Thermodynamics of a charged hard sphere in a compressible

Dec 1, 1981 - Jin-Kee Hyun, Keith P. Johnston, and Peter J. Rossky ... Keith P. Johnston, Gerald E. Bennett, Perla B. Balbuena, and Peter J. Rossky...
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J. Phys. Chem. 1981, 85,3944-3949

water solubilizing capacity of AOT in the latter solvents is reduced because of more favorable AOT-solvent interactions. In cyclohexane, motional differences in the AOT alkyl chains exist between the inverted micelles and the w/o microemulsions. Frequency-dependent data indicate a complex series of motions in the inverted micelles; motion at each carbon-carbon bond cannot be considered independent of other motional contributions. While NMR offers the promise of a complete description of the motional factors involved, in practice this may be impossible. In all three solvents the relaxation of the carbons nearest the head group is dominated by overall aggregate motion.

This correlation holds up to intermediate R values. At higher R, other motions, such as monomer diffusion, are contributing to the relaxation process. This work has further emphasized the distinction between Aerosol OT inverted micelles, and w/o microemulsions. Acknowledgment. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for the support of this research and to Dr. Ruth Inners and the South Carolina Magnetic Resonance Laboratory for acquiring the 100MHz spectra for us.

Thermodynamics of a Charged Hard Sphere in a Compressible Dielectric Fluid. A Modification of the Born Equation To Include the Compressibility of the Solvent Robert H. Wood,* Department of Chemistty, University of Delaware, Newark, Delaware 1971 1

Jacques R. Qulnt, and Jean-Plerre E. Groller Laboratoire de Thermodynamique et Cinetique Chimique, Unlversit6 de Clermont-Ferrand, AublGre, 63 7 70 France (Received: May 1, 198 1; In Final Form: July 28, 198 1)

Equations for the thermodynamic properties of a charged hard sphere in a compressible dielectric fluid have been derived. For small effects, a first-order correction term is accurate, but, when effects are large, for instance, near the critical point, the complete set of equations describing the thermodynamic properties must be solved by numerical techniques. For a fluid with the density and dielectric properties of water, the contribution from the compressibility of the solvent is very small at 300 K, but, as the critical point of the solvent is approached, the contribution becomes much larger. In a water-like dielectric fluid at 600 K and 17.7 MPa, the effect is very large; for example, for a 1-1 electrolyte with ions having radii of 0.2 nm, the calculated apparent molal heat capacity changes from -4900 to -2168 J mol-l K-l when the compressibility of the solvent is taken into account. This change results from the very large change in the solvent dielectric constant which rises from 17 at large distances from the ion to 28 at its surface. Similarly, the density near the ion is much higher and the effective pressure goes from 17.7 to 280 MPa. Thus, the compressibility of the solvent makes large contributions to the calculated thermodynamic properties of the ions, and these contributions cannot be ignored in a theory that is expected to be accurate near the critical point. However, since the theory presented here does not accurately represent the heat-capacity measurements of Smith-Magowan and Wood for aqueous NaC1, it seems likely that dielectric saturation must also contribute appreciably to the thermodynamic behavior and must be included in a more complete theory.

I. Introduction The Born equation1 is an equation for the electrostatic free energy of a charged hard sphere in an incompressible dielectric fluid. Ever since its inception, this equation has been used as a primitive model for the electrostatic contribution to the free energy of an ion in a dielectric solvent. Friedman and Krishnan2 have recently reviewed the application of the Born equation as model for electrolyte solutions and shown that it has not been very successful in water and other solvents a t 298 K. The basic reason for this failure is that neglected effects, resulting from the molecular nature of the solvent, are sufficiently important that the quantitative predictions of the model are not accurate. Refined models, which account for dielectric saturation or hydration in the first coordination sphere, (1) M.Born, 2.Phys., 1, 45 (1920). (2) H.L. Friedman and C. V. Krishnan in “Water; a Comprehensive Treatis”, Vol. 3, F. Franks, Ed., Plenum Press, New York, 1973, Chapter 1. 0022-3654/81/2085-3944$01.25/0

