12954
J . Phys. Chem. 1993,97, 12954-12958
Partial Molar Volumes of Ions in Nonpolar Solvents Harold A. Schwarz Chemistry Department, Brookhaven National Laboratory, Upton, New York 1 1 973-5000 Received: August 2, 1993; In Final Form: September 30, 1993'
A method of including density effects in electrostatic calculations of ions in nonpolar solvents has been developed which is useful down to a radius of 2 A. It is shown on the basis of free energy calculations that the first layer of solvent molecules around a small ion is most likely frozen in a high-density glasslike state. With this effect included, good agreement (f30 cm3/mol) is found with recent experimental data on five solvents, in which electrostriction volumes for C02-range from -180 to -265 cm3/mol. The effect on ionic mobility is discussed.
TABLE I: Partial Molar Volumes of COz- in Nonpolar Solvents.
Introduction
The theory of ionic electrostriction of solvents has been developed mainly for polar liquids because almost all experimental measurements have been made in these solvents. Recently, data on partial molar volumes of ions have become available for nonpolar liquids. Nishikawa, Itoh, and Holroyd1V2have measured pressure effects on several equilibria between electrons and negative ions in hydrocarbons. These reactions were initiated by pulses of X-rays and followed by measuring the decrease in conductivity which results from reaction of the highly mobile electron with a solute. A typical reactions is e;
+ CO, + CO;
TMS 2-MB 2,2-DMB 3-MP 2,2,4-TMF
P,bar
AV(1)b
liq
g l + liq
34 70 44 90 63
475 325 175 200 250
-179 -255 -254 -264 -190
-96 -119 -106 -136 -98
-197 -223 -235 -253 -238
In cm3/mol. * References 1 and 2. e Average of four temperatures
and pressures.
electrostriction volume obtained from eq 3 is7 (1)
which has been studied in tetramethylsilane (TMS),' 2-methylbutane (isopentane or 2-MB),2 3-methylpentane (3-MP),' 2,2dimethylbutane (2,2-DMB),2and 2,2,4-trimethylpentane(2,2,4TMP).' Partial molar volume changes for these reactions were found from the relation3
A P = (dAGo/dP)T
caIc t, T,OC
(2)
where AGO = -RT In Kq. Some values of AP for this reaction are given in Table I, and they range from -1 79 to -264 cm3/mol, an order of magnitude larger than for any comparable reaction in water. APfor reaction 1 is
Ap(1) = p(C0;) - V(C0,) - P(e;) The electron has been extensively investigated in hydrocarbon solvents. Solvation is slight, and AP between the "quasi-free" state, eqp, and the solvated state is small and n e g a t i ~ e . The ~ mobility of eqf-is about lo5 times larger than that of heavy ions such as C02-. Hence Vof e,f- must be negligible since it would be impossible for the solvent to relax around such a fast-moving species. Thus, the large APof reaction 1 must be attributed to ~ ( C O Z --)~ ( C O Zwhich ), is essentially theelectrostrictionvolume of the solvent around the ion, P,. When nonlinear effects such as density variation are ignored, the free energy of solvation of an ion, AG8, is596 (3)
where to is the dielectric constant at low electric field strength, e is the charge on the ion, and ro is the radius of the ion. The Abstract published in Advance ACS Absrrucrs, December 1,1993.
