On the Correlation of the Activity Coefficients in Aqueous Electrolyte

Feb 21, 2003 - The objective of this work is to extend the Kelvin mean spherical approximation (K-MSA) model to correlate the mean and individual ioni...
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Ind. Eng. Chem. Res. 2003, 42, 1279-1284

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On the Correlation of the Activity Coefficients in Aqueous Electrolyte Solutions Using the K-MSA Model C. Ghotbi,† G. Azimi,† V. Taghikhani,† and Juan H. Vera*,‡ Department of Chemical Engineering, Sharif University of Technology, Tehran, Iran, and Department of Chemical Engineering, McGill University, Montreal, Quebec, Canada H3A 2B2

The objective of this work is to extend the Kelvin mean spherical approximation (K-MSA) model to correlate the mean and individual ionic activity coefficients for symmetric and asymmetric electrolyte solutions at different concentrations and at different temperatures. Revised values of the parameters for the mean and individual ionic activity coefficients of 1:1 electrolytes and new values of the parameters for the mean ionic activity coefficients of asymmetric electrolytes are presented. The effect of the short-range electrostatic term (Pn) at different size ratios of the ions is examined. Assuming a constant anion diameter and a composition-dependent diameter for the cation, the K-MSA model gives realistic cation-hydrated diameters, in comparison with their crystallographic Pauling diameters. Notably, for both symmetric and asymmetric electrolytes, the like and unlike ionic radial distribution function at contact value are positive over all of the concentration and temperature ranges studied. The results obtained from the K-MSA model compare favorably with those obtained from the Boublik-Mansoori-Carnahan-StarlingLeland mean spherical approximation and Pitzer models for the mean ionic activity coefficients and with those obtained with the Khoshkbarchi-Vera model for the individual ionic activity coefficients. Introduction The mean spherical approximation (MSA) theory has proved to successfully represent activity coefficients in electrolyte solutions.1-3 Because this theory includes the excluded volume of the ions, as well as short-range and long-range interaction effects, it can be used up to high electrolyte concentrations. In this theory, the effect of the excluded volume of the ions is represented by a hard-sphere (HS) equation of state (EOS). Thus, the use of an accurate EOS for mixtures of hard spheres is the first essential step for the MSA to represent correctly the behavior of the electrolyte solutions. The Boublik-Mansoori-Carnahan-Starling-Leland (BMCSL) EOS for mixtures of hard spheres is commonly used as the reference part of the MSA model to correlate the mean ionic activity coefficients and the osmotic coefficients.3-6 The BMCSL-MSA model for electrolyte solutions has two major shortcomings. First, it gives negative values for the like ionic radial distribution functions (RDFs) at contact values at high concentration and large diameter ratio of the ions.7 Second, the values obtained for hydrated diameters of ions are inconsistent with the hydration phenomenon.8 To overcome these shortcomings, Taghikhani and Vera9 proposed the K-MSA model to correlate the activity coefficients of single 1:1 electrolytes in aqueous solutions. This model used the generalized formalism developed by Khoshkbarchi and Vera10 to extend the Kelvin EOS11 to mixtures of hard spheres. In this work, the K-MSA model is applied to correlate the mean ionic activity coefficients of single asymmetric electrolytes and the individual ionic activity coefficients * Corresponding author. E-mail: Fax: (514) 398-6678. † Sharif University of Technology. ‡ McGill University.

[email protected].

of some univalent-univalent electrolytes in aqueous solutions. The values for the parameters for 1:1 electrolytes, reported previously by Taghikhani and Vera,9 are revised. Equations arising from the analytical solution for the MSA theory obtained by Waisman and Lebowitz,12 based on the perturbation and integral equation theory,13 are not repeated here. They are exactly the same as those reproduced by Taghikhani and Vera.9 The effect of the Pn term, representing the short-range interactions, is studied as a function of the size ratio of the ions. Finally, the results obtained from the K-MSA model for asymmetric electrolyte solutions are compared with those obtained from some other commonly used models. To do a fair comparison, the calculations with the K-MSA model and those with the BMCSL-MSA and Pitzer models were performed using the same minimization procedure and the same experimental data of the mean ionic activity coefficients. Correlation of Experimental Data with the K-MSA Model In this work, following the same assumptions as previous publications,3,14-16 we consider the diameter of the anion to be constant and the hydrated diameter of the cation to be changed with the concentration of the electrolyte in the solution. The hydrated diameter of the cation σ+ is considered to be dependent on concentration according to the following relation:

