A Simple Model for Ion Solvation with Non Additive Cores *

Abstract : The solvation of monoatomic ions at infinite dilution in polar solvents is considered within the context of the mean spherical approximation (MSA). The role of both ion-dipole and dipole-dipole interactions in determining the Gibbs energy and entropy of solvation is considered in detail for the alkali metal and halide ions in water. The significance of the MSA parameters with respect to local interactions of water molecules with the ions and each other is discussed.

It is clear that in many problems the knowledge of the very complex details of the close range interactions are more than what one needs.In the interpretation of thermodynamic and kinetic parameters averages over large domains are predominant, and therefore, simple, yet controlled theories not only provide a way of fitting data, but also useful physical insight.
The success of the Marcus theory of electron transfer kinetics 4 in solution is a good example.
In this theory the polarization of the solvent is evaluated from a continuum model for the solvent, that is, the structure of the solvent is completely ignored.The recent rather spectacular calculations of Chandler and his group 5 have conclusively shown that within certain limitations, this is not unreasonable provided that the radius of the activated complex is chosen in a suitable manner.
There are also other examples where this is true.It was used by many investigators in the past, but in an entirely empirical way. 6 a series of recent papers 7 - 10 we have discussed the use of the Mean Spherical Approximation (MSA) in combination with a very simple asymmetric force model, which has a spherical repulsive core.The excellent correlations for the thermodynamic and kinetic parameters for a large number of solvents are more than mere coincidence.We believe that they are due to the fact that the MSA is not only a simple theory, but it also has some remarkable variational properties, that were discovered quite recently.Liquid state theories like the MSA and the HNC can be derived as variational problems of the free energy functional, which is written in terms of the Omstein-Zernike direct correlation function.This view was introduced by Rosenfeld 1- 13 for systems of hard objects in general, and hard spheres in particular.In that work the relation between the grand potentiai, the direct correlation function and die scaled particle theory was clearly shown.This analysis was extended to the case of charged hard spheres in general, and to the primitive model of electrolytes in particular. 14. 15It can be shown that the thermodynamics and the pair correlation functions can be derived from a simple nxoel in Ahich the lons are replaced by charged shells and the dipoles by charged dipolar shells.We have shown that the direct correlamon function in the MSA is just the electrostatic energy of the shells.Solving the MSA is thus equivalent to an electrostatics problem.
In the limit of high density and charge the Gibbs energy and the internal energy should diverge to the same order in the coupling parameter (charge/temperature ratio) while the entropy should diverge at a slower rate and therefore the Gibbs energy and the internal energy coincide in this limit.The mean spherical approximation (MSA) and the hypemented chain approximation (NBC) satisfy these bounds, discovered by Onsager some fifty years ago.(This is not true for the Debye Htickel theory, where the entropy also diverges).Therefore, in this limit, the HNC and the MSA Gibbs energy, and its derivatives, coincide.This is a very gratifying feature, because the HNC, which is the more accurate theory, is in general difficult to solve numerically, while the MSA is analytical in most cases, and of a rather simple form.
In the asymptotic limit the excess electrostatic energy is identical to the exact Onsager lower bound, which is achieved by immersing the entire hard core system in an infinite neutral and perfectly conducting (liquid metal) fluid. 1 4 In the case of the primitive model of electrolytes, this is equivalent to smearing the charge on the surface of the ion.It was also shown t 4 ,1 6 that the direct correlation function is equal to the interaction potential of two "smeared" ions of radius Ri, Rj where Ri is the radius of ion i plus some screening length.
The Onsager process of introducing the infinite conductor, naturally decouples all the different components in the system which may differ in size, shape, charge distribution and relative orientation in space.As a result, the variational free energy functional in the high coupling limit diagonalizes, and the mathematical solution of the asymptotic problem is given in term of the geometrical properties of the individual particles in the system.
The MSA Gibbs energy provides an appropriate functional from which the charge part of the direct correlation function is obtained by functional differentiation.
There is an actual numerical test case that proves the electrostatic analogy of the MSA.The very-high charge and density limit for the one component plasma (point ions 'in a neutralizing background) was investigated using the IINC equation by Ng  19 The median diameter of a non spherical object yields a surprisingly good equation of state, not only for hard dumbbells but also for real gases at extremely high pressures. 20. 2 1 Other examples are found in the extensive studies of Nezbeda and his group 22 . 23on the structure of molecular liquids.
The analytical solution of the MSA for mixtures of ions and dipoles with a non spherical sticky potential was obtained some time ago 24 -26 for the case of a sphere with a point dipole and a sticky potential of dipolar symmetry, which represents the specific non electrostatic interactions.
In this work it was shown that the contribution of the sticky potential was merely a shift in the polarization parameter, X.This work has been continued to include short ranged potentials of higher (tetrahedral) symmetry. 2 7 this spirit we have recently analyzed the solvation thermodynamics of monoatomic monovalent ions in water and in a number of polar solvents.7, 10,28 The basic idea comes from the observation that to a first approximation the corrections due to the molecular nature of the solvent result in a change in the screening length from l/r to 1/r + Xc./r O 0 s, the shift being different for each ion, according to its size; because of geometrical packing considerations the number of solvent molecules that can be the first neighbors of a given ion, i, vary according to the ratio ui/ias.
Consider now a realistic model of the solvent, with a soft repulsive core, and of non-spherical shape.Then, the distance of closest approach of a certain ion should be very different for the anion-solvent pair and the cation-solvent pair, even if the unsolvated anion and cation are of the same size.The correct statical mechanical model for such a situation is the mixture with nonadditive cores discussed in the literature. 29A simple way of accounting for this is to assume that at low concentration the effective radius of the solvent is rsi = rs+Asi() The Model In the MSA for a system of spherical ions and dipoles of arbitrary size 30 , the Gibbs energy of solvation in the limit of infinite dilution is 3 1 Gid Here, zi is the valence of the ion, co , the fundamental charge, e-, the static dielectric constant of the solvent, _., the permittivity of free space, ri, the radius of the ion, and No, the Avogadro constant.50 is equal to rs/k,, where rs is the radius of the solvent (the effective radius) and the MSA polarization parameter, X is calculated from the dielectric constant of the pure solvent Fes using the Wertheim relationship 32 )ý2 (X+1)4 = 16 Es (3) Although this value of the polarization parameter may be appropriate for estimating dipole-dipole interaction energy, it gives poor estimates of the Gibbs solvation energy when this is assumed to equal Gid, 7 ,10 ,28 Therefore, and in view of the above discussion we change eq. ( 2) to where 8A rs+AsA (7)   This separation and parameterization produces a much more sensible and physical model of Gid for ions in a number of solvents.
The second contribution to the Gibbs solution energy comes from repulsive dipole-dipole interactions 2 5 , 3 3 and is given by the equation

