1406
J. Phys. Chem. 1996, 100, 1406-1410
Use of Experimental Diffusion Coefficients To Probe Solute-Solute and Solute-Solvent Interactions in Electrolyte Solutions William E. Price* Department of Chemistry, UniVersity of Wollongong, Northfields AVenue, Wollongong, N.S.W. 2522, Australia
Reginald Mills and Lawrence A. Woolf Diffusion Research Unit, Atomic and Molecular Physics Laboratory, Research School of Physical Sciences, Australian National UniVersity, Canberra, A.C.T. 0200, Australia ReceiVed: August 10, 1995; In Final Form: September 27, 1995X
Interactions between ions have a very significant effect on the properties of electrolytes. This paper discusses two approaches to the study of interactions in electrolyte solutions using experimental transport coefficients. One is based on generating generalized transport coefficients from experimental data. A number of variants have appeared in the literature, the preferred ones, distinct diffusion coefficients, as formulated by Friedman are used here. Their usefulness and interpretation in terms of ionic interactions are discussed. One limitation of the above is that they are based on solvent-averaged potentials. The other approach deals with our attempts to use the water diffusion coefficient and shear viscosity data to gain insights into ion-solvent interactions and into hydration dynamics of multivalent cations.
Introduction One of the key questions in electrolyte theory which continues to attract attention concerns the role of interactions between the ions and how these affect the properties of aqueous electrolyte solutions. This paper examines two separate approaches to the study of these interactions using transport coefficients. Statistical mechanics provides methods for the calculation of thermodynamic properties of electrolytes from solventaveraged potentials between two ions.1,2 As exact interionic potentials are not known, these calculations are carried out using model potentials. Although the study of equilibrium properties of electrolyte solutions in terms of such model potentials is well advanced, thermodynamic (equilibrium) properties are often not sufficiently sensitive to changes in the model potential to provide useful tests. As a consequence, nonthermodynamic data have to be used to test the theories. Friedman and co-workers1,3,4 have been interested for a number of years in developing a theory that relates solvent-averaged potentials to the electrical conductance and closely related transport coefficients. Recent work has shown3-7 that certain pair interaction coefficients derived from transport data may be uniquely useful for probing ionic interactions as well as testing the validity of interionic potentials in models. The resulting generalized transport coefficients, denoted as velocity correlation coefficients,8 fij, or in a slightly different version of the theory3,4 as distinct diffusion coefficients, Dijd , have been promoted in experimental papers by some of the authors.6,7,9,10 In descriptive terms, the distinct diffusion coefficient, Dijd , is a measure of the displacement of particle i, due to the displacement of a particle, j, where the particles are distinct even if the species are the same. Friedman11,12 has argued that distinct diffusion coefficients (Ddij) are the preferred set of generalized experimental transport coefficients. This preference arises from the fact that they are directly related to the time integral of the ensemble average of particle velocity correlations. As Friedman13 has observed, “the * To whom correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, December 15, 1995.