0022-3654/93/2097- 12954$04.00/0
(4)
where do, XO, and €01 are the density, the compressibility,and the derivative of the dielectric constant with respect to density, all at zero field strength. The electrostriction volume arises from a radially-dependent increase in density which, to the same approximation as eqs 3 and 4, is given by8
where E is the electric field strength, e/er2. Ninomiya et al.2 have shown that if ro is taken as 2.3 A, as estimated from solid C02, eq 4 gives values of Pe which are less negative than the observed values by an average of 100 cm3/mol. If ro is taken as 2.0 A, the value that will be used here, then the deficiency is reduced to 50 cm3/mol, but thegain is at the expenseof unrealistic calculated densities of 1.6-1.8 g/cm3 at the surface of the ion. Thesevalues exceed the maximum density of hydrocarbonsbased on the van der Waals volume? which is 1.26 g/cm3. Equation 4 does not take into account the strong dependence of the compressibility on the density of the solvent. In fact, it is expected that a model which takes into account the density dependence would produce discrepancies even greater than 100 cm'/mol. There is no clear reason to suppose that such large discrepancies are due to a basic inadequacy of the electrostatic theory. It is the purpose of this paper to develop a more realistic electrostatic model which, in particular, incorporates the possibility of a phase change near the ion. Toevaluate the reasonableness of the model, it was necessary to incorporate density dependence into the free energy estimate. Other studies have considered this and other nonlinear effects for dipolar liquids,1° but the usual assumption, that the effects can be described by a single term in the square of the electric field strength, is not adequate for nonpolar liquids in which density changes are large. 0 1993 American Chemical Society
Volumes of Ions in Nonpolar Solvents
The Journal of Physical Chemistry, Vol. 97, No. 49, 1993 12955
Method Equation 3 is valid only if c is independent of the electric field strength, E. More generally, when E is not constant, the free energy of solvation is given by5
generate a series of equationswhosecoefficientscould beevaluated at zero E, but many derivatives would be necessary because the compressibilityvaries by about a factor of 10 with distance from the ion. An adequate approximation has been developed which uses eq 12 to bring in most of the information about (ad/8P)E:
Ad The dielectric displacement, D, is related to E by D = eE, and the integration limit, Dz,is e2/+. For the solutesbeing considered here, the dielectric constant is adequately expressed by the Clausius-Mossotti equation' e=
1
+2[P]d/M 1 - [P]d/M
(7)
do E2 -J 87r 0 c'dx[l + blAd + bz(Ad)2]dEZ
where e' and x are calculated from eqs 11 and 12 using the local density, d, which is do Ad. The coefficients bl and b2 are
+
and
where [PI is the molar polarizability, d is the density, and M is themolecular weight. For small density changes eqs 7 and 5 give
and so density effects contribute to the general phenomenon of dielectric saturation.8 For the hydrocarbons considered here the density contribution is 4-6 times larger than anisotropic polarizability contributions, as estimated from Kerr constants.'* This dominance is not found in aromatic solvents, in which the anisotropic polarizability termlZis several times larger than the density term. Only density effects will be considered here, and the treatment is not applicable to aromatics. The density of a liquid under pressure at zero E can be expressed by the Tait e q ~ a t i o n ' ~
dm (8) 1 - A In (1 P / B ) where A and Bare constants and dw is the density at zero E and pressure. For the liquid hydrocarbons considered here A lies between 0.08 and 0.1. The partial molar volume may be obtained from do =
The primed quantities are the correspondingderivativesof eo and xo with respect to density, obtained from eqs 7 and 8. These coefficients were determined by first expressing c as a power series in E2 to facilitate integration of eq 10, from which one gets
+
Ad = C2Ez+ C4@ C,E6
+
(9) where NA is Avogadro's number, Ad is the increase in density over do due to electrostriction, and the increase in mass around the ion is made up from bulk solvent at density do. Bottcher gives an equation for Ad8
(-)
A d = - do J fi ae dE2 8~ 0 a P E which can also be derived by applying = (dAG,/aP)TDto eq 6 , with some subsequent rearrangement. The first case to be considered is that in which the solvent does not form a glass; Le., the density is representedby the Tait equation all the way to the surface of the ion. The partial derivative in eq 10 can be replaced by c' (ad/aP)E. c' can be determined by differentiation of eq 7
ve
Xi'[
1 --x
x = xoox exp (Ax) where x = d / d w and xw is A/B, the compressibility at zero pressure and E. Equation 10can be solved numerically in principle by successivedifferentiations with respect to pressure. This would
x{
9-+-+37 xoco
xo
Ed" '0
1
Next, e', d , and x in eq 13 are expanded in Ad, eq 14 is used for Ad, and the new equation in EZis integrated. Comparison with eq 14 gives the above expressions for bl and bz. It may be seen from the outline of the derivation that eq 13 is only accurate to (Ad)z,and the question arises whether or not this formulation has any advantage over a simple expansion of (ae/aP)Eas eo'doxo[ 1 + BIAd + &(Ad)z]. The latter expansion is very ill-behaved. B1 is -3 times larger than bl, and 82 is -53 times larger than bz. The simple expansionis accurate only beyond 5
A.
