7704
J. Phys. Chem. 1996, 100, 7704-7709
Real Ionic Solutions in the Mean Spherical Approximation. 1. Simple Salts in the Primitive Model Jean-Pierre Simonin,*,† Lesser Blum,‡ and Pierre Turq† Laboratoire des Proprie´ te´ s Physico-Chimiques des Electrolytes (URA CNRS 430-Electrochimie), UniVersite´ Paris VI, Boıˆte no. 51, 8 rue CuVier, 75005 Paris, France, and Department of Physics, POB 23343, UniVersity of Puerto Rico, Rio Piedras, Puerto Rico 00931-3933 ReceiVed: December 1, 1995; In Final Form: February 14, 1996X
Departures from ideality in electrolytes are described in the framework of the primitive model of ionic solutions in which the solvent is a dielectric continuum, using the mean spherical approximation (MSA). To include solvation and solvent concentration effects, we consider that the permittivity of the solvent and the sizes of the ions are concentration-dependent parameters. New expressions are derived for the activity coefficients and the osmotic coefficient. They are applied to pure ionic aqueous solutions of 18 salts, taking simple functions for the adjusted parameters. Good fittings are obtained in the concentration range 0-6 mol/kg.
Introduction Because of its wide range of applications, the thermodynamic properties of electrolytes have been the subject of much interest even in recent literature.1-3 Certainly, among physical chemists, the most popular expressions have been the Debye-Hu¨ckel limiting laws (DHLL) and expressions derived therefrom.1 Among others, geochemists have extensively used Pitzer’s modifications of DHLL to describe departures from ideality in concentrated ionic mixtures (typically up to 6 mol/kg4 and up to 10-20 mol/kg, between 0 and 170 °C, for solutions of volatile weak electrolytes5). Also solubilities of minerals in natural waters can be predicted accurately.6 Pitzer’s treatment is based on the DH theory. It uses the DHLL plus a virial type series correction. Another theory that is fundamentally connected to the DHLL is the mean spherical approximation (MSA).7-10 In the DHLL the linearized Poisson-Boltzmann equation is solved for a central ion surrounded by a neutralizing ionic cloud. The main simplifying assumption of the DHLL is that the ions in the cloud are point ions. The MSA is the solution of the same linearized Poisson-Boltzmann equation but with finite size ions in the cloud. The mathematical solution of the proper boundary conditions of this problem is much more complex. However, simple variational derivations exist nowadays.11 The analytic solution of the MSA shares with the DHLL the remarkable simplicity of being a function of a single screening parameter Γ9 for an arbitrary neutral mixture of ions. The expressions of the thermodynamic excess functions are formally very similar to those of the DHLL. The MSA can be derived from first principles of statistical mechanics. Better approximations are the HNC equation and its improvements,12 but they need to be solved numerically for every individual system, which is often impractical. The MSA has been solved for the socalled “primitive” model,13 in which the solvent is regarded as a dielectric continuum, and for the “nonprimitive” model,14-19 in which the solvent is discrete and modeled as a collection of hard spheres with embedded point dipoles. Calculations of departures from ideality in ionic solution using the MSA have been published in the past by a number of authors. Effective ionic radii have been determined for the calculation of osmotic coefficients for concentrated salts,20 in †
Universite´ Paris VI. ‡ University of Puerto Rico. X Abstract published in AdVance ACS Abstracts, April 1, 1996.