σ+ ) σ+0 + σ+1c + σ+2c2

(1)

where c is the concentration of the electrolyte expressed as molarity. The parameters σ+j (j ) 0-2) are considered to be adjustable parameters. The diameter of the anion was considered constant and equal to the Pauling diameter. Experimental evidence,17 in particular neutron scattering studies,8 suggests a weak hydration of

10.1021/ie020790g CCC: $25.00 © 2003 American Chemical Society Published on Web 02/21/2003

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the halide ions. The use of a concentration-dependent cation diameter is justified by the fact that, in aqueous solutions, the cations hydrate and their sizes change with the electrolyte concentration.18,19 The values of the adjustable parameters were obtained by fitting the experimental data of the mean ionic activity coefficients available in the literature20-22 to minimize the average absolute relative deviation (AARD) between the calculated activity coefficients and the experimental data:

AARD (%) )

( )∑ 100 NP

NP |γ exp i

calc

- γi

| (2)

γiexp

i

where NP refers to the number of the experimental points. While the experimental data for the individual or mean ionic activity coefficients available in the literature are based on the molality scale, the activity coefficients calculated from the MSA model are based on the molarity scale. Thus, the experimental molalityscale activity coefficient data were converted to the molarity scale according to23 c

m

ln γi ) ln γi + ln

mid0 ci

(3)

In eq 3 the superscripts m and c denote the molality and molarity scales, d0 is the density of water, and m and c are the molality and molarity of the ions or the electrolyte in the solution, respectively. The ionic RDF at the contact surface of the ions arising from the K-MSA model takes the form

gij(σij) )

(

[

(

)

2σiσj ξ2 0.852ξ3 1 + + 1 - ξ3 σi + σj ξ3 (1 - ξ )2 3

)

2σiσj ξ2 σi + σj ξ3

2

ξ32(-0.232 + 0.061ξ32 + 0.479ξ39) (1 - ξ3)3

-

]

e2XiXj (4) DkT where

ξn )

x2 n Fσ 2∑ i i

(5)

and Xi is a parameter given by24

Xi )

Zi - [π/2∆]σi2Pn 1 + Γσi

(6)

For the fitting of the individual ionic activity coefficients, in addition to the cation diameter, it was necessary to introduce a composition-dependent anion diameter with the following form:

σ- ) σ-0 + σ-1c

(7)

The values of the adjustable parameters for the anion diameter were obtained by fitting the experimental data

of the individual ionic activity coefficients available in the literature.25-27 Results and Discussion The adjustable parameters of eq 1 for a number of symmetric and asymmetric electrolyte solutions together with the percent of average absolute deviation of the calculated mean ionic activity coefficients from the experimental data are presented in Table 1. This table also compares the results obtained from the K-MSA model with those obtained using the BMCSLMSA and Pitzer models up to saturation. The results presented in Table 1 were obtained by application of the K-MSA model, BMCSL-MSA, and Pitzer models to the experimental data of the mean ionic activity coefficients using the same data and fitting procedure for all of the models. As observed from Table 1 in the cases with low saturation molality, the K-MSA model produces a fitting of the mean ionic activity coefficients for electrolyte solutions that are almost the same as those produced by the BMCSL-MSA model. For the cases of highly soluble salts, however, the results obtained with the K-MSA model for some systems are more accurate than those obtained with the BMCSL-MSA model and also than those obtained with Pitzer’s model. The hydrated diameters at infinite dilution of the electrolytes, i.e., σ0+, obtained in this work are realistic and greater than the corresponding Pauling diameters.22 The values of the parameters for symmetric electrolytes presented here are different from those reported in a previous publication.9 The values previously reported by Taghikhani and Vera9 were in error because of the omission in the computer code of a term of the hard-sphere contribution. The missing term, however, was properly included in the equations presented in the text of the paper. Figure 1 shows the results of the correlation of the mean ionic activity coefficients in aqueous solutions, as a function of molality, obtained from the K-MSA model for the symmetric electrolytes NaCl and KCl and for the asymmetric electrolyte Ca(NO3)2. As seen from this figure, the model correlates well with the experimental values for the mean ionic activity coefficients of the electrolytes up to saturation. Figure 1 also shows the results obtained from the simplified version of the K-MSA model for KCl electrolyte solutions. This simplified version is obtained by setting to zero the electrostatic short-range term Pn in the MSA part of the K-MSA model. Clearly, this major simplification introduced in the model does not have a significant effect on the results. In general, the Pn term has a negative value, and it changes with the size ratio of the ions, i.e., with the electrolyte concentration. The absolute value of the Pn term decreases as the ratio of the ion diameters tends to unity. Because of the electroneutrality condition, the Pn term is equal to zero for the case of equal diameters of the anion and cation. Figure 2 shows a comparison of the results of the correlation of the mean ionic activity coefficients of a HCl solution obtained from the K-MSA model with those obtained with the other models. As seen from this figure, while all three models reproduce the mean ionic activity coefficients of the electrolyte with good accuracy at low concentrations, the BMCSLMSA and Pitzer models fail to accurately correlate the mean ionic activity coefficients at high concentrations. Table 2 presents the values of the hydrated diameters of cations for different alkali halides. As shown in this table, the cation hydrated diameters depend on the

