Explicit approximations of the mean spherical approximation model for

5403

J. Phys. Chem. 1993,97, 5403-5409

Explicit Approximations of the Mean Spherical Approximation Model for Electrolyte Solutions W. Sheng, N. Kalogerakis, and P. R. Bishnoi’ Department of Chemical and Petroleum Engineering, University of Calgary, 2500 University Dr. N. W.,Calgary, AB, Canada T2N 1 N4 Received: November 10, 1992; In Final Form: February 10, 1993 Two alternative explicit approximations to the mean spherical approximation (MSA) for electrolyte systems with ions of different sizes and electrostatic charges are presented. The proposed approximations enable easy derivation of other thermodynamic properties due to electrostatic interaction. Explicit expressions are desirable when an equation of state approach is used for electrolyte systems. In this study, the approximations are used to calculate osmotic coefficients and chemical potentials for a wide range of conditions. The deviations of the results from the full MSA are calculated and compared with those obtained by using the approximation of Harvey et a1.8 Both the proposed explicit forms are generally better, particularly for unequal charge ion mixtures with high ratio of ion diameters. One of the proposed forms gives chemical potentials very close to those obtained by using full MSA.

1. Introduction The mean spherical approximation (MSA) is a powerful tool for calculating the thermodynamic properties of simple electrolytes. Blum and co-workersl.2 applied the MSA to the primitive model of electrolytes. This theory has received extensive attention in recent years (Humffray,j Gering et a1.,4 Wu and Lee? Copeman and Stein:,’ Harvey et al.,8*9and CortiIO). All the previous work indicated that the MSA yields satisfactory results for activity and osmotic coefficients for dilute and moderate ion concentrations, and with some modifications it has been validated for very concentrated solutions (Landisl3). The correlation capabilities of MSA can be substantially improved if the ionic diameters are treated as adjustable parameters. The original MSA theory, solved by Blum and cO-workers,I~2 resulted in an implicit formulation of the screening parameter. However, the explicit solution of MSA is desirable for practical applications,l*especially when an equation of state approach is used for electrolyte systems. Theuse of a single equation of state (EOS) for both liquid and vapor phases has the advantage of describingvapor-liquid equilibrium (VLE) at high pressureswhen one or more supercritical components are present in the liquid phase. The equation of state approach also provides the model with thermodynamic consistency. It is well-known that iterative methods must be used for the VLE calculations. The computing time may become prohibitive for flowsheet calculations if the MSA parameters involved in an EOS are implicit. In addition, theimplicit equations for the screeningparameter result in implicit derivatives of the screening parameter withrespecttoeither density or composition. These implicit derivatives and all the thermodynamic properties have to be calculated numerically. It is therefore desirable to have an explicit approximation of the MSA model. Lee12 suggestedto ignorethe P, term in the screeningparameter when the ratio of ion diameters is close to unity or when the Bjerrum length is large. Although the removal of the P, term simplifies the equations considerably, the screening parameter expression still remains implicit. Therefore, even if conditions allow dropping of the P,,term, an explicit approximation of the MSA is still desirable. Three explicit approximations of the original MSA have been suggested in the literature. Copeman and Stein6.’ proposed an explicit approximation for the screening parameter based on an extension of the Debye-Hiickel radial distribution function. This approximation is good for all different ion sizes at low concen-

trations. Harvey et al.899published a simple approach based on the fact that for equal size ions, MSA resulted in a very simple explicit form. They used an effective diameter for mixtures. This approach gives reasonable results even at high concentrations but is very sensitive to the ratio of ion sizes. As Harvey et a1.8 concluded that Copeman and Stein’s approximation quickly becomes unsuitable at high concentrations, we will therefore not further study Copeman and Stein’s approximation. SanchezCastro and Blum14proposed two approximations on the basis of finding an appropriate mean ionic diameter. Their A1 approximation could approach the full MSA very closely with NewtonRaphson iteration. Their A2 approximation is explicit but does not give satisfactory estimates of the mean ionic diameter. Furthermore, they only examined their approximations at low concentration regions. In this work, we present two alternative explicit approximations of the original MSA screening parameter. The approximations are used to calculate osmotic coefficients and chemical potentials for a wide range of conditions. The results are compared with those obtained by using the original MSA and the approximation of Harvey et a1.8 2. MSA Theory aod Previous Explicit Approximations In the MSA theory, the contributions of ion-ion interactions to the thermodynamic properties of electrolyte solutions can be expressed as a function of the MSA screening parameter. Blum and co-workers1V2solved the MSA for the primitive model of a mixture of charged hard spheres and obtained the following equations for the screening parameter I’:

P, =

-c-+ ujr 1

PjQjzj

1

Q

3 T

PjQj

Q=l+-C2A 1 ujr

* Author to whom correspondence should bc addressed. 0022-365419312097-5403!§04.00/0 0 1993 American Chemical Society

+

(3) (4)

5404 The Journal of Physical Chemistry, Vol. 97, No.20, 1993

where e is the electronic charge, k is Boltzmann's constant, D is the static dielectric constant of the medium, T i s the absolute temperature, zi is the valence and ai the diameter of ion i, pi is the number density of ion i, and the summation is over all ions. The MSA screening parameter approaches half the value of the reciprocal Debye screening length at infinite dilution. The excess Helmholtz energy per unit volume, osmotic coefficient, and chemical potentials due to the electrical charges are given by Hoye and Blum2 as follows:

Sheng et al. very close to the solution of the implicit MSA. Hence, we propose the following approximation: --I

&Pi\

1

+ airo

where

P&=

1 -En, 1 + ajro PjQfj

and

n,=

(7)

1

3

+-E2A l + air, a

PjQj

In the above equations, I'o is expressed as

Equations 1 4 are implicit in I'. The order of eq 1with respect to I'depends on the number of ion species in the mixture. For a binary salt, the equation is of sixth order in I' and has to be solved numerically. LandisI3pointed out that only one physically meaningful root exits for the equation. The iterative solution for I' can be obtained by either the Newton-Raphson method or simple direct substitution. Copeman and Stein6,'proposed an explicit approximation based on an extension of the Debye-Hiickel radial distribution function. Although their approximationis good for ions with different sizes, it gives large deviations from the full MSA for moderate to high salt concentrations. This behavior is not surprising since Debye Hiickel theory is valid for low-density solutions only. Harvey et a1.* proposed another explicit approximation. For a mixture containing equal size ions, eq 1 simplifies to

r = -2a1[ ( I + 2 U K ) ' l 2 - 11

For a mixture of ions with unequal ionic diameter, we propose the use of the following equation for effective diameter as suggested by Blum:15

It is noted that for equalionic charge, eq 18 reduces to eq 11. For simplicity, the subscript "mixn in amixwill be omitted in the subsequent equations. Led3 investigated the effect of P,,term. He concluded that P,,is a negativequantityand thecontributionof P,,to thescreening parameter is only up to 6%. In view of this, we decided to examine the effect of dropping the Pmterm in the proposed approximation (eqs 14 to 18). Then the approximation simplifies to

(9)

where

Since for real systems the size of the ions is different from each other, they suggested the following effectivediameter for mixtures:

where p is the number density of the mixture and the summation is over all the species. For their approximation, Harvey et aL8 gave the following expressions for the osmoticcoefficientand the chemical potentials

For clarity, we denote eqs 14-18 as approximation I and eqs 17-19 as approximation 11. Harvey et aL8 made extensive comparisons of the deviations of their approximation from the full MSA for dimensionless excess Helmholtz energy, ionic chemical potentials, and osmotic coefficient. According to their work, the deviations in the osmotic coefficient and chemical potentials are two to three times as high as in excess Helmholtz energy. This is due to the fact that the osmotic coefficient and chemical potential are the first derivativesof the Helmholtz energy with respect to the density of a mixture and the species i. In this work, we present the comparisons of the deviations from the full MSA for the screening parameter, the osmotic coefficient, and the chemical potentials obtained using the proposed explicit approximations and that of Harvey et al. The osmotic coefficient for approximation I is obtained from eq 7 as

where I' is calculated from eq 14. For approximation 11, the osmotic coefficient is simplified to the following by dropping the Pd term where I' is obtained from eq 9. 3. hoposed Explicit Approximations and Required Equations

Our rationale behind the proposed explicit approximations of the MSA is based on the fact that the numerical solution of the implicit MSA by successive substitution converges very fast. Therefore, if we use a very good initial guess, such as that given by eq 9, one iteration of successive substitution should bring us

where r is given by eq 19. The LLQelff for approximation I is obtained from eq 6 as

where,'l P,,.-,,and

$20 are

obtained from eqs 14-16.