S0022-3654(95)03567-2 CCC: $12.00
solutions up to 1 mol/L,21 and for the computation of activity coefficients in ionic mixtures.22 In these studies, for a given salt, a unique hard sphere diameter was determined for the whole concentration range. Also, thermodynamic data were fitted with the use of one linearly density-dependent parameter (a hard core size σ(C) or dielectric parameter (C)), up to 2 mol/L, by leastsquares refinement,23-25 or quite recently with a nonlinearly varying cation size26 in very concentrated electrolytes. In previous work27 parametrization of the thermodynamic properties of pure electrolytes was done with the use of the density-dependent average diameter and dielectric parameter. Both are ways of including effects originating from the solvent, which do not exist in the primitive model. Obviously, they are not equivalent and they can be extracted from basic statistical mechanics arguments: it has been shown28 that, for a given repulsive potential, the equivalent hard core diameters are functions of the density and temperature; Adelman has formally shown29 (Friedman extended his work subsequently30) that deviations from pairwise additivity in the potential of average force between ions result in a dielectric parameter that is ion concentration dependent. Lastly, there is extensive experimental evidence31,32 for being a function of density. In the present work, the variation of these parameters with concentration is taken into account explicitly for the expression of the activity coefficient. A new derivation of the thermodynamic relations is given. The mean activity coefficient is fitted to experimental data with simple expressions for the sizes and the dielectric parameter. The method is illustrated with a numerical application to 15 alkali-halides and 3 acids. Theory The thermodynamic properties of electrolytes in the primitive MSA have been given elsewhere.33,34 For the sake of generality, we will discuss individual ionic excess thermodynamic properties. The single-ion activity coefficients for fixed diameters were discussed by several authors.35-37 In all previous work the implicit dependence of the sizes and dielectric constants on the concentration was not taken into account. The thermodynamic properties can be derived from the Helmholtz free energy A. We split the excess free energy into two terms. We define
β∆A ) β∆AMSA + β∆AHS (1) MSA where ∆A is the electrostatic contribution that we calculate in the MSA; ∆AHS is the ion hard sphere contribution. © 1996 American Chemical Society
Real Ionic Solutions in the MSA
J. Phys. Chem., Vol. 100, No. 18, 1996 7705
Electrostatic Contribution. Thermodynamic integration (equivalent to Guntelberg charging process) yields an expression for the MSA contribution to A. We get
∂β∆EMSA ∂Γ′
β∆AMSA ) β∆EMSA - ∫0 dΓ′ Γ
β∆AMSA ) β∆EMSA +
∆ ln γmMSA )
(3)
∆E
)∑
∆EiMSA
(4)
[
with
e2 FzN ii i
(5)
and
Ni )
η)
Γzi + ησi 1 + Γσi
(6)
∑ Ω 2∆ k 1 + Γσ
∆ ln γiMSA )
(7)
Ω)1+
]
∂∆AMSA ∂Fi
[
Γ)const
]
∂Fi
Fkσk3
∑ 2∆ k 1 + Γσ
)
[
π
Fkσk3 ∑ 6 k
The excess osmotic coefficient thermodynamic relation
∆φMSA ) Ft
∆φMSA
[
+
]
[] [ ] [] ∂σj
∂β∆EMSA ∂σj
Γ,Fk,σk(k*j),
∂
(9)
[ ]
∆ ln
γiMSA
)-
βe2
[
Γzi2
1 + Γσi
with
(11) qj )
i
In eq 10 the derivative is taken at constant Γ, because it is known11,13 that
∂∆AMSA )0 ∂Γ
(12)
+ ησi
∑j Fjqj
Γ)const
Ft ) ∑ Fi
[
]
∂β∆AMSA ∂Fi
(
1 + Γσi
[] ∂σj
∂Fi
+
)] []
ησi2
(18)
β∆EMSA ∂
∂Fi
]
[
+
3
ησj2(2 - Γ2σj2) - 2zj Γ2zj2 βe2 + η (1 + Γσ )2 (1 + Γσj)2 j
(19)
(20)
From this result we find also, with eq 14,
∆ ln γmMSA )
β∆EMSA Ft
-
βe2 2 η2 π Ft
+
1 Ft
∑j FjqjD(σj) β∆EMSA D() Ft
with the notation
Let us define the mean activity coefficient of the mixture by