Ind. Eng. Chem. Res., Vol. 42, No. 6, 2003 1281 Table 1. K-MSA Parameters for Cations and AARD % of the Calculated Activity Coefficients from the Experimental Data Obtained with the K-MSA, BMCSL-MSA, and Pitzer Models for Symmetric and Asymmetric Single Electrolyte Solutions at 298.15 K K-MSA parameters

AARD (%)

electrolyte

σ+0 (Å)

σ+1 (Å mol-1 L)

σ+2 (Å mol-2 L2)

mmax

K-MSA

BMCSL-MSA

Pitzer

LiCl LiBr LiI NaCl NaBr NaI NaNO3 KCl KBr KI KOH KClO3 KBrO3 RbCl RbBr CsCl CsBr CsI HCl Li2SO4 Na2SO4 K2SO4 Rb2SO4 Cs2SO4 MgCl2 MgBr2 CaCl2 CaBr2 Ca(NO3)2 BaCl2 BaBr2

4.115 4.272 5.431 3.486 3.692 3.711 3.796 2.983 3.115 3.186 4.024 1.907 2.990 2.860 2.815 1.969 2.269 2.030 4.519 4.770 4.052 4.047 4.625 4.944 5.921 6.581 5.438 6.089 5.951 5.321 5.439

-0.006 0.105 -0.660 -0.265 -0.107 0.102 -0.658 -0.339 -0.261 -0.149 -0.015 -1.015 0.137 -0.040 0.041 0.397 0.266 0.345 -0.096 -0.761 -1.335 -1.809 -1.693 -1.490 -0.058 -0.009 0.058 0.064 -0.404 -0.314 0.057

-0.005 -0.013 0.177 0.035 0.018 -0.001 0.097 0.048 0.044 0.029 -0.003 0.771 -3.313 0.026 0.048 -0.032 0.061 -0.001 0.000 0.172 0.360 0.859 0.786 0.806 -0.015 -0.017 -0.027 0.055 0.056 0.110 -0.027

19.2 20.0 3.0 6.0 9.0 12.0 11.0 5.0 5.5 4.5 20.0 0.7 0.5 7.8 5.0 11.0 5.0 3.0 16.0 3.0 4.0 0.7 1.8 1.8 5.0 5.0 6.0 6.0 6.0 1.8 2.0

1.48 3.24 0.67 0.85 0.60 1.05 1.85 0.16 0.21 0.30 0.59 0.22 0.15 0.13 0.13 0.62 0.63 0.71 0.97 0.76 1.15 1.82 0.19 0.99 0.79 0.50 1.32 0.66 0.67 0.21 1.32