0022-3654/96/20100-1406$12.00/0
lack of secular concentration dependence is helpful if one wishes to fill by inspection whether there are specific interactions among the system’s components”. A further advantage of the Dd formalism is the inherent symmetry (e.g. Dijd ) Dijd ) which is not present in the fij’s. The relevant equation for Dijd ’s in electrolyte solutions is
Dijd ) {(Ni + Nj)∫0 〈V1i(t) V2j(0)〉dt/3}Vd∞∑Ni/V fixed ∞
i
(1)
where Ni is the number of particles of i, 〈...〉 is an equilibrium ensemble average, V1i is the velocity of particle 1 of species i, V2j is the velocity of particle 2 of species j, V is the total volume, and t is the time. i can be equal or not equal to j. The original generalized coefficients for electrolyte solutions, the mass-fixed Ωij of Onsager, have also been widely used after transformation to solvent-fixed lij’s as developed by Miller14 for systems consisting of a single electrolyte in an ionizing solvent. The connection between the lij and experimental quantities is
rirjDv titjΛ lij ) 3 2 + 3 N 10 F z z 10 RTrr z (d(mφ)/dm) i j 1 1
(2)
Here N is normality, ti is the transference number of species i, zi is the charge of species i (i ) 1, cation; i ) 2, anion), ri is the stoichiometric coefficient of ion i (r ) r1 + r2), F is the Faraday constant, R is the gas constant, and T is the absolute temperature. Dv is the experimental mutual diffusion coefficient, φ is the molal osmotic coefficient, and Λ is the equivalent conductance. In the present context the lij/N have been calculated for one of the systems that are discussed as they provide a very useful path to the Dijd through the equation
Dijd ) (RT(lij/N)[|zi| + |zj|]2δijDis where δij is the Kronecker delta. © 1996 American Chemical Society
(3)
Solute-Solute and Solute-Solvent Interactions
J. Phys. Chem., Vol. 100, No. 4, 1996 1407
TABLE 1: Generalized Transport Coefficients (105 cm2 s-1) for NiCl2 Solution as a function of Concentration at 25 °C c, mol L-1
conductance, cm2 S mol-1
t(Ni2+)
D(Ni2+)
D(Cl)
D(H2O)
Dm
l11/N
l12/N
l22/N
Dd11
Dd12
Dd22
0 0.001 0.003 0.005 0.007 0.01 0.02 0.03 0.05 0.07 0.1 0.2 0.5 1 1.5 2 2.5 3 3.5 4
258.7 249.9 235.3 229.2 224.7 219.8 209.5 202.6 193.5 187.1 180.3 164.72 139.83 113.41 92.72 75.73 60.37 47.09 35.9 26.1
0.41 0.409 0.407 0.405 0.403 0.401 0.391 0.385 0.376 0.370 0.366 0.346 0.313 0.271 0.255 0.242 0.23 0.216 0.205 0.195
0.705 0.705 0.704 0.704 0.704 0.703 0.701 0.700 0.696 0.693 0.688 0.671 0.623 0.542 0.46 0.391 0.322 0.261 0.207 0.157
2.0322 2.030 2.027 2.023 2.019 2.014 1.996 1.978 1.941 1.905 1.85 1.775 1.58 1.317 1.101 0.914 0.748 0.598 0.436 0.337
2.3 2.300 2.299 2.299 2.298 2.298 2.296 2.293 2.289 2.285 2.278 2.134 1.975 1.736 1.405 1.102 0.979 0.801 0.63 0.433
1.248 1.250 1.253 1.255 1.257 1.260 1.263 1.257 1.246 1.231 1.206 1.145 0.978 0.752 0.580 0.453 0.361 0.294 0.240 0.190
1.423 1.402 1.366 1.349 1.336 1.320 1.278 1.249 1.205 1.172 1.135 1.035 0.842 0.617 0.471 0.364 0.286 0.227 0.182 0.141
0.000 0.059 0.160 0.204 0.238 0.276 0.358 0.402 0.456 0.484 0.499 0.539 0.508 0.409 0.307 0.237 0.199 0.181 0.166 0.146
8.193 8.049 7.812 7.730 7.677 7.630 7.569 7.497 7.398 7.299 7.137 6.864 6.176 5.259 4.324 3.556 2.894 2.344 1.864 1.420
0.000 -0.020 -0.054 -0.070 -0.083 -0.097 -0.135 -0.161 -0.197 -0.223 -0.250 -0.316 -0.411 -0.472 -0.453 -0.421 -0.361 -0.297 -0.234 -0.174
0.000 0.044 0.119 0.152 0.177 0.206 0.266 0.299 0.339 0.360 0.371 0.401 0.378 0.304 0.228 0.176 0.148 0.134 0.123 0.108
0.000 -0.070 -0.181 -0.214 -0.233 -0.245 -0.239 -0.238 -0.215 -0.191 -0.162 -0.147 -0.098 -0.027 -0.058 -0.065 -0.061 -0.034 0.052 0.030
In the first section of this paper, distinct diffusion coefficients will be used to describe ionic interactions in a series of 1-2 and 2-2 electrolytes which have recently been under investigation in the laboratory. This approach largely ignores the solvent and the valuable information that can be gleaned about solutesolvent interaction from solvent diffusion data. In the second part of this paper we discuss some recent attempts by us to use diffusion data as a probe for solute-solvent interaction and hydration dynamics in multivalent electrolyte solutions.9,10,15-19 Sources of Data Values of lij and Dijd as given in Table 1 were obtained by reevaluation of available transport property measurements and thermodynamic activity data.20,21 Other Dijd data have been taken predominantly from previous work in these laboratories and are referenced appropriately.