The Journal of Physical Chemistry, Vol. 97, No. 20, 1993 5405

Explicit Approximations of the MSA Model Chemical potential can be derived by differentiating eq 22 per eq 8 as follows:

A,?,

-=

kT PjQjzj

Figure 1. Three-dimensional percent deviation surfaces of osmotic coefficientcalculated by approximation I and Harvey et al. approximation from the full MSA for 2:l electrolyte at r = 0.2. / /

an0

lr

ap,

6A

-= -(no - 1)u; +

2

0.001

-

-=?rUiP, 2

0.01

0.1

1

Reduced Number Density

Figure 2. Percent deviations of osmotic coefficient calculated by approximations I and I1 from the full MSA for 2:l electrolyte with r = B = 10; (- * -) B 50. 0.5: (-) B = 2, (- - -) B 5; (**e)

4. Results add Discussions

For approximation 11, the above equations are simplified to

where

For approximation I and 11,eqs 20 and 2 1were used to compute the osmotic coefficients and eqs 23 and 29 were used for the analytical calculations of the chemical potentials, The chemical potentials, ~ ~for these l ~ approximations ~ , were also computed numerically by differentiating eqs 22 and 28 as per eq 8. The results obtained by the numerical differentiation differed within computer round off errors. For Harvey et al. approximation, equations (12) and (1 3) were used for calculating the osmotic coefficients and the chemical potentials, pelmi. For full MSA, we used the equations reported by Blum and Hoye for the osmotic coefficient calculations and obtained the chemical potentials, pdCCi, by adopting the Harvey et al. approach of numerically differentiating eq 6 according to eq 8. A mixture of equal charge ions of two species can be characterized by three dimensionless parameters. They are the reduced Bjerrum length, the reduced number density, and the ratio of ion sizes. In our comparisons, these three independent variables were chosen. The reduced Bjerrum length B is defined as

the reduced number density as

Sheng et al.

5406 The Journal of Physical Chemistry, Vol. 97, No. 20, 1993

TABLE I: Average and Maximum Absolute DeViPtio~(96) of the Screening Parameter, Osmotic Coefficient,and Chemical Potentials for the Three Approximations for 1:l Electrolyt- from tbe Full MSA Metbod. Harvey et al. approx I approx I1 r

B

0.2

2

0.2

5

0.2

10

0.2

50

0.2

100

0.5

2

0.5

5

0.5

10

0.5

50

0.5

100

0.8

2

0.8

5

0.8

10

0.8

50

0.8

100

AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD

ip

r

18.52 35.62 21.53 40.23 25.03 43.94 34.71 5 1.67 39.25 54.38 4.96 9.48 5.67 10.70 6.59 11.84 9.33 14.56 10.71 15.62 0.56 1.07 0.64 1.20 0.74 1.34 1.06 1.69 1.22 1.80

3.30 7.64 6.09 12.20 8.26 15.20 13.25 20.68 15.46 22.48 0.67 1.42 1.35 2.60 1.88 3.38 3.10 4.83 3.64 5.33 0.07 0.14 0.15 0.27 0.20 0.36 0.34 0.53 0.40 0.58

U+

U-

2.65 6.98 8.49 16.73 12.44 22.12 20.41 30.59 23.64 33.08 3.25 6.20 0.15 0.38 1.54 2.48 4.30 6.30 5.24 7.28 1.93 3.77 0.92 1 .80 0.42 0.87 0.26 0.33 0.43 0.55

21.90 39.76 18.57 34.67 17.59 32.22 18.93 30.53 20.65 31.03 9.45 17.77 6.98 13.24 5.99 1 1.09 5.46 8.90 5.73 8.67 2.63 5.05 1.68 3.23 1.26 2.38 0.83 1.41 0.78 1.24

,a

r

2.42 6.54 5.57 13.80 8.84 20.84 19.75 42.05 26.42 52.88 0.67 1.83 1.63 4.01 2.56 5.87 5.32 10.42 6.82 12.35 0.07 0.21 0.19 0.46 0.29 0.67 0.59 1.13 0.75 1.34

