- Email: [email protected]
The role of dipole—dipole interactions in the solvation of monoatomic monovalent ions in water on the basis of the mean spherical approximation
253
.I. Electroanal. Chem., 355 (1993) 253-263 Elsevier Sequoia S.A., Lausanne
JEC 02527
The role of dipole-dipole interactions in the solvation of monoatomic monovalent ions in water on the basis of the mean spherical approximation * W. Ronald Fawcett Department of Chembtry, University of California, Davis, CA 95616 (USA)
Lesser Blurn Department of Physics, P.O. Box 23343, University of Puerto Rico, Rio Piedras, PR 00931 (USA) (Received
12 October
1992; accepted
20 October
1992)
Abstract
The role of dipole-dipole interactions in determining the Gibbs the alkali metal halides in water is considered within the context of When these interactions are included, one obtains a much improved the thermodynamic functions on ionic size over a wide range of this present results for solvation data in water and other polar solvents
energy and entropy of solvation of the mean spherical approximation. description of the dependence of parameter. The significance of the is discussed.
INTRODUCTION
We have recently [l-3] analyzed thermodynamic data for the solvation of monoatomic monovalent ions in water and a number of other polar solvents on the basis of the mean spherical approximation (MSA) [4-61. The attractive feature of the MSA is that it provides simple analytical results which can easily be used by those analyzing experimental data. An important departure of our work from that carried out earlier is that the polarization parameter A which characterizes ion-solvent interactions is regarded as an adjustable parameter whose value is estimated from the experimental Gibbs solvation energy. This leads to the conclusion that A depends not only on the nature of the solvent but also on the sign of
l Dedicated to Professor Brian E. Conway his contribution to electrochemistry.
0022-0728/93/$06.00
0 1993 - Elsevier
on the occasion
Sequoia
of his 65th birthday
S.A. All rights reserved
and in recognition
of
254
the charge on the ion. Although this result is perhaps not surprising, it emphasizes that a model which considers only point charge-point dipole interactions cannot describe experimental results adequately. In other words, the statistical mechanical description of an electrolyte solution must include consideration of higher order moments, and the polarizability of the solvent. These features of the system are incorporated in a ‘sticky’ parameter [7] which reflects the fact that anions and cations bind differently to the solvent. Anions disrupt water structure very little, for example, whereas cations which are attracted to the electronegative oxygen atom cause significant local disruption of the solvent. It can be shown [7] that the effect of this sticky interaction is to change the value of the parameter A so that instead of a single value of A, which is derived for a mixture of ions in a solvent of hard dipoles, there are different values for each ionic species, Ai. All deviations from the MSA will be lumped into this effective polarization parameter. The present model of the electrolyte solution consists of a mixture of charged hard spheres of radius ri (the ions) and hard spheres with a permanent point dipole of radius rs (the solvent). These molecules can also have a sticky dipole [7] interaction which may bind preferentially the anions or cations. Analysis of this system within the MSA leads to explicit expressions for the thermodynamic functions, in particular, the Gibbs energy of solvation. Our previous analysis [l-3] was based on the assumption that the dipole-dipole contribution could be neglected, the Gibbs energy of solvation being equated to the MSA estimate of the ion-dipole contribution. This assumption definitely affects the value of the polarization parameter estimated from experimental data, but confirmation that it is a reasonable approach was obtained when it was noted [2] that it leads to a reasonable separation of the components for cations and anions with experimental data for the alkali metal halides. Nevertheless, a better description of the system is obtained when dipole-dipole interactions are considered. In the present manuscript, the model used previously is extended to include consideration of dipole-dipole interactions. The Gibbs energy and entropy of solvation of the alkali metal and halide ions in water are reconsidered using the MSA model and the results are compared with those obtained earlier. THE MODEL
In the MSA for a system of spherical hard ions and dipoles of different sizes, the following expression is derived [S-12] for the ion-dipole contribution to the Gibbs energy of solvation in the limit of infinite dilution: G,=
-
NO(zieO>” 87r6,ri
(1)
Here, zi is the valence of the ion, e, the fundamental charge, E, the static dielectric constant of the solvent, l0 the permittivity of free space, ri the radius of
255
the ion, and N, the Avogadro constant. The dimensionless parameter depends on the nature of the solvent and the ion, is given by ti
=
rs/(
Ari)
ti, which
(2)
where rs is the radius of the solvent molecule represented as a sphere, and A the MSA polarization parameter. In earlier work 16-91, A was calculated from the dielectric constant of the pure solvent E, using the Wertheim relationship [13]: h2( A + 1)4 = 165
(3)
Although this value of the polarization parameter may be appropriate for estimating the dipole-dipole interaction energy, it gives poor estimates of the Gibbs solvation energy when this is assumed to equal G, [l-3]. Thus, values of A appropriate for cations and for anions were determined separately on the basis of experimental data for the alkali metal and halide ions, respectively. Thus, for a monovalent monoatomic cation, G, is given by
NotzieO)2 Gid(C+)
=
-
8TE 0
l i l-c
I
1
(4)
(ri+&)
where
(5)
4 = r,/A. A, being the corresponding lent monoatomic anions G,(A-)
=
_
cationic polarization
N”;z[o” 0
parameter.
