Nonprimitive Model of Mean Spherical Approximation Applied to

For ion−dipole mixtures, one important development in the integral equation theories is the method of MSA for the primitive model by Blum (1975), wh...
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Ind. Eng. Chem. Res. 1998, 37, 4183-4189

4183

Nonprimitive Model of Mean Spherical Approximation Applied to Aqueous Electrolyte Solutions Wen-Bin Liu, Yi-Gui Li,* and Jiu-Fang Lu Department of Chemical Engineering, Tsinghua University, Beijing 100084, People’s Republic of China

The mean ionic activity coefficients of 46 single 1:1 electrolytes in aqueous solutions are successfully correlated by a nonprimitive model of mean spherical approximation in which the diameters of cation and anion are treated as two adjustable parameters. The ion-ion, iondipole, and dipole-dipole interactions are represented by three implicit parameters, Γ, B10, and b2, respectively. The water molecule parameters can be obtained by fitting the saturated vapor pressure data from 283.15∼573.15 K with an equation of state derived from the mean spherical approximation and statistical associating fluid theory. Both the correlated accuracy and the range of electrolyte concentrations are acceptable. Introduction Understanding the thermodynamic properties of aqueous electrolyte systems is essential for the design and simulation of a variety of important industrial processes, such as inorganic, organic, and biological separations, hydrometallurgy, geochemistry, corrosion, and pollution. Recently, considerable advances have been made in the statistical mechanical approaches for electrolyte solutions, such as molecular simulation, perturbation theory, and integral equation theory. The integral equation theories, including three different approachess hypernetted chain (HNC), Percus-Yevick (PY), and mean spherical approximation (MSA)sare derived from the solutions of the Ornstein-Zernike equation. The most accurate theory in dealing with the long-range forces seems to be the HNC theory. But this theory involves the resolution of highly nonlinear equations and is very difficult in its practical applications. However, MSA is attractive for chemical engineers to examine the thermodynamic properties of electrolyte solutions for its simple analytical solution, which is reasonably accurate for ionic solutions, and has been applied extensively to real electrolyte solution systems. For ion-dipole mixtures, one important development in the integral equation theories is the method of MSA for the primitive model by Blum (1975), which leads to the analytical expressions for the thermodynamic and structural properties of electrolyte solution. Lu et al. (1993) have used the analytical solution of the primitive MSA in real electrolyte solutions with success. For the nonprimitive model of electrolyte solution, Blum and Wei (1987), Wei and Blum (1987), and Blum et al. (1992) gave the analytical expressions for the thermodynamic properties of equal sizes and arbitrary sizes of hard iondipole mixtures. But very few studies have been conducted to develop the nonprimitive model to real electrolyte solutions. Planche and Renon (1981) introduced an extension to Blum’s primitive MSA as the Coulombic interactions incorporated with a temperature-dependent short-range interaction term as the derivative of a δ function. For each neutral species, the short-range contribution includes two adjustable parameters in addition to the ionic diameters. This equa* To whom correspondence should be addressed. E-mail: [email protected].

tion could successfully be correlated with the experimental osmotic coefficient data of electrolyte solutions. Lvov and Wood (1990) used the Blum-Wei (1987) equation to correlate the densities of the NaCl-H2O system over a wide range of temperatures and pressures and the correlated results are satisfactory. Li et al. (1996) attempted to apply the nonprimitive model of MSA to 14 real 1:1 electrolyte solutions by fitting the diameters of cation and anion simultaneously. The correlated average relative deviations for the mean activity coefficients were not satisfactory and the correlated maximum electrolyte concentrations are only up to 3M. The defect of these equations is that they did not consider the association interaction between the solvent molecules. Therefore, there is a need to develop a nonprimitive model by use of MSA to calculate the thermodynamic properties of electrolyte solutions. In this paper, we revisit the performance of the correlation of mean ionic activity coefficients in real electrolyte aqueous systems by introducing the statistical associating fluid theory (SAFT) (Chapman et al., 1989), which solves the association interaction between the water molecules, into the nonprimitive electrolyte model. Nonprimitive MSA Equation For an electrolyte solution containing an ion and dipole mixture, the Ornstein-Zernike equation can be expressed as follows: n