-
∂Fi
(13) Γ)const
Γ,Fk,σk
2zi - ησi2
This result simplifies greatly all the expressions derived hereafter. The single-ion activity coefficient reads
∆ ln γiMSA )
∂
After some tedious but straightforward algebra we find that
(10)
where
+
∂Fi
∂β∆EMSA
(8)
is calculated from the
∂ ∆AMSA ∂Ft Ft
(17) Γ)const
Γ,Fk(k*i),σk,
k
∆)1-
]
∂∆EMSA ∂Fi
∂β∆EMSA
∑j
k
π
(16)
Remember that both the diameters and the dielectric parameter are a function of the concentration: σj({Fi}), ({Fi}), for i from 1 to n, where n is the number of ions. Using standard implicit function differentiation, we get
Fkσkzk
1 π
(15)
We observe that we need only one derivative in all of our calculations:
i
∆EiMSA ) -
β∆AMSA + ∆φMSA Ft
∂∆ ln γmMSA ∆φMSA ∂∆φMSA ) + ∂Ft Ft ∂Ft
The expression for the excess MSA internal energy ∆EMSA (per unit volume)34 can be rewritten in the following different form. MSA
(14)
and by differentiation with respect to Ft, we get the GibbsDuhem relation in the form
3
Γ 3π
∑i Fi∆ ln γiMSA
Ft
It is easy to show38 that
(2)
We obtain13
1
∆ ln γmMSA )
D ) ∑ Fj j
∂ ∂Fj
(21)
7706 J. Phys. Chem., Vol. 100, No. 18, 1996
Simonin et al. obtain
and using eqs 3, 15, and 21, we get
Γ3
∆φMSA ) -
-
βe2 2 η2 π Ft
3πFt
+
1
∑ FiqiD(σi) F i
∆ ln γiHS ) Mi + ∑ QjFj j
t
β∆EMSA D() Ft
(22)
(30)
∂Fi
with
Mi ) -ln x + σiF1 + σi2F2 + σi3F3
These are new expressions that include the contributions due to the density variation of the diameters and the dielectric constant. Consider now the particular case of a pure binary salt, of density FS. Let us suppose that one mole of the salt releases ν+ moles of cations and ν- moles of anions. Then, considering that the diameters and the permittivity are a function of FS, we have that
Qi ) F1 + 2σiF2 + 3σi2F3 and
F1 ) F2 )
∂ D ) FS ∂FS If the average diameter approximation, σj ) σ, is used then η ) 0. Denoting by γ( the mean activity coefficient of the salt, we get the simpler relations
(
F3 ) X0 -
3X2 x
3X1 X22 X22 1 + 3 ln x +3 x X3 x2 X2 3
)
X23 1 3X1X2 - X23/X32 X23 X23 + 2 2 ln x + X2 x x2 X x3 X3 3
3
3
where
MSA ∆ ln γ( ) MSA
β∆E Ft
-
MSA
(
)
∂σ 1 ∂ Γ β∆E + (23) ν+ + ν- 1 + Γσ ∂FS ∂FS
π
Fkσkn ∑ 6 k
Xn )
x ) 1 - X3
with
β∆EMSA ) -
2
Γ βe F (ν z 2 + ν-z-2) 1 + Γσ S + +
(24)
At low densities (X3 , 1) we have that
F1 = 3X2(1 + X3 + X32 + ...)
and
∆φ
∂σj
MSA
(
∂σ 1 ∂ Γ Γ3 β∆EMSA )+ 3πFt ν+ + ν- 1 + Γσ ∂FS ∂FS
)
(25)
9 F2 = 3X1 + X22 + X3(3X1 + 8X22) + 2
(
)
(
X32 3X1 +
We recall that Γ satisfies the closure equation
Γ ) 2
πβe2
∑i Fi [(zi - ησi )/(1 + Γσi)] 2
2
(26)
Usually, eq 26 is easily solved by iteration starting with the initial value 2Γ1 ) κ, where κ is the Debye screening parameter:
κ)
(
4πβe2
)
∑ Fizi2 i
1/2
(
π β∆AHS ) 6 X23
(
)
X3
(
)
)
From eq 30 we calculate the mean hard sphere activity coefficient in the same way as in eq 14,
∆ ln
γmHS
)
( ) X23
X0X32
X3
- 1 ln x +
+
X23(1 + 2X3 - X32)
x
X0x2
+
X0X3x2
3X1X2(2 - X3)
(28)
+
1 Ft
∑j FjQjD(σj)
(34)
and since eq 15 also holds for the hard sphere part, we find HS
)
X3 1 - X3
+
3X1X2 X0(1 - X3)2
3X1X2 X23 + (29) 1 - X3 X (1 - X )2 3 3
The corresponding contribution to the activity coefficient is calculated in a way similar to eq 18. After some algebra we
)
(
∆φ
- X0 ln(1 - X3) + 2
)
(27)
Hard Sphere Contribution. We calculate the hard sphere contribution β∆AHS from the Carnahan-Starling approximation.39,40 We have that
45 2 X + ... (32) 4 2
8 15 F3 = X0 + X23 + 3X1X2 + X3 X0 + X23 + 6X1X2 + 3 2 72 3 2 X3 X0 + X2 + 9X1X2 + ... (33) 5
With the mean ionic diameter approximation σi ) σ eq 26 becomes
Γ ) [(1 + 2κσ)1/2 - 1]/(2σ)
(31)
+
X23(3 - X3) X0(1 - X3)3 1 Ft
+
∑j FjQjD(σj)
(35)
Applications to Experiment In the present study, only the case of pure salts is considered.