Similarly, for monova-
(6)
where 6, = r,/A,
(7)
and A, is the polarization parameter for the anion. In our simple theory, 6, and 8, are clearly adjustable parameters. The difference between them is due to the fact that real solvent molecules are non-spherical, and that there are specific chemical bonds between the ions and the solvent which are different for the cations and the anions. The difference could be interpreted as resulting from an attractive or repulsive sticky interaction, and/or a different distance of closest approach, or both. Thus, &
=
rs+ ‘si
i=C,A
___
A
Or, in other words, hi = -&A
s
i=C,A s1
256
The second contribution to the Gibbs solution energy comes from repulsive dipole-dipole interactions [lO,ll] and is given by the equation G
= dd
(Zie0)*(E,-l)*
4+5i
64~~o~,ri( 1-t 5i)*
(10)
E + *(* ’ 3, S
2(A+l)*
Here, the parameter A is assumed to be that given by the Wertheim equation, that is, the value appropriate for dipole-dipole interactions [lO,ll]. This expression may be simplified considerably when one considers the range of values typical for A. For water whose dielectric constant is 78.3 at 25°C A is equal to 2.65. The corresponding value of the ratio A(A + 3)/2(A + l)* which appears in the denominator of the term in square brackets in eqn. (10) is 0.56. This is negligible with respect to the value of E, and may be neglected. The ratio appearing in the numerator of this term, namely (3A + 2)/(A + l), is equal to 2.7. Thus, the expression for Gdd may be rewritten as
NO(‘ieO)* Gdd
=
(11)
647re,,ri
The simplified expression has the advantage that A does not appear on its own but only in the ratio ti. This fact is important in applying the expression to experimental data. Finally, on the basis of previous work [ll] only a fraction 8 of the dipole-dipole term is used in estimating the Gibbs energy of solvation so that the final expression for G, becomes G, = G, + f3Gdd
(12)
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 1 8Te
Sid(C+)
=
No(Zieo)* I 0
-cf
1
1 de, dT
(13) (ri+ac)
-
(ri+~c)*
where dS,/dT is the temperature derivative of 6, which is obtained from the temperature derivative of A,. Comparing eqns. (4) and (131, it is easily shown that Gid(C+)
l1
The corresponding
-d’s
dT
87re,G;( C+) dS, N,( zieo)*
expression for an anion is
dT
(14)
2.57
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. From the expression for G,,, one obtains the following equation for the corresponding entropy:
The derivative dh/dT obtains dh -= dT
[
A(1 +A) 3A + 1
may be found from the Wertheim equation from which one
1
1 de, -2~, dT
Assuming that the fraction 8 is independent expression for the entropy of solvation is S, = Sip + BS,,
(17) of temperature,
the resulting
(18)
This model is now examined with respect to data for monoatomic monovalent ion solvation in water. 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 theory [2], namely, the values given by Pauling 1141 and those extracted from neutron and X-ray diffraction experiments [El. The 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 [16]. Values 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 their Pauling radii. In our previous analysis of the Gibbs solvation energy [1,2] the contribution of dipole-dipole interactions was ignored (0 = 0 in eqn. (12)). Analysis of the data for the alkali metal halides showed that the best values of 6, and 6, (eqns. (4) and (611, assuming Pauling radii for the ions, were 82.4 and 18.0 pm, respectively [2]. The quality of the fit between theory and experiment is really excellent, the standard deviation for the calculated values being 1.5 kJ mol-‘, that is, significantly less than 1%. This result is illustrated in Fig. 1 in terms of a plot of -G, against l/(ri + Si). It is apparent that the values of G, fit very well on a straight line in the region where data exist but that the slope of the best straight line through these data is slightly higher (75.2 kJ nm mol-‘1 than that predicted by
258 TABLE 1 Pauling radii and standard thermodynamic parameters of solvation of monoatomic monovalent ions in water at 298 K No.