hij(r12) ) cij(r12) +

∑ Fk∫cik(r13)hkj(r32) drj3 k)1

(1)

where hij(r12) and cij(r12) are the total and direct correlation functions for particles i and j at positions r1 and r2 . These functions will depend on orientation for dipolar particles. Fk is the number density of particle k. The integral is over all coordinations of particle 3. hij(r) ) gij(r) - 1, gij(r) is the radial distribution function. This equation relates the direct correlation function to the pair radial distribution function and also represents the definition of the direct correlation function. To solve the O-Z equation, a so-called closure relation between the hij(r) and the cij(r) is required. Our interest is to use the mean spherical approximation, which is

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4184 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998

based on the following closure relation of the O-Z equation:

hij(r) ) -1

rij < σij

(2)

cij(r) ) -βUij(r)

rij > σij

(3)

where σij ) (σi + σj)/2 and σi and σj are the diameters of particles i and j, respectively. β ) 1/kT, where k is the Boltzmann constant, and Uij(r) represents the pair potential between i and j. An analytic solution of primitive model for this closure relation was derived in the early 1970s by Waisman and Lebowitz (1972) and was used to correlate the activity coefficient, osmotic coefficient, and other thermodynamic properties by some research groups. For the nonprimitive model, Blum and Wei (1987) and Blum et al. (1992) presented an approximate analytical solution for equal ionic sizes or arbitrary sizes of ions and solvents of point dipoles. Now, we consider a system which consists of a mixture of hard spheres of diameter σi, with number density Fi and charge zie, where e is the elementary charge, and hard cores of diameter σd, with number density Fd and point dipole µd of solvent molecule. Using the invariant expansion in rij > σij, the total pair correlation h(r12) can be expanded as (Vericat and Blum, 1982) 000 000 + h000 + h(r12) ) h000 ii (r12)Φ id (r12)Φ 011 000 011 000 110 hid (r12)Φ + hdd (r12)Φ + hdd (r12)Φ110 + 112 h112 dd (r12)Φ

Fourier transformation, Laplace transformation, Baxter-Wertheim factorization, and some manipulations, three well-known nonlinear equations of MSA have been deduced by Blum and Wei (1987) and Blum et al. (1992) as follows:

{

∑i Fi(a0i )2 + Fd(a1d)2 ) d02 -

∑i Fia0i k10di + a1d(1 - Fdk11dd) ) d0d2 ∑i Fi(k10di )2 ) y12 + Fdd22

2 (1 - Fdk11 dd) + Fd

The definitions of all the variables in the above equations are shown in the Appendix. By solving the above equations, three implicit characteristic parameters, Γ, B10, and b2, can be obtained by an iteration method. The detailed computational procedures will be described later. The magnitudes of these three parameters are dependent on the electrolyte concentration in the solution and are related to the interaction strength between ion-ion, ion-dipole, and dipole-dipole terms, respectively. The thermodynamic properties can be formulated without knowing the explicit expression of the pair correlation functions. According to this approximation method, the excess internal energy of system with respect to the hard-sphere repulsion part is

βUMSA 2 (d0 V (4)

where hii, hid, and hdd are the coefficients of the invariant expansion, respectively. The rotational invariants Φmnl, which depend only on the mutual orientations of the molecules, are defined as follows:

(12)

∑i FiziNi - 2d0d2FdB10 - 2d22Fdb2/σd3)/4π

(13)

The excess Helmholtz free energy is given by

-

βAMSA V

) (-2do2

∑i FiziNi + 2d0d2FdB10)/12π + J′

(14)

where

Φ

112

Φ000 ) 1

(5)

Φ011 ) (ω2rˆ 12)

(6)

Φ110 ) -x3(ω1ω2)

(7)

) xe/10[3(ω1rˆ 12)(ω2rˆ 12) - (ω1ω2)]

J′ ) (π/3)

(15)