0.74 2.05 1.73 4.29 2.74 6.39 6.00 12.30 7.90 15.11 0.22 0.60 0.53 1.31 0.84 1.91 1.73 3.35 2.21 3.95 0.02 0.07 0.06 0.15 0.10 0.22 0.20 0.37 0.25 0.44

P-

,a

r

0.60 1.92 1.34 4.20 1.59 5.12 1.09 4.44 0.63 3.25 0.04 0.12 0.14 0.46 0.17 0.54 0.12 0.40 0.08 0.29 0.00 0.00 0.00 0.01 0.00 0.02 0.00 0.01 0.00 0.01

2.65 3.84 4.64 12.69 8.43 20.74 20.05 42.74 26.78 53.58 0.82 1.62 1.46 4.27 2.51 6.21 5.39 10.69 6.89 12.56 0.10 0.21 0.17 0.50 0.29 0.71 0.60 1.16 0.76 1.36

3.27 8.02 3.49 8.46 4.05 9.51 6.59 13.80 8.31 16.17 0.96 2.46 1.05 2.56 1.22 2.81 1.89 3.74 2.32 4.22 0.11 0.28 0.12 0.30 0.14 0.32 0.21 0.42 0.26 0.47

U+

0.13 0.28 0.31 1.18 0.54 1.87 0.97 2.80 1.17 3.01 0.00 0.01 0.08 0.28 0.1 1 0.36 0.12 0.34 0.12 0.30 0.00 0.00 0.00 0.01 0.01 0.02 0.01 0.01 0.00 0.01

P+

P-

6.39 13.04 3.42 6.78 2.00 3.81 0.20 0.30 0.46 1.03 5.31 10.77 2.93 5.84 1.85 3.66 0.59 1.18 0.34 0.69 2.15 4.23 1.21 2.37 0.78 1.52 0.27 0.53 0.17 0.33

16.72 30.05 10.72 21.17 7.49 15.84 3.44 8.22 2.81 6.64 7.89 14.62 4.73 9.06 3.14 6.15 1.16 2.37 0.77 1.56 2.44 4.65 1.41 2.71 0.92 1.78 0.32 0.64 0.20 0.40

Twenty data points were taken from the computation of the deviation at the given B and r. AAD, average absolute deviations, 96

-= I 1

AAD = MAD, maximum absolute deviations, 96

NDP

NDP i = l

calcd from approx - calcd from full MSA calcd from full MSA

calcd from approx - calcd from full MSA calcd from full MSA

-12 : c

-14 0.001

'

'

' ' . . ' . '

'

' " . ' ' . '

0.01

'

'

'',',,I

0.1

1

0.001

Reduced Number Density

Figure 3. Percent deviations of osmotic coefficient calculated by approximation I and Harvey et al. approximation from the full MSA for 2:l electrolytewith r = 0.5: (-) B = 2,(- - -) B = 5; B = 10; (- -) (-e)

B = 50.

.

and the size ratio of ions as

+

r = U+/IJ-

0.01

0.1

1

Reduced Number Density

(34)

where a = (u+ u-)/2. Besides the abovethree characteristicvariables,another factor, the charge of ions, is required to define electrolyte systems with unequal charge ions. Two cases are investigated in this work,

Figure 4. Percent deviations of osmotic coefficient calculated by approximations I and I1 from the full MSA for 1:l electrolyte with r =

0.5: (-) B

2;(- - -) B = 5; (-*) B = 10;(- * -) B

50.

i.e., 1:l and 2:l electrolyte solutions. Three different size ratios of r = 0.2,0.5, and 0.8 are chosen for each case. The ranges of other two independent variables are chosen as 1 C B C 100.0 and 0.001 C C 1.O. These conditions cover most of the real systems and the typical regions of physical interestfor electrolytesolutiom. For example, for a typical aqueous4ectrolyte solution, is approximately 0.2 and B is near 2. Only single salts with one cationic species and one anionic species are presented here.