Ion
Pauling radius ri /rim
Gibbs solvation energy G, /kJ mol-’
Solvation entropy S, /J K-’ mol-’
Li+ Naf K+ Rb+ cs+ FClBrI-
0.060 0.095 0.133 0.148 0.169 0.136 0.181 0.195 0.216
-
- 141.0 - 109.7 - 74.2 -62.1 - 59.1 - 136.8 - 79.9 - 64.4 -41.4
487.5 387.6 314.0 292.7 260.5 456.8 339.7 325.9 279.5
theory (69.45 kJ nm mol-‘). As a result, an ion with infinite radius is predicted to have a Gibbs solvation energy of 35.7 kJ mol-’ 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 G,. The same data were re-examined on the basis of eqn. (12) using the expression for Gdd given by eqn. (11). The fit was carried out so that the resulting estimates of
450 -
-i
350 -
z
E
3
250 -
\ 150 -
0
2
4
( ri + 6 )-‘/
6
8
nm-’
Fig. 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, r,, plus the MSA parameter 6. The value of 6 for cations was 82.4 pm and for anions, 18.0 pm. The solid line shows the least-squares fit to these data, and the broken line the value of the ion-dipole contribution to the Gibbs energy according to eqns. (4) and (6). The numbers refer to individual ions listed in Table 1.
259
TABLE
2
Mean spherical
approximation
parameters
for the alkali metal and halide
ions in water
Parameter
Value
Radius correction for cations, 8, /nm Radius correction for anions, 6, /nm Dipole-dipole fraction for cations, Bc Dipole-dipole fraction for anions, tIa Temperature derivative of radius correction for cations and anions, (da /dT)/pm K-
0.0692 0.0011 0.31 0.26
’
at 25°C
0.032
G, and Gdd would go to zero for an ion of infinite radius. The resulting best values of S, and 6, were much smaller than those obtained by the simpler analysis, and are recorded in Table 2. The corresponding estimates of G, are plotted against l/(ri + Si) in Fig. 2. As expected these estimates are larger in magnitude than the experimental values of G, also plotted on the same graph. Both plots are linear in l/(ri + SJ and have zero intercepts within the experimental standard deviation. Estimates of BG,, for the same nine ions are also plotted in this figure. For the cations, this quantity varies from a high of 41.2 kJ mol- ’ for Li+ to a low of 22.7 kJ mol-’ for Cs+, and is approximately 8% of the magnitude of G,. In the case of the anions it represents a slightly higher fraction of the total
500
300
100
-100
1 0
2
6
4
( ri + 6
I 6
>-‘/ ml-’
Fig. 2. Plot of the single ion Gibbs energy of solvation G,, dipole-dipole contribution Gdd, against the reciprocal of the for the alkali metal cations (0, 0) and halide anions (+, 0). and for anions, 1.1 pm. The solid lines show the least-squares to individual ions listed in Table 1.
the ion-dipole contribution Gi,, and the ion radius ri plus the MSA parameter 6 The value of 6 for cations was 69.2 pm, fit to each set of data. The numbers refer
260
Gibbs energy and varies from a high of 42.7 kJ mol-’ for F- to a low of 30.8 kJ mol - ’ for the I - ion. The values of 0G,, are also approximately linear in l/(ri + Si) with an intercept corresponding to zero for an infinitely large ion. The value of 6 used in estimating Gdd is that estimated on the basis of the Wertheim equation, namely, 51.6 pm. When the contributions G, and 0G,, are added, one obtains estimates of G, which agree with those obtained from experimental data to within a few kJ mol-‘. Values of the parameters used in our calculations including the fraction 8 are summarized in Table 2. A successful 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 a,, 6, and A. The latter was calculated using eqn. (1.5) which gives a value of - 2.48 X 10e8 K-’ for dh/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 da/dT is 0.048 pm K-l. Values of S, and 6, were determined from the single ion estimates of S, given by Conway [15]. In the case of cations on the basis of eqns. (14) and (18), one may write
(19) where X= -
87re,G;(C+) NO( ‘ielI>’
and l-’
Y=s,(c+)
(
+ s
1
Gid(C+)
dEs dT
+ %s
(21)
4
S&C+) being the solvation entropy of the cation. Similar equations can be written for anions using eqns. (15) and (18). A plot of Y against X for both ions is shown in Fig. 3. These data were fitted to a straight line passing through zero (oneparameter fit) as shown; the resulting slope, which equals both da,/dT and da,/dT, is 0.032 pm K-‘. 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&/dT and da,/dT is about 65% of that found on the basis of the Wertheim equation (eqn. (17)). Using the above coefficients, values of S, were estimated for the alkali metal and halide ions and are plotted in Fig. 4 together with estimates of S,, and experimental values of S, against the experimental Gibbs solvation energy G,. 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
261
0
1 lOO--
50--
01 0
1
2
X /
4
3
PN mol-’
Fig. 3. Plot of the function Y (eqn. (21)) against the function X (eqn. (20)) for the alkali metal cations (0) and halide anions (a). 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.