have met with more ~ u c c e s s . ~ - ~ In contrast to the situation near room temperature, where it is not very useful, the Born equation has recently been found to provide a surprisingly accurate model for the heat capacities of aqueous solutions at temperatures of 500-600 K and a pressure of 17.7 MPaa6 The apparent molal heat capacities a t constant pressure (17.7 MPa) of infinitely dilute solutions are very large and negative (C,,# N -3000 J mol-l K-l a t 600 K). The fact that the Born equation is very good a t predicting and correlating these very large effects indicates that the effects are primarily electrostatic in origin. Presumably, the Born equation is valid because chemical solvation effects are small compared to electrostatic effects a t this temperature and pressure. (3) M. H. Abraham, J. Lisze, and L. MBszhos, J. Chern. Phys., 70, 2491 (1979). , (4) P. R. Tremaine and S. Goldman, J. Phys. Chern., 82,2317 (1978). (5) D.Smith-Magowan and R. H. Wood, J. Chem. Thermodyn., in press.

0 1981 American Chemical Society

The Journal of Physical Chemistry, Vol. 85, No. 25, 198 I

Thermodynamics of a Charged Hard Sphere

The Born equation gives the electrostatic free energy of a hard-sphere ion in an incompressible, dielectric solvent but, as the critical point is approached, the compressibility of a fluid approaches infinite values. Thus, it seems clear that the Born equation which neglects compressibility might be expected to be a poor model near the critical point. The present investigation was undertaken to determine whether the model of hard spheres in a compressible dielectric fluid is a better model for real solutions near the critical point. In a compressible dielectric fluid, the density of the fluid becomes greater near an ion because of the effective pressure increase created by the electrostatic field. The present results show that, for water at 600 K, this change in density leads to an increase in dielectric constant (an effect in the opposite direction to dielectric saturation) which in turn has a large effect on thermodynamic properties. In contrast, differences between a hard-sphere model in a compressible vs. incompressible fluid are negligible for a water-like fluid near 298 K. Many authors have presented modifications of the Born equation to include the effects of dielectric saturationa2g3 However, effects due to compressibility of the solvent have been neglected since they are small for most solvents at ordinary temperatures. The results given below indicate that both dielectric saturation and solvent compressibility contribute appreciably to the thermodynamic properties near the critical point. 11. Derivation of Equations

( A ) Basic Equations. Our starting point is Frank's6 classic exposition of the thermodynamics of a fluid substance in an electrostatic field. We assume that the dielectric constant is a function only of the temperature and the density of the fluid or K(p,T). Thus, we assume there are no dielectric saturation effects ((dK/dE2)v,T= 01, where dielectric saturation is defined as a dependence of the dielectric constant on the electric-field strength at constant volume and temperature. However, if the electric field changes the local pressure and density, there will be a corresponding change in the local dielectric constant. An alternate assumption would be that the dielectric constant does not depend on the field strength at constant pressure and temperature. Because the correlation of the molecular dipoles should be more directly connected with the distance between them, we prefer the first assumption about dielectric saturation. This is also the implicity assumption made by Frank.6 We want to calculate the electrostatic energy of a hard-sphere ion of radius a in a continuous dielectric medium. The dielectric medium is compressible and has a dielectric constant, K, that changes with both density and temperature. The ion is surrounded by an infinite bath of dielectric fluid, and the external pressure on the fluid is po. Because of the electric field near the ion, the pressure, p , at this point will be above po.6Because of this pressure increase (as well as electrostriction), the fluid will have a greater density and the local dielectric constant will be different (normally larger because of the greater density). Note that this effect is the opposite to that of dielectric saturation. We commence by calculating the dependence of the density, p , of the fluid in the electric field, E, using (~P/@),T

= ( ~ P ' / ~ ) ( ~ K / ~ P ) E (1) ,T

which is Frank's6 eq 15. Here K is the relative permittivity or dielectric constant, eo is the vacuum permittivity, and (6)H.S. Frank, J. Chem. Phys., 23, 2023 (1955).