Equation 13 was solved at each radius by RungeKutta numerical integration. The integration was terminated when E2 = ez/ez+ was reached, with t computed from eq 7 using the local density. The electrostriction volume was then determind by Simpson integration of eq 9. The accuracy of the approximation in eq 13 can be monitored by computing the ratio b2Ad/bl,which must be < 1 , the smaller the better. For typical parameters it was 0.15 at 2.5 A and reached 0.38 at r = 2 A, at which radius the factor 1 + blAd b2(Ad)2was 1.6. If the assumption is made that the error is less than the contribution of the last term, then the total error in from this approximation is less than 2 cm3/ mol. The choice of the ion radius ro deserves some consideration. It should be sufficiently small to excludeall compressible volume. Whenliquidsarecompressed,evento thedensities near thesurface of the ion, bond lengths and van der Waals radii are very little affected. It is the free volume which contracts, and so all volume outside of the van der Waals volume of the ion should be compressible. Consequently, ro was taken here as the gas-phase collision radius of CO2,14 2.0 A. Note that ro has no relation to ~ ( C O Z )which , is about 30 cm3/mol larger than the van der Waals volume. It is assumed that this excess volume compresses
+
ve
At low E, (ad/aP)Emust approach the product d x , where x is the compressibilityat zero E, ( l / d ) ( a d / a P ) . By differentiation of the Tait equation, x is
(14)
where
Schwarz
12956 The Journal of Physical Chemistry, Vol. 97,No. 49, 1993
TABLE Ik Parameters Used in Calculations
.-
---
TMS 2-MB 2,2-DMB 3-MP 2,2,4-TMP 0
.-
I
0.6314 0.5662 0.6275 0.5967 0.6557
0.0985 0.0943 0.0836 0.0885 0.0827
358 254 366 237 356
31.2 25.3 29.9 29.9 39.1
8 (7.05) 9 (7.10) 8 (7.07) 8 (7.14) 7 (7.32)
Reference 17. * Calculated from ref 15. From additivity rules, ref
11.
with the same compressibilityas the solvent. (It is, if anything, slightly more compressible.) The parameters required for the integration are given in Table 11, and the calculated electrostriction volumes are given in the column labeled “liq” of Table I. The calculated values are an average of 120 cm’/mol more positive than the observed values, worse than the simple treatment as was predicted earlier. The explanation of the lack of agreement may lie in the formation of a glasslike layer near the ion. This phase change would be accompanied by a substantial density increase, which on the molecular level can be attributed to loss of rotation. The description at the molecular level, however, involves many unknown parameters. Fortunately, a phenomenological description involving macroscopic parameters is possible. If pressure is applied to a liquid which contracts upon freezing, at a temperature somewhat above the melting point, the density will be adequately described by the Tait equation until a pressure is reached at which the volume decrease due to freezing, -AV,, results in sufficiently large work, -PAV,, to overcome the loss in free energy due to the freezing. At this point the sample freezes with a sudden density increase,and above this pressurethe density can again be adequately described by a Tait equation but with different parameters (smaller compressibility).ls An analogous effect will occur around an ion in a nonpolar solvent. Glass formation will cause the calculated electrostatic free energy (eq 6) to become more negative (increase the stabilization) because the density increase will cause an increase in the dielectric constant. If the magnitude of the stabilization is large enough to offset the free energy required to form the glass, then a glass will be formed. Since most of the electrostatic stabilization comes at small radii ( 0 2 in eq 6 varies with l/+) and since the energy required to form the glass increases with the volume (r3), then the overall free energy of the system should be minimized if there is glass formation within some small radius while the solvent remains a liquid outside the radius. Liquidglass transitions are second-order,so the solidification will occur over a range of radii, not at a single radius, The determining parameter, E2, varies inversely with and so this range should be small. It will be assumed here that glass formation occurs suddenly at a radius r, and that inside this radius the density can be computed from
+,
where subscripts 1 and g refer to liquid and glass, E, is the field strength at r,, and Ad, is the increase in density due to glass formation at radius r,. A form more amenable to calculation is that inside r,, Ad = Ad
+ do jgc’dxg[ 1 + 6,,Ad + 62,(Ad)z] dE2 87r 0
(1 5 )
where x, is calculated from eq 12 with A = A, and xm = A,/B,, bl, and b~ are evaluatedas above but for do, eo, and xm appropriate to the glass, and Adais the change in density expected for glass formation at the applied temperature and pressure, a parameter which is necessarily extrapolated from low-temperature data. The electrostriction volume is computed by numerically
integrating eq 9 from the upper limit down to r, from eq 13 and continuing the integration from rgto ro with Ad from eq 15. The dielectric constant is computed from the density at each radius, and the free energy of solvation is computed by numerical integration of eq 6 (by Simpson’s rule).