2(X+ 1)2 j
This expression may be simplified considerably when one considers the range of values typical for X•.For water whose dielectric constant is 78.3 at 25 'C, X is equal to 2.65.The corresponding value of the ratio %(),+3)/2(X+ 1)2 whicn appears in the denominator of the term in square brackets in eq. ( 8) is 0.56.This is negligible with respect to the value of es and may be neglected.The ratio appearing in the numerator of this term, namely, (3X+2)/(%+1) is equal to 2.7.Thus, the expression for Gdd may be rewritten as Gdd co FES) (ri+si)2 The simplified expression has the advantage that the parameter X does not appear on its own but only in 8i.This fact is important in applying the expression to experimental data.Finally, on the basis of previous work 2 5 , only a fraction 0 of the dipole-dipole term is used in estimating the Gibbs energy of solvation so that the final expression for Gs becomes Gs = Gid + 0 Gdd (10)   The expressions for the corresponding entropies are obtained from the temperature derivatives of the above Gibbs energies.For the ion-dipole term, the result for a cation is where d&,/dT is the temperature derivative of 8c which is obtained from the temperature derivative of kc.Comparing eqs.( 4) and (11), it is easily shown that "es) = " s2 No(zie 0 ) 2

dT
The corresponding expression for an anion is dT From these equations it follows that two additional parameters are required to estimate the entropy of solvation of a salt, namely, the temperature derivatives of the polarization parameters for the cation and anion.10, 2 8 From the expression for Gdd, one obtains the following equation for the corresponding entropy: Sdd =Gdi[E_2 des / (--1 L 2.7 8} dX, ].2±cA The derivative dX/dT may be found from the Wertheim equation from which one obtains dX [X(1+X)1 Assuming that the fraction 0 is independent of temperature, the resulting expression for the entropy of solvation is Ss = Sid + 0 Sdd (16)   This model is now examined with respect to data for monoatomic monovalent ion solvation in water. x