Figure 1. Cation-cation experimental distinct diffusion coefficients (Dd11) as a function of square root of ionic strength at 25 °C for a range of electrolyte systems. Key: filled square, NiCl2; filled circle, MgCl2;9 open square, ZnCl2;9 filled triangle, ZnSO4;7 open triangle, Na2SO4.6
Distinct Diffusion Coefficients as a Probe for Ionic Interactions The first electrolyte solution to be examined using the Dijd formalism is the NiCl2/H2O system at 25 °C. This system was chosen because some years ago Enderby and Neilson22 interpreted their neutron diffraction studies on the basis of an abnormal partial structure factor in concentrated NiCl2 solutions. They suggested that there could be Ni2+-Ni2+ ordering. It is of interest here that Friedman and Dudowicz23 used computer simulation modeling to show that such a conclusion was unjustified. At about the same time Geiger, Hertz, and Mills24 applied the fij formalism to NiCl2 solutions and demonstrated that Ni-Ni ordering was not shown by those data. The Dijd approach is used here to examine this system. In applying this formalism to nonelectrolyte systems,25 the data for a given system are compared to a standard system which approaches thermodynamic ideality. In electrolyte systems, however, the comparison is best made to a strong electrolyte which is fully dissociated at all concentrations and where there is a minimum of complexation. In the case of NiCl2 the most suitable reference system is a MgCl2 solution, an archetypal strong electrolyte. Figures 1-3 show plots of Dd11, Dd22, and Dd12 (where 1 ) cation and 2 ) anion) for various electrolytes against the square root of the ionic strength. The Dijd for NiCl2 are taken from Table 1. The other systems are solutions of MgCl2, ZnCl2 (a
Figure 2. Cation-anion experimental distinct diffusion coefficients (Dd12) as a function of square root of ionic strength at 25 °C for a range of electrolyte systems. Key and data sources as for Figure 1.
system with complexes of the form ZnClx), Na2SO4, and ZnSO4 (where there is evidence of ion pairing and higher-order clusters). In interpreting the concentration dependence of the Ddij, some characteristics of the time integrals should be borne in mind. In reference systems, where there is little interaction, all the Ddij (where i * j) will have negative values. If there is dimerisation, the Diid will tend to be positive, and if complexes are formed, then both the Diid and Dijd can show positivity.
1408 J. Phys. Chem., Vol. 100, No. 4, 1996
Price et al. viscosity, is not required in the Friedman formulation for electrolyte solutions. Development of a Hydration Model
Figure 3. Anion-anion experimental distinct diffusion coefficients (Dd22) as a function of square root of ionic strength at 25 °C for a range of electrolyte systems. Key and data sources as for Figure 1.
