Corrected DebyeHÃ¼ckel Theory of Salt Solutions: Size Asymmetry and

J. Phys. Chem. B 2002, 106, 1403-1420

1403

Corrected Debye-Hu1 ckel Theory of Salt Solutions: Size Asymmetry and Effective Diameters Zareen Abbas,* Magnus Gunnarsson, Elisabet Ahlberg, and Sture Nordholm Department of Chemistry, Go¨ teborg UniVersity, SE-412 96 Go¨ teborg Sweden ReceiVed: May 30, 2001; In Final Form: October 5, 2001

A recently developed extension of the Debye-Hu¨ckel theory, the corrected Debye-Hu¨ckel theory, is applied to salt solutions as described in Monte Carlo (MC) simulations and experiment. On the basis of modern density functional analysis, the theory accounts for both long-range electrostatic interactions in a linear response approximation and short-range ion size effects by a generalized van der Waals analysis. In its simplest implementation, the corrected Debye-Hu¨ckel restricted primitive model (CDH-RPM) theory, all ions are taken to be of the same diameter d in which case the excluded volume effects vanish in the linear response domain. Comparison with MC simulations verify the a priori application of the CDH theory to RPM electrolytes over a wide range of concentrations. Here, we investigate the relationship between the appropriate effective diameter d and the real diameters d1 and d2 in simulated binary salt solutions. Moreover, in the experimental studies of salt solutions, the relationship between d and the crystallographic or hydrated ionic diameters was investigated. Calculations have been performed for 67 salts. The optimized diameters are found to be in accord with current understanding of hydration effects. The range of concentrations in which a good fit is obtained for mean ionic activity coefficients and osmotic coefficients is typically 0-1 M. Comparison is also made with the Specific Interaction theory and the Pitzer and Bromley models of salt solutions.

1. Introduction Salt solutions impose themselves on us through their occupation of two-thirds of the surface of the earth and through the vital role they play in living organisms. The history of the study of salt solutions is commensurately long. The development of a theory of salt solutions was long hampered by the long range of the Coulomb interaction. Although Milner may have been first to construct a sound theory in 1912,1 the most recognized cornerstone was laid by Debye and Hu¨ckel in 1923.2 They resolved the problem of long-range interaction in a brilliantly simple linear response analysis of screening in electrolyte solutions which retains its relevance today. Our purpose here is, in fact, to explore a modern extension of the Debye-Hu¨ckel (DH) theory and shed new light on the remarkable utility of their original analysis. The theory of electrolyte solutions has evolved in several different strands since the work of Debye and Hu¨ckel. Considering first the most mathematically rigorous strand, it was continued from the base of the linear DH theory to the application of the nonlinear Poisson-Boltzmann theory which was well known to Debye and Hu¨ckel but simplified to achieve analytical solvability. A major advancement in the theory was made by Mayer in 19503 when he reformulated his cluster expansion theory of nonideal gases to apply to the solutions of electrolytes. He was able to put the approximations of Debye and Hu¨ckel on a more general statistical mechanical footing. The limitations in terms of neglect of not only nonlinear electrostatic response but also finite ion size effects and electrostatic correlation effects could be seen. To account for these effects, the use of integral equation theory from the theory of liquids was proposed. Most successful was the HNC (hypernetted chain) integral equation4 * Corresponding author. Tel: +4631-7722886. Fax: +4631-7722853. E-mail: [email protected].

based on a closure of the hierarchy of coupled correlation functions suggested by the Ornstein-Zernike analysis of direct and indirect correlations.5 The HNC theory was used by Rasaiah and co-workers6 in an extensive series of calculations on electrolyte solutions in the restricted primitive model (RPM). This model was introduced by Friedman7 to represent a hypothetical electrolyte where the ions are rigid hard spheres of equal size dissolved in a structureless solvent described by a dielectric constant. The HNC theory was generally able to accurately reproduce the numerically exact simulation results for primitive model electrolytes up to concentrations of 2 M. Waisman and Lebowitz8 introduced a new closure of the Ornstein-Zernike correlation analysis, the mean spherical approximation (MSA), which greatly simplified the calculations reducing them to analytical evaluations in most cases. The MSA is a linear theory based on a mean field treatment of the electrostatic interactions as in the Debye-Hu¨ckel and Poisson-Boltzmann theory. It was originally applied to RPM electrolyte solutions but later extended by Blum9 to unrestricted primitive models. The MSA mitigates the most significant drawback of the HNC theory, that is, its complex analytical form. The price for this simplification is loss of accuracy, but the MSA has been found quite accurate enough to be of great practical utility.10 Other modern forms of electrolyte theory were developed by Outhwaite (modified Poisson-Boltzmann theory)11 and Andersen and Chandler (mode expansion theory)12 which further improved the understanding of correlation effects in electrolyte solutions. The most accurate and general theory is, however, the numerical simulation by Monte Carlo (MC) or molecular dynamics (MD) techniques. Valleau and co-workers13 used MC simulation to resolve the properties of RPM electrolyte solutions. Simulations have since provided a rich store of data on electrolyte solutions in not only the RPM representation but also the unrestricted primitive model (UPM),14,15 the solvent primi-

10.1021/jp012054g CCC: $22.00 © 2002 American Chemical Society Published on Web 01/19/2002

1404 J. Phys. Chem. B, Vol. 106, No. 6, 2002 tive model (SPM),16 and “civilized models” including a realistic representation of the water solvent.17 In parallel with the evolution of the fundamental theory reviewed above, the Debye-Hu¨ckel theory gave rise to a continuing series of more pragmatic and empirical extensions. Robinson and Stokes extended the applicability by taking into consideration the hydration of ions and obtained a two-parameter equation with adjustable ion size and a hydration number.18 The hydration numbers obtained for different salts were unreasonably large. Glueckhauf19 noted that hydration numbers could be improved by use of volume rather than mole fraction statistics in the theory. Stokes18 concluded that the theory was nevertheless incomplete in that it did not account properly for the nonelectrostatic mechanisms. Guggenheim20,21 used the Brønsted theory of specific interactions of ions to include such shortrange interactions. Scatchard22 and Ciavata23 showed that the resulting Specific Interaction (SI) theory could with proper choice of parameters be used over a wide range of electrolyte concentrations. Pitzer24 developed a very useful extension of the DH theory where hard sphere effects are incorporated via a virial expansion including up to three virial coefficients. Pitzer obtained these virial coefficients by applying the series expansion to the pressure equation of the solution and using an appropriate interaction potential for the ionic solution. This introduces virial coefficients which are functions of the ionic strength of the salt solution in accord with a suggestion of Scatchard.22 Pitzer argued that a decrease in the second virial coefficient with respect to ionic strength arises from a similar decrease in the average radial distribution function (RDF) at contact. This behavior of the RDF for ionic solutions was also seen in Monte Carlo simulations13 and in the HNC calculations of Ramanathan and Friedman25 on a model with a soft repulsive potential. To fit the experimental data on mean ionic activity and osmotic coefficients, the virial coefficients are treated as adjustable parameters. For example, for 1:1 electrolyte two parameters and for 2:2 electrolyte three parameters are adjusted. Along with these adjusted parameters, Pitzer assumed a value of 1.2 for Ba in the DH term of his equation. Pitzer has generated a very useful set of the values of these fitting parameters.26 By using these values, one can calculate the activity and osmotic coefficients for single and mixed electrolytes up to very high concentrations, that is, usually >4 molal. However, because of the large number of fitting parameters, there is a loss in convenience and physical understanding. Bromley27 has proposed a simplification of the Pitzer model. He omitted the third virial coefficient and simplified the ionic strength dependence of the second virial coefficient. Other empirical models can be found in the literature, for example, the local composition model of Chen,28 the Meissner equations,29 and several others, but they are usually less accurate than the Pitzer model. For reasons of simplicity and practical utility, the semiempirical strand of electrolyte theory is likely to dominate among users despite the advances made by ion-ion correlation analysis and modern liquid state theory. If numerical prediction suffices, the MC simulation method also recommends itself because of its combination of accuracy, generality, and algebraic simplicity. As noted above, the MC simulation method has been applied to wide range of electrolyte models because it can most easily accommodate complexity in interaction and system geometry. Nevertheless, there is still a need for theory which combines physical and numerical simplicity with the highest achievable accuracy. The corrected Debye-Hu¨ckel (CDH) theory is a significant step in this direction occupying a place between the