Results and Discussion
In order to apply the above model to experimental data one must choose radii for the ions and an extrathermodynamic assumption for extracting single ion solvation energies from experimental values for the salts, namely, the alkali metal halides.Two sets of ionic radii give good fits between experiment and theoryI 0 , namely, the values given by Pauling 34 and those extracted from neutron and X-ray diffraction experiments. 35The Pauling values are used here simply because they are more familiar.The extrathermodynamic assumptions used to separate experimental quantities into those for cations and anions were examined in detail for aqueous systems by Conway.36Values of the Gibbs energy and entropy of solvation for the alkali metal cations and halide anions estimated by Conway are summarized in Table 1 together with the Pauling radii for these ions.
In our previous analysis of the Gibbs solvation energy 7T ,1 0 , the contribution of dipole-dipole interactions was ignored ( 0 = 0 in eq. ( 10)).Analysis of the data for the alkali metal halides showed that the best values of 8c and 5A (eqs.( 4) and ( 6)) assuming Pauling radii for the ions were 82.4 and 18.0 pm, respectively. 10The quality of the fit between theory and experiment is really excellent, the standard deviation for the calculated values being 1.5 kW mol-1 , that is, significantly less than 1 percent.This result is illustrated in Fig.I in terms of a plot of -Gs against l/(ri+Sj).It is apparent that the values of Gs fit very well on a straight line in the region where data exist but that the slope of the best straight line througl these data is slightly higher (75.2 Id nm mol-1 ) than that predicted by theory (69.45 Id nm mol-1).As a result, an ion with infinite radius is predicted to have a Gibbs solvation energy of 35.7 Id mo[-' when this simple model is applied.
This can be attributed to the failure to account for dipole-dipole interactions which constitute a positive contribution to Gs.
The same data were re-examined on the basis of eq. ( 10) using the expression for Gad given by eq. ( 9).The fit was carried out so that the resulting estimates of Gid and Gad would go to zero for an ion of infinite radius.The resulting best values of k and SA were much smaller than those obtained by the simpler analysis, and are recorded in Table 2.The corresponding estimates of Gid are plotted against 1I(ri+Ai) in A successfal model for ionic solvation must also be able to estimate correctly the entropy of solvation.In order to make this calculation one must have estimates of the temperature derivatives of the parameters Sc, SA and A.. The latter was calculated using eq.( 15) which gives a value of -2.48 x 10-8 K-1 for dX/dT on the basis of the dielectric properties of water.Assuming rs, the radius of a water molecule, is equal to 137 pm the corresponding value of dS/dT is 0.048 pm K-1 .
Values of 8c and 8 A were determined from the single ion estimates of Ss given by Conway. 13In the case of cations on the basis of eqs.( 12) and ( 16), one may write

