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Bjerrum's theory includes approximations that are not fully justified: the ions are considered to be spheres, the dielectric constant in the vicinity of the ion is considered to be equal to that in the pure solvent, the possibility of interactions between ions other than pair formation (e.g. the formation of hydrogen bonds) is neglected and the effect of ion solvation during formation of ion pairs is not considered (the effect of the solvation on ion-pair structure is illustrated in Fig. 1.7). Although the Bjerrum theory is thus not in general quantitatively applicable, the concept of ion association is very useful. It has assisted in an explanation of various phenomena observed in the study of homogeneous
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Fig. 1.7 Possible hydration modes of an ion pair: (A) contact of primary hydration shells, (B) sharing of primary hydration shells, (C) direct contact of ions
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catalysis, electrical mobility, the behaviour of polyelectrolytes, etc. Ion pairs (A+B~) are formed in fused salts through a process in which the cations or anions of the solvent and of the solute exchange positions in the solvent lattice until the cations and anions occupy neighbouring positions. If the solvent is denoted as XY, then this process can be expressed by the scheme:
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( Y X + Y" X + Y"X +Y") +Y" + + + + + X B~X Y"A B "X ^rx + Y"X+ Y~ X + Y~X CY"
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According to M. Blander the association constant depends on the change in the Gibbs energy, AG, during exchange of a solvent anion Y~, next to a cation of the dissolved salt A + , for an anion of the dissolved salt B~ according to the relationship lnl += ^ (1-2.20)
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where n is the coordination number of the cation A + . The energy AG is proportional to the combination of the reciprocal values of the distances between the centres of the corresponding pairs of ions (i.e. the sum of the radii of these ions): (1.2.21) Clearly, it is the size of ions that is decisive in ion-pair formation. Moreover, the coulombic interactions can extend even to more distant neighbours. The Blander equation is then, of course, no longer applicable. References
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Ben-Nairn, A., Hydrophobic Interactions, Plenum Press, New York, 1980. Bloom, H., and J. O'M. Bockris, Molten electrolytes, MAE, 2, 262 (1959). Burger, K., Solvation, Ionic and Complex Formation Reactions in Non-aqueous Solvents, Elsevier, Amsterdam, 1983. Case, B., Ion solvation, 2 of Reactions of Molecules at Electrodes (Ed. N. S. Hush), Wiley-Interscience, London, 1971. Conway, B. E., Ionic Hydration in Chemistry and Biophysics, Elsevier, Amsterdam, 1981. Davies, C. W., Ion Association, Butterworths, London, 1962. Debye, P., Polar Molecules, Dover Publication Co., New York, 1945. Denison, J. T., and J. B. Ramsey, /. Am. Chem. Soc, 11, 2615 (1955). Desnoyers, J. E., and C. Jolicoeur, Ionic solvation, CTE, 5, Chap. 1 (1982). Dogonadze, R. R., E. Kalman, A. A. Kornyshev, and J. Ulstrup (Eds), The Chemical Physics of Solvation, Elsevier, Amsterdam, 1985. Eisenberg, D., and W. Kautzmann, The Structure and Properties of Water, Oxford University Press, Oxford, 1969.
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28 Fuoss, R. M., and F. Accascina, Electrolytic Conductance, Intersceince Publishers, New York, 1959. Gilkerson, W. R., /. Chem. Phys., 25, 1199 (1956). Inmann, D., and D. G. Lovering (Eds), Ionic Liquids, Plenum Press, New York, 1981. Klotz, I. M., Structure of water, in Membranes and Ion Transport (Ed. E. E. Bittar), Vol. I, John Wiley & Sons, New York, 1970. Mamantov, G. (Ed.), Characterization of Non-aqueous Solvents, Plenum Press, New York, 1978. Marcus, Y., Ion Solvation, John Wiley & Sons, Chichester, 1986. Papatheodorou, G. W., Structure and thermodynamics of molten salts, CTE, 5, Chap. 5 (1982). Samoilov, O. Ya., Structure of Aqueous Electrolyte Solutions and Hydration of Ions, Consultants Bureau, New York, 1965. Skarda, V., J. Rais, and M. Kyrs, /. Nucl. Inorg. Chem., 41, 1443 (1979). Sundheim, G., Fused Salts, McGraw-Hill, New York, 1964. Tanaka, N., H. Ohtuki, and R. Tamamushi (Eds), Ions and Molecules in Solution, Elsevier, Amsterdam, 1983. Tanford, C , The Hydrophobic Effect: Formation of Micelles and Biological Membranes, 2nd ed., John Wiley & Sons, New York, 1980. Tremillon, B., Chemistry in Non-aqueous Solvents, Reidel, Dordrecht, 1974. 1.3 Interionic Interactions
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Thermodynamics describes the behaviour of systems in terms of quantities and functions of state, but cannot express these quantities in terms of model concepts and assumptions on the structure of the system, intermolecular forces, etc. This is also true of the activity coefficients; thermodynamics defines these quantities and gives their dependence on the temperature, pressure and composition, but cannot interpret them from the point of view of intermolecular interactions. Every theoretical expression of the activity coefficients as a function of the composition of the solution is necessarily based on extrathermodynamic, mainly statistical concepts. This approach makes it possible to elaborate quantitatively the theory of individual activity coefficients. Their values are of paramount importance, for example, for operational definition of the pH and its potentiometric determination (Section 3.3.2), for potentiometric measurement with ionselective electrodes (Section 6.3), in general for all the systems where liquid junctions appear (Section 2.5.3), etc. The expression for the chemical potential of a component of a real solution can be separated into two terms: & -tf= AjU, = RT \nxi + RT In y,= A^, i d +A M f (1.3.1)
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where Afiild = RT\nXi is the difference in the chemical potentials between the standard and actual states, for ideal behaviour, and Apf is the correction for the real behaviour, which can be identified with the expression containing the activity coefficient. It can further be written for all
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