density of Water + Sodium Chloride + Potassium Chloride Solutions at 298.15 K. J. Chem. Eng. Data 1996, 41, 516-520. (29) Reilly, P. J.; Wood, R. H.; Robinson, R. A. The Prediction of Osmotic and Activity Coefficients in Mixed-Electrolyte Solutions. J. Phys. Chem. 1971, 75, 1305-1315. (30) Linke, W. F.; Seidell, A. Solubilities of Inorganic and Metal Organic Compounds; American Chemical Society: Washington, DC, 1965.

Received for review April 25, 2005 Revised manuscript received July 8, 2005 Accepted July 18, 2005 IE050488I