Abbas et al. rigorous liquid state theories and the semiempirical extensions of Debye-Hu¨ckel theory. It is based on a modern density functional analysis30 which makes it quite straightforward to recover the DH theory and add what appears to be the most important ion size effects. Recalling that the DH limiting law assumed point ions and was then extended to allow for ion size in the interaction between a screening ion and the central ion it screens, the CDH theory takes one step further and accounts also for the ion size in the interaction between two screening ions. This is done by recourse to the van der Waals analysis of particle interactions in ordinary simple fluids. Two new mechanisms are then introduced: the excluded volume effect and the hole correction of electrostatic interactions. A very fortunate cancellation arises of the excluded volume effect if we assume that all ions are of equal size as in the RPM representation of salt solutions. The linear response assumption means that counterion adsorption and coion desorption precisely cancel in the total particle density. It follows that the excluded volume effect decouples from the screening mechanism which is changed from its simple Debye-Hu¨ckel form only by the analytically integrable hole correction. The result is a CDHRPM theory which is ab initio for RPM salt solutions and applicable to realistic UPM models or real salt solutions on the basis of either a mixing rule for the mapping of two known ionic diameters on one RPM diameter or an empirical fit of the RPM diameter to the data. Thus, the theory is both parameter free and fully predictive and is a one-parameter semiempirical theory in the mold of the DH theory for hard sphere ions or for the many extensions of the DH theory discussed above. We have already shown31 that the accuracy is quite good for the ab initio application to RPM representations of 1:1, 2:1, and 3:1 salt solutions but not for 2:2 salt solutions where the coupling is outside the linear response domain. In this work, we focus on the application to UPM electrolytes and real salt solutions and investigate the mapping of different diameters onto a corresponding RPM diameter. We shall present extensive comparisons with UPM data from MC simulations14,15 and HNC calculations32 reported in the literature and thereby put the use of the theory for real salt solutions on a firm basis. Our CDH-RPM calculations are carried out both ab initio on the basis of a mixing rule for the diameter and semiempirically by optimizing the RPM diameter. We have also performed wide ranging calculations for some 67 salt solutions for which we report optimized CDH-RPM diameters and corresponding ranges of validity generally of about 0-1 M. We also compare with alternative semiempirical theories, that is, the SI, Pitzer and Bromley extensions of the DH theory. This paper is organized as follows. In the next section (section 2), the CDH theory of bulk electrolytes is summarized. In section 3, MC simulation and HNC results for unrestricted primitive model (UPM) 1:1 and 2:1 electrolytes are compared with the results of CDH-RPM theory. The results for the fitting of experimental data are given in Section 4. In section 5, we shall compare the CDH results with the results of Specific Interaction theory (SI), Pitzer and Bromley’s semiempirical models. Finally, discussion and conclusions are given in section 6. 2. The CDH-RPM Theory The CDH-RPM theory has been described some years ago in a more limited form for symmetric salts30 and currently in a more general form incorporating the screening of not only salt solutions but solute macroions or planar surfaces.31 Here, we shall focus on the theory as applied to binary salt solutions in the primitive model, that is, the solvent has been subsumed in

Corrected Debye-Hu¨ckel Theory of Salt Solutions

J. Phys. Chem. B, Vol. 106, No. 6, 2002 1405

the dielectric constant . The ions are taken to be hard spheres with imbedded charges z+e and z-e. The hard sphere diameters d1 and d2 will not in the experimental salt solutions or in the corresponding primitive models studied by Monte Carlo simulation be identical, but a key simplification will follow in the CDH-RPM theory by mapping the realistic solution on corresponding restricted primitive models (RPM). Just how this is done and the relation between d1 and d2 and the effective diameter d in the CDH-RPM theory will be the focus of this paper. The CDH theory is based on a combination of the DebyeHu¨ckel analysis of linear screening of charged particles in electrolyte solutions and the van der Waals or, more precisely, GvdW analysis of short-range ion-ion interactions. The derivation is accomplished within the density functional variational approach of the GvdW theory where the ion densities are assumed to have the well-known exponential form of DebyeHu¨ckel theory, that is,

n1(r) ) n1B + c(r) n2(r) ) n2B - c(r)

(1)

c(r) ) F(r)/(z+e - z-e)

(2)

2

1 κ F(r) ) -Zpe exp(-κ(r - d)) 4π(1 + κd) r

(3)

Here, n1 and n2 are the ion densities, n1B and n2B are their bulk limit values, c is a convenient scaled ion density, F is the charge density, κ is a variational parameter corresponding to the Debye-Hu¨ckel inverse screening length, and Zpe is the charge of the ion at the origin. In our present work, we will have Zp ) z+ or z- since the central ion is one of the salt ions. In the traditional Debye-Hu¨ckel theory, only the interaction between the central ion and the screening ions is allowed to reflect the steric size of the ions represented by the hard sphere diameter d. The interaction between screening ions is without hard sphere truncation. In the CDH theory, on the other hand, hard sphere truncation applies to both central-screening and screeningscreening ion interactions. The neglect of ion size in the screening atmosphere combined with linearization of the entropy in c leads to the traditional DH result with the inverse screening length κ0 defined by

(z2+e2n1B) + (z2-e2n2B) κ20 ) okBT

(4)

Allowing for ion size in the treatment of the screening atmosphere leads in general to two types of effects: i) excluded volume effects and ii) hole correction of the electrostatic energy. In the RPM approximation, however, the excluded volume effects vanish in the local entropy limit since the linear screening mechanism maintains the total particle density n ) n1 + n2 unchanged at its bulk value n ) nB. With the excluded volume effect of a mixture of equal-sized hard spheres only related to n, this effect vanishes leaving only the hole correction of the electrostatic energy to reflect ion size in the screening mechanism. This simplification of the screening mechanism is the reason for us to propose the CDH-RPM theory as a particularly useful account of the bulk thermodynamic properties of salt solutions. The change in free energy associated with the screening of salt ions in the linear CDH-RPM approximation is

∆Fsc )

kBT Z2p κ3 nB z+(-z-) 16π(1 + κd)2

[

]