Results The problem with introducing a glass layer is that none of the four new parameters (Ad,, r,, AB,and B,) have been measured directly. Fortunately, enough is known about solids at Low temperatures to make reasonable extrapolations of densities and compressibilitiesto the temperatures used in the electrostriction measurements. All liquids considered here, except 3-MP, crystallize as well as form glasses. Densities, heat capacities, and compressibilitiesare nearly the same for the two solid phases, and to this extent data on the two phases will be used interchangeably. The radius r, depends on the free energy of glass formation, which is the most difficult information to obtain or estimate. For the present r, will be assumed to be one molecular layer, and this assumption will be verified later in terms of free energy changes. Estimation of Ad* The total change in density upon melting has been measured for three of the liquidsdiscussedhere,l6 2-MB, 2,2-DMB, and 2,2,4-TMP, and is about 10%. (This includes changes during solid transitions which allow free rotation in the solid.) Thedensityof the liquid is known at higher temperatures,” but the density of the solid must be extrapolated from the freezing point. The thermal expansion coefficients of the solids can be estimated from Klemm’s rule9 that CYT= (0.1 f O.02)/Tm,where Tmis the melting point. The expansion of the solid is only about one-third that of the liquid, so liquid expansion dominates. On this basis the extrapolated ratios of solid to liquid densities at the temperatures given in Table I are 1.26, 1.20, and 1.23 for 2-MB, 2,2-DMB, and 2,2,4-TMP. Accurate measurements of the densities of 2-MB and 3-MP glasses have been made at 77 K,18 and extrapolation of these glass densities to room temperature gives glass to liquid density ratios of 1.19 and 1.16. It is assumed that Ad, = 0.2d0 for all solvents. Compressibility of Classes. Little is known about the compressibility of hydrocarbons at low temperatures, but estimates can be made. It is assumed that the Tait A parameters for glasses are equal to those measured for the liquid. Actually, there is very little variation of the parameter for any compound or phasea9In one high-pressure study n-pentane was found to solidify at 23 “C and 17 500 bar with a 3.7% density in~rease.1~ The change in density with pressure is the result of different compression of the solid and liquid. The liquid parameters are known, and if, as explained above, the extrapolateddensityincreaseat zero pressure is assumed to be 20%, then B,/BI is 3.1. A value of 3 will be used for all solvents. Support for a ratio of 3 as a generalization comes from the fact that the ratio of the thermal expansion coefficient to the compressibilitytends to be constant and equal in the two p h a ~ e s . ~ The ratio of the measured thermal expansion coefficient of the liquid to that estimated for the solid by Klemm’s rule, used above, is 2.5 & 0.5 for the compounds in Table I, and the ratio B,/B] should be similar. Choiceof r,. A simple model based on cross sections and surface area gives the number of nearest neighbors as 3[(2 + rdW)/ rvdWlz,rounded to the nearest integer, for spheres with van der Waals radii rvdW packed around a 2-A sphere. The value of rs was chosen (by making several trials) as the radius which contained a mass equivalent to this number of solvent molecules. The number of neighbors and the correspondingr, are given in Table 11. The results of the volume calculations with glass formation included are given in Table I in the column labeled ‘gl liq”. It is seen that there is reasonable agreement with the observed
+
Volumes of Ions in Nonpolar Solvents
The Journal of Physical Chemistry, Vol. 97, No. 49, 1993 12957
TABLE III: Free Energies of Solvation'
TMS 2-MB 2,2-DMB
-43.0 -40.1 -41.0
3-MP
-40.1
2,2,4-TMP
-42.0
-2.1 -2.0 -2.6 -2.2 -2.6
2.4 1.8 3.3 2.4 3.6
In kcal/mol. No solid phase. Stabilization due to one glass layer.
