KCl in water at supercritical temperatures,3 made use of an expression in which

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1 A REPRESENTATION OF ISOTHERMAL ION-ION-PAIR-SOLVENT EQUILIBRIA INDEPENDENT OF CHANGES IN DIELECTRIC CONSTANT* By WILLIAM L. MARSHALL AND ARVIN S. QUIST REACTOR CHEMISTRY DIVISION, OAK RIDGE NATIONAL LABORATORY, OAK RIDGE, TENNESSEE Communicated by Raymond M. Fuoss, July 17, 1967 Changes in isothermal equilibrium constants (K) involving electrolytes in aqueous-organic or organic-organic solvents have almost universally been correlated with changes in the dielectric constant (D) of the solvent mixtures.1 In accord with this concept, a plot of log K against 1/D should yield a straight line with a slope that is a function of the ion size parameter. However, such linear behavior is not observed when precise values for K are tested, for example, those for waterdioxane solvent mixtures varying over a wide range of composition.2 We have noted, however, that Franck, in studies 11 years ago of equilibrium constants for KCl in water at supercritical temperatures,3 made use of an expression in which the concentration of water was included to express its participation as a reactant in the equilibrium process. His values of KO = a[k(aq)+] a [Cl(aq)-]/a [KC(aq) ] [a(h2) ]n (1) = K/ [a(h2) ]n = K/ [C(H2) ln, where activities are in moles/liter, K is the value at infinite dilution of electrolyte, and a(h2) = C(H2) were nearly independent of density. More recent work in this laboratory has extended these results to cover the range of 4 to 8C for KHS4 and NaCl and solvent densities from.3 to.75 gm/cm3,4' 6 and our values of KO have been found constant to better than ± 12 per cent at constant temperature over nearly the entire range of density.5 Corresponding values of the conventional constant, K, change by a factor of 3. From the constancy of KO over the wide range of density, it became evident to us that KO was also independent of changes in dielectric constant. We believe that Franck's earlier observation is a manifestation of a much more general principle-that at constant temperature, correct values of K are independent of changes in dielectric constant not only for water systems at high temperature but for all other solvent-electrolyte systems. The currently accepted theory of equilibria involving ion-pair formation1 is based on a sphere in a continuum model and considers that the dielectric constant of the solvent is the controlling factor in association. However, in recent years evidence has been accumulating6' 7 to indicate that ion-solvent interaction must be considered and that the dielectric constant of the solvent mixture may not be the only significant factor in ion association. The ion-solvent interactions depend upon the charge on the ion and the dipole moment of the individual solvent molecules. Since the ionic charge is invariant and the dipole moments of the solvent molecules would be expected to remain constant (at constant temperature), then the ion-solvent interaction energy would remain constant (at constant temperature) over wide ranges of solvent concentrations. In ion-pair formation, the expected solvation of the ion pair would 91

2 92 CHEMISTRY: MARSHALL AND QUIST PROC. N. A. S. involve dipole-dipole interactions, which would also be expected to remain constant at constant temperature. The complete equilibrium in ion-pair formation thus includessolvation of ions and ion pairs, as well as the interaction between the solvated ions to form the solvated ion pair. With this concept, the correct constant for the equilibrium should indeed be a constant (at constant temperature) and would correspond to the KO in equation (1) where [C(H2) I" could be replaced by Cn(reactive solvent) for some other reactive solvent. To test this principle with respect to water-containing solvents, we have examined the extensive data for electrolytes at 25C in organic-water solvent mixtures, particularly dioxane-water mixtures where dioxane may be assumed to be an "inert" or unreactive diluent. Dioxane does not appear to affect the hydration of ions since water is much more polar than dioxane, and consequently water molecules should preferably (selectively) solvate the ions.8'9 Thus, a plot of log K against log C(H2) (in moles/liter) in dioxane-water mixtures at constant temperature should yield a straight line with a slope corresponding to n, the number of moles of water in the reaction equilibrium (see eq. (1)). The KO obtained from this plot would then be independent of changes in dielectric constant. The examination of data for some 4 different electrolytes in various solvent mixtures verifying this concept, with additional interpretation and references, will be submitted for publication shortly. As one example of the typical agreement attained, a plot of the log K values of Kunze and Fuoss2 for the dissociation constant of NaCl at 25C in water-dioxane mixtures (54-8 wt. % dioxane) against log C(H2) yields a straight line with an approximate random deviation of ±.5 pk units (41% in K) and a slope, n, of +6.4 as shown in Figure 1. The corresponding plot2 of log K against 1/D (Fig. 1) yields a pronounced curve with a maximum deviation from a straight line (54-8 wt. % dioxane) of.25 pk unit (a factor of 1.8 in K). If the principle is correct, then it should apply also to other solvent systems. This proposition was tested for several organic solvent systems where one solvent component is relatively nonpolar and essentially acts as a diluent. The values for the dissociation constants of several quarternary ammonium salts in nitrobenzene-carbon tetrachloride solutions obtained by Hirsch and Fuoss' are shown in Figure 2 plotted as log K against the logarithm of the molar concentration of nitrobenzene. Thus, a linear relationship is again observed for these examples where nitrobenzene as the highly polar solvent behaves in the same fashion as water in equation (1). The deviation at very low nitrobenzene concentrations was accounted for by including minor solvation also by CCl4; this procedure gave a fit to the values of K for tetrabutylammonium picrate of about ±15 per cent over the entire set. On the same figure are shown the comparative nonlinear plots of log K against 1/D. The solubility of salts in water at high temperatures and pressures and in waterorganic solvent mixtures also depends primarily on the concentration of the solvating species. Morey," Kennedy,'2 Jasmund,13 and Franck'4 have shown this relationship between solubility at high temperatures and the density of the solvent, water. From the measured solubilities of CaSO4 in water at 25C at pressures to 1 bars, 15 we have found a linear relationship between the logarithm of the molar solubility and the logarithm of the molar concentration of water. A similar re-

