J. Phys. Chem. 1996, 100, 4251-4255
4251
Role of Solvent Permittivity in Estimation of Electrolyte Activity Coefficients on the Basis of the Mean Spherical Approximation W. Ronald Fawcett* and Alex C. Tikanen Department of Chemistry, UniVersity of California, DaVis, California 95616 ReceiVed: August 15, 1995; In Final Form: December 1, 1995X
A modified version of the model for electrolyte solutions based on the restricted version of the mean spherical approximation is applied to estimate mean ionic activity coefficients for aqueous solutions of the alkali metal halides. The modification involves using the actual dielectric permittivity of the solution rather than the permittivity of the pure solvent. In the case of sodium and potassium salts, fits between theory and experiment are possible over wide concentration ranges up to 2.5 M using one adjustable parameter, namely, the single effective ionic diameter. Limitations of the model, especially with respect to systems with weak ion pairing, are discussed.
Introduction Although the Debye-Hu¨ckel theory1 of electrolyte solutions gives a good description of their physical properties when they are very dilute, it fails in a serious way at higher concentrations, especially for electrolytes involving polyvalent ions. In these systems, neglect of the finite size of all constituent ions is an important defect and leads to overestimation of the effects of ion-ion interactions. One way of overcoming this defect is to represent the electrolyte as a collection of hard spheres corresponding to the constituent cations and anions. These spheres are imagined to be immersed in a dielectric continuum which represents the solvent. This approach is clearly primitive because it ignores the discrete nature of the molecules which make up the solvent. However, the properties of such a system may be derived on the basis of the mean spherical approximation (MSA), first proposed by Lebowitz and Percus,2 to yield analytical expressions which give a simple extension of the Debye-Hu¨ckel model. The MSA was originally applied to electrolyte solutions by Waisman and Lebowitz3 for a symmetrical electrolyte with ions of equal size. The “unrestricted” general solution of the MSA was presented by Blum et al.4-6 and was used by Triolo and co-workers7-9 to estimate osmotic and mean activity coefficients for aqueous electrolyte solutions. In the simplest version of the MSA which is termed “restricted,” the ions are assumed to all have the same size. When the restricted MSA model is applied to 1-1 electrolyte solutions, such as NaCl in water, one is able to obtain a reasonable fit to experimental activity coefficient data for concentrations up to ∼0.3 M using only one adjustable parameter, namely, the single effective ionic radius. On the other hand, the extended DebyeHu¨ckel theory applied to the same data can only be fit up to ∼0.1 M using a different but optimized size parameter. Attempts have been made to extend the MSA treatment to higher concentrations by allowing the ion size parameter to be a function of ionic strength.6,10 Such an approach is not realistic because of the sharp reduction of effective cation size with increasing electrolyte concentration. Moreover, it does not address the fundamental reasons for breakdown of the MSA model. In an early paper by Blum et al.,8 it was pointed out that the fit of the MSA model to experimental data could be improved if the change in the dielectric permittivity of the solvent with X
Abstract published in AdVance ACS Abstracts, February 15, 1996.
0022-3654/96/20100-4251$12.00/0
electrolyte concentration is considered. It is well-known, on the basis of early experimental work by Hasted et al.,11,12 that the permittivity of electrolyte solutions is less than that of the pure solvent due to the disruption of solvent structure in the immediate vicinity of the ions and the fact that the finite sized ions do not have an orientational contribution to the permittivity. Hasted13 gave an approximate linear expression for this change and demonstrated that the rate of change of the permittivity with concentration depends on the nature of the electrolyte. He then presented arguments for estimating the effects of the individual ions in the electrolyte so that his data could be used to calculate the change in solvent permittivity in electrolyte solutions which had not been studied experimentally, but whose constituent ions had been studied in other electrolytes. More recently, Wei et al.,14,15 have studied the permittivity of aqueous solutions of LiCl, RbCl, and CsCl over a wide concentration range and obtained information about dielectric relaxation in these systems. Similar studies have been reported for NaF and KF solutions by Buchner et al.16 Our interest was to examine the role of ion size in the MSA with careful consideration of the change in dielectric permittivity which accompanies increasing electrolyte concentration. Our work has involved both the restricted and unrestricted versions of this theory.4,5 The individual hard sphere and electrostatic contributions calculated by the restricted version disagree with those from the unrestricted version by more than 10% when the electrolyte concentration is above 1 M and the ratio of the ionic diameters is greater than 4:1.17 However, we have found for the restricted case of a 1-1 electrolyte that the differences in the electrostatic and hard sphere contributions cancel one another to a great extent and produce almost the same results as the unrestricted calculations. Thus, one may use a single effective ionic size in the restricted version which is almost identical to the average of the effective ionic radii used in the unrestricted calculations. The restricted version represents a simple extension of the Debye-Hu¨ckel model and therefore can easily be used without resorting to complicated computer calculations. Our results for the restricted version are presented here because they are easily used by physical chemists who require an improved model for the thermodynamic properties of an electrolyte solution. We will show that when the experimental dielectric permittivity is used in the model, one can estimate the activity coefficients of simple 1-1 electrolytes for concentrations up to 2.5 M using one adjustable parameter, © 1996 American Chemical Society
4252 J. Phys. Chem., Vol. 100, No. 10, 1996
Fawcett and Tikanen
namely, the single effective ionic diameter. Results for the alkali metal halides are given in the present paper.