The Journal of Physical Chemistry, Vol. 97, No. 20, 1993 5407

Explicit Approximations of the MSA Model

TABLE II: Aven e and Maximum A h l u t e Devhtians (%) of the Scr Purmeter, Osmotic Cocliiciemt, and CBCmicrl ~otentiaisfor the Are Approximations for 2 1 Electrolytes from the Full A Metbod' Harvey et al. approx I approx I1

2

r 0.2

B 2

AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD AAD MAD

0 36.20 56.26 41.74 61.23 46.32 64.50 56.40 70.31 60.40 72.13 16.83 26.15 19.56 28.73 21.80 30.55 26.82 34.12 28.91 35.35 5.10 7.84 5.98 8.63 6.66 9.15 8.14 10.20 8.74 10.56

r

P+

12.23 20.75 15.88 24.99 18.62 27.73 24.52 32.66 26.96 34.26 5.28 8.36 6.65 9.92 7.67 10.94 9.86 12.79 10.75 13.41 1.64 2.51 1.98 2.86 2.24 3.08 2.78 3.50 3.00 3.64

U-

21.13 34.14 27.58 41.07 32.02 45.12 40.84 5 1.30 44.24 53.69 7.89 11.66 11.38 16.11 13.60 18.53 17.75 22.21 19.28 23.26 1.97 2.54 3.19 4.22 3.93 5.1 1 5.23 6.39 5.68 6.73

35.42 55.22 33.76 52.17 33.88 50.76 37.00 50.1 1 39.20 50.67 18.62 30.38 17.06 26.61 16.79 24.92 17.85 23.57 18.76 23.64 6.28 10.63 5.60 8.83 5.44 8.04 5.65 7.33 5.89 7.28

0

r

2.44 7.04 5.16 13.39 8.02 19.54 17.55 37.65 23.36 46.71 0.66 1.88 1.49 3.73 2.29 5.29 4.65 9.10 5.93 10.70 0.07 0.18 0.17 0.40 0.26 0.59 0.52 1.01 0.66 1.18

0.77 2.31 1.63 4.29 2.53 6.14 5.39 11.23 7.07 13.62 0.22 0.62 0.49 1.22 0.75 1.73 1.52 2.94 1.93 3.44 0.02 0.06 0.05 0.13 0.09 0.20 0.17 0.34 0.22 0.39

u+ 0.08 0.39 0.26 0.91 0.37 1.18 0.50 1.36 0.63 1.53 0.02 0.07 0.05 0.18 0.07 0.21 0.06 0.17 0.06 0.14 0.00 0.00 0.00 0.01 0.00 0.01 0.00 0.01 0.00 0.01

u1.03 3.16 1.63 4.71 1.71 5.06 0.40 2.44 0.77 1.67 0.08 0.29 0.18 0.56 0.19 0.60 0.10 0.36 0.04 0.21 0.00 0.00 0.00 0.02 0.01 0.02 0.00 0.01 0.00 0.01

0 2.43 4.12 4.42 11.74 7.62 18.44 17.63 37.16 23.48 46.35 0.76 1.72 1.35 3.86 2.24 5.48 4.70 9.24 5.98 10.81 0.09 0.18 0.15 0.43 0.26 0.63 0.53 1.03 0.67 1.19

r

U+

2.43 5.81 2.77 6.74 3.35 7.98 5.76 12.14 7.32 14.28 0.79 1.96 0.88 2.12 1.03 2.38 1.64 3.22 2.01 3.63 0.09 0.24 0.10 0.25 0.12 0.28 0.19 0.37 0.23 0.41

3.81 7.25 5 0.2 2.07 3.98 10 0.2 1.22 2.37 0.2 50 0.16 1.12 100 0.2 0.20 0.41 2 0.5 3.30 6.43 0.5 5 1.84 3.58 0.5 10 1.17 2.28 0.5 50 0.38 0.78 0.5 100 0.23 0.48 2 0.8 1.39 2.71 5 0.8 0.79 1.53 10 0.8 0.51 0.98 50 0.17 0.8 0.34 100 0.8 0.1 1 0.22 Twenty data points were taken for the computation of the deviation at the given B and r. AAD, average absolute deviations, 96.

U-

19.57 34.03 12.82 25.53 9.18 20.43 5.03 13.75 4.79 12.89 9.77 17.52 5.91 11.12 3.94 7.67 1.48 3.10 1.01 2.11 3.16 5.96 1.84 3.51 1.20 2.31 0.42 0.84 0.27 0.53

MAD, maximum absolute deviations, % MAD = maximum

calcd from approx - calcd from full MSA calcd from full MSA

-5

-25 0.001

0.01

0.1

o.mi

1

Reduced Number Densky

Figure 5. Percent deviations of osmotic coefficient calculated by approximationI and Harvey et al. approximation from the full MSA for 1:l electrolyte with r = 0.5: (-) B = 2; (- - -) B = 5; (-) B = 10;(- -) B = 50.