3 VOL. 58, 1967 CHEMISTRY: MARSHALL AND QUIST r.8 4/(dielectric constant) e -2 5 z c -3.-._ ' o log CH2O(moles/liter) 4.5 FIG. 1.-Log K (dissociation) for sodium chloride vs. log C(H2) (moles/liter) in dioxane-water mixtures; also compared vs. 1/(dielectric constant, D); K values of Kunze and Fuoss.2' lationship was observed when the solubilities of NaCi at 6C1i were treated in a similar manner. Ricci and co-workers have measured the solubilities of several salts in water-dioxane mixtures.8' 17 By plotting the logarithm of molar solubility against the logarithm of water molarity, linear relationships are observed. For most examples of solubility, a constant slope is observed at low water concentrations corresponding to the change in waters of hydration upon dissolution to form the ion pair, the significant equilibrium under these conditions. Over a small range of increasing water concentration, this slope increases sharply to another constant value corresponding (at high water concentrations) to the extra waters of

4 94 CHEMISTRY: MARSHALL AND QUIST PROC. N. A. S. - U) cn.2 - * = Bu4N PICRATE o =Bu4N IODIDE a= Bu4N NITRATE VALUES OF K FROM -2 _ *HIRSCH AND FUOSS, [J. AM. CHEM. SOC., 82,448 (496O -8r _ l' z z V(dielectric constant) /~ <7 47' ox X.,/I X, (pl~ ifz" 7t KO.4 IIA PURPUEE NITROBENZENEEN `SLOPE (n)=6.4 OBU.N+ AA a an BU4NA NITROBENZENE NITROBENZENE I CNITROBENZENE (moles/liter).8 4. FIG. 2.-Log K (dissociation) for several tetrabutylammonium salts vs. log nitrobenzene molarity in carbon tetrachloride-nitrobenzene mixtures; also compared vs. 1/(dielectric constant, D); K values of Hirsch and Fuoss.1 hydration in forming the hydrated ions. The significant equilibrium here is that involving complete dissociation. Our generalized concept supports the findings of Ricci and Davis'8 that the mean activity coefficients of certain slightly soluble electrolytes in their saturated solutions in several different solvent systems were constant as calculated from Debye- Huckel theory. The concept also appears to be related to temperatures of formation of liquid-liquid immiscibility in aqueous uranyl sulfate solutions at 28-4C.'9 2 The principle can be used to calculate values for d, the important ion size parameter of the extended Debye-Huckel theory. From plots such as Figure 1, the value of n can be estimated. For NaCl, this value was 6.4. From the solubility