one obtains the result
∆Aes ) ∆Ues +
The Model The MSA results for the restricted system are obtained by solving the Ornstein-Zernicke equation for a system of charged hard spheres in a dielectric continuum. The closure conditions for the direct correlation function are
zizjβe02 cij(r) ) -βuij ) , r>σ 4π0SOr
gij(r) ) 0, r < σ
(2)
(ln yi)es )
Γ)
2
(
zi2Fi
∑i
(3)
)
1/2
(1 + Γσi)2
(4)
where R2 ) βe02/0SO and Fi is the concentration of ion i expressed as ions m-3. For the case that the ions all have the same size this reduces to
Γ ) κ/2(1 + Γσ)
4π0SO(1 + Γσ)
(ln yi)hs ) (ln y()hs )
for dilute solutions. In these expressions, e0 is the fundamental electronic charge, zi, the valence of the cation, zj, that of the anion, SO, the relative permittivity of the pure solvent, 0, the permittivity of free space, and β ) 1/kT. The distance r is measured from the center of a reference ion which has a diameter σ. Since the spheres are hard, the distance of closest approach of any two ions is σ. Γ is a parameter which is equal to κ/2 in the limit of zero ionic strength, where κ is the DebyeHu¨ckel shielding parameter. The parameter Γ is given by
R
-βe02zi2Γ
(12)
In order to complete the MSA estimate of ln yi, one must add the hard sphere contribution which is obtained from the Percus-Yevick model for noninteracting hard spheres.18 This is given by
Solution of the integral equation gives the result that
zizjβe02 exp(-2Γr) gij(r) ) 4π0SOr
(11)
Differentiating with respect to Fi, one obtains the expression for the electrostatic contribution to the single ion activity coefficient which is
(1)
and for the pair correlation function
Γ3 3πβ
6η 3η2 2η + 2 + 3 ∆ ∆ ∆
(13)
where
∆)1-η η)
πσ3 6
∑i Fi
(14) (15)
The resulting expression for the single ion activity coefficient is
ln yi ) (ln yi)es + (ln yi)hs
(16)
The mean ionic activity coefficient is the quantity which is directly comparable to experimental data. Its electrostatic contribution may be obtained from eq 12 after the usual thermodynamic manipulations, the result being
(ln y()es )
-βe02|z+z-|Γ 4π0SO(1 + Γσ)
(17)
(5) Since the ions are assumed to have the same size, (ln y()hs is also given by eq 13 and
where
κ ) (2FβIe0/SO0)1/2
(6)
ln y( ) (ln y()hs + (ln y()es
I ) 1/2∑Fizi2
(7)
In order to estimate values of ln yi or ln y( as a function of electrolyte concentration, only one adjustable parameter is required, namely, the single effective ionic diameter σ. The electrostatic contribution to y( decreases with an increase in concentration due to a corresponding increase in ionic strength and thus, the parameter Γ. On the other hand, the hard sphere contribution increases with an increase in concentration due to an increase in the packing fraction η. These two trends combine to give values of y( which initially decrease, reach a minimum, and then increase. These qualitative features of the experimental results are predicted by the theory but the fit of theory to experiment using the dielectric permittivity of the pure solvent is limited to low concentrations once the optimum choice of σ has been made. The MSA was applied in the present study with a variable permittivity for the solvent continuum. On the basis of both theoretical19 and experimental studies,20 the dependence of the relative permittivity of the solvent on electrolyte concentration cS is given by
i
F is the Faraday constant. Solving eq 5 for Γ, one obtains
(1 + 2σκ)1/2 - 1 (8) 2σ This has the correct assymptotic form in which Γ approaches κ/2 in the limit that σ goes to zero. The thermodynamic properties of the electrolyte solution are derived by first calculating the excess internal energy due to electrostatic interactions, ∆Ues. For the case that the ions all have the same size, ∆Ues is given by Γ)