180 ,
I
140
2
0 5
60
5 -_ 0
100
200
G, /
300
400
500
kJ mole’
Fig. 4. Plot of the single ion entropy of solvation S,, the ion-dipole contribution Si,, and the dipole-dipole contribution S,,, against the Gibbs energy of solvation G, for the alkali metal cations (0, 0) and halide anions (+, 0). The curves are fitted to data for Si, and S,,. The numbers refer to individual ions listed in Table 1.
262
solvation entropy is clearly the larger, varying from - 161 J K-i mol-’ for Li+ to - 55 J K-’ mall ’ for Cs+; in the case of the anions, this contribution is - 145 J K-’ mol-’ for F- and - 65 J K-’ mol-’ for II. The positive dipole-dipole contribution is generally less than 10% of that from ion-dipole interactions. The estimate of S, agrees with the experimental value quite well except for the heavier halide ions. Thus, in the case of I- ion, the estimate of S, is higher in magnitude by 20 kJ mol- ‘. This may be a result of some oversimplification used in the present analysis, especially the assumption that dS,/dT and da,/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 [1,3]. It is obvious that the values of A for these solvents that would be obtained using the present analysis would be quite different from those reported earlier [1,3]. However, preliminary calculations indicate that they follow the same trend. Thus, the correlation observed between A, and the Gutmann donor number (DN) [17] for the solvent, and A, and the Dimroth-Reichardt acidity parameter E, [18] should be maintained. This provides an important connection between the MSA A and these empirical solvent parameters which are often used by chemists in assessing solvent properties [19,20]. CONCLUSIONS
The present study has shown that the description of the variation in Gibbs energy and entropy with ion size is significantly improved when dipole-dipole interactions for the solvent molecules are taken into consideration. This is especially important when considering 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 polarization 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 description applied here for water to other polar solvents. ACKNOWLEDGEMENT
The support of the Office of Naval Research is gratefully acknowledged. REFERENCES 1 2 3 4
L. Blum and W.R. Fawcett, J. Phys. Chem., 96 (1992) 408. W.R. Fawcett and L. Blum, J. Electroanal. Chem., 328 (1992) 333. W.R. Fawcett and L. Blum, J. Chem. Sot. Faraday Trans., 88 (1992) 3339. H.L. Friedman and W.D.T. Dale, in B.J. Berne, (Ed.), Modern Theoretical Chemistry: Mechanics, Vol. 5, Plenum, New York, 1976. 5 L. Blum, Chem. Phys. Letters, 26 (1974) 200; J. Chem. Phys., 61 (1974) 2129.
Statistical
263
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
S.A. Adelman and J.M. Deutch, J. Chem. Phys., 60 (1974) 3935. D.Q. Wei and L. Blum, J. Chem. Phys., 89 (1988) 1091. L. Blum, J. Stat. Phys., 18 (1978) 451. D.Y.C. Chan, D.J. Mitchell, and B.W. Ninham, J. Chem. Phys., 70 (1979) 2946. P. Garisto, P. Kusalik, and G.N. Patey, J. Chem. Phys., 79 (1983) 6294. D. Wei and L. Blum, J. Chem. Phys., 57 (1987) 2999. L. Blum, F. Vericat, and W.R. Fawcett, J. Chem. Phys., 96 (1992) 3039. MS. Wertheim, J. Chem. Phys., 55 (1971) 4291. L. Pauling, J. Am. Chem. Sot., 49 (1927) 765. Y. Marcus, Chem. Rev., 88 (1988) 1475. B.E. Conway, J. Solution Chem., 7 (1978) 721. V. Gutmann and R. Schmid, Coord. Chem. Rev., 12 (1974) 263. C. Reichardt, Solvent Effects in Organic Chemistry, Verlag Chemie, Weinheim, Y. Marcus, Ionic Solvation, Wiley-Interscience, New York, 1985. R.G. Makitra, Ya.N. Pirig, and R.B. Kiveliuk, private communication.