Real Ionic Solutions in the MSA
J. Phys. Chem., Vol. 100, No. 18, 1996 7707
The experimental data were taken from Robinson and Stokes.1 In contrast to Pitzer’s work, which is given in molalities (Lewis-Randall theory (LR)), the MSA naturally expresses thermodynamic quantities in terms of concentrations, in the framework of the McMillan-Mayer (MM) theory of solutions.41 Thus, the data have to be converted from Lewis-Randall to McMillan-Mayer scales. So, any molal mean activity coefficient can be converted into a molar mean activity coefficient, according to1
γMM exp ) γexp(1 + mM)
d0 d
(36)
with m the molality of the salt, M its molecular weight (in kg), d0 the density of the pure solvent (in kg/L), and d that of the solution. In this first study, we do not include in eq 36 all of the corrections of the LR to MM conversion.42 The drawback of this simplification is that osmotic coefficients, converted at the same level of approximation, do not satisfy rigorously the Gibbs-Duhem relation. For this reason, we present here results for activity coefficients, and we will discuss the case of osmotic coefficients in more detail subsequently. Activity coefficients can be calculated theoretically by using eqs 21 or 23 and 34 MM MSA HS ) ∆ ln γ( + ∆ ln γ( ln γcal
(37)
For a given molality, the concentration C and the density d of the solution are calculated from
C ) md/(1 + mM)
(38)
d ) d0 + d1C - d2C3/2
(39)
and
where the values for d1 and d2 are tabulated.43 We apply the present treatment to the case of solutions of alkali-halides MX, in which M and X are respectively either Li, Na, K, Rb, or Cs and Cl, Br, or I, and also to HCl, HBr, and HI. Calculations have been performed up to the saturation point for each salt. Among these salts, data at very high concentrations (above 6 mol/kg) are available:1 for HCl (up to 16 mol/kg), for LiCl and LiBr (up to 20 mol/kg for both salts), and for CsCl (up to 11 mol/kg). Activity coefficients calculated from eq 37 have been fitted to experimental ones, obtained from eq 36, by a least-squares numerical method. In all cases the permittivity was taken concentration dependent. Concerning the hard core diameters, a density-dependent average diameter was first fitted. Another fitting was tried in which the size of the halide ion was taken constant, equal to its crystallographic value, and the size of the cation was adjusted. This procedure has also been used in refs 23 and 24. It is related to the assumption that the hydration number of halide ions is low. This current assumption is supported by experimental evidence.44 In particular, neutron scattering studies45 point to the weak hydration of the chloride ion. For each procedure the following simple laws were taken,
σ ) σ(0) + σ(1)C )
0 1 + RC
with 0 the dielectric constant of pure water, taken as 78.38 at
Figure 1. Mean activity coefficients for the salts HCl (]), LiCl (+), NaCl (0), KCl (×), and CsCl (4) vs concentration. The lines are the results from the least-squares adjustment of average salt diameters.