3945

p is the isothermal compressibility. Note that Frank's equation has been converted to SI units. At constant solvent chemical potential, N, and temperature, T , we have dp = (dp/dE2),,,~dE2 = ( G # P ~ / ~ ) ( ~ K / ~dE2 P ) E , T(2) or dp/p2 = toP(aK/dp)E,T dE2/2

(3)

where p, /3, and ( d K / d ~ are ) ~ known ,~ from experimental measurements on the pure solvent. Integration of this equation allows calculation of the density of the solvent as a function of the electric field at constant solvent activity and temperature. For a model fluid of the type considered here, in which the dielectric constant is a function only of temperature and density, dielectric saturation is not permitted, and K, therefore, depends on field strength only when the field strength changes the density. In other words, at constant T, K varies with density alone, but the change in the density can be brought about by either an external pressure increase directly or by a local pressure change resulting from an increase in field strength. Next, we calculate the electrostatic work done on the fluid, dw,', when charging a volume element, u = l / p , associated with unit mass of the fluida6The result is a slight modification of Frank's eq 26 dw,' = E d(uP) = t& d[(K - l)E/p] (4) (5)

where P is the polarization and once again the dielectric constant and the density of the fluid must be known at various field strengths so that the expression can be integrated. The free energy associated with a charged hard sphere immersed in a fluid can be thought of as residing in the surrounding fluid, and we therefore calculate next the total electrostatic work of charging the sphere by integrating the free-energy density surrounding the sphere over the entire volume of the fluid. The contribution per unit volume of fluid in the field is just w,'p, and to this must be added the contribution per unit volume that results from the space (in vacuo) occupied by the fluid. The result is

In this expression, both density and w,' vary with the electric field so it is necessary to have an expression for the electric field as a function of the distance from the ion, R, in order to integrate the expression. The necessary equation is

E = q/(4nc6(R2) (7) where the dielectric constant, K , changes with the electric-field strength so that this is a transcendental equation. Using this equation in eq 3 and 5 allows us to express the work and the density as a function of the radius and to integrate eq 7 to obtain the total electrostatic free energy, G". Numerical techniques can be used to solve the above set of equations, given the experimental properties of the fluid. This approach is explored in the following section. For some purposes, it is sufficient to calculate only the firstorder correction to the Born equation due to the compressibility of the solvent. In a later section, this fist-order correction to the Born equation is derived.

3946

The Journal of Physical Chemistry, Val. 85, No. 25, 198 1

( B )Numerical Integration. The numerical integration of eq 6 using eq 3, 5, and 7 is quite straightforward by considering a series of fixed densities greater than the zero electric-field density, po. For each chosen density, eq 3 is integrated (using the trapezoid rule) to obtain a value of the electric field necessary to produce this density. Then, from eq 7, the distance from the ion corresponding to this field is found. In any of these calculations, the dielectric constant must be determined at the known density and temperature. We have used Bradley and Pitzer's equation' for the dielectric constant of water as a function of pressure and temperature, so it was first necessary to find the pressure on pure water which would produce the same density increase and then to calculate the dielectric constant at this pressure. In all calculations, the equation of state of water given by Keenan et ala8was used. Once the values of R giving these densities, electric fields, and dielectric constants are known, then integration of eq 5 followed by integration of eq 6 is straightforward. Both integrations used the trapezoid rule (a spline integration would have been better). Finally, the electrostatic contribution to the free energy of hydration was calculated by substracting the free energy in a vacuum. Ace1 = Gel - Gelvac = Gel - q 2 / ( 8 ~ € o a ) (8) Numerical differentiation of this equation gives the electrostatic contribution to the solution enthalpy, He', and the solution heat capacity at constant pressure, Cpel AHe' = d(AGel/T)/d(l/T) = He' - Helvac = He' - Gel

6;)

Wood et al.