1979.
.I. Electroanal. Chem., 355 (1993) 253-263 Elsevier Sequoia S.A., Lausanne
JEC 02527
The role of dipole-dipole interactions in the solvation of monoatomic monovalent ions in water on the basis of the mean spherical approximation * W. Ronald Fawcett Department of Chembtry, University of California, Davis, CA 95616 (USA)
Lesser Blurn Department of Physics, P.O. Box 23343, University of Puerto Rico, Rio Piedras, PR 00931 (USA) (Received
12 October
1992; accepted
20 October
1992)
Abstract
The role of dipole-dipole interactions in determining the Gibbs the alkali metal halides in water is considered within the context of When these interactions are included, one obtains a much improved the thermodynamic functions on ionic size over a wide range of this present results for solvation data in water and other polar solvents
energy and entropy of solvation of the mean spherical approximation. description of the dependence of parameter. The significance of the is discussed.
INTRODUCTION
We have recently [l-3] analyzed thermodynamic data for the solvation of monoatomic monovalent ions in water and a number of other polar solvents on the basis of the mean spherical approximation (MSA) [4-61. The attractive feature of the MSA is that it provides simple analytical results which can easily be used by those analyzing experimental data. An important departure of our work from that carried out earlier is that the polarization parameter A which characterizes ion-solvent interactions is regarded as an adjustable parameter whose value is estimated from the experimental Gibbs solvation energy. This leads to the conclusion that A depends not only on the nature of the solvent but also on the sign of
l Dedicated to Professor Brian E. Conway his contribution to electrochemistry.
0022-0728/93/$06.00
0 1993 - Elsevier
on the occasion
Sequoia
of his 65th birthday
S.A. All rights reserved
and in recognition
of
254
the charge on the ion. Although this result is perhaps not surprising, it emphasizes that a model which considers only point charge-point dipole interactions cannot describe experimental results adequately. In other words, the statistical mechanical description of an electrolyte solution must include consideration of higher order moments, and the polarizability of the solvent. These features of the system are incorporated in a ‘sticky’ parameter [7] which reflects the fact that anions and cations bind differently to the solvent. Anions disrupt water structure very little, for example, whereas cations which are attracted to the electronegative oxygen atom cause significant local disruption of the solvent. It can be shown [7] that the effect of this sticky interaction is to change the value of the parameter A so that instead of a single value of A, which is derived for a mixture of ions in a solvent of hard dipoles, there are different values for each ionic species, Ai. All deviations from the MSA will be lumped into this effective polarization parameter. The present model of the electrolyte solution consists of a mixture of charged hard spheres of radius ri (the ions) and hard spheres with a permanent point dipole of radius rs (the solvent). These molecules can also have a sticky dipole [7] interaction which may bind preferentially the anions or cations. Analysis of this system within the MSA leads to explicit expressions for the thermodynamic functions, in particular, the Gibbs energy of solvation. Our previous analysis [l-3] was based on the assumption that the dipole-dipole contribution could be neglected, the Gibbs energy of solvation being equated to the MSA estimate of the ion-dipole contribution. This assumption definitely affects the value of the polarization parameter estimated from experimental data, but confirmation that it is a reasonable approach was obtained when it was noted [2] that it leads to a reasonable separation of the components for cations and anions with experimental data for the alkali metal halides. Nevertheless, a better description of the system is obtained when dipole-dipole interactions are considered. In the present manuscript, the model used previously is extended to include consideration of dipole-dipole interactions. The Gibbs energy and entropy of solvation of the alkali metal and halide ions in water are reconsidered using the MSA model and the results are compared with those obtained earlier. THE MODEL
In the MSA for a system of spherical hard ions and dipoles of different sizes, the following expression is derived [S-12] for the ion-dipole contribution to the Gibbs energy of solvation in the limit of infinite dilution: G,=
-
NO(zieO>” 87r6,ri
(1)
Here, zi is the valence of the ion, e, the fundamental charge, E, the static dielectric constant of the solvent, l0 the permittivity of free space, ri the radius of
255
the ion, and N, the Avogadro constant. The dimensionless parameter depends on the nature of the solvent and the ion, is given by ti
=
rs/(
Ari)
ti, which
(2)
where rs is the radius of the solvent molecule represented as a sphere, and A the MSA polarization parameter. In earlier work 16-91, A was calculated from the dielectric constant of the pure solvent E, using the Wertheim relationship [13]: h2( A + 1)4 = 165
(3)
Although this value of the polarization parameter may be appropriate for estimating the dipole-dipole interaction energy, it gives poor estimates of the Gibbs solvation energy when this is assumed to equal G, [l-3]. Thus, values of A appropriate for cations and for anions were determined separately on the basis of experimental data for the alkali metal and halide ions, respectively. Thus, for a monovalent monoatomic cation, G, is given by
NotzieO)2 Gid(C+)
=
-
8TE 0
l i l-c
I
1
(4)
(ri+&)
where
(5)
4 = r,/A. A, being the corresponding lent monoatomic anions G,(A-)
=
_
cationic polarization
N”;z[o” 0
parameter.