TABLE 1: Values of Adjusted Parameters (Average Hard Core Diameter; m < 6 mol kg-1) salt
σ(0) (Å)
102 σ(1) (Å mol-1 L)
102 R (mol-1 L)
AARDa (%)
HCl HBr HI LiCl LiBr LiI NaCl NaBr NaI KCl KBr KI RbCl RbBr RbI CsCl CsBr CsI
4.30 4.63 5.20 4.18 4.10 5.30 3.76 3.25 3.59 3.48 2.95 3.18 3.22 2.76 2.74 2.73 2.76 2.82
-4.57 -4.90 -9.28 -3.36 -2.69 -9.35 -1.53 -40.8 -60.0 -0.44 -28.4 -33.9 0.13 -32.5 -34.3 1.64 2.64 1.36
5.41 10.3 23.0 6.08 0.38 35.6 8.16 -15.8 -22.0 7.73 -9.74 -12.1 4.89 -9.07 -9.92 1.58 2.61 3.37
0.09 0.04 0.23 0.26 0.69 0.23 0.22 0.52 0.47 0.12 0.56 0.57 0.09 0.47 0.50 0.25 0.28 0.17
a AARD denotes the average relative deviation of the calculated activity coefficients against the experimental data: AARD(%) ) (100/ N N) ∑k)1 |γcal(k) - γexp(k)|/γexp(k), with N the number of concentration points.
25 °C, and it is expected that
σ(1) < 0 and
R>0 In this procedure, three parameters therefore need to be adjusted. Results and Discussion The ability of the method to describe activity coefficients in the range 0-6 mol/kg is exemplified in Figure 1, for the case of the calculation of average salt diameters; for clarity, only the result for salts containing chloride is presented and the curve for RbCl is not shown. For these salts the average relative deviation (AARD) of the fitting is less than 0.17%. Complete results for the adjusted parameters are shown in Table 1 for the determination of mean salt diameters (procedure 1) and in Table 2 for the adjustment of the cation’s size (procedure 2). In both tables the accuracy of the least-squares procedure is mentioned through the AARD between the calculated activity coefficient and the experimental data.
7708 J. Phys. Chem., Vol. 100, No. 18, 1996
Simonin et al.
TABLE 2: Values of Adjusted Parameters (Determination of the Cation’s Hard Core Diameter; m < 6 mol kg-1 salt HCl HBr HI LiCl LiBr LiI NaCl NaBr NaI KCl KBr KI RbCl RbBr RbI CsCl CsBr CsI a
σPa (Å)
σ(0) (Å)
102σ(1) (Å mol-1 L)
102R (mol-1 L)
AARD (%)
1.20 1.20 1.20 1.90 1.90 1.90 2.66 2.66 2.66 2.96 2.96 2.96 3.38 3.38 3.38
5.00 5.36 6.06 4.76 4.30 6.25 3.90 3.99 4.42 3.34 3.13 3.22 2.82 2.51 2.09 1.89 1.71 1.42
-8.83 -9.67 -17.8 -6.60 -5.30 -18.1 -3.03 -3.92 -6.57 -0.93 -3.09 -7.08 0.00 -1.18 -1.15 2.42 3.80 -0.88
6.76 11.5 25.0 6.96 0.33 37.9 8.18 8.60 14.3 7.75 6.51 7.51 5.03 5.06 5.29 2.12 3.44 4.51
0.07 0.04 0.24 0.26 0.69 0.25 0.22 0.05 0.16 0.12 0.07 0.13 0.09 0.03 0.1 0.24 0.27 0.16
σP is the Pauling diameter of the cation.