Z2pe2 κd (2 + 3κd + exp(-κd)) (5) 4πod 4(1 + κd)2 Here, the first term is the cost in entropy of adsorbing counterions and desorbing coions around each of the salt ions. The second term gives the lowering of the electrostatic energy achieved by the screening rearrangement of ions. Minimization of this screening free energy with respect to κ yields a larger value than κ0, that is, a shorter screening length as a result of less repulsion among the ions in the screening atmosphere. Since both terms are proportional to Z2p, the κ- value in a charge asymmetric electrolyte (e.g., 2:1, 3:1, etc.) will be the same for both ions. Once the variationally optimized κ- value has been obtained, we know the form of the charge density and can integrate over it to find the internal energy U. It satisfies the relation 2 2 2 2 3 (x1z +e + x2z -e )κ U ) UI ) UIk + UIc ) NkBT 2 8πo(1 + κd)kBT

(6)

Here, the first term on the right is the kinetic contribution UIk to the internal energy while the second term UIc accounts for the electrostatic interactions with x1 and x2 denoting the mole fractions, that is,

xi )

niB , i ) 1, 2 n1B + n2B

(7)

The other properties we shall be concerned with are the Helmholtz free energy per ion f, the osmotic coefficient φ, and the mean ionic activity coefficient γ. The electrostatic contributions to these properties can be obtained by the charging method from UIc. There are, however, also important contributions to φ and γ coming from the background hard sphere mixture. We have chosen to treat this hard sphere mixture by the GvdW(HS-B2) free energy functional and corresponding equation of state.33 The corresponding free energy per hard sphere is

[

fhs ) - kBT x1 ln n1B + x2 ln n2B +

(2π3 -1)n

B

]

- ln(1 - nBd3) (8)

and the corresponding chemical potential of an ion of species j is

µj ) µ(id)j + µ(ex)j(0) ) kBT[3 ln Λj + ln njB + 1] + k BT

[

nBd3

1 - nBd

3

- ln(1 - nBd3) + 2

(2π3 - 1)n d

]

3

B

(9)

Here, Λj is the thermal wavelength which will be an unimportant constant in the present application. The kinetic contribution to the free energy is

fk ) 3kBT[x1 ln Λ1 + x2 ln Λ2]

(10)

The electrostatic contribution to the free energy is obtained as

fsc )

∫01 dλ2UIc(λ)/λ

(11)

1406 J. Phys. Chem. B, Vol. 106, No. 6, 2002

Abbas et al.

Here, λ is the charging parameter such that zj(λ) ) λzj for all ionic charges. It is possible to transform the integral to be over the square root of the concentration C if we set s ) λxC and note that

fsc )

∫0xC ds2sUIc(s)/C

(12)

The pressure can now be obtained as

P)-

∂ ∂ f ) - (fk + fhs + fsc) ) Phs + kBTUIc/V - fsc/V ∂V ∂V (13)

where V ) 1/nB. The corresponding osmotic coefficient is now found as

φ)

fsc PV PhsV ) + UIc kBT kBT kBT

(14)

The hard sphere pressure in our GvdW(HS-B2) theory is given by

[

]

1 2π + -1 nBd3 , (GvdW(HS-B2)) Phs ) kBTnB 3 3 1 - nBd (15)

(

)

Finally, we obtain the logarithm of the mean ionic activity coefficient from the relation

ln γ ) [x1(µ(ex)1 (0) + ∆µ (ex)1) + x2(µ (ex)2 (0) + ∆µ (ex)2)]/kBT (16) Here, the contribution due to the screening mechanism satisfies the simple relation

x1∆µ(ex)1 + x2∆µ(ex)2 )

∂ Nf ) kBTUIc ∂N sc

(17)

The thermodynamic properties are nearly as easily obtained as in the original Debye-Hu¨ckel theory, but there are substantial differences arising because of both hole corrections of the electrostatic energies and treatment of the background hard sphere mixture by the GvdW(HS-B2) theory. This theory is very simple and quite accurate enough at the low-packing densities we will encounter here. Although it will not be used here, the CDH-RPM theory can be straightforwardly extended to the so-called “solvent primitive model” (SPM) wherein the solvent has been added as neutral hard spheres of the same size as the ions. In this type of application, however, we recommend the use of a more accurate hard sphere theory such as the GvdW(I) theory which particularly improves the accuracy at highpacking densities as arise when the solvent has been added bringing the whole system to liquid densities. Verification of these suggestions can be found in our preceding study.31 3. Comparison with MC Simulations and HNC Calculations The “Restricted Primitive Model” (RPM) of electrolyte solutions has been tested extensively by Monte Carlo simulations13 and integral equation theories.6 The conclusion from these studies is that by using the RPM the essential features of real electrolytes (salt solutions) can be captured. However, in salt solutions the ions are of different sizes and the solvent has a specific structure. A next stage in the modeling is to consider an electrolyte having ions of different sizes but the solvent is

still considered as a structureless dielectric continuum. Such a model can be denoted “Unrestricted Primitive Model” (UPM). In recent years, bulk thermodynamic data, that is, internal energy (UI), activity (γ), and osmotic (φ) coefficients from MC simulations,14,15 HNC,32 MSA,34 and MPB35 calculations for UPM electrolytes have become available in the literature. Since the CDH theory in its present form does not allow explicit representation of different cationic and anionic sizes, we shall map the simulated or calculated UPM results for 1:1 and 2:1 electrolytes onto the results of CDH-RPM theory by using a single effective ionic diameter d. As mentioned earlier, the charge density in the RPM case is decoupled from the number density which leads to the effect that excluded volume effects vanish. Thus, the mapping of the UPM results at different ion size ratios onto the CDH-RPM theory will provide a critical test of the sensitivity of our theory to the ion size asymmetry. We shall first calculate a single mean ionic diameter from the diameters of the cation and the anion used in the MC simulations or in the HNC calculations. The ion size ratio is defined as follows:

R)

d+ d-

(18)

Here, d+ and d- are the diameters of the cation and the anion, respectively. In the MC simulations and in the HNC calculations, an ion size ratio is obtained by varying the cation size while keeping the anion size constant. Thus, at each ion size ratio, the size of the cation can easily be calculated. For example, at ion size ratio R ) 0.4, the calculated cation diameter is 0.168 nm, while the anion diameter is fixed at 0.42 nm. In the MC simulations for 1:1 electrolytes and in the HNC calculations for 2:1 electrolytes the anion size is fixed at 0.42 nm, whereas in the HNC calculations for 1:1 electrolyte it is fixed at 0.425 nm. We use the Lorentz mixing rule to obtain a mean diameter dh from the diameters of the cations and anions, that is,

dh )

(d+ + d-) 2

(19)