5 VOL. 58, 1'367 CHEMISTRY: MARSHALL AND QUIST 95 information to be presented in the larger article, a value of two molecules of H2 has been estimated for the number involved in the hydration of the neutral NaCl ion pair. The average ion volume then can be expressed as 7ave = [VNa+ + VC1 + (n + 2) (2VH+ + V2-)]/2, (2) where VNa +, VCI -, VH +, and Vo2 - are ionic volumes obtained from a tabulation of crystallographic radii2' (rna+ =.95 A, rcl- = 1.81 A, ro2- = 1.4 A), and 2VH+ is considered negligible compared with Vo2-. Since Vave = 1/6ir(d)3, then it follows that d for sodium chloride solutions is 4.9 A. This value is in very good agreement with that of 5.2 A given by Robinson and Stokes and obtained from the evaluation of transport numbers by extended Debye-Huckel theory.22 The constancy of the slope, n, of Figure 1 produces an ion size parameter that is independent of the added "inert" solvent. This constancy of n and KO also implies that any isothermal changes in the structure of water may be unimportant to the electrolyte-solvent equilibrium. The proposal presented here does not contradict Debye-Huckel theory and its use of the dielectric constant of the solvent in calculating mean ionic activity coefficients. Long-range interionic effects on ions are certainly still dependent on the solvent dielectrics. The concept indeed appears to support the extended Debye- Huckel theory by providing comparable hydrated ion size parameters. The actual value of KO will depend upon the dipole moments (and ionic charges) of the various reactants and products in the equilibrium. These values will change with temperature. Knowledge of these quantities might provide a means for an a priori calculation of KO. Although linear relationships between conventional equilibrium constants (not including waters of hydration) and some function of the concentration of water have been reported in the literature23-26 (usually applied to equilibria involving protons), we believe that this communication and subsequent article present for the first time a generalized concept applying over a wide range of temperature, pressure, and solvent composition, and used with molar concentration units. Thus, the inclusion of the solvent species as an active participant in equilibria appears to lead to expressions of considerably greater generality than those previously used for the description of electrolyte systems. * Work sponsored by the U.S. Atomic Energy Commission under contract with Union Carbide Corporation. 1 Fuoss, R. M., J. Am. Chem. Soc., 8, 559 (1958). 2 Kunze, R. W., and R. M. Fuoss, J. Phys. Chem., 67, 911 (1963). 3 Franck, E. U., Z. physik. Chem. (Frankfurt), 8, 17, 192 (1956). 4 Quist, A. S., and WV. L. Marshall, J. Phy8. Chem., 7, 3714 (1966). 5 Quist, A. S., and W. L. Marshall, "The electrical conductances of aqueous sodium chloride solutions to 8'C and 4 bars," J. Phys. Chem., in press. 6 D'Aprano, A., and R. M. Fuoss, J. Phys. Chem., 67, 174, 1722 (1963). 7Atkinson, G., and S. Petrucci, J. Am. Chem. Soc., 86, 7 (1964); J. Phys. Chem., 7, 3122 (1966). 8 Davis, T. W., J. E. Ricci, and C. G. Sauter, J. Am. Chem. Soc., 61, 3274 (1939). 9 Fratiello, A., and D. C. Douglass, J. Chem. Phys., 39, 217 (1963). 1 Hirsch, E., and R. M. Fuoss, J. Am. Chem. Soc., 82, 118 (196). 11 Morey, G. W., and J. M. Hesselgesser, Tram. Am. Soc. Mech. Eng., 73, 865 (1951). 12 Kennedy, G. C., Econ. Geol., 45, 629 (195).

6 96 CHEMISTRY: MARSHALL AND QUIST PROC. N. A. S. 13 Jasmund, K., Heidelberger Beitr. Mineral. Petrogr., 3, 38 (1953). 14 Franck, E. U., Z. physik. Chem. (Frankfurt), 6, 345 (1956). 15 Dickson, F. W., C. W. Blount, and G. Tunell, Am. J. Sci., 261, 61 (1963). 16 Sourirajan, S., and G. C. Kennedy, Am. J. Sci., 26, 115 (1962). 17 Ricci, J. E., and G. J. Nesse, J. Am. Chem. Soc., 64, 235 (1942). 18 Ricci, J. E., and T. W. Davis, J. Am. Chem. Soc., 62, 47 (194). 19 Secoy, C. H., J. Am. Chem. Soc., 72, 3343 (195). 2Marshall, W. L., and J. S. Gill, J. Inorg. Nucl. Chem., 25, 133 (1963). 21 Pauling, L., The Nature of the Chemical Bond (Ithaca, N. Y.: Cornell University Press, 1945). 22 Robinson, R. A., and R. H. Stokes, Electrolyte Solutions (London: Butterworth and Co., Ltd., 1965), 2d ed. rev., p Harned, H. S., and L. D. Fallon, J. Am. Chem. Soc., 61, 2377 (1939). 24 Hudson, R. F., and B. Saville, J. Chem. Soc. (1955), p Feakins, D., and C. M. French, J. Chem. Soc. (1957), p Aksnes, G., Acta Chem. Scand., 16, 1967 (1962).