∆Ues )
-02IΓ 2π0SO(1 + Γσ)
(9)
The corresponding excess Helmholtz energy is estimated using the thermodynamic relationship
∂β∆Aes/∂β ) ∆Ues
(10)
where ∆Aes is the excess Helmholtz energy. After some algebra,
S ) SO - δScS + bcS3/2
(18)
(19)
Estimation of Electrolyte Activity Coefficients
J. Phys. Chem., Vol. 100, No. 10, 1996 4253
TABLE 1: Values of the Constant A in Equation 19 Which Relates the Solution Density to Its Molality A for given anion cation
F
-
Li+
Na+ K+ Rb+ Cs+
1.0260 0.8471 0.9710 1.2595
Cl-
Br-
I-
0.5667 0.6938 0.6323 0.7748 0.8308
0.8248 0.8388 0.7624 0.9077 0.8536
0.8697 0.9146 0.8310 0.8487 0.6420
where δS is the dielectric decrement for the electrolyte and b, a parameter describing the curvature of this dependence. Since S decreases with an increase in ionic strength, the parameters κ and Γ both increase. As a result, the effects of ion-ion interactions on the estimation of (ln y()es are greater than estimated on the basis of the relative permittivity of pure water, SO. Our calculations show that introduction of this change in the MSA model, that is, replacement of SO with S in the above equations, results in a significant improvement in the concentration range over which theory can be fitted to experimental results. Method of Data Analysis The activity coefficient data used were those of Robinson and Stokes21 which have been compiled by Lobo.22 These data are tabulated in terms of molality, and conversion to a molarity scale is required in order to make the MSA calculations. Density data were obtained from the compilations of Lobo.22 The density of the electrolyte solution, F, is related to the molality, m, by the following empirical expression23
ln F ) ln F0 +
Am(MW) m(MW) + 1000
(20)
where F0 is the density of the pure solvent, MW is the molecular weight of the electrolyte, and A is a constant. Excellent fits were obtained for 19 alkali metal halides, the results being summarized in Table 1. The concentration range over which the fit was made depended on the available data, but it was usually up to 2 or 3 m. It should be noted that the molality scale mean activity coefficient, γ( , must be converted to a molarity scale mean activity coefficient which is usually denoted as y( using the relationship21
y( ) γ(mF0/cs
(21)
The variation in the permittivity with electrolyte concentration was examined on the basis of eq 19 using data reported recently for LiCl,14 RbCl,15 CsCl,15 NaF,16 and KF.16 The older data by Hasted et al.11,12 were also examined, but less weight was given to the results. Assuming that SO is known, the parameters δS and b can be found from a two parameter least squares fit. As a result, the curvature parameter b was determined to be 5 ( 1 L3/2 mol-3/2 independent of the nature of the 1-1 electrolyte considered. A similar conclusion is made using only the data reported by Buchner et al.16 Thus, assuming b ) 5 L3/2 mol-3/2, further analysis of the dielectric permittivity data requires that the dielectric decrement be determined as a function of electrolyte nature. On the basis of the analysis presented by Hasted,13 the parameter δS can be assumed to be made up of additive cationic and anionic contributions. Thus, one expects trends in δS with change in anion size when one examines values of δS for salts with a common cation or with change in cation size for salts with a common anion. Using the available experimental data and this principle, values of δS were estimated for all 19 alkali metal halides considered. The results, which