The mean diameter calculated at R ) 0.4 by using the Lorentz rule when d- ) 0.42 nm is 0.294 nm. This value of dh was used in the CDH calculations, and results were compared with the MC or HNC data of UPM electrolytes. This procedure is in accord with the traditional practice in the use of the original Debye-Hu¨ckel theory, but it may not be the best mapping of an UPM onto a RPM representation. When we use the Lorentz mean ionic diameters in our theory below, we will denote it CDH-RPM(L). Recently, Harvey et al.36 have tested the accuracy of this linear approximation for a mean ionic diameter, that is, the Lorentz rule, in 1:1 UPM electrolytes at different ion size ratios. The values of internal energy and osmotic coefficients calculated by unrestricted MSA (ions have different sizes) were in good agreement with the results obtained when a linear mean ionic diameter was used. At electrolyte concentrations 1 at the highest concentration. This means that at low concentrations, the electrostatic repulsive forces dominate, but as the concentration increases, charge oscillations begin to appear and the function g++(r) which is increasing at low concentrations changes into a decreasing function at highest concentration. The contact values of the g- - radial distribution function for 1:1 electrolytes are independent of the ion size ratios from 0.4 to 0.8, whereas for 2:2 electrolytes the contact values of this radial distribution function show pronounced R and concentration dependence. This implies that in 1:1 electrolytes there will not be any large ionic clusters where a large anion is held in contact with a smaller cation. These structural differences between 1:1 and 2:2 electrolytes are reflected in our fitting scheme where we have obtained good fits for 1:1 UPM electrolytes but have failed to fit the 2:2 UPM electrolytes.

3.2. Asymmetrical Electrolytes. HNC calculations for 2:1 UPM electrolytes with a doubly charged cation (z+ ) +2, z) -1) have been carried out by Sloth and Sørensen32 at ion size ratios R ) 0.5, 1, and 2. They calculated the configurational internal energy (UIc), single ionic activity coefficients (ln γ+ and ln γ-), and osmotic coefficients (φ) at each ion size ratio. The electrolyte parameters for the 2:1 system were d- )0.42 nm and d+ was varied to obtain the required R value in the electrolyte concentration range 0.009441 to 2.7793 mol/liter. Comparison of the HNC data and the CDH-RPM(L) predictions (when Lorentz mean ionic diameters are used) for UIc and the mean ionic activity coefficients (ln γ) are given in Figures 5 and 6. The results for the osmotic coefficient are not given here because they were very similar to the results obtained for the mean ion activity coefficient. The mean ionic diameters obtained at ion size ratios 0.5, 1, and 2 are 0.35, 0.42, and 0.56 nm, respectively. Again the CDH-RPM(L) results agree closely with the HNC results for R ) 1, whereas at R ) 0.5 and 2 the difference between the results of the two theories is larger. At R ) 0.5 the CDH-RPM(L) theory underestimates UIc, ln γ, and φ, whereas at R ) 2 it overestimates the HNC results. As has been explained for the 1:1 electrolyte, we found that at R values less than unity the Lorentz mean ionic diameters are too small which leads to the underestimation of the thermodynamic properties of electrolytes. On the other hand, at R values higher than 1 the CDHRPM(L) theory overestimates these thermodynamic properties because of the large mean ionic diameters. For 2:1 electrolytes, the deviations between the CDH-RPM(L) and the HNC results are much larger as compared to the 1:1 electrolyte. This is because there is charge and ionic size asymmetry in the 2:1 electrolytes which leads to larger deviations at R ) 0.5 and 2. At R ) 1, there is just charge asymmetry and that can be handled nicely in the CDH theory. Thus, the CDH results agree well with the HNC results and also with the MC simulations as shown in the previous article.31 In Figures 7 and 8, the results of fitting the HNC data for UIc and ln γ at different ion size ratios are presented. Again the results for osmotic coefficient are not presented here because of the above-mentioned reasons. One optimized diameter was used to fit these three thermodynamic quantities. The best-fitted


Abbas et al.

Figure 5. The configurational internal energy (UIc) obtained from the Hypernetted Chain (HNC) calculations of Sloth and Sørensen32 for 2:1 electrolyte are compared with the CDH results at ion size ratios 0.5, 1, and 2. The mean ionic diameters (dh) used in CDH calculations are 0.35, 0.42, and 0.56, respectively, for ion size ratios 0.5, 1, and 2.

Figure 6. As in Figure 5 but for the natural logarithm of mean ionic activity coefficients (ln γ).

diameters obtained at different ion size ratios are given in Table 1. To fit the data at R ) 0.5 the ionic diameter was increased as compared to the Lorentz mean ionic diameter, whereas at R ) 2 the best-fitted diameter was less than the mean ionic diameter. The best-fitted ionic diameter for R ) 1 is the same as the mean ionic diameter. As described earlier, the curves in Figures 7 and 8 are the best-fitting curves as obtained by visual inspection. For the ion size ratios 1 and 2, good fits were obtained for both ln γ and φ. On the other hand, at R ) 0.5 the fitting was good at low concentrations but at higher concentrations, that is, >1 M, the fit was poor. The configurational energy (UIc) was fitted nicely up to 1 M concentrations at ion size ratios 1 and 2, whereas at concentrations >1 M the CDHRPM(O) theory overestimates UIc. On the other hand, at R ) 0.5 the fit was poor almost in the whole concentration range except at very low electrolyte concentration. Good fits were

also obtained for UIc at R ) 0.5 but for a smaller diameter (0.45 nm) as compared to the best general value, that is, 0.48 nm. The data for ln γ and φ were, however, not satisfactorily fitted with d ) 0.45 nm. Again a qualitative understanding of the results shown in Figures 5-8 can also be inferred from the behavior of RDFs with respect to ion size ratio and electrolyte concentration. We shall first consider when R ) 1, that is, the RPM. The radial distribution functions for 2:1 RPM electrolytes were obtained by Rasaiah.6c It was observed that contact values of g( decrease as the concentration increases. Thus, at low concentrations there is a tendency to form ion pairs but not so strong a tendency as was seen in the 2:2 electrolytes. The contact values for g++ RDF were small in the concentration range up to 1.3 M, but the contact values of g- - increase significantly with the concentration. This implies that at low concentrations there will



Figure 7. The configurational internal energy data (UIc) obtained from the Hypernetted Chain (HNC) calculations of Sloth and Sørensen32 for 2:1 electrolyte are fitted by using the single optimized ionic diameter in CDH-RPM(0) calculations at ion size ratios 0.5, 1, and 2. The best-fitting ionic diameters obtained for ion size ratios 0.5, 1, and 2 are 0.48, 0.42, and 0.41, respectively.