ES s2
+ Ss( 9Sc(C+) being the solvation entropy of the cation.Similar equations can be written for anions using eqs.( 13) and (16).A plot of Y against X for both ions is shown in Fig. 3.These data were fit to a straight line passing through zero (one parameter fit) as shown, the resulting slope which equals both d8s/dT and d8A/dT being 0.032 pm K-1.It is important to note that the present analysis suggests that these temperature coefficients are equal in contrast to our previous conclusion based on an analysis which ignored dipole-dipole interactions, The estimated error in the slope is not large being about 4%.It should also be noted that the magnitude of d~c/dT and d8AdT is about 65 percent of that found on the basis of the Wertheim equation (eq.( 15)).
Using the above coefficients values of Sid were estimated for the alkali metal and halide ions and are plotted in Fig. 4 together with estimates of Sdd and experimental values of Ss against the experimental Gibbs solvation energy Gs.It is clear that the entropy of solvation is approximately a quadratic function of the Gibbs energy as suggested by the MSA model.The ion-dipole contribution to the solvation entropy is clearly the larger varying from -161 J K-1 mol 1 -for Li+ to -55 J K-'mol- 1 for Cs+; in the case of the anions, this contribution is -145 J K-1mol-1 for Fand -65 J K-1 mol-1 for E. The positive dipole-dipole contribution is generally less than ten percent of that from ion-dipole interactions.The estimate of Ss agrees with the experimental value quite well except for the heavier halide ions.Thus, in the case of Iion, the estimate of Ss is higher in magnitude by 20 kJ mol-1 .This may be a result of some oversimplification used in the present analysis, especially the assumption that d&c/dT and djA/dT are equal.Nevertheless, the agreement between theory and experiment is really quite good and demonstrates that the MSA provides a simple and powerful model for ion solvation in water.
The present analysis may easily be extended to the other polar solvents considered in our previous work. 7. 28It is obvious that the values of X for these solvents that would be obtained using the present analysis would be quite different from those reported earlier. 7. 2 8However, preliminary calculations indicate that they follow the same trend.Thus, the correlation observed between 4c and the Gutmann donor number, DN 37 for the solvent, and XA and the Dimroth-Reichardt acidity parameter ET 38 should be maintained.This provides an important connection between the MSA X and these empirical solvent parameters which are often used by chenusts in assessing solvent properties. 39,40 In conclusion, the present study has shown that ,he description of the variation in Gibbs energy and entropy with ion size is significantly improved when dipole-dipole mteractdons for the solvent molecules are taken into consideration.This is especially important when considenng the entropy of solvation which is very poorly estimated by other models.The simplicity of the MSA makes it an attractive model for general use by physical chemists.Furthermore, the polanzation parameter has practical significance as a measure of the solvents' ability to act as a Lewis acid or a Lewis base.In future papers, we will expand the descrption applied here for water to other polar solvents.shows the least squares fit to these data, and the broken line the value of the iondipole contribution to the Gibbs energy according to eqs. ( 4) and (6).The numbers refer to individial ions listed in Table 1.17)) against the function X (eq.( 18)) for the alkali metal cations (e) and halide anions (*).The solid line was determined by a one parameter least squares fit going through the origin.The numbers refer to individual ions listed in Table 1.

4 nr
. ,,,(_ f f Of THIS PAGf or :.'.I. .unclassified INTRODUCTION Ionic solvation has been the subject of current interest for at least a century, I Modern experimental techniques 2 and the availability of powerful computers 3 have produced real progress in the understanding of this complex problem.

Figure 2 .
As expected these estimates are larger in magnitude than the experimental values of Gs also plotted on the same graph.Both plots are linear it, l/(ri+6i) and have zero intercepts within the experimental standard deviation.Estimates of OGad for the same nine ions are also plotted in this figure.For the cations, this quantity varies from a high of 41.2 Id mol-I for Li+ to a low of 22.7 Id mol-[ for Cs+, and is approximately eight percent of the magnitude of Gid.In the case of the anions it represents a slightly higher fraction of the total Gibbs energy and varies from a high of 42.7 kU mol[ for F to a low of 30.8 kd mol-[ for I-ion.The values of OGdd are also approximately linear in 1/(ri+S) with an intercept corresponding to zero for an infinitely large ion.The value of 8 used in estimating Gdd is that estimated on the basis of the Wertheim equation, namely, 51.6 pm.W1en the contributions, Gid and OGdd are added one obtains estimates of Gs which agree with those obtained from experimental data to within a few kJ mol-1 .Values of the parameters used in our calculations including the fraction 0 are summarized in Table Figure 1.Plot of the single ion Gibbs energy of solvation for the alkali metal cations (0) and halide anions (* ) against the reciprocal of the ion radius, rl, plus the MSA parameter &.The value of 5 for cations was 82.4 pm and for anions, 18.0 pm.The solid line

Figure 2 .
Figure 2. Plot of the single ion Gibbs energy of solvation, Gs, the ion-dipole contribution, Gi, and the dipole-dipole contribution, Gdd, against the reciprocal of the ion radius ri plus the MSA parameter 8 for the alkali metal cations (o, o) and halide anions (*, 0).The value of 5 for cations was 69.2 pm, and for anions, 1.1 pm.The solid lines show the least squares fit to each set of data.The numbers refer to individual ions listed inTable 1.

Figure 4 .
Figure 4. Plot of the single ion entropy of solvation, Ss, the ion-dipole contribution, Sid, and the dipole-dipole contribution, Sdd, against the Gibbs energy of solvation Gs for the alkali metal cations (o, o) and halide anions (#, 0).The curves are fitted to data for Sid and Sdd.The numbers refer to individual ions listed in Table 1.

Table 1 .
Pauling radii and standard thermodynamic parameters of solvation of mnoatornic

Table 2 .
MSA Parameters for the Alkali Metal and Halide Ions in Water at 25 °C