In Figure 1, the Dd11 for NiCl2 and MgCl2 are negative and virtually coincide, and as it is known that the latter is a fully dissociated electrolyte, the absence of Ni-Ni ordering is confirmed. It will be observed that the Dd11 values for Na2SO4 and ZnSO4 are positive and those of the latter salt markedly so. This gives a clear indication of the occurrence of like-ion configurations which could arise from Na+‚‚SO42-‚‚Na+ (or higher) clusters. This has been discussed in detail by Weingartner et al.6 Cation-anion interactions, represented by Dd12, are shown in Figure 2. First these increase and then reach a plateau before diminishing. However, the values for the two sulfate systems and for zinc chloride are much larger and reach a maximum at higher concentrations that with the other divalent salts. This is consistent with stronger correlation of motions due to cationanion interactions because of ion-pair configurations in the sulfate and complexation in the case of zinc chloride. It should be noted that the nickel chloride values are similar in both magnitude and concentration dependence (shape of the curve) to those for the magnesium chloride, again emphasizing the lack of evidence of unusual behavior in the nickel chloride system. Anion-anion interactions characterized by Dd22 are given in Figure 3 for the various electrolyte systems. Again, Dd22 values for NiCl2 and MgCl2 are negative and similar over the whole concentration range. Similar behavior is shown for the other electrolytes at low concentrations, but with increasing concentration, large positive Dd12 are observed. For zinc and sodium sulfate systems this has been used to indicate the presence of clusters of ions containing more than one sulfate ion.6,7 Positive Dd22 values are even more prominent in the zinc chloride case. The first part of this paper has shown that calculation of the Dd requires the use of a wide range of transport data and their combination with thermodynamic data. Although the Dd provide a valuable insight into interionic attraction and repulsion, they have been formulated in terms which obscure the role of ion-solvent interactions. An examination of Dd’s for a wide range of types of nonelectrolyte solutions25 has shown that the number-fixed reference frame provides the most direct physical insight into intermolecular interactions. The Friedman formulation of the Dd for electrolyte solutions in terms of a solventaveraged potential does not use the diffusion coefficient of the major component, the solvent, which is required for the most accurate interpretation of neutron scattering studies of the solvation of ions. Another transport coefficient which strongly reflects the influence of ion-solvent interactions, the shear
Measurements of a quasielastic neutron scattering and X-ray diffraction in concentrated aqueous solutions containing di- or trivalent cations such as nickel or chromium give strong supporting evidence for the long-held concept that such ions have additional hydration outside the primary coordinated shell.26,27 Recent infrared vibrational spectroscopic data28 and molecular dynamics calculations29 also give credence to this model. The secondary hydration is pictured as comprising water molecules with a less-established association than those of the primary layer coordinated with the central ion; they are in more rapid exchange with the bulk water environment. Generally, neither X-ray nor neutron scattering results give a quantitative estimate of secondary hydration.27,30 Cr3+ is one of the exceptions to this generalization; when it does not form complexes, a value of 12 has been determined from X-ray data.30 Our role in measuring diffusion data for the ions and solvent comprising electrolyte solutions for both neutron scattering and Dd studies has led to the development of a simple model of hydration15,17,31 which expresses the diffusion coefficient of the solvent in terms of those for the ions and also uses the shear viscosity to provide information on the secondary hydration of polyvalent cations. The model is based on that proposed by Salmon and co-workers32 to interpret neutron scattering results but does not use scattering data. It uses information on primary hydration such as that assessed by Friedman.33 It is convenient to describe the model by reference to diffusion in a solution of a strong electrolyte in water. If the tracer diffusion coefficients of the cation (Dm), the anion (Da), and the water (Dw) are measured as a function of composition, the data can be used to model the dynamics of the water within the system. At any instant the water is assumed to be distributed in four distinct categories: coordinated to the metal ion in the primary hydration sphere, in a secondary hydration layer around the cation, in a hydration layer around the anion, and in the bulk water. Typically, the diffusion coefficient of the water in the solution is measured using tritium (3H) as a label. The tritium is assumed to be statistically distributed between all the water molecules within the solution.15 The measured diffusion coefficient for 3HHO (corrected to that for H2O using the factor introduced by Mills34) will reflect the amount of bulk (noncoordinated) and coordinated water. A diffusion coefficient must be estimated for the water molecules in each of these environments. All the metal ions under consideration have a lifetime for a water molecule coordinated to them that is long compared with the time for a single diffusive step.33 These water molecules can therefore be considered to move through the solution with the cation as a single kinetic entity. Water molecules within the secondary hydration layer around the cation do not have a well-defined lifetime before exchange with other molecules. Evidence from neutron scattering data suggests that the overall lifetime is very short.32 The secondary hydration molecules are therefore assigned a diffusion coefficient Ds intermediate between Dm and Db, the bulk water value. This is given by the equation
Ds ) XDm + (1 - X)Db
(4)
where X is a fraction (0 < X < 1). Because of a lack of data to define the calculations, a value of X ) 0.5 is usually chosen; the effect of varying X is discussed below. The hydration layer surrounding the anion is also not regarded as strongly bound.33
Solute-Solute and Solute-Solvent Interactions
J. Phys. Chem., Vol. 100, No. 4, 1996 1409
TABLE 2: Total Cation Hydration Numbers from Diffusion Measurements from 0-2 m for a Number of Polyvalent Metal Salt Solutions salt
Nh
Nh* a
data sources
Fe(ClO4)3 FeCl3 CrCl3 Cr(ClO4) AlCl3 LaCl3 Zn(ClO4)2 NiCl2 MgCl2 Fe(ClO4)2 ZnCl2
14 16 14 19 14 14 9 9 9 9 17b
19 19 22 22 22 21 12 12 12 12
17 16 7 7 7 7 9 20 42 15 10
a N * represents hydration numbers obtained from the model h assuming no hydration contribution from the anion (i.e. Na ) fa ) 0). b Obtained over a restricted concentration range 0-1 m.