Figure 8. As in Figure 7 but for the natural logarithm of mean ionic activity coefficients (ln γ).

be strong repulsion between the coions, whereas at the highest concentrations charge structuring can occur because of ionion correlations. At higher concentrations, the hard sphere packing effects will also play a role because there will be twice as many anions as cations which will create a packing problem. For the UPM 2:1 electrolyte, Sloth and Sørensen32 have provided the radial distribution functions at different ion size ratios. At electrolyte concentrations e1 M, the contact values of g(, g++, and g- - are independent of ion size ratio. This behavior is similar to that for 1:1 UPM electrolytes. However, at higher concentrations these RDFs are dependent on the ion size ratios. Large oscillations in g( were seen for R ) 0.5 at highest concentration (see Figure 6 of ref 32) as compared to R ) 2. For coions, the oscillations at highest concentration are larger for anions as compared to cations. Thus, in 2:1

electrolytes there will be packing effects and charge oscillations at higher concentrations. On the other hand, at concentrations Ca2+ > Sr2+ > H+ > Li+ > Ba2+ > Na+ > K+ > Rb+ > Cs+ Samoilov50 proposed a new approach for ionic hydration where he studied the exchange of water molecules in the immediate vicinity of ions. By calculating a change (due to ionic influence) ∆E in the activation energy of the closest water molecules, he could discern the different patterns of ionic hydration for the alkali salts. He concluded that water molecules are tightly bound to ions such as Li+, Na+, Ca2+, and Mg2+. For large ions such as K+, Cs+, Cl-, Br-, and I- the water molecules are more mobile than in pure water and he referred to this phenomenon as “negative hydration”. In a very recent study, Koneshan et al.51 have investigated the mobilities of alkali metal halides and Ca2+ as charged and uncharged species by using molecular dynamics (MD) simulations. The mobilities of Li+, Na+, K+, Rb+, and F- increase on discharge, whereas Cl, Br, and I have smaller mobilities than the corresponding anions. On the other hand, the residence time of water molecules in the first solvation shell decreases on discharging the small cations while it increases for the large neutral I. The radial distribution curves obtained by these authors clearly show that the second hydration shell is present for most of the ions studied. For anions, the second and even third hydration shells are clearly present. A coherent picture of ionic hydration appears from their study, that is, the small highly charged ions are strongly hydrophilic, and this character decreases with increasing size of the cation. On the other hand, the hydration of large cations and anions can be described as hydrophobic hydration.52 Thus, once again the model of ionic hydration put forward by Frank and Gurney seems to be strongly supported by these studies.


Abbas et al.

TABLE 3: Best-Fitting Diameters and the Ranges of their Applicabilitya

HCl HBr HI HClO4 HNO3 LiCl LiBr LiI LiOH NaF NaCl NaBr NaI NaOH KF KCl KBr KI KOH RbF RbCl RbBr RbI CsF CsCl CsBr CsI NH4Cl NH4Br NH4I MgCl2 MgBr2 MgI2 CaCl2 CaBr2 CaI2 SrCl2 SrBr2 BaCl2 BaBr2 BaI2 MgSO4 CdSO4 AlCl3 LaCl3 LiClO4 NaClO4 NH4Cl O4 Mg(ClO 4)2 Ca(ClO 4)2 Sr(ClO4 )2 Ba(ClO 4)2 Cd(ClO 4)2 Cu(ClO 4)2 LiNO3 NaNO3 KNO3 RbNO3 CsNO3 AgNO3 NH4NO 3 Mg(NO 3)2 Ca(NO3 )2 Sr(NO3) 2 Cd(NO3 )2 Cu(NO3 )2 Pb(NO3 )2

Pauling type diameter (nm)

hydrated-1 diameter (nm)

hydrated-2 diameter (nm)

fitted diameter (nm)

conc range (moles/liter)

hydrated size ratio (d+/d-)

0.293 0.308 0.332 0.352 0.291 0.255 0.270 0.294 0.207 0.235 0.282 0.298 0.322 0.235 0.271 0.318 0.335 0.359 0.271 0.282 0.329 0.345 0.369 0.303 0.350 0.366 0.390 0.329 0.344 0.368 0.289 0.309 0.340 0.308 0.328 0.360 0.325 0.345 0.332 0.352 0.384 0.302 0.325 0.298 0.331 0.314 0.342 0.358 0.368 0.387 0.403 0.415 0.384 0.369 0.253 0.281 0.317 0.328 0.349 0.294 0.297 0.287 0.305 0.322 0.302 0.287 0.318

0.594 0.612 0.640 0.646 0.592 0.527 0.545 0.573

0.414 0.421 0.436 0.459 0.413 0.465 0.472 0.487 0.453 0.430 0.442 0.449 0.464 0.430 0.424 0.436 0.443 0.458 0.424 0.425 0.437 0.444 0.459 0.431 0.443 0.450 0.465 0.437 0.444 0.459 0.498 0.507 0.527 0.479 0.489 0.509 0.474 0.483 0.468 0.477 0.497 0.577 0.551 0.525 0.490 0.510 0.487 0.482 0.558 0.539 0.534 0.528 0.541 0.557 0.464 0.441 0.435 0.436 0.442 0.435 0.436 0.497 0.478 0.473 0.479 0.495 0.471

0.420 0.440 0.465 0.430 0.390 0.405 0.410 0.46 0.230 0.280 0.348 0.365 0.385 0.355 0.33 0.317 0.330 0.350 0.380 0.365 0.300 0.300 0.305 0.365 0.270 0.276 0.285 0.325 0.335 0.325 0.480 0.480 0.500 0.440 0.453 0.458 0.435 0.445 0.430 0.430 0.470 0.340 0.330 0.500 0.468 0.450 0.345 0.23 0.530 0.510 0.480 0.460 0.500 0.520 0.390 0.280 0.170 0.160 0.160 0.145 0.245 0.46 0.4 0.375 0.430 0.430 0.230

0-1.4 0-1.0 0-1.2 0-1.2 0-0.8 0-0.9 0-1.6 0-0.7 0-2 0-1 0-0.8 0-0.9 0-1 0-1 0-1.4 0-1 0-1 0-0.8 0-1.8 0-1 0-1.4 0-0.8 0-1 0-1.2 0-1.6 0-1 0-2b 0-0.8 0-0.7 0-0.8 0-0.5 0-1.4 0-1.6 0-1 0-1.8 0-1.6 0-0.6 0-1 0-0.6 0-0.8 0-1.4 0-1.8b 0-1.4b 0-0.8 0-0.4 0-1.2 0-0.8 0-1.8b 0-0.8 0-0.6 0-1 0-0.8 0-1 0-1 0-1.4 0-1 0-1.4 0-1.4 0-1 0-1 0-1 0-0.5 0-0.5b 0-0.5b 0-0.6 0-1 0-1

0.82 0.85 0.77 0.71 0.85 1.08 1.04 0.98 1.14 1.03 0.97 0.94 0.87 1.03 1 0.95 0.92 0.86 1 1.01 0.95 0.92 0.87 1.03 0.98 0.95 0.89 0.95 0.92 0.87 1.34 1.294 1.22 1.21 1.17 1.1 1.17 1.14 1.13 1.1 1.03 1.08 0.98 1.68 1.38 0.9 0.81 0.79 1.11 1 0.98 0.94 1.02 1.1 1.08 0.98 0.96 0.955 0.98 0.95 0.955 1.34 1.22 1.18 1.22 1.33 1.17

0.499 0.554 0.573 0.600 0.543 0.599 0.617 0.645 0.552 0.608 0.626 0.654 0.577 0.633 0.651 0.679

0.564 0.589 0.626 0.588 0.611 0.648 0.601 0.626

0.591 0.612 0.672 0.605 0.578 0.606 0.633 0.655 0.687 0.711 0.647 0.634 0.524 0.552 0.596 0.605 0.630 0.558 0.561 0.583 0.597 0.575 0.562

a In column two the average crystallographic diameters for salt solutions are given. Hydrated-1 denotes the average hydrated diameters calculated from the data given in Marcus review article44 and Hydrated-2 means the hydrated diameters according to the empirical model of Marcus.45 Fitted diameters obtained in the CDH-RPM(O) theory are given in the fifth column whereas the electrolyte concentration ranges where these diameters are valid are given in the sixth column. In the last column are shown the ion size ratios calculated from the hydrated diameters of the ions as given by the Marcus model.45 b Only activity coefficients were successfully fitted.