Thus the excess pressure can be obtained:

βPMSA ) β(UMSA - AMSA)/V (8)

The pair potential Uij(r) can be simply expressed as

Uid(r) ) -

zieµ 011 Φ r2

Udd(r) ) -

βµiMSA ) zi(d02Ni - d0d2Fdmi)/4π (9)

(10)

2

µ 112 Φ r3

(16)

From Hoye and Stell (1977), the ionic excess chemical potentials are as follows:

2

zizje Uii(r) ) r

FkFjσkj3(2l + 1)-1[hkjmnl(r)σkj)]2 ∑ ∑ kj mnl

(11)

Since the internal energy of the hard sphere mixture is zero, the chemical potentials of the hard sphere mixture are described by the MCSL equation (Mansoori et al., 1971):

πPhsσi3β 3σi(ζ2 + ζ1σi) 9ζ22σi2 + + + 6 ∆ 2∆2 ζ3 1 ζ3 2 ζ2σi 3 ln ∆ + 2 ln ∆ + ∆ 2 ∆ ζ3 ζ3(1 + ∆) (18) ∆

βµihs ) -ln ∆ + 3

where i and d denote ion and dipolar solvent, respectively. ω1 and ω2 represent the unit vectors along the dipolar moment and rˆ is the unit vector along r. With the invariant expansion, the O-Z equation becomes a set of matrix equations. By performing the

(17)

with

( )[ ζ2σi ζ3

2

( )] ( )[

]

Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4185 3

πPhsβ ζ0 3ζ1ζ2 (3 - ζ3)ζ2 + ) + 6 ∆ ∆2 ∆3 ζl )

π

(19)

n

Fiσil ∑ 6 i)1

(l ) 0, 1, 2, 3)

∆ ) 1 - ζ3

(20)

where M is the number of association sites on each water molecule. In this work, we choose M ) 4. The summation is over all the association sites in a water molecule, and XA is the mole fraction of molecules not bonded at site A, which is given by

XA )

(21)

-1 + x1 + 8Fd∆AB 4Fd∆AB

(30)

where P and V are the pressure and volume of the system, respectively. Then the excess chemical potential of ion i for this electrolyte solution system is

where ∆AB is the association strength between sites A and B given by

βµi ) βµiMSA + βµihs

∆AB ) ghs(d)[exp(AB/kT) - 1]σd3κAB

(22)

Evaluation of the Water Molecule Parameter Water is a typical polar associating solvent and its molecule parameters play a very important role in the calculation of thermodynamic properties for electrolyte solutions. Since our equation neglects the contributions of Lennard-Jones dispersion, induced dipole moment, and quadrupole interaction terms, the diameter and dipole moment of water must be fitted from its experimental saturated vapor pressure data. To accurately describe the properties of water, we introduce the statistical associating fluid theory (SAFT) to deal with the association interaction between water molecules. It has been shown that this theory can significantly improve the correlated accuracy of the mean ionic activity coefficient in electrolyte solutions by use of the perturbation theory-based equation of state (Li et al., 1998). For pure water, since there are no contributions from the interaction of ions, then eqs 13 and 14 can be reduced to

Here AB and κAB are the association energy and volume for the interaction between sites A and B and ghs(d) is the hard-sphere radial distribution function given by

ghs(d) )

-

βA V

2

) J′ )

2

[(β3/β6) - 1] 3πσd3

+

Z ) ZMSA + Zid + Zhs + Zassoc

(34)

with hs

βA )

(23) Zhs )

(b2β24/β122)2 3 6πσ

(24)

4ζ3 - 2ζ32

(37)

(1 - ζ3)3

Zid ) 1

(38)

(26)

(A + PV)L ) (A + PV)V

+ 3

2πσd

b22β242 (λ2 - 1)2 + 18ζ3 36ζ β 4

P

3 12

[ln X ∑ A

A

A

- X /2] + M/2

(28)

d

sat

)

6ζ3Lζ3V(AL - AV) πσd3(ζ3L - ζ3V)

(40)