Similarly, for monova-
(6)
where 6, = r,/A,
(7)
and A, is the polarization parameter for the anion. In our simple theory, 6, and 8, are clearly adjustable parameters. The difference between them is due to the fact that real solvent molecules are non-spherical, and that there are specific chemical bonds between the ions and the solvent which are different for the cations and the anions. The difference could be interpreted as resulting from an attractive or repulsive sticky interaction, and/or a different distance of closest approach, or both. Thus, &
=
rs+ ‘si
i=C,A
___
A
Or, in other words, hi = -&A
s
i=C,A s1
256
The second contribution to the Gibbs solution energy comes from repulsive dipole-dipole interactions [lO,ll] and is given by the equation G
= dd
(Zie0)*(E,-l)*
4+5i
64~~o~,ri( 1-t 5i)*
(10)
E + *(* ’ 3, S
2(A+l)*
Here, the parameter A is assumed to be that given by the Wertheim equation, that is, the value appropriate for dipole-dipole interactions [lO,ll]. This expression may be simplified considerably when one considers the range of values typical for A. For water whose dielectric constant is 78.3 at 25°C A is equal to 2.65. The corresponding value of the ratio A(A + 3)/2(A + l)* which appears in the denominator of the term in square brackets in eqn. (10) is 0.56. This is negligible with respect to the value of E, and may be neglected. The ratio appearing in the numerator of this term, namely (3A + 2)/(A + l), is equal to 2.7. Thus, the expression for Gdd may be rewritten as
NO(‘ieO)* Gdd
=
(11)
647re,,ri
The simplified expression has the advantage that A does not appear on its own but only in the ratio ti. This fact is important in applying the expression to experimental data. Finally, on the basis of previous work [ll] only a fraction 8 of the dipole-dipole term is used in estimating the Gibbs energy of solvation so that the final expression for G, becomes G, = G, + f3Gdd
(12)
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 1 8Te
Sid(C+)
=
No(Zieo)* I 0
-cf
1
1 de, dT
(13) (ri+ac)
-
(ri+~c)*
where dS,/dT is the temperature derivative of 6, which is obtained from the temperature derivative of A,. Comparing eqns. (4) and (131, it is easily shown that Gid(C+)
l1
The corresponding
-d’s
dT
87re,G;( C+) dS, N,( zieo)*
expression for an anion is
dT
(14)
2.57
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. From the expression for G,,, one obtains the following equation for the corresponding entropy:
The derivative dh/dT obtains dh -= dT
[
A(1 +A) 3A + 1
may be found from the Wertheim equation from which one
1
1 de, -2~, dT
Assuming that the fraction 8 is independent expression for the entropy of solvation is S, = Sip + BS,,
(17) of temperature,
the resulting
(18)
This model is now examined with respect to data for monoatomic monovalent ion solvation in water. 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 theory [2], namely, the values given by Pauling 1141 and those extracted from neutron and X-ray diffraction experiments [El. The 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 [16]. Values 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 their Pauling radii. In our previous analysis of the Gibbs solvation energy [1,2] the contribution of dipole-dipole interactions was ignored (0 = 0 in eqn. (12)). Analysis of the data for the alkali metal halides showed that the best values of 6, and 6, (eqns. (4) and (611, assuming Pauling radii for the ions, were 82.4 and 18.0 pm, respectively [2]. The quality of the fit between theory and experiment is really excellent, the standard deviation for the calculated values being 1.5 kJ mol-‘, that is, significantly less than 1%. This result is illustrated in Fig. 1 in terms of a plot of -G, against l/(ri + Si). It is apparent that the values of G, fit very well on a straight line in the region where data exist but that the slope of the best straight line through these data is slightly higher (75.2 kJ nm mol-‘1 than that predicted by
258 TABLE 1 Pauling radii and standard thermodynamic parameters of solvation of monoatomic monovalent ions in water at 298 K No.