Figure 10. The natural logarithm of mean ionic activity coefficient is plotted as function of concentration in moles per liter. CDH denotes the Corrected Debye-Hu¨ckel theory in its RPM(O) form and Exp denotes the experimental results for solutions of LiCl, LiBr, and LiI from Robinson and Stokes.41 The best-fitting diameters used in the CDH calculations are 0.405, 0.41, and 0.46 nm for LiCl, LiBr, and LiI, respectively.

Figure 11. The natural logarithm of the mean ionic activity coefficient for magnesium halide salts is plotted as function of concentration in moles per liter. CDH denotes the Corrected Debye-Hu¨ckel theory in its RPM(O) form and Exp denotes the experimental results for solutions of MgCl2, MgBr2, and MgI2 from Hamer and Wu.42 The best-fitting diameters used in the CDH calculations are 0.48, 0.48, and 0.50 nm for MgCl2, MgBr2, and MgI2, respectively.

Dispersion forces also play a role. The dispersion interaction potential of ions is of the form (λ/R6), where R is the distance between the ion and the water molecule. These forces will be significant in the first hydration shell while in the region outside they will die out quickly. These forces are sensitively dependent on the molecular weight of the molecules. Thus, dispersion forces will be most important for the large solute molecules. To represent the 2:1 salts we have chosen the halide salts of magnesium, and the experimental data fitted by a single effective ionic diameter are presented in Figure 11. A good fit to the experimental data for the MgCl2 solution was obtained only up to 0.5 M, whereas for MgBr2 and MgI2 satisfactory fits were obtained up to 1.4 and 1.6 M concentrations, respectively. The

best-fitting diameters for these three salts are 0.48, 0.48, and 0.50 nm, respectively (Table 3). The halide salts of Ca, Sr, and Ba showed the same behavior as of Mg though the range of satisfactory fit varies with the salt type. The values of the bestfitting diameters and the ranges of their applicability are given in Table 3. In Figure 12, the results of our CDH-RPM(O) fits of some experimental data for strong acids and bases are presented. A very good fit to the experimental data is obtained in the whole concentration range. Moreover, the best-fitting diameters for these salts are reasonable, that is, they are larger than the Pauling type diameters and less than the hydrated diameters. In LiOH, they are very close to the Pauling type diameter. For other strong


Abbas et al.

Figure 12. The osmotic coefficient is plotted as function of concentration in moles per liter for solutions of HCl, HClO4, LiOH, and KOH. CDH denotes the Corrected Debye-Hu¨ckel theory in its RPM(O) form and Exp denotes the experimental results from Hamer and Wu.42 The best-fitting diameters used in the CDH calculations are 0.42, 0.43, 0.23, and 0.38 nm for HCl, HClO4, LiOH, and KOH, respectively.

Figure 13. The osmotic coefficient is plotted as function of concentration in moles per liter for solutions of LiNO3, NaNO3, KNO3, and Mg(NO3)2 salts. CDH denotes the Corrected Debye-Hu¨ckel theory in its RPM(O) form and Exp denotes the experimental results from Robinson and Stokes.41 The best-fitting diameters used in the CDH calculations are 0.39, 0.28, 0.17, and 0.46 nm for LiNO3, NaNO3, KNO3, and Mg(NO3)2, respectively.

acids such as HBr and HNO3 and the strong base NaOH, the fitted diameters are reasonable and the range of applicability is specific to the individual acid or base. Nitrate salt solutions are known to have some weak ion association. The activity coefficients of these salts are usually low lying for alkali halides. This may be because the nitrate ion is a large oxyion and naturally it will show a hydrophobic behavior in its hydration. The ion is also rather asymmetric in water so in the ionic interaction of salt solutions the hydration is likely to play a strong role. In Figure 13, some examples of the 1:1 and 2:1 nitrate salt solutions are given. LiNO3 behaves normally in that the osmotic coefficient data show an upward curvature and the best-fitted diameter obtained in CDH theory is reasonable. As the cation size increases, the osmotic coef-

ficient data show a significant trend, that is, low lying behavior. To fit these data, we have to choose very small ionic diameters in the CDH-RPM(O) theory and for KNO3 the bestfitting diameter is already smaller than the corresponding average crystallographic diameter. This again illustrates the trend discussed above that two structure-breaking ions will attract each other strongly and will induce some structure when they come close to each other. Similarly, for Mg(NO3)2 the experimental data behaves normally and the best-fitting diameter is reasonable. The low-lying behavior starts at Ca, and it is strong in Sr(NO3)2, and the best-fitting diameters are reduced to smaller values. Again for cadmium and copper the nitrate salt solutions behave normally, whereas Ag and Pb nitrates have low-lying behavior. The perchlorate 1:1 and 2:1 salts behave

Corrected Debye-Hu¨ckel Theory of Salt Solutions normally, that is, the experimental data show an upward curvature and the fitted diameters are reasonable, except in NH4ClO4 (Table 3). The trends in the results which we have obtained in this extensive study of salt solutions can be qualitatively explained by the considerations of hydration discussed above. The range of applicability of our theory is generally approximately 0-1 M, and the CDH theory overestimates the experimental data at concentrations >1 M. This overestimation is largest in 3:1 salts as compared to the 2:1 and 1:1 salt solutions. Hence, the range of applicability for 3:1 and 2:1 salt solutions is less than in the 1:1 case. This is because the CDH theory is based on linear response to electrostatic interaction and equal ion sizes. These assumptions are most applicable to 1:1 salt solutions and become more doubtful for 2:1, 3:1, and 2:2 salts. The packing effects become prominent in the 2:1 and 3:1 salt solutions. The 2:2 salt system is so strongly coupled that our linear theory cannot resolve it properly. In the CDH theory, the best-fitted diameters generally follow the pattern of ionic hydration. To model the activity data of a salt of two structure-breaking ions, the fitted diameter has to be reduced and in some cases values are obtained which are less than the corresponding crystallographic values. On the other hand, in well-behaved solutions, the fitted diameters are usually larger than the crystallographic diameters. In the CDH theory, the ionic size is a distance of closest approach between the ionic species. Thus, when we adjust d in CDH theory to fit the experimental data, we are including the nonlinear or short-range effects (electrorestriction, reduction in the dielectric constant of water near an ion, hydrophobic effects, or dispersion force effects) by just adjusting the distance of closest approach in the form of the effective ionic diameter. Marcus45 proposed a simple model to calculate the thermodynamic properties of hydrated ions. He derived the radius of the first hydration shell and the solvation number of different ions. The predictions of his theory agree well with the experimental data for enthalpy of solvation, entropy of solvation, and so forth. We have used the reported values of ionic radius plus the width of the first hydration shell to calculate the radius of a hydrated ion. The corresponding values of mean ionic hydrated diameters are given in Table 3. Our best-fitted diameters for most of the salts follow the trends of Marcus diameters. For example, in the alkali and alkaline earth metal chlorides the hydrated size of the cation decreases as the size of the anion increases. Thus, the mean diameters decrease. Furthermore, in the model of Robinson and Stokes, as discussed in the Introduction, the effective ionic diameters obtained are quite close to our values and our range of applicability is quite close to that of their model in many cases (see Table 1 of ref 18). To fit the experimental data, Robinson and Stokes arrived at unreasonable values of the hydration number. If we compare with our theory, we have just included the hole correlations in the ionic atmosphere and the background excluded volume effect due to the finite size of the ion. Nevertheless, we have with reasonable diameters obtained results similar to those of Robinson and Stokes. Recently, Lund et al.53 have calculated the activity coefficients in seawater by using MC simulations in which the ionic diameters were optimized. If the radius of the halide ion is fixed at the crystallographic value, then the individual cation radii obtained in our study are very close to the cation radii obtained by MC simulations. For example, if the radius of the Cl- ion is fixed at 0.17 nm,53 the radii of H+, Na+, K+, NH4+, Mg2+, and Ca2+ obtained from the fitted ionic diameters (Table 3) are 0.25, 0.178, 0.147, 0.155, 0.31, and 0.27 nm, respectively. The