Figure 1. Plot of the dielectric permittivity of RbCl solutions, S, against their concentration. The data (b) were obtained by Wei et al.15 and (() by Hasted et al.11,12
TABLE 2: Values of the Permittivity Decrement δS for the Alkali Metal Halides δS for given anion/(L mol-1) -
cation
F
Li+ Na+ K+ Rb+ Cs+
17 16 15 14
Cl-
Br-
I-
20 19 18 17 16
21 20 19 18 17
22 21 20 19 18
are summarized in Table 2, fall in the range from -14 for CsF to -22 for LiI and have an estimated accuracy of (1 L mol-1. On the basis of this analysis, the value of S was assumed in all estimates of the activity coefficient to be given by
S ) 78.45 - δScS + 5cS3/2
(22)
The result of this analysis for the RbCl system is shown in Figure 1. The plot shows the scatter in the data which are available both from the early work of Hasted11,12 and from the more recent work of Wei et al.15 It is clear that the curve fitted on the basis of eq 22 provides a good description of the behavior of S in the concentration range studied here. Activity coefficient data were examined in the electrolyte concentration range from 0.1m to 2.5m. The value of the single effective ionic diameter σ was adjusted until a fit between theory and experiment was found with a deviation in y( less than (0.005 at all tabulated22 concentrations. The concentration at which the deviation became greater than (0.005 was noted. This was used to define the range over which a successful fit between theory and experiment is possible. The results of our analyses are reported below. Results and Discussion Data for 19 alkali metal halides electrolytes were analyzed using the restricted MSA model with both a varying permittivity for the solvent and the assumption of a constant value equal to that of the pure solvent. In all cases, a significant improvement in the fit between the calculated and experimental values of the mean activity coefficient y( was found using the experimental value of the solvent’s permittivity in the electrolyte solution. For seven electrolytes, namely, NaCl, KCl, RbCl, NaBr, KBr, NaI, and KI, an exact fit was obtained for concentrations up to 2.5m by adjusting the single effective ionic diameter σ. Results for the NaBr system are shown in Figure 2. The fit obtained is impressive when one considers the fact that other models which describe the activity coefficient over such a wide concentration range require more than one adjust-
4254 J. Phys. Chem., Vol. 100, No. 10, 1996
Fawcett and Tikanen
Figure 2. Plot of the mean ionic activity coefficient for aqueous NaBr solutions against the square root of the molarity. The solid line throughout the experimental points shows the MSA estimates with an optimized value of 407 pm for σ and a varying solvent permittivity. The dotted curve was calculated with a constant permittivity equal to that of the pure solvent and an optimized value of 366 pm for σ.
Figure 3. Plot of the mean ionic activity coefficient for aqueous RbF solutions against the square root of the molarity. The solid line shows the MSA estimates assuming σ ) 389 pm and varying solvent permittivity.
TABLE 3: Values of the Single Effective Ionic Diameter σ Obtained from the Best Fit of the Model to Experimenta σ for given anion/pm cation
F-
Cl-
Br-
I-
328 (1 M)b 365 (1.4 M) 389 (0.9 M) 408 (0.5 M)
435 (1.6 M) 388 (3 M) 362 (2.5 M) 349 (2.5 M) 317 (1 M)
448 (1.5 M) 407 (2.5 M) 376 (3 M) 349 (1.2 M) 318 (0.9 M)
489 (0.5 M) 427 (2.5 M) 394 (2.5 M) 351 (0.8 M) 311 (0.6 M)
Li+ Na+ K+ Rb+ Cs+ a
The concentration in parentheses gives the range over which a successful fit between theory and experiment was obtained (see text). b Experimental data available only up to 1 m.