-

Tables I and I1 summarize the results of the average percent absolute deviations of the screening parameter, osmotic coefficients, and chemical potentials for the three approximationsfrom the full MSA method. Three different ion size ratios, r = 0.2, 0.5,and0.8, are used in each table. The reduced number density ranges from 0.001 to 0.5. The results are also shown in Figures 1-13.

0.01

0.1

1

Re&md Number Dmdty

Flgum 6. Percent deviationsof cclcccalculated by three approximations from the full MSA for 1:l electrolyte with r = 0.5: (-) B = 2; (- -) B = 10.

-

Although not shown in the tables, it was observed that approximation I gave all positive deviations for the screening parameter and the osmotic coefficient, while the approximation of Harvey et 01. gave all negative deviations. This behavior can also be seen in Figure 1, where deviations for the osmotic coefficient are plotted for the size ratio, r, of 0.2 for a 2:l electrolyte. For approximation 11, however, it was observed that the deviations were negative at low electrolyte concentrations and positive at high electrolyte concentrations.

Sheng et al.

5408 The Journal of Physical Chemistry, Vol. 97, No. 20, 1993 51 0

0

-2

ae 6a '

8 - 4

$ 4 3

g

-8

-10

5

1

-10

-15

-..'I

[

-5

6

'.,

2

-12

-20 -25

-14

0.m

0.01

-35 0.001

1

0.1

' ' , , . . , I

'

,

' , ' ' , ' , I

"""'

0.1

0.01

1

Reduced Number Density

RedNed Number Density

Figure 7. Percent deviations of pelec-calculated by three approximations from the full MSA for 1:l electrolyte with r = 0.5: (-) B = 5 ; (- * -) B = 50. 14

'

Figure 10. Percent deviations of pelcccakulated by three approximations from the full MSA for 2:l electrolyte with r = 0.5: (-) B = 2; (- -) B = 10.

1

L

I

5, 0

8

.-6

-5 -10

a

-15 u

3

2

-20 -25 -30

-4-

0.01

0.001

0.1

1

-35 0.001

Reduced Number Density

Figure 8. Percent deviations of pelec+calculated by three approximations from the full MSA for 1:l electrolyte with r = 0.5: (-) B = 2; (- * -) B = 10. 8

'1

-

0.01

0.001

1

7

I

I Approx. I II --ox. II H-Herveyotal

Figure 11. Percent deviations of pels- calculated by three approximations from the full MSA for 2:l electrolyte with r = 0.5: (- -) B = 5 ; (.-) E = 50. 10

,

1

t

-81

0.1

Reduced Number Density

1

' .

'

, , ' . " ' '

0.01

'

' , . , . . ' I

'"""I

0.1

1

Reduced Number Density

-20t

0.001

' ' . . ' . , . I

0.01

' ' , ' . . ' . '

'

0.1

' , . A 1

Reduced Number Density

Figure 9. Percent deviations of pelec+catilated by three approximations from the full MSA for 1:l electrolyte with r = 0.5: B = 5 ; (- -) B = 50.

Figure 12. Percent deviations of pels+ calculated by three approximations from the full MSA for 2:l electrolyte with r = 0.5: (-) B = 2; (- -) B = 10.

For clarity, Figures 2-5 show the deviation distribution in two dimensions for four typical B values. From these figures we can see that the deviations from the full MSA increase as the rduced number density is increased. As seen from Figure 2, approximations I and I1yield small deviationsfor theosmotic coefficients from the full MSA and the results are comparable. Figure 3 shows the comparison of percent deviations for the osmotic coefficientsfrom the full MSA for approximation I and the Harvey et al. approximation. The maximum percent deviation of approximation I is about one-third of that given by the Harvey et al. approximation under the same conditions. The results for 1:l electrolytes, shown in Figures 4 and 5, are similar to those for 2:l electrolytes shown in Figures 2 and 3.