To calculate the total electrostatic free energy, substitute eq 15 and 13 into eq 6 to get

Gel,

we

Now we need to express the electric field, E, as a function of radius. Substituting eq 14 into eq 7 E = 4/(4?r€o[Ko+ f/z~o~K12p02E2]R2) (17) Equation 17 can be simplified by expanding the denominator, retaining only the E2 term, and approximating the latter with E = 4/(47r4Z2)

Finally, substituting eq 18 into eq 16, keeping only the lowest terms and integrating, gives

ACpe1= d(AHel)/dT = Cpel- Cpelvac= Cpel (9b) where we have used the fact that the electrostatic contribution to the vacuum free energy Gelvac is not a function of temperature. Thus, Selvac = 0; GeIVac = He]',; and C;IVac = 0. At each step the accuracies of the numerical approximations were estimated and shown to be adequate ( ~ 1 % in AC el). ( C ) First-&der Correction. The calculation of the first-order corrections to the Born equation starts with the expansion of the dielectric constant in a Taylor series. Keeping only the linear term gives K = KO+ Ki(p - PO) (10) with (dK/dp)E,T = K1 (11) where KOand po are the dielectric constant and the density at Po and T. Substituting this value of ( d K / d p ) ~into , ~ eq 2 and integrating gives (12) P = PO/O - €oPK1PoE2/2) Also, since the specific volume, u, is given by u = l / p , we have u = uo(1 - ~ o P K i ~ c & ~ / 2 ) (13) We now calculate the dielectric constant, K , as a function of E by substituting eq 12 into eq 10, dropping higher-order terms in E , to give K = KO+ f/2€oPK12p:E2 (14) Next, we calculate the work of charging by substituting eq 12 and 14 for p and K into eq 5 and integrating while keeping only terms up to order E2. The result is (7)D. J. Bradley and K. S. Pitzer, J. Phys. Chern., 83, 1599'!1979). (8) J. H..Keenan, F. G. Keyes, P. G. Hill, and J. G. Moore, Steam Tables", Wiley-Interscience, New York, 1978.

(19) where a is the radius of the hard-sphere ion. We now write K1 = (dK/ap)Tg = (aK/dP)T/(aP/dP)T Kp/(PoP) (20) where K p = (dKO/dp)T,Ewhich is useful since it is easier to calculate K p than K1. The electrostatic free energy of hydration of the ion, AGel, is then the free energy of charging the ion in the fluid minus the value for charging it in a vacuum. Subtracting the latter ( q 2 / ( 8 ~ ~ ofrom a)) eq 19 and substituting eq 20 yields

The first term in eq 21 is the Born equation1 while the second is the first-order correction due to the compressibility of the solvent. Since this correction term depends on a-5, it will only be important near the ion. The other thermodynamic properties are calculated by taking the appropriate partial derivatives of eq 21. The results are

where KT = (dK0/dT),, and ASe1equals the electrostatic

The Journal of Physical Chemistry, Vol. 85, No. 25, 1981

Thermodynamics of a Charged Hard Sphere

TABLE I : Electrostatic Contribution to the Gibbs Free Energy of Solution a t 1 7 . 7 MPa of Two Hard-Sphere Ions of Radius a in a Water-Like Dielectric Medium

TABLE 11: Electrostatic Contribution to t h e Enthalpy at 1 7 . 7 MPa of Two Hard-Sphere Ions of Radius a in a Water-Like Dielectric Medium

Af@/(kJ mol-')

AGel/(kJ mol-') TIK

alnm

0.2

0.3

0.4

3947

T/K

0.5

a/nm

300 Born' Born + I b Born + Lc 400 Born' Born + I b Born -I-L" 500 Born' Born + I b Born + Lc 550 Born' Born + I b Born + Lc 600 Born' Born + Ib Born + Lc

0.2

0.3

0.4

0.5

-698.0 -697.8 -697.8 -706.4 -705.4 -705.0 -727.2 -721.2 -712.9 -754.9 -738.3 -693.6 -859.5 -788.8 -324.7