1.99 3.18 0.62 0.95 0.76 1.15 1.79 0.18 0.23 0.20 0.53 0.24 0.23 0.21 0.13 0.60 0.68 0.74 1.09 0.85 1.22 1.83 0.23 1.02 0.89 0.61 1.44 0.83 0.37 0.19 1.88

4.73 7.17 0.56 0.53 0.40 0.67 2.60 0.04 0.05 0.05 1.98 0.30 0.25 0.16 0.14 0.68 0.78 0.78 1.72 0.38 0.39 1.94 0.31 1.03 0.43 0.26 1.22 0.89 1.26 0.27 1.20

Table 2. Variation of the Hydrated Diameter of Cations for Different Alkali Halides cation diameter at infinite dilution, σ0 (Å)

Figure 1. Correlation of the mean ionic activity coefficients of NaCl, KCl, and Ca(NO3)2 in aqueous solutions as a function of molality: (b) experimental data for NaCl;20 ([) experimental data for KCl;20 (2) experimental data for Ca(NO3)2;22 (s) the K-MSA model; (O) simplified version of the K-MSA model.

halide anion

Li+

Na+

K+

Cs+

ClBrI-

4.115 4.272 5.431

3.486 3.692 3.711

2.983 3.115 3.186

1.969 2.269 2.030

decrease in the charge density of the counterion. In addition, for each anion, the hydration phenomenon is more important in the presence of small cations. The parameters reported in Table 1 for the K-MSA model show that the hydrated diameter of a cation decreases as its concentration increases. This is due to the decrease in the number of water molecules oriented around the cation. For the case of small cations, the decrease in the hydrated diameter is more important. The variation of the hydration layer thickness λ for the sodium ion in aqueous NaCl solutions is shown in Figure 3. This hydration layer thickness is defined by the relation

σ ) σp + λ

Figure 2. Correlation of the mean ionic activity coefficients of a HCl solution as a function of molality: (2) experimental data;20 (s) the K-MSA model; (- ‚ -) the Bromley model;21 (‚‚‚) the Meissner model;21 (- ‚‚ -) the Pitzer model;21 (- -) the Chen model;21 (O) the BMCSL-MSA model.

nature of their counterions. For each particular cation, the size parameter increases as the size of the counterion increases, which is to be expected because of the

(8)

where σp and σ are the Pauling diameter and the hydrated diameter of the cation, respectively. Figure 4 shows the effect of temperature on the variation of the mean ionic activity coefficients, as a function of molality, for KCl in aqueous solution. As seen from this figure, an increase in the temperature results in a decrease of the hydration diameter and, in turn, in an increase in the mean ionic activity coefficient.28 This phenomenon is more significant at higher concentrations of the electrolyte.

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Figure 3. Variation of the hydration layer thickness for Na+ in the aqueous solution of NaCl as a function of molality.

Figure 5. Correlation of the activity coefficients in a KCl solution as a function of molality: ([) experimental data for K+;27 (b) experimental data for Cl-;27 (2) experimental data for KCl;27 (s) the K-MSA model.

Figure 4. Variation of the mean ionic activity coefficient as a function of molality for a KCl solution at different temperatures: (2) experimental data at 25 °C;28 ([) experimental data at 40 °C;28 (]) experimental data at 50 °C;28 (s) the K-MSA model. Table 3. K-MSA Parameters for the Individual Ions and the Average Value of the Relative Deviation, in Percent, with Respect to the Experimental Data of the Activity Coefficients of the Ions Produced by the K-MSA and Khoshkbarchi-Vera (K-V) Models AARD (%) electrolyte

ion

σ0

σ1

LiCl

Li+

6.669 0.816 7.22 0.03 2.327 2.383 5.323 1.381 4.790 2.443 1.954 4.409 2.505 4.138 2.890 4.456

-0.578 0.072 -0.64 0.84 4.564 0.430 -0.489 0.145 -0.089 0.035 0.158 -0.085 -0.354 -0.228 -2.080 0.398