The implicit parameter b2 can be obtained with the Newton iteration method:

β62 4πβFdµd2 β32 ) 43 β β 4 6

d

(39)

and the saturated vapor pressure at vapor-liquid equilibrium can be calculated:

(27)

4Fd∆AB - (1 + 8Fd∆AB - x1 + 8Fd∆AB) 4 2 (29) XA 4F ∆ABx1 + 8F ∆AB

)

(36)

β24 ) 1 - b2/24

Zassoc )

(

(35)

(25)

b2d22

)

(1 - ζ3)2

β12 ) 1 + b2/12

The contribution of association interaction to Helmholtz free energy and compressibility factor can be described by the following expression (Huang and Radosz, 1990):

βA

4ζ3 - 3ζ32

When the vapor-liquid equilibrium is reached, the chemical potentials of water in two phases are equal; then

Thus the excess compressibility factor can be obtained:

assoc

(32)

(33)

βAid ) 6ζ3/π

where

ZMSA ) -

(1 - ζ3)3

βA ) βAMSA + βAid + βAhs + βAassoc

d

MSA

1 - 0.5ζ3

Then the equation of state for pure water can be written as

2

Fdd2 b2 βUMSA )V 2πσ 3

(31)

(41)

12

Based on the equation of state of water, the parameters σd, µd, AB/k, and κAB for a water molecule were obtained by fitting saturated vapor pressures of water from 298.15 to 573.15 K. The average relative deviation of the fitted saturated vapor pressure is within 1.5%.

4186 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998

These regressed parameter values are σd ) 0.2415 nm, µd ) 2.37 D, AB/k ) 1269.27 K, and κAB ) 0.01234. Correlated Results of Mean Ionic Activity Coefficients If we take the infinite dilution as the reference state for ions, then the ionic activity coefficient of the electrolyte, fi, which is in mole fraction scale, can be calculated from the following relation:

ln fi ) β[µi(xi) - µi(xif0)]

(42)

where the expressions for µi is the chemical potential of ion i (see eq 22), and xi is the mole fraction of ion i. The mean ionic activity coefficient γ(, which is in the molality scale, can be calculated by use of the following definition:

γ( ) (γ+ν+γ-ν-)1/ν

(43)

where the subscripts + and - denote the cation and anion and ν is a stoichiometric coefficient (ν ) ν+ + ν-). The mean ionic activity coefficients in the molality scale, γ(, from literature can be converted into the mean ionic activity coefficients in the mole fraction, f(, to make them compatible with the scale of our present equations, according to the following relation:

f( ) γ((1 + 0.018νm)

(44)

with m the molality of the electrolyte. Using the above equations and the regressed parameters of water molecule, we correlate the mean ionic activity coefficients of 46 real 1:1 electrolyte solutions. In this work, the diameters of cation and anion are adjustable parameters that can be obtained by fitting the experimental mean ionic activity coefficients. Equation 12 can be solved by the following assumptions: (a) The Pauling diameters of ions are taken as the initial diameters of cation and anion. (b) The ranges of Γ, B10, and b2 suggested by Lvov and Wood (1990) are adopted and the initial values are 4.5, 0.015, and 1.2, respectively. (c) The average relative deviation of mean ionic activity coefficient is taken as the objective function in the calculation. According to the above assumptions, the variables in eq 12 can be computed with the initial values of ionic diameters (σ+, σ-) and Γ, B10, and b2 to generate a new set of σ+, σ-, Γ, B10, and b2, which are regarded as the new initial values, and then recompute these defined variables until the minimum average relative deviation of mean ionic activity coefficient is reached. By this iteration procedure, we can obtain the optimal values of σ+ and σ-, which are listed in Table 1. In this work, the maximum electrolyte concentrations are 3.0 mol/kg for each aqueous electrolyte system. The molecular and ionic parameters thus obtained can be used to predict the mean ionic activity coefficients up to mmax ) 6.0 mol/kg without any changes of the adjustable parameters. All the correlated (mmax ) 3.0 mol/kg) and the predicted (mmax ) 6.0 mol/kg) average relative deviations of mean ionic activity coefficients, together with the regressed diameters of cations and anions, are listed in Table 1. The correlated (mmax ) 3.0 mol/kg) and also the predicted (mmax ) 6.0 mol/kg)