partial molar volumes of reaction 1 with a standard deviation of 30 cm3/mol. Free Energy of Solvation. The criterion for the formation of a glasslike layer around an ion is that the loss in free energy upon freezing the layer must be less than or equal to the gain in solvation free energy from the increasein dielectric constant upon freezing. The free energy of solvation can be calculated by numerical integration of eq 6 with e computed for the densities determined from eq 13 (no glass layer) or eqs 13 and 15 (with a glass layer). Free energies calculated for the case in which there is no glass layer are given in Table I11 along with the changes resulting from inclusion of a layer of glass. The free energy loss on forming a glass from the liquid is not known. For crystalline solids, the free energy change at the melting point is zero, so at higher temperatures the liquid is more stable than the solid by -(T-Tm)ASm,where subscript m refers to melting. At the molecular level, the entropy of melting is determined by loss of crystal structure and gain of rotational degrees of freedom. On the more macroscopic level, the entropy change is expressed as a configurational entropy and an entropy proportional to thevolumechange, (aS/av)TAV,. For condensed phases3(aS/aV)Tis aT/xo.The configurationalentropy is about half the entropy of melting. The enthalpy of melting is large as the structure that thesolid assumes tends to minimize the enthalpy. Since the liquid-glass transition is second-order, there are no enthalpy, entropy, or volume changes at the glass transition temperature. Instead,thereisan abrupt increase in heat capacity2' which approaches that extrapolated from the solid over the next 80 "C or so. The glass is softening in this region as free rotation gradually sets in, and thermal expansion is larger than for the liquid. Thus, the glass mimics melting in this range, and at higher temperatures there is an apparent AH,and ASr It will be assumed by analogy with melting that there is a temperature T,, about 50 K above T, at which AH,= TxASgso that AG, = -( T- Tx)AS,. For the temperatures given in Table I, Tx is about half of T, so AG, will be approximated as -'/,TAS,. AS, can be estimated as a~AV,/xo where AV, is the difference in electrostriction volumes in Table I calculated with and without a glass layer. Thequantity-TAS,/Z is tabulated in Table I11 for comparison with the stabilization gained from glass formation. The fact that the magnitudes are very similar suggests that it is reasonable to assume,thatthe extra stabilization gained from one layer of glass formation is sufficient to overcome the free energy required to form the glass. If a second layer is added in the calculation, the stabilization is smaller than AG, by a factor of about 20, and so glass formation is restricted to one layer. Other Effects. The optoelectrical Kerr effectl1yz0in hydrocarbons and TMS can be interpreted as indicating that the polarizability is anisotropicto 10% or 15%. If the glass is formed with favorable orientations frozen in, then there will be an additional volume change and free energy of stabilization due to this orientation. To test the effect, the molar polarizabilityof the glass layer was increased by lo%, which resulted in a very small volume change, about -1 to -2 cm3/mol. The change in free energy from this effect is comparable to the change due to the density increase on glass formation given in Table 111, but generating favorable orientations would decrease the entropy of the glass and thus increase AG,. Thus, it is difficult to predict whether or not the glass molecules will be oriented.
The Clausius-Mossotti equation, eq 7, is derived with the assumption that a spherewith unit dielectric constant is embedded in a fluid with constant dielectric constant and constant E. The stronglydivergentfield around an ion is perhaps the only situation in which this approximation is not valid. A simple, though not rigorous,method of introducing the divergenceeffect is to modify the Lorentz derivation,Il which places a virtual sphere with unit dielectric constant in a constant electric field in a fluid with dielectric constant e. The sphere is virtual because it is assumed not to distort the field. If the Lorentz derivation is followed but the sphere is given the dimension of a solvent molecule, r,, and placed in the field of an ion at a distance r (E = e/&), then
where
and s = r,/r. As was the case with polarization effects, there is very little change in electrostriction volume upon introducing this effect (-2 to -5 cm3/mol). AG, is more negative by about 2 kcal/mol, but the difference with and without a glass layer is negligible.