As these water molecules are similar to those in the secondary cation layers, they may be assigned a diffusion coefficient Da* intermediate between the experimental value of the anion Da and Db using an expression analogous to eq 4. The problem remaining is the assignment of a value to Db. An oversimplistic view would use the value for D0 ()2.3 × 10-9 m2 s-1 at 298.15 K), the intradiffusion coefficient of pure water, independent of the salt concentration and effects represented by the solution viscosity; this is obviously incorrect. To account for the effects measured by viscosity, one may use the Stokes-Einstein rule. However, there is ample evidence15,31,35 that this is an overcorrection. Instead, a modified form is preferred17,36 as shown in eq 5
Db ) D0(η0/η)R; 0 < R < 1
(5)
where η0/η is the viscosity ratio between the value for pure water and the value in the salt solution, and R is a variable fractional parameter. For a solution of molality m, of a general electrolyte MqXr, the fractions of water molecules in each of the primary (fm) and secondary (fs) cation hydration layers and in the hydration layer of the anion (fa) are fm ) Npmq/mw, fs ) Nsmq/mw, and fa ) Namr/mw, where mw is the molality of the water, and Np and Na are primary hydration numbers for the cation and the anion. Ns is the number of water molecules in the secondary hydration layer around the cation, which is to be estimated. Hence, the measured diffusion coefficient for water (H2O), Dw, is made up of four contributions as shown in eq 6
Dw ) fmDm + fsDs + faDa* + fbDb
(6)
where fb is the fraction of water in the bulk environment (fb ) 1 - fa - fm - fs; fb g 0). By using experimental diffusion data at each composition, an estimate for the total effective cation hydration number may be obtained (Nh ) Np + Ns). By fitting the experimental data over a concentration range, an average value may be calculated. Table 2 shows the results of applying the above model to a large number of 2:1 and 3:1 electrolytes. For the systems ZnCl2 and FeCl3 complexation of the metal ion by chloride ligands is significant enough to show differences in the diffusion data compared to similar noncomplexed systems. A modified hydration model allowing for the presence of complexes was used in these cases, and the reader is referred to earlier papers for details.10,16 Chromium chloride is thought to form complexes as evidenced by X-ray diffraction data.30 However, comparison of diffusion coefficient data7,37 for ions in Ga(ClO4)3, Cr(ClO4)3, and CrCl3 reveals very little difference. In
addition, recent experiments studying the intradiffusion of chloride and perchlorate ions in solutions of chromium chloride indicate no evidence of the effect of complexation of chloride upon the diffusion properties.38 Consequently, the hydration model used for CrCl3 ignores effects of complexation. In all cases studied, the best fit to the experimental data occurred for an R value in eq 5 of 0.4 ( 0.05; the values in the table are for R ) 0.40 for ease of comparison. In the full model, the effect of varying R by (12% changed the calculated value of Nh by about (2-3. A change in X of (0.1 changed the value of Nh by (1.5-2.5. When the hydration of the anion is set at zero, deviation of the experimental water diffusion coefficient from the best fit averaged over the entire concentration range 0-2 m was always less than 5%. The values for the primary hydration of the cation and the chloride ion are both taken as six, whereas the primary hydration for the perchlorate ion is taken to be four.39 Further manipulation of the parameter X to make the secondary shell water molecules more like the bulk environment (i.e. X > 0.7) does lead as one might expect to greater values for the secondary hydration number. For example, for nickel chloride solutions values of Nh ) 12, 26, and 33 are obtained from the model for X values of 0.7, 0.8, and 0.9, respectively. Similar values are found for other systems. However, the quality of the fit of the experimental water diffusion coefficient data to the predicted values is dramatically poorer across the concentration range (rmsd ) 10-30%). It should be remembered that the effective metal ion hydration obtained is an average over a concentration range. Metal ion hydration numbers