Table 1. Correlated and Predicted Average Relative Deviation (ARD%) of Mean Ionic Activity Coefficients for 46 1:1 Electrolyte Solutions and Regressed Diameter Values of Cations (σ+ ) and Anions (σ- ) at 25 °Ca system

σ+

σ-

mmax

HCl + H2O HBr + H2O HI + H2O HNO3 + H2O LiCl + H2O LiBr + H2O LiI + H2O LiNO3 + H2O NaF + H2O NaCl + H2O NaBr + H2O NaI + H2O NaNO3 + H2O KF + H2O KCl + H2O KBr + H2O KI + H2O KNO3 + H2O RbF + H2O RbCl + H2O RbBr + H2O RbI + H2O RbNO3 + H2O CsF + H2O CsCl + H2O CsBr + H2O CsI + H2O CsNO3 + H2O NaOH + H2O KOH + H2O NaClO3 + H2O NaClO4 + H2O NaBrO3 + H2O NaAc + H2O NaSCN + H2O NaH2PO4 + H2O KClO3 + H2O KBrO3 + H2O KSCN + H2O KH2PO4 + H2O NH4Cl + H2O NH4NO3 + H2O LiClO4 + H2O HClO4 + H2O TlNO3 + H2O AgNO3 + H2O

1.773 1.765 1.775 1.769 1.753 1.785 1.797 1.780 1.748 1.759 1.770 1.790 1.798 1.754 1.800 1.821 1.830 1.815 1.760 1.802 1.820 1.839 1.811 1.799 1.808 1.823 1.842 1.818 1.733 1.714 1.791 1.816 1.798 1.775 1.800 1.820 1.812 1.823 1.826 1.757 1.796 1.818 1.820 1.810 1.930 1.796

4.017 4.100 4.263 4.062 4.000 4.113 4.241 4.079 3.653 3.920 4.012 4.167 3.961 3.848 3.993 4.073 4.220 3.960 3.875 4.030 4.100 4.231 4.017 4.000 4.092 4.160 4.298 4.052 3.756 3.885 4.068 4.199 3.941 4.109 4.200 3.979 3.964 3.874 4.231 3.939 4.084 4.124 4.338 4.316 4.090 3.848

3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 1.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 1.4 3.0 3.0 3.0 3.0 2.5 3.0 3.0 3.0 0.7 0.5 3.0 1.8 3.0 3.0 3.0 3.0 0.4 3.0

ARDcorr ARDpre (%) mmax (%) 3.071 2.855 3.029 2.179 3.071 1.666 0.495 2.454 3.647 2.706 2.019 2.046 1.726 3.365 2.096 2.208 1.656 0.865 4.344 1.311 1.456 1.877 1.823 3.502 1.170 1.259 0.312 1.512 5.134 3.487 1.544 0.815 1.219 2.034 0.871 0.823 1.687 0.938 0.486 0.612 1.325 0.584 0.956 0.626 0.344 1.430

6.0 6.0 6.0 6.0 6.0 6.0

11.24 6.511 4.327 5.708 8.614 5.409

6.0

7.047

6.0 6.0 6.0 6.0 4.5 4.5 6.0 4.5

7.703 5.820 3.473 4.524 8.023 3.325 2.993 1.943

6.0 5.0 5.0 4.5

3.533 1.864 2.824 1.325

6.0 5.0

4.425 1.739

4.5

8.125

6.0

1.819

6.0

2.315

4.5

1.803

6.0 6.0

3.834 1.231

6.0

2.321

6.0

3.342

a

mmax is the maximum molality of electrolyte for correlation and prediction. The original data for mean ionic activity coefficients are taken from literature(Robinson and Stokes, 1955). ARDcorr and ARDpre represent the correlated and predicted average relative deviation of the mean ionic activity coefficients, respectively.