Ion
Pauling radius ri /rim
Gibbs solvation energy G, /kJ mol-’
Solvation entropy S, /J K-’ mol-’
Li+ Naf K+ Rb+ cs+ FClBrI-
0.060 0.095 0.133 0.148 0.169 0.136 0.181 0.195 0.216
-
- 141.0 - 109.7 - 74.2 -62.1 - 59.1 - 136.8 - 79.9 - 64.4 -41.4
487.5 387.6 314.0 292.7 260.5 456.8 339.7 325.9 279.5
theory (69.45 kJ nm mol-‘). As a result, an ion with infinite radius is predicted to have a Gibbs solvation energy of 35.7 kJ mol-’ 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 G,. The same data were re-examined on the basis of eqn. (12) using the expression for Gdd given by eqn. (11). The fit was carried out so that the resulting estimates of
450 -
-i
350 -
z
E
3
250 -
\ 150 -
0
2
4
( ri + 6 )-‘/
6
8
nm-’
Fig. 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, r,, plus the MSA parameter 6. The value of 6 for cations was 82.4 pm and for anions, 18.0 pm. The solid line shows the least-squares fit to these data, and the broken line the value of the ion-dipole contribution to the Gibbs energy according to eqns. (4) and (6). The numbers refer to individual ions listed in Table 1.
259
TABLE
2
Mean spherical
approximation
parameters
for the alkali metal and halide
ions in water
Parameter
Value
Radius correction for cations, 8, /nm Radius correction for anions, 6, /nm Dipole-dipole fraction for cations, Bc Dipole-dipole fraction for anions, tIa Temperature derivative of radius correction for cations and anions, (da /dT)/pm K-
0.0692 0.0011 0.31 0.26
’
at 25°C
0.032
G, and Gdd would go to zero for an ion of infinite radius. The resulting best values of S, and 6, were much smaller than those obtained by the simpler analysis, and are recorded in Table 2. The corresponding estimates of G, are plotted against l/(ri + Si) in Fig. 2. As expected these estimates are larger in magnitude than the experimental values of G, also plotted on the same graph. Both plots are linear in l/(ri + SJ and have zero intercepts within the experimental standard deviation. Estimates of BG,, for the same nine ions are also plotted in this figure. For the cations, this quantity varies from a high of 41.2 kJ mol- ’ for Li+ to a low of 22.7 kJ mol-’ for Cs+, and is approximately 8% of the magnitude of G,. In the case of the anions it represents a slightly higher fraction of the total
500
300
100
-100
1 0
2
6
4
( ri + 6
I 6
>-‘/ ml-’
Fig. 2. Plot of the single ion Gibbs energy of solvation G,, dipole-dipole contribution Gdd, against the reciprocal of the for the alkali metal cations (0, 0) and halide anions (+, 0). and for anions, 1.1 pm. The solid lines show the least-squares to individual ions listed in Table 1.
the ion-dipole contribution Gi,, and the ion radius ri plus the MSA parameter 6 The value of 6 for cations was 69.2 pm, fit to each set of data. The numbers refer
260
Gibbs energy and varies from a high of 42.7 kJ mol-’ for F- to a low of 30.8 kJ mol - ’ for the I - ion. The values of 0G,, are also approximately linear in l/(ri + Si) with an intercept corresponding to zero for an infinitely large ion. The value of 6 used in estimating Gdd is that estimated on the basis of the Wertheim equation, namely, 51.6 pm. When the contributions G, and 0G,, are added, one obtains estimates of G, which agree with those obtained from experimental data to within a few kJ mol-‘. Values of the parameters used in our calculations including the fraction 8 are summarized in Table 2. A successful 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 a,, 6, and A. The latter was calculated using eqn. (1.5) which gives a value of - 2.48 X 10e8 K-’ for dh/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 da/dT is 0.048 pm K-l. Values of S, and 6, were determined from the single ion estimates of S, given by Conway [15]. In the case of cations on the basis of eqns. (14) and (18), one may write
(19) where X= -
87re,G;(C+) NO( ‘ielI>’
and l-’
Y=s,(c+)
(
+ s
1
Gid(C+)
dEs dT
+ %s
(21)
4
S&C+) being the solvation entropy of the cation. Similar equations can be written for anions using eqns. (15) and (18). A plot of Y against X for both ions is shown in Fig. 3. These data were fitted to a straight line passing through zero (oneparameter fit) as shown; the resulting slope, which equals both da,/dT and da,/dT, is 0.032 pm K-‘. 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&/dT and da,/dT is about 65% of that found on the basis of the Wertheim equation (eqn. (17)). Using the above coefficients, values of S, were estimated for the alkali metal and halide ions and are plotted in Fig. 4 together with estimates of S,, and experimental values of S, against the experimental Gibbs solvation energy G,. 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
261
0
1 lOO--
50--
01 0
1
2
X /
4
3
PN mol-’
Fig. 3. Plot of the function Y (eqn. (21)) against the function X (eqn. (20)) for the alkali metal cations (0) and halide anions (a). 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.