of 30 are rather meaningless at concentrations of 2 m where water molecules are undoubtedly shared between ions. In addition, it should be noted that if a lower range of concentration is used (0-1 m) over which the average value of Nh is calculated, then higher estimates are obtained. For example, a calculated value of Nh ) 16 is found for Zn(ClO4)2. Whether this is a meaningful difference is arguable given the simplicity of the model. It is, however, consistent with notions of sharing of water molecules by ions at higher concentrations (Gurney cospheres) when there is less free water available. Comparisons, where possible, between the hydration numbers from the diffusion data and X-ray and neutron scattering results are good. For example, from X-ray scattering data30 the secondary hydration of chromium ions is found to be 12. From Table 2 a value of 13 is obtained from Cr(ClO4)3 diffusion data. In addition, there is similar good agreement for divalent species. X-ray data30 yields a secondary hydration of 5-6 for uncomplexed Ni2+; this is similar to the value in Table 2. In conclusion, it is found that this simple model fits the experimental data well and produces physically reasonable values for the effective hydration of the cations. The significant findings from the model may be summarized below: (i) The model is simple yet physically realistic and is based on the interpretation of neutron scattering evidence in terms of different water environments within a salt solution. (ii) The total (effective) cation hydration numbers are consistent for each of the valence types. The values obtained for La3+ and Mg2+ agree well with those obtained from compressibility and activity coefficient data.40 Where X-ray and neutron scattering data are available, the values in Table 2 are within the experimental uncertainty.22,30,32,41 The only anomaly appears to be NiCl2 where the neutron data yields an effective cation hydration of 21 ( 2 for Ni2+; this is much larger than the present values. (iii) Where the anion hydration is ignored (effectively making fa ) 0 in eq 5), the difference between Nh for perchlorate
1410 J. Phys. Chem., Vol. 100, No. 4, 1996 solutions and chloride solutions of the same metal ion is not significant. The same is true for the hydration numbers obtained from the full model. (iv) The specific inclusion of anion hydration has a strong influence on the values of the cation Nh. It is likely that use of the anion data may bias the calculated Nh value, particularly at higher concentrations where competition between the ions for water molecules becomes significant in the model. For example, in a 2 m solution of CrCl3 the fraction of water associated with the anions for a time is 0.65. At concentrations where competition for water molecules is high, it is possible that the contribution of the anion toward the experimental water diffusion coefficient is less than that given by the model. If a value of X(anion) ) 0.4 is adopted in eq 5, then a higher value of Nh ) 16-17 is obtained for Cr3+ in CrCl3. It is likely that the lifetime of a water molecule within a hydration layer (particularly in that of the anion and the secondary hydration layer of the cation) is concentration dependent. It is possible that computer simulation can provide better estimates of Da and Ds. (v) The use of a value between that of the ion and the bulk water for the diffusion coefficient of the water in the shortlived secondary hydration layer of the cation and the primary anion layer is phyically quite realistic in that it allows for sharing of water molecules at high concentration. References and Notes (1) Friedman, H. L. Annu. ReV. Phys. Chem. 1981, 32, 179. (2) Friedman, H. L. Faraday Discuss. Chem. Soc. 1988, 85, 1. (3) Friedman, H. L.; Raineri, F. O.; Wood, M. D. Chem. Scr. 1989, 29A, 49. (4) Zhong, E. C.; Friedman, H. L. J. Phys. Chem. 1989, 92, 1685. (5) Mills, R.; Hertz, H. G. J. Chem. Soc., Faraday Trans. 1 1982, 78, 3287. (6) Weingartner, H.; et al. J. Phys. Chem. 1993, 97, 6289. (7) Price, W. E.; Weingartner, H. J. Phys. Chem. 1991, 95, 8933. (8) Hertz, H. G. Ber. Bunsen-Ges. Phys. Chem. 1978, 81, 657. (9) Price, W. E.; Woolf, L. A.; Harris, K. R. J. Phys. Chem. 1990, 94, 5109. (10) Price, W. E.; Woolf, L. A. J. Solution Chem. 1995, 24, 211.