form a fairly consistent set of values. In the case of the iodides, the diameters deviate more significantly from the former values. Anyhow, results agree with the current notions that the smaller cations are more hydrated than the bigger ones. However, it is seen in Table 2 that the fitted diameter of Rb+ falls below the Pauling diameter, and this anomaly is still more pronounced for Cs+. This feature was emphasized a long time ago,46,47 and a similar finding was obtained in ref 23. It may be explained by the existence of attractive non-Coulomb short-range forces between a big alkali cation and a halide, which lead to the use of a distance of closest approach that is somewhat smaller than the actual distance. Lastly, we find that the influence of the corrective terms (containing D(σj) and D()) on the electrostatic contribution, arising from the concentration dependence of the parameters, is important at moderate and high concentration. The effect on the hard sphere contribution is much smaller. For instance, in the case of NaCl, the corrections to the excess energies (eqs 21 and 34) are respectively about 1% and -0.1% at 0.1 m, 8% and -1% at 1 m, and 30% and -4% at 6 m. These results apply to the molality range 0-6 mol/kg. Excellent fittings of the activity coefficients are obtained in this range. Adjustments above this limit up to the saturation point, for HCl, LiCl, LiBr, and CsCl, yield poorer agreement with experiment. This phenomenon may be due to the simplification adopted for the LR-MM conversion, which is expected to be significant at high density. However, we find that data at very high concentration, above 10 m typically, can be described with good precision. But since the values for the parameters are not consistent with those below 6 m, we do not present them here. Conclusions
Figure 2. Parameter σ(0) (average diameter description) for the salts MX studied: abscissa ) cation M, and X ) Cl (]), Br (4), I (0).
It is shown that activity coefficients of concentrated (typically up to 6 m) simple pure aqueous electrolytes can be described accurately in the primitive MSA with the use of only three adjusted parameters. In contrast with other descriptions, such as Pitzer’s treatment, variation of these parameters can be interpreted in terms of ionic hydration effects. The next step will be the application of our expressions to mixtures of electrolytes and to unsymmetrical ionic solutions. Solutions of ions that give rise to association phenomena should require more refined suitable methods. Acknowledgment. L.B. thanks the Department d’Electrochimie of the Universite´ P. et M. Curie for their warm hospitality and support. He gratefully acknowledges support through ONR and EPSCOR EHR-91-08775.
Figure 3. Parameter σ(0) (individual diameter description) for the salts MX studied: abscissa ) cation M, and X ) Cl (]), Br (4), I (0).
It is found that the second procedure yields values for the parameter σ(1) which increase monotonously with the halide cation size. However, it becomes positive for cesium chloride and cesium bromide. At the same time, R is positive for all salts. In contrast, procedure 1 yields many unrealistic values for σ(1) and R. From Table 2 one finds that the decrease in the cation’s diameter is typically about 1% per mol/L. The decrease in the permittivity amounts to several percent per mol/L. In the case of NaCl for instance we get = 55 at 6 mol/kg. The dependence of the parameter σ(0) on the nature of the cation is shown in Figure 2 for procedure 1 and in Figure 3 for procedure 2. In the latter case, it is satisfying to notice that, for a given cation, the sizes for the chlorides and the bromides
References and Notes (1) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.; Butterworths: London, 1959. (2) Pytkowica, R. M., Ed. ActiVity Coefficients in Electrolyte Solutions; CRC Press: Boca Raton, 1979; Vols. I and II. (3) Horvath, A. L. Handbook of Aqueous Electrolyte Solutions; Ellis Horwood: Chichester, 1985. (4) Pitzer, K.; Mayorga, G. J. Phys. Chem. 1973, 77, 2300. (5) Edwards, T. J.; Maurer, G.; Newman, J.; Prausnitz, J. M. AIChE J. 1978, 24, 966. (6) Harvie, C. E.; Moller, N.; Weare, J. H. Geochim. Cosmochim. Acta 1984, 48, 723. (7) Percus, J. K.; Yevick, G. Phys. Rev. 1966, 110, 251. (8) Waisman, E.; Lebowitz, J. L. J. Chem. Phys. 1970, 52, 4307. (9) Blum, L. Mol. Phys. 1975, 30, 1529. (10) Blum, L.; Høye, J. S. J. Phys. Chem. 1977, 81, 1311. (11) Blum, L.; Rosenfeld, Y. J. Stat. Phys. 1991, 63, 1177. (12) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids, 2nd ed.; Academic Press: London, 1986.