mean ionic activity coefficients of MBr and MI, in which M is Li, Na, K, Rb, and Cs, are shown in Figures 1-6. Conclusion The statistical associating fluid theory (SAFT) is first introduced into the Blum-Wei’s nonprimitive MSA equation (σ+ * σ- * σd). This new nonprimitive electrolyte MSA model has successfully been applied to correlate the mean ionic activity coefficient of 46 single strong 1:1 electrolytes in aqueous solutions with only two adjustable parameters for each electrolyte. The ion-ion, ion-dipole, and dipole-dipole interactions are represented by three parameters, Γ, B10, and b2, respectively. It is proved that this model can accurately correlate the mean ionic activity coefficients in a wider range of electrolyte concentrations except for a few electrolytes that yield rather larger deviations only in

Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4187

Figure 1. Comparison between the correlated and experimental mean ionic activity coefficients. Experimental data: (b) LiBr, (O) NaBr, (9) KBr, (0) RbBr, (2) CsBr; (s) correlated results.

Figure 4. Comparison between the predicted and experimental mean ionic activity coefficients. Experimental data: (b) LiI, (O) NaI, (9) KI, (0) RbI, (2) CsI; (s) predicted results.

Figure 2. Comparison between the predicted and experimental mean ionic activity coefficients. Experimental data: (b) LiBr, (O) NaBr, (9) KBr, (0) RbBr, (2) CsBr; (s) predicted results.

Figure 5. Comparison between the correlated and experimental mean ionic activity coefficients. Experimental data: (b) LiCl, (O) NaCl, (9) KCl, (0) RbCl, (2) CsCl; (s) correlated results.

Figure 3. Comparison between the correlated and experimental mean ionic activity coefficients. Experimental data: (b) LiI, (O) NaI, (9) KI, (0) RbI, (2) CsI; (s) correlated results.

Figure 6. Comparison between the correlated and experimental mean ionic activity coefficients. Experimental data: (b) HCl, (O) HBr, (9) HI, (0) HNO3; (s) correlated results.

higher electrolyte concentrations. The reason for this phenomenon can be attributed to the oversimplified approximations of this nonprimitive model compared to the real electrolyte systems. From Table 1 and Figures 1-6, we can see that the calculated results of this new equation are superior to

our previous work both in the correlated accuracy of the mean ionic activity coefficients and in the concentration range of electrolytes for correlation. Li et al. (1996) applied the nonprimitive model of MSA to 14 real 1:1 electrolyte solutions by fitting the diameters of cation and anion simultaneously. However, the correlated

4188 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998

average relative deviations for the mean ionic activity coefficients were 5.5% and the correlated maximum electrolyte concentrations are only up to 3.0 mol/kg, while with our equation, the correlated average relative deviation is only 1.46% with the maximum electrolyte concentrations as 3.0 mol/kg and the predicted deviation is 3.87% with 6.0 mol/kg. In addition, although LvovWood(1990) adopted the Blum-Wei (1987) equation to correlate the densities of the NaCl-H2O system over a wide range of temperatures and pressures and the correlated results are satisfactory, the adjustable parameters in their equations are up to 12, including 10 parameters for ions and two parameters for water molecules. There are only two adjustable parameters (the diameters of cation and anion) in our equation. Table 1 also shows that the fitted ionic diameters depend on the specified salts. It can be seen that the diameter of the same cation, such as Na+, has a little difference in various salts with different anions (such as NaCl and NaBr). The same phenomenon happens with the same anion in various salts with different cations. This can be explained by the fact that the contributions of ionic solvation, Lennard-Jones dispersion, ionic induced dipole moment, and multipolar interactions are neglected in our nonprimitive model. Thus the thermodynamic properties among different electrolytes with the same ion are reflected only by the diameters of ions. This phenomenon can also be found in the calculation of mean ionic activity coefficients in aqueous electrolytes with the electrolyte equation of state based on perturbation theory (Jin and Donohue, 1988). The advantage for this nonprimitive MSA equation is that the vacuum dielectric constant can be taken in the whole calculation, while in the nonprimitive perturbation theory equation (Jin and Donohue, 1988) the solvent dielectric constant must be used in the longrange electrostatic interaction term. Another advantage for this work is that the ionic parameters, such as the dispersion energy parameter i/k and polarizability Ri, are unnecessary in correlating the mean ionic activity coefficients. However, because there are no literature data available for some ions, such as H2PO4-, ClO4-, ClO3-, SCN-, COOH-, BrO3-, and so on, which limits the perturbation theory to some electrolytes consisting of these ions. In addition, although this equation seems to be rather complicated, it converges rapidly during the iteration procedure and is not difficult in the correlation of the mean ionic activity coefficients, if the appropriate initial values of parameters are chosen.