LiBr NaF NaCl NaBr KF KCl KBr

ClLi+ BrNa+ FNa+ ClNa+ BrK+ FK+ ClK+ Br-

σ2 0.068 0.003 -4.120 0.046 0.015 -0.028 0.132 0.414

K-MSA

K-V

0.32 0.63 2.41 1.89 0.42 0.76 1.96 1.11 1.64 1.44 1.12 2.06 1.25 0.30 1.16 2.70

1.00 0.80 3.00 2.00 0.90 1.30 9.00 3.40 7.70 4.30 1.00 2.00 5.40 2.00 1.00 2.00

The K-MSA parameters for eqs 1 and 4, for different ions, are reported in Table 3. In addition, Table 3 gives the average values of the absolute percent error, with respect to experimental data25-27 obtained using the K-MSA and Khoshkbarchi-Vera models.27 Notably, the results obtained from the K-MSA model are generally more accurate than those obtained with the Khoshkbarchi-Vera model.27 A comparison of the values for the parameters reported in Tables 1 and 3 shows that a different set of parameters is required once the activity coefficients are considered separately. Figure 5 shows the correlation of the individual ionic activity coefficients in KCl aqueous solutions as a function of molality. Clearly, the K-MSA model fits well the experimental data of the individual ionic activity coefficients. Figure 6 presents the RDF of the cation at the contact value, as a function of concentration, for NaCl and

Figure 6. Variation of the cationic RDFs at contact values for Na+ and Ca2+ in NaCl and Ca(NO3)2 in aqueous solutions, respectively, as a function of the concentration: (‚‚‚) Na+; (s) Ca2+. Table 4. Comparison of the AARD % Values with Respect to the Experimental Data of the Mean Ionic Activity Coefficients Calculated from the K-MSA Model with Parameters Reported Previously and Parameters Reported in This Study for 1:1 Electrolyte Solutions calculation up to previous mmax electrolyte LiCl LiBr LiI NaCl NaBr NaI NaNO3 KCl KBr KI KOH KClO3 KBrO3 RbCl RbBr CsCl CsBr CsI HCl

calculation up to new mmax

previous previous new new mmax AARD % AARD % mmax 16/6 9/6 3 6 9/6 8/6 9/6 5 4.5 4.5 9/6 0.7 0.5 7.8/6 5 9/6 5 3 9/6

0.82/0.18 0.37/0.19 0.59 0.80 0.70/0.38 0.68/0.34 0.26/0.20 0.20 0.53 0.19 0.74/0.60 0.09 0.03 0.39/0.25 0.24 0.18/0.15 0.16 0.02 0.17/0.10

0.76/0.43 0.49/0.47 0.67 0.85 0.60/0.15 0.44/0.38 0.91/0.61 0.16 0.16 0.30 0.32/0.38 0.22 0.15 0.13/0.12 0.13 0.56/0.62 0.63 0.71 0.80/0.68

19.2 20.0 3.0 6.0 9.0 12.0 11.0 5.0 5.5 4.5 20.0 0.7 0.5 7.8 5.0 11.0 5.0 3.0 16.0

previous AARD %

new AARD %

401.55 146.73 108.02 118.34 78.13 57.71 107.35 69.02 51.96 51.33 33.91 26.19 16.19 141.74 73.65 95.31 163.60 70.54 2 × 1018

1.48 3.24 0.67 0.85 0.60 1.05 1.85 0.16 0.21 0.30 0.59 0.22 0.15 0.13 0.13 0.62 0.63 0.71 0.97

Ca(NO3)2 solutions. It is important to note that the values of RDF for cations are positive, and thus physically meaningful, over the whole concentration range. To close, we comment on the effect of the error in the calculations reported by Taghikhani and Vera.9 Columns 3 and 4 of Table 4 compare the results for symmetrical electrolytes obtained with the parameters reported in this work with the results previously reported by Taghikhani and Vera.9 The results of ref 9 were obtained by omitting the hard-sphere contribution and, notably, in the dilute region considered in the