Real Ionic Solutions in the MSA (13) Blum, L. In Theoretical Chemistry, AdVances and PerspectiVes; Eyring, H., Henderson, D., Eds.; Academic Press: New York, 1980; Vol. 5. (14) Blum, L. J. Stat. Phys. 1978, 18, 451. (15) Vericat, F.; Blum, L. J. Stat. Phys. 1980, 22, 593. (16) Blum, L.; Wei, D. Q. J. Chem. Phys. 1987, 87, 555. (17) Blum, L.; Vericat, F.; Fawcett, W. R. J. Chem. Phys. 1992, 96, 3039. (18) Blum, L.; Fawcett, W. R. J. Phys. Chem. 1992, 96, 408. (19) Blum, L.; Fawcett, W. R. J. Phys. Chem. 1992, 97, 7185. (20) Watanasiri, S.; Brule´, M. R.; Lee, L. L. J. Phys. Chem. 1982, 86, 292. (21) Ebeling, W.; Scherwinski, K. Z. Phys. Chem. 1983, 264, 1. (22) Corti, H. R. J. Phys. Chem. 1987, 91, 686. (23) Triolo, R.; Grigera, J. R.; Blum, L. J. Phys. Chem. 1976, 80, 1858. (24) Triolo, R.; Blum, L.; Floriano, M. A. J. Phys. Chem. 1976, 67, 5956. (25) Triolo, R.; Blum, L.; Floriano, M. A. J. Phys. Chem. 1978, 82, 1368. (26) Sun, T.; Le´nard, J. L.; Teja, A. S. J. Phys. Chem. 1994, 98, 6870. (27) Simonin, J. P.; Blum, L. J. Chem. Soc., Faraday Trans., in press. (28) Andersen, H. C.; Chandler, D.; Weeks, J. D. AdV. Chem. Phys. 1976, 34, 105. (29) Adelman, S. A. J. Chem. Phys. 1976, 64, 724. (30) Friedman, H. L. Kinam 1981, 3A, 101. (31) Hasted, J. B.; Ritson, D. M.; Collie, C. H. J. Chem. Phys. 1948, 16, 1.
J. Phys. Chem., Vol. 100, No. 18, 1996 7709 (32) Barthel, J.; Buchner, R.; Bachhuber, K.; Hetzenhauer, H.; Kleebauer, M.; Ortmaier, H. Pure Appl. Chem. 1990, 62, 2287. (33) Blum, L.; Høye, J. S. J. Phys. Chem. 1977, 81, 1311. (34) Sanchez-Castro, C.; Blum, L. J. Phys. Chem. 1989, 93, 7478. (35) Høye, J. S.; Blum, L. Mol. Phys. Chem. 1978, 35, 299. (36) Vericat, F.; Grigera, J. R. J. Phys. Chem. 1982, 86, 1030. (37) Humffray, A. J. Phys. Chem. 1983, 87, 5521. (38) Cartailler, T.; Turq, P.; Blum, L.; Condamine, N. J. Phys. Chem. 1992, 96, 6766. (39) Mansoori, G. A.; Carnahan, N. F.; Starling, K. E.; Leland, T. W. J. Chem. Phys. 1971, 54, 1523. (40) Salacus, J. J.; Stell, G. J. Chem. Phys. 1982, 77 (7), 3714. (41) McMillan, W. G.; Mayer, J. E. J. Chem. Phys. 1945, 13, 276. (42) Friedman, H. L. J. Solution Chem. 1972, 1, 387. (43) Sohnel, O.; Novotny, P. Densities of Aqueous Solutions of Inorganic Substances; Physical Sciences Data 22; Elsevier: Amsterdam, 1985. (44) Hinton, J. F.; Amis, E. S. Chem. ReV. 1971, 71, 627. (45) (a) Hertz, H. G. Water: A ComprehensiVe Treatment; Franks, F., Ed.; Plenum: New York, 1973; Vol. 3. (b) Hewish, N. A.; Enderby, J. E.; Howells, W. S. J. Phys. C: Solid State Phys. 1983, 16, 1777. (c) Neilson, G. W.; Enderby, J. E. Proc. R. Soc. A 1983, 390, 353. (46) Gurney, P. W. Ionic Processes in Solution; Dover: New York, NY, 1953. (47) Frank, H. S. In Chemical Physics of Ionic Solutions; Conway, B. F., Barradas, R. G., Eds.; Wiley: New York, NY, 1966.
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