able parameter.24 If the solvent permittivity is set equal to that of the pure solvent, the concentration range over which the fit is successful using a new optimized value of σ is in all cases much less. For example, the data for NaCl can only be fit up to 0.3 M using the dielectric constant of the pure solvent. Values of σ and the range over which a successful fit was obtained are summarized in Table 3. The fit was very sensitive to the single effective ionic diameter, which is reported to the nearest picometer. The best value of σ decreased with an increase in cation atomic number for a given anion except in the case of the fluoride salts. The latter observation is attributed to the presence of ion pairing in the fluoride solutions.16 As an example, results for the RbF system are shown in Figure 3. At concentrations above 1 M, the estimated value of y( is higher than that observed experimentally. When ion pairing is present, calculations based on the stoichiometric electrolyte concentration overestimate the effects of nonideality due to ions. If the error predominantly affects the hard sphere contribution, the estimated value of y( lies above the experimental value. Careful analysis of dielectric relaxation data for the fluorides16 suggests that solvent separated ion pairs are formed in these systems. This feature of the system may be the reason for the positive deviations of the theoretical estimates from the experimental values at higher concentrations. Values of the single effective ionic diameter, σ, found for the chlorides, bromides, and iodides are plotted against the sum of crystallographic radii for the component ions, rc + ra, as estimated by Shannon and Prewitt,25 in Figure 4. Linear
Figure 4. Plot of the mean ionic diameter σ used to obtain the MSA estimates against the sum of the crystallographic radii for cation and anion, rc + ra.
correlations are found between σ and rc + ra for varying cation with the nature of the anion being held constant. The negative slope of these lines demonstrates that the effective radius of the smallest cation, Li+, is much larger than its crystallographic radius due to strong hydration. The degree of cation hydration decreases with an increase in atomic number. However, the relationship between σ and rc + ra is not simple due to the fact that the slopes of the lines shown in Figure 4 vary with the nature of the anion. Thus, other factors such as ion pairing and imperfection in the model used to describe the concentration dependence of the solvent permittivity play a role in determining the value of σ obtained in the present analysis. The Li+ salts are similar to the F- salts in that the model fits the experimental data over a limited concentration range. A typical system is LiI for which the estimates of y( by the present model lie above the experimental data for concentrations greater than 0.5 M (see Figure 5). Thus, all systems with small highly solvated ions show similar behavior at high electrolyte concentrations. Presumably, solvent separated ion pairs are also formed in these solutions. Quite different results are obtained for the Cs+ salts and for RbBr and RbI. For these systems, the value of σ which provides the best fit to y( data at low concentrations predicts values of the activity coefficient which fall below the experimental results at higher concentrations. A good example is provided by the CsCl system for which a reasonable fit was obtained up to ∼1 M (see Figure 6). The failure of the model for these systems may also be due to ion pairing but under circumstances where the failure to describe the electrostatic contribution is more important than that to describe the hard sphere contribution. Ion pairs in Cs+ electrolytes are expected to be contact in nature
Estimation of Electrolyte Activity Coefficients
Figure 5. Same as in Figure 3 but for aqueous LiI solutions with σ ) 489 pm.
J. Phys. Chem., Vol. 100, No. 10, 1996 4255 aspect of analyzing these results is separation of dynamic from static contributions to the permittivity decrement.20 This feature of the experimental data was not considered in early work.11,12 On the other hand, eq 22 is expected to give a correct qualitative description of the concentration dependence of S in the range considered. Refinement of this description is expected to result in a change in the best value of σ and perhaps extension of the range over which a good fit can be achieved. In summary, a very significant improvement in the MSA description of electrolyte solutions is obtained when one includes the concentration dependence of the solvent permittivity in the model. The theory is quantitatively simple and can be used by experimentalists who need thermodynamic properties of electrolyte solutions without resorting to complex computations. An improvement in the model would require that one consider the effects of ion pairing. Work in this direction is currently under way. Acknowledgment. The financial support of the Office of Naval Research, Washington, D.C., is gratefully acknowledged. References and Notes
Figure 6. Same as in Figure 3 but for aqueous CsCl solutions with σ ) 317 pm.
as opposed to the solvent separated ion pairs anticipated in Li+ and F- electrolytes. If one attempts to improve the extended Debye-Hu¨ckel model by changing the constants to reflect changes in solvent permittivity, no improvement in the concentration range over which the model describes experimental data is found. This fact suggests that the important feature of the MSA is that it considers the finite size of all of the ions in the system, not just that of the central ion. Our analysis also emphasizes that the features of the system which should be considered beyond the electrostatic effect of ion-ion interactions are the finite size of all ions and the change in dielectric permittivity with electrolyte concentration. Further improvement in the theoretical description should be provided by considering the effects of ion pairing. It is obvious that the description of the change in dielectric permittivity given here is approximate. This is mainly due to the fact that only a few 1-1 electrolytes have been examined using modern dielectric relaxation experiments. An important
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