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previous study, the adjustable parameters were able to absorb the error in the equations. In some cases, the AARD % deviation obtained using the same number of parameters with the complete equation is even higher than the deviation obtained with the incomplete equation used in ref 9. For this comparison, we have used the same data and the same maximum molality as those used in ref 9. The maximum molalities used in these calculations are shown in the second column of Table 4. Another comparison was also made between the two sets of results obtained from the K-MSA model. This comparison is shown in the sixth and last columns of Table 4, and the maximum molality considered is indicated in column 5. Column 6 of Table 4 reports the AARD % that would result if the parameters reported in ref 9 were used with the correct equations to calculate the mean ionic activity coefficients. The reason for these large errors is that the parameters were adjusted by omitting the hard-sphere terms. On the other hand, as shown in column 7 of Table 4, the complete K-MSA model accurately correlates the experimental data for the mean ionic activity coefficients up to saturation. Conclusions Corrected size parameters of the cations in univalent-univalent electrolyte solutions obtained from the K-MSA model are presented. The K-MSA model is extended to accurately correlate the mean ionic activity coefficients of the asymmetric electrolytes in aqueous solutions by fitting the size parameters of the cations to the experimental mean ionic activity coefficient data available in the literature. It is shown that in some cases the K-MSA model produces a better fitting of the mean ionic activity coefficients for the electrolytes with high solubility in comparison with the results obtained with the BMCSL-MSA and Pitzer models. In other cases, almost the same results are obtained using the K-MSA, BMCSL-MSA, and Pitzer models. The effect of shortrange electrostatic interaction, temperature, and size of the cation and its counterions on the values of the activity coefficients was investigated. It was concluded that the effect of the short-range electrostatic interactions can be safely neglected. The variation of the adjusted diameter of the hydrated cation in dilute concentration with temperature and with the size of the cation and their counterions produced physically meaningful results. The K-MSA model was also used to correlate the activity coefficients of the individual ions of the electrolytes studied in this work. In this study it was necessary to consider that the diameter of the anion was concentration-dependent. With this assumption, the K-MSA model correlated accurately the experimental data of the activity coefficients of the individual ions. The results obtained using the K-MSA model are better than those obtained using the Khoshkbarchi-Vera model. The most important result found in this work is that the values for the RDFs at contact values for cations are positive over the whole concentration range. List of Symbols AARD ) average absolute relative deviation c ) molarity d ) density D ) dielectric constant e ) basic electric charge

g(σ) ) radial distribution function at contact value k ) Boltzmann constant m ) molality NP ) number of experimental data points Pn ) parameter in eq 6 T ) absolute temperature X ) parameter in eq 6 Z ) charge of ion Greek Letters γ ) activity coefficient λ ) hydration layer thickness σ ) ionic size parameter Γ ) inverse shielding length F ) number density ξn ) function defined by eq 5 Subscripts i, j, k ) component index 0 ) pure solvent ( ) mean ionic P ) Pauling Superscripts c ) molarity scale m ) molality scale calc ) calculated exp ) experimental

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(13) Barker, J. A.; Henderson, D. Perturbation Theory and Equation of State for Fluid. II. A Successful Theory of Fluids. J. Chem. Phys. 1967, 47, 4714. (14) Blum, L. Simple Method for the Computation of Thermodynamic Properties of Electrolytes in the Mean Spherical Approximation. J. Phys. Chem. 1988, 92, 2969. (15) Sanchez-Castro, C.; Blum, L. Explicit Approximation for the Unrestricted Mean Spherical Approximation for Ionic Solutions. J. Phys. Chem. 1989, 93, 7478. (16) Sheng, W.; Kalogerakis, N.; Bishnoi, P. R. Explicit Approximation of the Mean Spherical Approximation Model for Electrolyte Solutions. J. Phys. Chem. 1993, 97, 5403. (17) Hinton, J. F.; Amis, E. S. Solvation Numbers of Ions. Chem. Rev. 1971, 71, 627. (18) Samoilov, O. Ya. Structure of Aqueous Electrolyte Solutions and the Hydration of Ions; Consultants Bureau Enterprises Inc.: New York, 1965. (19) Chong, S. H.; Hirata, F. Ion Hydration: Thermodynamic and Structural Analysis with an Integral Equation Theory of Liquids. J. Phys. Chem. 1997, 101, 3209. (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. (21) Zemaitis, J. F., Jr.; Clark, D. M.; Rafal, M.; Scrivner, N. C. Handbook of Aqueous Electrolyte Thermodynamics; DIPPERAIChE Publications: New York, 1986.

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Received for review October 7, 2002 Revised manuscript received January 17, 2003 Accepted January 21, 2003 IE020790G