It is also seen in Tables I and I1 that approximation I gives an excellent representation of the chemical potential for both the ions. Both approximation I and I1 are sensitive to the ratio of ion sizes like the Harvey et al. approximation. All the three approximations yield the same results as the full MSA for an equal ion size mixture. For small ion size difference, e.g., r = 0.8, Harvey et al. approximationgives satisfactory results although approximations I and I1 are better. Figures 6-1 3 givetwo-dimensional distribution of thedeviations fromthefullMSAfor thechemicalpotential. Fromthesefigures, we can see that approximation I gives deviations close to zero line for all the cases. It can also be seen that although approximation I1 gives high deviations at low Bvalues, it is generally better than the Harvey et al. approximation.

(e-)

The Journal of Physical Chemistry, Vol. 97, No. 20, 1993 5409

Explicit Approximations of the MSA Model I

.-.

-I-._

Nomenclature AA excess Helmholtz energy per unit volume B reduced Bjerrum length D static dielectric constant e electronic charge k Boltzmann constant P,, defined by eq 2 r ratio of ion size T temperature z electronic charge Subscripts species i, j i, j 0 approximate values mix mixture property

The maximum deviations of screening parameter, osmotic coefficient, and chemical potentials from the full MSA model calculated by the three approximations are also included in Tables I and 11. As shown in Table I, the maximum deviations obtained from approximationsI and I1aregenerally less than thoseobtained from the Harvey et al. approximation. For example, for r = 0.2 and B = 5, the Harvey et al. approximation yielded maximum deviations for osmotic coefficient, screening parameter, and chemical potentials for cation and anion of 40, 12, 17,and 35% respectively, while those obtained from approximation I are 14, 4,1,and 496,respectively. As shown in Table 11,for 2:lelectrolyte systems, the maximum deviations obtained from Harvey et al. approximation are much higher than those given by approximations I and 11. This indicates that the effective diameter used by Harvey et al. is not suitable for electrolytes with unequal electroniccharges. Useof eq 18 for such electrolytes may possibly improve the results from the Harvey et al. approximation.

5. Conclusiom Two explicit approximations (I and 11) of the MSA model for electrolytes have been proposed that do not deviate significantly, except at high value of B and low value of r, from the full MSA model for screeningparameter, osmoticcoefficient, and chemical potential. Both approximations yield similar results for the screening parameter and osmotic coefficient. Approximation I, however, gives a much better representation for the chemical potentials compared to the mathematicallysimpler approximation 11. Both the proposed approximations are better than that proposed by Harvey et ala8 Acknowledgment. This work was carried out as a part of the "Solid Deposition in Hydrocarbon Systems" Project, Contract 5091-260-2138, funded by the Gas Research Institute, Chicago, IL, and the Gas Processors Association, Tulsa, OK.

Superscript elec electric charge contribution Greeks a

I? A K

fi

p

u

d

defined byeq 5 screening parameter defined byeq 4 defined as in eq 10 chemical potential number density of ion diameter of ion osmotic coefficient

References and Notes (1) Blum, L. Mol. Phys. 1975,35, 299. (2) Hoye, J. S.;Blum, L. J . Phys. Chem. 1977,81, 1311. (3) Humffray, A. A. J . Phys. Chem. 1983,87, 5521. (4) Gering, K.L.; Lee, L.; Landis, L. H.;Savidge, J. FluidPhase Equilib. 1989, 48, 1 11. (5) Wu, R. S.; Lee, L. Fluid Phase Equilib. 1992, 78, 1. (6) Copeman, T. W.; Stein, F. P. Fluid Phase Equilib. 1986, 30, 237. (7) Copeman, T. W.; Stein, F. P. Fluid Phase Equilib. 1987, 35, 165. (8) Harvey, A. H.; Copeman, T. W.; Prausnitz, J. M.J . Phys. Chem. 1988, 92,6432. (9) Harvey, A. H.; Prausnitz, J. M. AIChE J . 1989, 35, 635. (10) Corti, H. R. J . Phys. Chem. 1987, 91, 686. (1 1) Henderson, D.; Blum, L.; Tani, A. Equation ofstare: Theories and

Applications; ACS Symposium Series; Chao, K. C., Robinson, R. L., Us.; American Chemical Society: Washington, DC, 1986; p 281. (12) Lee, L. J . Chem. Phys. 1983, 78, 5270. (13) Landis, L. H. Ph.D. Thesis, University of Oklahoma, 1985. (14) Sanchez-Castro, C.; Blum, L. J . Phys. Chem. 1989, 93, 7378. (15) Blum, L. J . Phys. Chem. 1988, 92, 2969.