-465.3 -465.3 -465.3 -470.9 -470.7 -470.7 -484.8 -483.4 -482.9 -503.3 -498.4 -495.2 -573.0 -546.4 -502.5

-349.0 -349.0 -349.0 -353.2 -353.2 -353.2 -363.6 -- 363.2 -363.2 -377.4 -375.8 -375.5 -429.7 -419.2 -413.0

-279.2 -279.2 -279.2 -282.6 -282.6 - 282.6 -290.9 -290.8 -290.8 -302.0 -301.4 -301.4 -343.8 -339.4 -338.3

a The enthalpy for two ions of radius a calculated from the Born equation ( t h e first terms of eq 21 and 22 substituted into eq 23). The enthalpy calculated by numerical integration (see section IIB). c The enthalpy ealculated by using the linear correction term, eq 21-23.

TABLE 111: Electrostatic Contribution to the Apparent Molal Heat Capacity a t a Constant Pressure of 17.7 MPa of Two Hard-Sphere Ions of Radius a in a Water-Like Dielectric Medium

\

Cpel/(J mol-' K ' )

r

i y

2000

T /K

r

-2000

a/nm

300 Born' Born + Born + 400 Born' Born + Born 500 Born' Born t Born + 550 Born' Born + Born -+ 600 Born' Born + Born +

Ib Lc

Ib

+ Lc

-4000

0.2

0.3

0.4

0.5

Rlnm

Figure 1. Plot of apparent molal heat capacity, C p s , vs. radius, I?, calculated for three approximations: B is the Born equation; L is the Born equation plus linear correction term (see section IIC); I is the Born equation plus numerical integration of the correction term (section 115). All calculations are for water-like fluid at 600 K and 17.7 MPa.

contribution to the entropy of the ion in solution (Sel) because the electrostatic contribution to the entropy of the gas-phase ion is zero. Also AH= A G + T A S (23)

where KTT = (a2K/dP),. Finally

111. Results and Discussion It is interesting to examine the difference between the Born model and the present model which includes com-

Ib Lc

Ib Lc

Ib

Le

0.2

0.3

0.4

0.5

-57 -38 -28 -23 -54 -38 -28 -23 -53 --38 -28 -23 -123 -82 -61 -49 -105 -78 -60 -49 -92 -78 -60 -49 -353 -235 -176 -141 -236 -202 -167 -137 + 1 6 -187 -165 -137 -873 -582 -437 -349 -488 -446 -388 -330 + 1 1 9 1 -310 372 -328 -4900 -3267 -2450 -1960 - 2 1 6 8 -2064 -1924 -1724 + 2 6 0 8 7 + 8 1 4 -1482 -1643

' The apparent molal heat capacity, C,o,

calculated from the Born equation (the first term in eq 24) for two The apparent molal heat capacity, ions of radius a. calculated by numerical integration (see section The apparent molal heat capacity, C p q $ ,calculated by using the linear correction term, eq 24.

%?:

pressibility effects. The difference between the two models, the correction, can be calculated by two methods: (1)numerical integration (section IIB) and (2) the linear correction (section IIC). The results of calculations using both numerical integration and the linear correction term are given in Tables 1-111 and Figure 1. An examination of these tables shows that, when the correction for the effect of compressibility is small, both numerical integration and the linear correction term give the same answers. However, when the correction is large, the linear term grossly overestimates the correction. For example, at 600 K and a radius of 0.2 nm, the linear term overestimates the correcton to the heat capacity by a factor of 11. Tables 1-111 also show that the correction for the effect due to the compressibility of the solvent is very small at 300 K and increases as the critical point is approached. The correction also increases as the ionic radius gets smaller. The results in Tables 1-111 are at 17.7 MPa.