N ) total molecule numbers P ) pressure r ) distance R ) gas constant T ) absolute temperature U ) pair potential V ) volume x ) mole fraction XA ) mole fraction of molecule not bonded at site A z ) ionic charge Z ) compressibility factor Greek Symbols β ) 1/kT β3 ) 1 + b2/3 β6 ) 1 - b2/6 β12 ) 1 + b2/12 β24 ) 1 - b2/24 ∆AB ) association strength between sites A and B AB ) association energy between sites A and B Φmnl ) rotational invariant γ ) ionic activity coefficient in molality scale Γ ) MSA parameter κAB ) association volume between sites A and B µ ) chemical potential, dipole moment ν ) stoichiometric coefficient F ) number density σ ) diameter Subscripts d ) dipole dd ) interaction between dipole and dipole i,j ) component indices id ) interaction between ion and dipole ii ) interaction between ion and ion Superscripts assoc ) association corr ) correlation hs ) hard sphere id ) ideal MSA ) mean spherical approximation pre ) prediction

Appendix The definitions of all the variables in eq 12 are as follows:

d02 ) 4πβe2 d22 ) 4πβµd2/3

Acknowledgment

β3 ) 1 + b2/3

We gratefully acknowledge the support of this work by the National Natural Science Foundation Commission of China under Grant 29576250.

β6 ) 1 - b2/6 λ ) β3/β6

Nomenclature A ) Helmholtz free energy b2, B10 ) MSA parameters c ) direct pair correlation function e ) elementary charge f ) ionic activity coefficient in mole scale g ) radial distribution function h ) total pair correlation function k ) Boltzmann constant m ) molality concentration M ) numbers of association sites on each molecule

y1 )

w1 )

4 β6(1 + λ)2

∑i β (σ 6

1 w2 ) Fdσd2B10 2

d

∑i

Fizi2 + λσi)(1 + Γσi) Fizi2σd2

[2β6(σd + λσi)(1 + Γσi)]2

Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4189

νη ) [-w1/2 + x(w1/2)2 + 2B10w2/β62]/w2 ∆Γi )

νηFdσd2σi2B10 8β6(σd + λσi)

Γis ) [(1 + σiΓ - ∆Γi)D - 1]/σi DiF ) ziβ6/[2(1 + σiΓ - ∆Γi)] Fiσi2(DiF)2

∑i

νη2Fdσd2

D)1+

Dac )

[2β6(σd + λσi)]2

∑i Fi(DiF)2 Fiσi(DiF)2

∑i σ

Ω10 ) νη

d

+ λσi

a0i ) β6ΓisDiF/Dac a1d ) -k10 di ) 1-

(

σd2DiF 2Dβ62

Fdk11 dd

(

)

Dβ6 σdB10 Ω10λ + 2Dac 2 Dβ6

)

Ω10Γis σd3B10a0i νη + + σd + λσi Dac 12β6

(

)

Fdσd2Ω10a1d Fdσd3B10a1d 1 ) λ+ + Dβ6 12β6 2β 2 mi )

Ni )

[

6

νηDiF (σd + λσi)

]

2DiF Fdσd3B10νησi 1+ - zi/σi β6σi 24(σd + λσi)

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Received for review January 20, 1998 Revised manuscript received June 8, 1998 Accepted June 23, 1998 IE980035W