Discussion The method used here treats the problem as one of free energy minimization with the constraint that at least the first layer of solvent is indivisible. Without this constraint, the radius r, would be chosen so that the glass-to-liquid difference in work done on an infinitesimal mass of solvent, am, at that radius would be equal to the free energy released on melting that much solid. The work done per unit volume is
(compare with eq 6) where D,2 is e*/rg4. Since the dielectric constant for either the glass or the liquid varies by less than 4% in this range, this integral is well representedby e2/8aer,4. Hence, the difference in work done in the two phases is
The free energy released on melting is, as before, approximately (l/2TaT/x)bV, or '/2TaT(l/d1 - l/d,)bm/x. These equations can be solved for r, by using the densities and extinction coefficients derived from eqs 13 and 15 and give (for 2-MB) re = 5.6 A, which is typical of the hydrocarbons. Thus, the thickness of the shell is 3.6 A compared to 5 A for a complete layer. It may be seen that free energy minimization gives a very similar conclusion to the method used above, but it was felt that it would be difficult to freeze 72% of a molecular layer while the remaining 28% remained liquid, and so the treatment as one complete layer of glass is preferred. The electrostatic free energies of solvation of the ions are affected relatively little by inclusion of density effects in the integration. The average increase in stability over the Born equation, eq 3, is -2.5 kcal/mol, or about 6% when no glass is included. Another -2.3 kcal/mol is added on when the glass is included, but this effect is offset by the AG, for glass formation, so the calculation without glass is probably the better representation of the free energy. The Born equation has been used to calculate the effect of solvent on photoionization thresholds,which are kinetic processes. If a neutral molecule is photolyzed, then a pair of ions is formed. The electron is ejected before the solvent can relax, though electronic polarization can follow it. Thus, the ionization really
12958 The Journal of Physical Chemistry, Vol. 97,No. 49, 1993
is accomplished at constant density and the Born equation is the correct description. On the other hand, if a negative ion is being photolyzed, then the solvent is already relaxed around the ion. But, again, the electron is ejected before the solvent relaxes, and so energy is still stored in the unstable density distribution of the solvent. This energy probable just about cancels the extra -2.5 kcal/mol stabilization the ion has more than the Born equation, so again, the Born equation is probably a good representation. Ionic Diffusion. The freezing of solute around an ion should affect ionic diffusion. For a given solvent, the product of diffusion constant and particle radius, Dir, is expected to be approximately constant, according to the Stokes-Einstein equation.22 If the ion moves with a glasslike layer of eight solvent molecules, its radius will be about (9r8)I/3,or about twice that of a solvent molecule, and so the ratio of ion diffusion constant to the self-diffusion constant of the solvent should be about 0.5. If solvent can exchange with the cluster on the same time scale as diffusion, then the factor could be somewhat larger. The experimental evidence on ionic diffusion is not in good agreement with this prediction. Positive and negative ion mobilities, which are equal to Di/kT, can be measured separately in nonpolar liquids, because the ion concentrations required are so small that the spacecharge field resulting from their separation is negligible compared to the applied potential gradient. Positive ion diffusion constants in 2,2,4-TMP,23~yclohexane,23.~~ and cisand t r a n s - d e ~ a l i nare ~ ~factors of 0.25-0.3 smaller than selfdiffusion, but in TMS2S*26 the ratio is 0.58 and in n-hexane2’*2* theratiois0.48. Negativeion mobilities (other than theelectron) were measured in one study23and found to be all about the same and more than twice as large as the positive ion mobilities. The values of Di for C1-, B r , I-, 0 2 - , and SF6- are 70% and 90% of self-diffusion in 2,2,4-TMP and cyclohexane. Electrostatic clustering around ions should be independent of the charge on the ion, and so there should be no difference in diffusion constants of positive and negative ions. In fact, it is very difficult to explain the difference observed in hydrocarbons. Certainly, there is no precedence for it when comparing ions of similar size but opposite charge in polar solvents. A possible explanation arises from the fact that the positive ions in these studies are very poorly characterized. They start as parent ions of the solvent, but during the drift time across the cell (>0.02 s) much can happen to them. For instance, if the sample contains >lo-’ M olefinic impurity, ionic polymerization may occur (terminated by a charge-transfer reaction). While it would be surprising to find such similar effects in four different solvents, it is worth noting that when biphenyl was added, which reacts with both electrons and positive ions to produce well-characterized ions, the positive and negative ion mobilities were the same.23 Also, the highest ratios of positive ion diffusion coefficient to self-diffusion, in TMS and n-hexane, were found in exceptionally pure solvents. If such condensation chemistry is going on in the solutions, the effect on the separate determination of the two mobilities is not clear. Thus, at present there is some agreement with experiment, but further experimental work in this area is desirable.