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The Journal of Physical Chemistty, Vol. 85,No. 25, 1981 27

I

0.9

I

r 240

25

200

23

160

m

I b

21 I

a 120

19

80

17

40

15

1L

0 1 3 02

\-I 0 3

'

0 I4

'

Wood et al.

the solvent at 17.7 MPa for the calculation, and this is not far off at 0.5-nm radius, but is quite far off at 0.2 nm, where the density, effective compressibility, and the dielectric constant of the solvent are quite different. Our calculations at 600 K and 0.2 nm are not as accurate as at lower temperatures and larger radii. This is because we have used the equation of Keenan et aL8 for the solvent properties, and this equation is not accurate above 100 MPa. Examination of the results shows that no serios errors were introduced, but an equation of state accurate at higher pressures would be necessary for the highest accuracy. Most theoretical models for electrolyte solutions are based on the McMillan-Mayer standard state and usually assume a dielectric constant that is independent of distance from the ion? The present results indicate that near the critical point this will become a very poor approximation. Already at 600 K, the change in dielectric constant is quite large and increasing extremely rapidly. Models assuming a constant dielectric constant predict separation of salt solutions into two liquid phases with different concentrations at temperatures near the critical pointVgb These phase separations have not been observed experimentally, and it may well be that the increase in dielectric constant near the ion reduces the attractive force at short range and prevents phase separation at these temperatures. The electrostatic contribution to the heat capacity of the solution per mole of salt at infinite dilution calculated from the Born equation is a contribution to the experimental apparent molal heat capacity, Cp,#,or parJia1 molal heat capacity, Cp,2(at infinite dilution CP,? C,,,), so we can directly compare experimental values urlth our calculations. The calculations given in Tables 1-111 and Figure 1were made at 17.7 MPa so the results could be compared with experimental measurements of the heat capacity of aqueous NaCl recently published by Smith-Magowan and Wood.5 These results indicate that the apparent molal heat capacity of aqueous NaCl a t infinite dilution at 600 K and 17.7 MPa is --3000 J mol-l kg-l. From Figure 1, we see that, at a radius of 0.327 nm, the Born equation gives the experimental result for apparent molal heat capacity. However, Figure 1 shows that, when the compressibility of the solvent is taken into account, there is no radius which will give the experimental result! In fact, the calculated heat capacity is practically constant at -2000 J mol-l K-l, from 0.2 to 0.4 nm, when the compressibility of the solvent is included in the calculation. This failure of the model means that there must be another effect in the opposite direction from the compressibility effect, and the most likely candidate at this point is dielectric saturation. Dielectric saturation reduces the value of the dielectric constant of the solvent as the electric field increases and thus acts to cancel increases in dielectric constant due to compressibility of the solvent. To explore this further, both theoretical and experimental measurements will be necessary. An appropriate theory should include, in a rigorous way, both the effects of compressibility of the solvent and dielectric saturation of the solvent. We are preseiatly working on such a development. Recently, an apparatus for measuring the dielectric saturation of water at room temperature has been developed.1° A very crude extrapolation of these measurements to 600 K indicates that dielectric-saturation effects give rise to decreases in the dielectric constant, which are ap-

-4" 0 5

Rlnm

Flgure 2. Plots of dielectric constant, e (X), density, p (0),and effective pressure, P" (0) as functions of the radius, R , for an ion in a continuous. water-like dielectric fluid at 600 K and 17.7 MPa.