J. Phys. Chem. B, Vol. 106, No. 6, 2002 1417 corresponding values obtained by MC simulations are 0.25, 0.18, 0.15, 0.15, 0.3, and 0.28 nm. Moreover, for strongly dissociating salts a consistent set of cation radii can be obtained by considering the anion radii fixed at their crystallographic values. For example, if the radii of Cl-, Br-, and I- are fixed at 0.17, 0.195, and 0.216 nm, the sets of radii obtained for H+, Li+, Na+, and K+ in these halide salt solutions are (0.250, 0.245, 0.249), (0.235, 0.215, 0.244), (0.178, 0.17, 0.169), and (0.147, 0.135, 0.129), respectively. Another approach which is conceptually close to our CDHRPM theory is based on the Mean Spherical Approximation (MSA). In several studies the MSA has been used to calculate the mean ionic activity and osmotic coefficients of electrolyte solutions in restricted40,54 and unrestricted primitive models.55,56 As discussed in the previous article,31 it appears that when experimental data is fitted in the restricted version of MSA, that is, data is fitted by adjusting a single effective ionic diameter at a constant dielectric constant, the range of good fit is very limited. For example, in NaCl a good fit was obtained only up to 0.3 M. In the CDH-RPM theory we have obtained a good fit for NaCl up to 0.8 M concentration. The best-fitting diameters for alkali metal halides obtained in this study are in good agreement with the single effective ionic diameters obtained by Simonin et al..56 Also, Simonin et al. have fitted the experimental data by making the ionic diameters and permittivity of the solvent electrolyte concentration dependent. In a recent work, they have also included ion association.57 In the last column of Table 3, the ion size ratios of hydrated ions according to the Marcus model45 are given. In the limit of infinite dilution, we see that for 1:1 salt solutions the ion size ratio varies between 0.7 and 1.1 and for 2:1 salts it varies between 0.8 and 1.4. These ion size ratios will be different at higher electrolyte concentrations but at sufficiently low concentrations they should be valid as listed. This compact range of size ratios may be the reason that our fitting scheme (with the assumption that both the ions are of same size) has performed well especially in those salt solutions where ion association is considered to be small. We have noticed in fitting the model electrolytes that at ion size ratios near unity a good fit to the thermodynamic data was obtained for both 1:1 and 2:2 electrolytes despite the strong electrostatic coupling of the latter. 5. Comparison with SI Theory and Pitzer and Bromley’s Models We shall compare here results obtained by fitting the experimental data on salt solutions with the SI theory, the Pitzer model, the Bromley model, and the CDH theory. As pointed out in the Introduction, in SI theory, as well as in the Pitzer and Bromley models, the long-range electrostatic interactions are modeled by the DH term which includes the ion size a. In all these models, ion size is considered to be fixed and independent of salt in the range of electrolyte concentrations studied. In the SI theory, a is fixed at 0.45 nm and in the Pitzer and Bromley models, a has values 0.37 and 0.31 nm, respectively. On the other hand, in the CDH-RPM(O) theory finite ion size is introduced in the form of background hard sphere effects and hole corrections to the electrostatic energy. Ion size is varied for each salt to fit the experimental data. Pitzer included the hard sphere effects by virial expansion but his virial coefficients are ionic strength dependent. In our CDH theory, the hard sphere effects are included as background terms which are not dependent on the ionic charges because the charge density is decoupled from the number density. The “number


Abbas et al.

Figure 14. The calculated natural logarithm of the mean ionic activity coefficient of an LiBr solution is plotted as function of concentration in moles per liter for four theories. CDH denotes the Corrected Debye-Hu¨ckel theory in its RPM(O) form, Pitzer (Pitzer model), SI (Specific ion interactions theory), and BL (Bromley model). The best-fitting diameter used in the CDH calculations is 0.41 nm.

Figure 15. As in Figure 14 but for a solution of NH4ClO4 salt solution. The best-fitting diameter used in the CDH calculations is 0.23.

density contributions” are obtained from an equation of state (GvdW(HS-B2) in the present case) applied to the uncharged hard sphere fluid. We shall compare the results of CDH theory and the three earlier theories keeping these inherent differences between the theories in mind. The experimental data for a large number of salts were fitted with these theories. An almost perfect fit to the experimental data with the Pitzer model was obtained in nearly all cases. The rank order of these three earlier theories with respect to the fit to the experimental data is as follows: Pitzer > Bromley > SI theory. In Figures 14 and 15, the results for LiBr and NH4ClO4 are presented. As shown in Figure 14, the CDH theory agrees nicely with the Pitzer model up to 1.8 M electrolyte concentration. On the other hand, both the SI and Bromley models are less accurate in the intermediate concentration range while at low and high concentrations they agree with the Pitzer

model. Since the Pitzer model fits the experimental data nearly perfectly, the experimental data is not shown in the figures. An interesting feature was observed in the case of more strongly coupled salts such as nitrates, fluorides, NH4ClO4, and so forth, that is, the qualitative behavior of the SI theory is wrong as shown in Figure 15. Figure 15 is produced with a constant value of interaction coefficient β as suggested originally by Guggenhiem.20 However, the value of the interaction coefficient used in this figure is from a recent article.58 At very low concentrations the agreement is good, but at intermediate concentrations the SI theory overestimates the experimental data (Pitzer curve). On the other hand, at higher concentrations the SI theory and Bromley models underestimate the experimental data. The CDH theory agrees nicely with the experimental data in almost the whole concentration range. At very high concentration the CDH theory overestimates a little because packing