Schwarz
Conclusions The free energy which is calculated for an ion in a nonpolar solvent is minimized if approximately one layer of solvent molecules around the ion is frozen in a glasslike state. The principal physical manifestation of this effect is a much larger electrostriction volume than is expected if the solvent remains liquid. The calculated volumes are in very good agreement with the experimental data. A second prediction, that small ions of either charge should diffuse with diffusion coefficients about half that of self-diffusion in the liquid because the frozen layer doubles the size, is in disagreement with some data on positiveions though it does agree with the data on negative ions. Acknowledgment. The author thanks M. Newton for his comments and thanks R. Holroyd for many discussions and for suggesting the application to diffusion constants. This research was carried out a t Brookhaven National Laboratory under Contract DE-AC02-76CH00016 with the U.S. Department of Energy and supported by its Division of Chemical Sciences, Office of Basic Energy Sciences. References and Notes (1) Nishikawa, M.; Itoh, K.; Holroyd, R. J . Phys. Chem. 1988,92,5262. (2) Ninomiya, S.;Itoh, K.; Nishikawa, M.; Holroyd, R. J. Phys. Chem. 1993. - - - , 97. 9488. (3) Lewis: G. N.; Randall, M. Thermodynamics, 2nd ed.; McGraw-Hill: New York, 1961; pp 203, 104-107. (4) Munoz, R. C.; Holroyd, R. A.; Itoh, K.;Nakagawa, K.; Nishikawa, M.; Fueki, K. J. Phys. Chem. 1987, 91,4639. (5) Lewis, G. N.; Randall, M. Thermodynamics, 2nd cd.; McGraw-Hill: New York, 1961; Chapter 31. (6) The CGS formulation of electrostatic equations will be used throughout. (7) Drude, P.; Nernst, W. 2.Phys. Chem. (Munich) 1894, 15, 79. (8) Bottcher, C. J. F.; Van Belle, 0. C.; Bordewijk, P.; Rip, A. Theory of Electric Polarization, 2nd ed.; Elsevier: Amsterdam, 1973; p 320. (9) Bondi, A. Physical Properties of Molecular Crystals, Liquids, and Glasses; Wiley: New York, 1974; pp 453, 381-390. (10) Whalley, E. J . Chem. Phys. 1963, 36, 1400. (1 1) Bottcher, C. J. F.; Van Belle, 0. C.; Bordewijk, P.; Rip, A. Theory of Electric Polarization, 2nd ed.; Elsevier: Amsterdam, 1973; pp 166168. (12) Hellmans, L.; DeMaeyer, M. J . Chem. Soc.,Faraday Tram. 2 1982, 78, 401. Battaglia, R. Chem. Phys. Lert. 1978, 54, 124. (1 3) Bradley, R. S . High Pressure Physics and Chemistry; Academic: London, 1963; Vol. 1, p 151. (14) Hirschfelder, J. 0.;Curtis, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1964. (15) Bridgman, P. W. The Physics ofHigh Pressure; Bell: London, 1949; pp 190, 129. (16) Wurflinger, A. Ber. Bunsen-Ges. Phys. Chem. 1975, 79, 1195. (17) Rossini, F. D. Selected Values of Physical and Thermodynamic Properties of Hydrocarbons; Carnegie Press: Pittsburgh, 1953. (18) Rosengren, K. J. Acta Chem. Scand. 1962, 16, 1421. (19) Gelles, S.H. J . Chem. Phys. 1968,48, 526. (20) Buckingham, A. D. J . Chem. Phys. 1956,25,428. Hellemans, L.; De Maeyer, M. J. Chem. SOC.,Faraday Trans. 2 1982, 78,401. (21) Douslin, D. R.; Huffman, H. M.J. Am. Chem. Soc. 1946,68,1704. Prentice-Hall: Englewood (22) Moore, W. J. Physical Chemistry, 3rd 4.; Cliffs, NJ, 1962; p 766. (23) Allen, A. 0.;de Haas, M. P.; Hummel, A. J . Chem. Phys. 1976,154, 2587. (24) Gee, N.; Freeman, G. R. J . Chem. Phys. 1992, 96, 586. (25) Schmidt, W. F. Proceedingsof the InternationaIConferenon~quid Radiation Detectors; Waseda University: Tokyo, 1992; p 252. (26) Parkhurst, Jr., H. J.; Jonas, J. J. Chem. Phys. 1975, 63, 2698. (27) Itoh, I.; Nishikawa, M.; Holroyd, R. J . Phys. Chem. 1993.97, 503. (28) McCall, D. W.; Douglass, D. C.; Anderson, E. W. J . Chem. Phys. 1959, 31, 1555.