Price et al. (11) Friedman, H. L. Friends of Diffusion, 1988, private communication. (12) Friedman, H. L.; Mills, R. J. Solution Chem. 1986, 12, 69. (13) Friedman, H. L.; Mills, R. J. Solution Chem. 1981, 10, 395. (14) For references, see: Miller, D. G. J. Phys. Chem. 1981, 85, 1137. (15) Easteal, A. J.; Price, W. E.; Woolf, L. A. J. Phys. Chem. 1989, 93, 7517. (16) Easteal, A. J.; et al. J. Solution Chem. 1991, 20, 319. (17) Price, W. E.; Woolf, L. A. Ber. Bunsen-Ges. Phys. Chem. 1990, 94, 381. (18) Price, W. E.; Woolf, L. A. J. Solution Chem. 1993, 22, 873. (19) Price, W. E.; Woolf, L. A. J. Solution Chem. 1992, 21, 239. (20) Stokes, R. H.; Phang, S.; Mills, R. J. Solution Chem. 1979, 8, 489. (21) Rard, J. A.; Miller, D. G.; Lee, C. M. J. Chem. Soc., Faraday Trans. 1 1989, 85, 3343. (22) Howe, R. A.; Howells, W. S.; Enderby, J. E. J. Phys. (Paris), Lett. 1974, L111. (23) Friedman, H. L.; Dudowicz, J. B. The Problem of Long-Range Forces in Computer Simulation. Report of NRCC Workshop, 1980; p 8. (24) Geiger, A.; Hertz, H. G.; Mills, R. J. Solution Chem. 1981, 10, 83. (25) Mills, R.; Malhotra, R.; Woolf, L. A.; Miller, D. G. J. Phys. Chem. 1994, 98, 5565. (26) Hewish, N. A.; Enderby, J. E.; Howells, W. S. J. Phys. C 1983, 16, 1777. (27) Johansson, G. AdV. Inorg. Chem. 1992, 39, 159. (28) Bergstrom, P.-A.; Lindgren, J.; Read, M.; Sandstrom, M. J. Phys. Chem. 1991, 95, 7650. (29) Danf, L. X.; et al. J. Am. Chem. Soc. 1991, 113, 2481. (30) Magini, M.; et al. X-ray Diffraction of Ions in Aqueous Solutions; CRC Press: Boca Raton, FL, 1988. (31) Easteal, A. J.; Mills, R.; Woolf, L. A. J. Phys. Chem. 1989, 93, 4968. (32) Salmon, P. S.; Howells, W. S.; Mills, R. J. Phys. C 1987, 20, 5727. (33) Friedman, H. L. Chem. Scr. 1985, 25, 42. (34) Mills, R. J. Phys. Chem. 1973, 77, 685. (35) Stokes, R. H.; Mills, R.; Woolf, L. A. J. Phys. Chem. 1957, 61, 1634. (36) Zwanzig, R.; Harrison, A. K. J. Chem. Phys. 1985, 83, 5861. (37) Price, W. E.; Woolf, L. A. J. Solution Chem. 1993, 22, 837. (38) Woolf, L. A. Unpublished data. (39) Bergstrom, P.-A.; Lindgren, J. J. Mol. Struct. 1991, 245, 221. (40) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Butterworths: London, 1959. (41) Neilson, G. W.; Howe, R. A.; Enderby, J. E. Chem. Phys. Lett. 1975, 33, 284. (42) Mills, R.; Lobo, V. M. M. Self-Diffusion in Electrolyte Solutions; Elsevier: Amsterdam, 1989.
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