Corrected Debye-Hu¨ckel Theory of Salt Solutions effects start to play a role. The ionic diameter obtained by fitting the experimental data for this case is 0.26 nm, which is less than the corresponding average crystallographic diameter. Thus, we know that in these systems nonlinear effects play a role and have to be compensated by reducing the distance of closest approach, that is, ionic size in the CDH case. This divergence of SI theory at higher electrolyte concentrations was also seen for NaF, KF, LiOH, NaNO3, KNO3, and NH4NO3. The Bromley model also showed this underestimation for KNO3, RbNO3, CsNO3, Mg(NO3)2, Sr(NO3)2, and Ba(NO3)2. The effect was more prominent in osmotic coefficients as compared with activity coefficients. As for as SI theory is concerned, it has been suggested58 that two interaction coefficients should be used and that the second interaction coefficient is allowed to be concentration dependent. When these interaction coefficients were used, the qualitative behavior of the SI theory became correct. However, in NaF and KF the SI theory still underestimates the experimental data at higher electrolyte concentrations. This may indicate that the approximation of retaining only the second virial coefficient is not so good at high concentrations. That is why in the Pitzer equation a third virial coefficient has to be included to model strongly coupled salt solutions. On the other hand, the CDH theory is transparent in the sense that we know that small ionic diameters mean strongly coupled solutions and the deviation at high concentrations is mainly due to packing effects. In the Pitzer model, it is not so obvious in which way the nonlinear effects are modeled at higher concentrations. 6. Concluding Remarks The CDH-RPM theory is an a priori extension of the Debye-Hu¨ckel theory of bulk salt solutions given that the RPM representation is applicable and that the hard sphere diameter is known or determined empirically by fitting experimental data. We have learned from earlier work that our theory works very well for RPM salt solutions over a range of concentrations up to 1 M or more except for strongly coupled 2:2 salts. In real experimental salt solutions, the mapping to RPM representation and the determination of the common ionic diameter d is necessarily to some extent empirical and nontrivial. The main purpose of this work has been to show that this RPM mapping can be done reliably. We have divided the task in two steps. First, we have studied unrestricted primitive model electrolytes (UPMs) where the cations are smaller or larger than the anions and have shown that a mapping onto an RPM representation is possible. In this case, we compare with the data from MC simulation studies. We found that if the ion size ratio is close to unity one can use a linear average diameter dh. On the other hand, if the size ratio is outside a range of, say, 0.8-1.2 then an optimization with respect to the fit of bulk thermodynamic properties yields a significantly non-Lorentzian value which noticeably improves the fit. The comparison with the MC and HNC results for UPM salt solutions has led to the conclusion that mapping of unequal to common ionic diameter is not a problem as long as the ionic size ratios are not too large. The range of hydrated ion size ratios appearing in our 67 salt solutions is 0.71-1.68 with a typical deviation from unity by only about 10%. By reference to the much wider range of ion size ratios encountered in our comparison with MC and HNC results, the RPM representation of salt solutions is well justified with respect to the first step, that is, UPM to RPM mapping. The second step is the mapping from reality to the primitive model. In this step, we subsume the solvent in the form of a dielectric constant and model the

J. Phys. Chem. B, Vol. 106, No. 6, 2002 1419 ions as charged hard spheres. These great simplifications are more difficult to test, but in this work, successful fit of experimental data with ionic diameters in accord with the independent experimental determinations constitutes a justification of our model. The nature of our CDH-RPM theory places it close to the semiempirical Pitzer theory which is able to produce even better agreement with the experimental data. We can, however, significantly reduce the empirical parametrization with only a marginal loss of accuracy. In the Pitzer theory, there appear a global ion size parameter and two or three virial coefficients optimized for each salt. By comparison, we have only our diameter d as an empirical parameter for salt solutions and for RPM models our theory is a priori. Thus, with respect to its derived nature the CDH theory is more akin to the MSA theory but with respect to its simplicity and practical utility it is closer to the Pitzer extension of the Debye-Hu¨ckel theory. The MSA is based on a simplified integral equation for the pair correlation function of a fluid. The CDH theory has a more traditional physical basis, that is, the combined analysis of van der Waals for short-range interactions in simple fluids and of Debye and Hu¨ckel for long-range electrostatic effects. We acknowledge that such a theory may seem late in arrival but it is in the most important sense both modern and very practical. We have thus shown that a modern density functional approach working directly with the free energy simply encapsulates both longrange electrostatic and short-range ion size effects of electrolyte solutions including a wide range of salt solutions. Further applications to mixed salt solutions, macroion screening, and surface complexation are planned or on the way. Ion association can be incorporated in the CDH theory as, for example, in the binding mean spherical approximation theory of Bernard et al.57 The prospects seem good that the simple basis shall continue to generate predictions of high accuracy. Acknowledgment. The financing of this work by the Swedish National Board for Industrial and Technical Development and A° ngpannefo¨reningen is gratefully acknowledged. Stimulating discussions about ionic hydration processes with Professor Itai Panas have helped in formulating this article. References and Notes (1) Milner, S. R.. Philos. Mag. 1912, 42, 223. (2) (a) Debye, P.; Hu¨ckel, E. Phys. Z. 1923, 24, 185. (b) Debye, P.; Huckel, E. Phys. Z. 1924, 25, 97. (3) Mayer, J. E. J. Chem. Phys. 1950, 18, 1426. (4) Rasaiah, J. C.; Friedman, H. L. J. Chem. Phys. 1968, 48, 2742. (5) Ornstein, L. S.; Zernike, F. Proc. Akad. Sci. (Amsterdam) 1914, 17, 793. (6) (a) Rasaiah, J. C.; Friedman, H. L. J. Chem. Phys. 1969, 50, 3965. (b) Rasaiah, J. C.; Card, D. N.; Valleau, J. P. J. Chem. Phys. 1972, 56, 248. (c) Rasaiah, J. C. J. Chem. Phys. 1972, 56, 3071. (7) Friedman, H. L. J. Chem. Phys. 1960, 32, 1134. (8) Waisman, E.; Lebowitz, J. L. J. Chem. Phys. 1972, 56, 3086. (9) Blum, L. J. Mol. Phys. 1975, 30, 1529. (10) Planche, H.; Renon, H. J. Phys. Chem. 1981. 85, 3924. (11) (a) Outhwaite, C. W. J. Chem. Phys. 1969, 50, 2277. (b) Burley, D. M.; Huston, V. C. L.; Outhwaite, C. W. J. Mol. Phys. 1974, 27, 225. (c) Outhwaite, C. W. J. Chem. Soc., Faraday Trans. 2 1987, 83, 949. (12) (a) Andersen, H. C.; Chandler, D. J. Chem. Phys. 1970, 53, 547. (b) Andersen, H. C.; Chandler, D. J. Chem. Phys. 1972, 57, 1918. (13) (a) Card, D. N.; Valleau, J. P. J. Chem. Phys. 1970, 52, 6232. (b) Valleau, J. P.; Cohen, L. K. J. Chem. Phys. 1980, 72, 5935. (c) Valleau, J. P.; Cohen, L. K.; Card, D. N. J. Chem. Phys. 1980, 72, 5942. (14) Abramo, M. C.; Caccomo, C.; Malescio, G.; Pizzimenti, G.; Rogde, S. A. J. Chem. Phys. 1984, 80, 4396. (15) Rogde, S. A. Chem. Phys. Lett. 1983, 103, 133. (16) Wu, G. W.; Lee, M.; Chan, K. Y. Chem. Phys. Lett. 1999, 307, 419. (17) Lyubartsev, A. P.; Laaksonen, A. Phys. ReV. E 1995, 52, 3730. (18) Robinson, R. A.; Stokes, R. H. J. Am. Chem. Soc. 1948, 70, 1870.

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Corrected DebyeHÃ¼ckel Theory of Salt Solutions: Size Asymmetry and

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