180 ,
I
140
2
0 5
60
5 -_ 0
100
200
G, /
300
400
500
kJ mole’
Fig. 4. Plot of the single ion entropy of solvation S,, the ion-dipole contribution Si,, and the dipole-dipole contribution S,,, against the Gibbs energy of solvation G, for the alkali metal cations (0, 0) and halide anions (+, 0). The curves are fitted to data for Si, and S,,. The numbers refer to individual ions listed in Table 1.
262
solvation entropy is clearly the larger, varying from - 161 J K-i mol-’ for Li+ to - 55 J K-’ mall ’ for Cs+; in the case of the anions, this contribution is - 145 J K-’ mol-’ for F- and - 65 J K-’ mol-’ for II. The positive dipole-dipole contribution is generally less than 10% of that from ion-dipole interactions. The estimate of S, agrees with the experimental value quite well except for the heavier halide ions. Thus, in the case of I- ion, the estimate of S, is higher in magnitude by 20 kJ mol- ‘. This may be a result of some oversimplification used in the present analysis, especially the assumption that dS,/dT and da,/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 [1,3]. It is obvious that the values of A for these solvents that would be obtained using the present analysis would be quite different from those reported earlier [1,3]. However, preliminary calculations indicate that they follow the same trend. Thus, the correlation observed between A, and the Gutmann donor number (DN) [17] for the solvent, and A, and the Dimroth-Reichardt acidity parameter E, [18] should be maintained. This provides an important connection between the MSA A and these empirical solvent parameters which are often used by chemists in assessing solvent properties [19,20]. CONCLUSIONS
The present study has shown that the description of the variation in Gibbs energy and entropy with ion size is significantly improved when dipole-dipole interactions for the solvent molecules are taken into consideration. This is especially important when considering 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 polarization 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 description applied here for water to other polar solvents. ACKNOWLEDGEMENT
The support of the Office of Naval Research is gratefully acknowledged. REFERENCES 1 2 3 4
L. Blum and W.R. Fawcett, J. Phys. Chem., 96 (1992) 408. W.R. Fawcett and L. Blum, J. Electroanal. Chem., 328 (1992) 333. W.R. Fawcett and L. Blum, J. Chem. Sot. Faraday Trans., 88 (1992) 3339. H.L. Friedman and W.D.T. Dale, in B.J. Berne, (Ed.), Modern Theoretical Chemistry: Mechanics, Vol. 5, Plenum, New York, 1976. 5 L. Blum, Chem. Phys. Letters, 26 (1974) 200; J. Chem. Phys., 61 (1974) 2129.
Statistical
263
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
S.A. Adelman and J.M. Deutch, J. Chem. Phys., 60 (1974) 3935. D.Q. Wei and L. Blum, J. Chem. Phys., 89 (1988) 1091. L. Blum, J. Stat. Phys., 18 (1978) 451. D.Y.C. Chan, D.J. Mitchell, and B.W. Ninham, J. Chem. Phys., 70 (1979) 2946. P. Garisto, P. Kusalik, and G.N. Patey, J. Chem. Phys., 79 (1983) 6294. D. Wei and L. Blum, J. Chem. Phys., 57 (1987) 2999. L. Blum, F. Vericat, and W.R. Fawcett, J. Chem. Phys., 96 (1992) 3039. MS. Wertheim, J. Chem. Phys., 55 (1971) 4291. L. Pauling, J. Am. Chem. Sot., 49 (1927) 765. Y. Marcus, Chem. Rev., 88 (1988) 1475. B.E. Conway, J. Solution Chem., 7 (1978) 721. V. Gutmann and R. Schmid, Coord. Chem. Rev., 12 (1974) 263. C. Reichardt, Solvent Effects in Organic Chemistry, Verlag Chemie, Weinheim, Y. Marcus, Ionic Solvation, Wiley-Interscience, New York, 1985. R.G. Makitra, Ya.N. Pirig, and R.B. Kiveliuk, private communication.
1979.