Other calculations at 50 MPa show that the correction is smaller a t higher pressures. Presumably, the correction for the effect of compressibility reaches a maximum at the critical temperature and pressure (where compressibility is infinite). At temperatures and pressures far from the critical temperature and pressure, the correction should be much less important. Table I shows that the compressibility of the solvent has only a small effect on the free energy of the system. The maximum effect a t 600 K and a radius of 0.2 nm is 0.8% while the enthalpy and heat-capacity effects are higher (ca. 8% and 50%, respectively) for the same conditions (see Tables I1 and 111). Thus, although the compressibility of the solvent has only a small effect on the free energy of the ion, it has an enormous effect on its heat capacity. This is reminiscent of the Born equation itself. The free energy calculated by the Born equation changes very little with temperature and pressure. However, both entropy and enthalpy have very strong temperature and pressure dependences and the heat capacity an even stronger dependence (see Tables 1-111). It is instructive to look at the physical effects of the electric field on the solvent as a function of the radius at 600 K and 17.7 MPa. While performing the numerical integration (section IIB), one calculates values of the density of the solvent and the dielectric constant as a function of radius. One also has the values of the effective pressure, PX;that is, the pressure necessary, in the absence of an electric field, to create the same density. These quantities are plotted as a function of the radius for a 0.2-nm ion in water at 600 K and 17.7 MPa (see Figure 2). The effects are enormous. P* starts at 17.7 MPa at an infinite distance from the ion and quickly climbs to 280 MPa at 0.2 nm. At the same time, the dielectric constant increases from 17 to 28, and the density increases from 0.64 to 0.9 g ~ m - ~Thus, . the solvent near the ion is quite different from bulk solvent, and any continuum model of the solvation of ions in a dielectric field must take these effects into account when the temperature and pressure are near the critical point. Far from the critical point, these effects are much smaller. For a 0.2-nm ion in a water-like fluid at 300 K and 17.7 MPa, the dielectric constant, E , increases from 78 to 81 near the ion, while the effective pressure, P, increases from 17.7 to -90 MPa near the ion. It is clear from an examination of Figure 2 why a linear correction term is not adequate at 600 K and 17.7 MPa. The reason is that the linear term uses the properties of

(9) (a) H. L. Friedman and W. T. Dale, "Modern Theoretical Chemistry", Vol. IV, part A, B. J. Berne, Ed., Plenum Press, New York, 1977; (b) H. L. Friedman and B. Larsen, J. Chem. Phys., 70,92 (1979). (10)A. E. Davies, M. J. van der Sluijs, G. P. Jones, and M. Davies, J. Chem. SOC.,Faraday Trans. 2, 74, 571 (1978).

Thermodynamics of a Charged Hard Sphere

proximately canceled by increases in dielectric constant due to compressibility of the solvent at 600 K and 17.7 MPa. Thus, we apparently have the familiar situation where an approximate equation is more accurate than its refined version. However, the present results indicate that, although the Born equation gives qualitatively the correct order of magnitude for the effects, it cannot be trusted as a quantitative equation since it is unlikely that exact cancellation of the effects of compressibility and dielectric saturation will occur at all temperatures and pressures. Thus, it seems very likely that the detailed pressure and temperature dependence of the Born equation will be incorrect and that the errors will increase as the critical point is approached. This is an important conclusion because recent discussions of high-temperature thermodynamic properties have made extensive use of the Born equation. In particular, Smith-Magowan and Wood5have shown that the Born equation qualitatively predicts the correct temperature and pressure dependence of the heat capacity of aqueous NaCl. However, they noted that deviations between the Born equation results and the experimental results appeared to be systematic. In a similar way, Helgeson and Kirkhaml' have used the Born equation to (11) H. C. Helgesen and D. H. Kirkham, Am. J. Sci., 276, 97 (1976).

The Journal of Physical Chemistty, Vol. 85,No. 25, 1981 3949

extrapolate volumetric data for electrolytes to temperatures as high as 800 K. Although these are probably the best extrapolations available, the present study indicates that substantial errors are possible. There is one way in which the present results are in accord with experimental evidence; Table I11 and Figue 1 show that, when the compressibility is taken into account, the heat capacity at high temperatures is practically independent of the radius of the ion. This is in agreement with recent experimental evidence that a variety of 1-1 electrolytes have similar heat capacities at high temperatures.12 Acknowledgment. R.H.W. acknowledges support from the National Science Foundation under Grant No. CHE 8009672. We are grateful to Peter Thompson for helpful discussions and to the Franco-American Exchange Commission for their support of the interuniversity exchange between the University of Delaware and the University of Clermont-Ferrand 11. (12) The papers presented by W. Lindsay, Jr., by R. C. Murray, Jr. and J. W. Cobble, and by C. M. Criss at the 41st Annual Meeting of the International Water Conference, Pittsburg, PA, Oct 1980. All give evidence for this.