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300 CHAPTER 3

the positive and negative ions and is the hydration number of the
electrolyte.
It has been found that, in the case of several electrolytes, the values of the
hydration numbers obtained by fitting the theory [Eq. (3.130)] to experiment are in
reasonable agreement with hydration numbers determined by independent methods
(Table 3.14). Alternatively, one can say that, when independently obtained hydration
numbers are substituted in Eq. (3.130), the resulting values of show fair
agreement with experiment.
In conclusion, therefore, it may be said that the treatment of the influence of
ion–solvent interactions on ion–ion interactions has extended the range of concentra-
tion of an ionic solution which is accessible to theory. Whereas the finite-ion-size
version of the Debye–Hückel theory did not permit theory to deal with solutions in a
range of concentrations corresponding to those of real life, Eq. (3.130) advances theory
into the range of practical concentrations. Apart from this numerical agreement with
experiment, Eq. (3.130) unites two basic aspects of the situation inside an electrolytic
solution, namely, ion–solvent interactions and ion–ion interactions.

3.7. THE SO-CALLED “RIGOROUS” SOLUTIONS OF THE
POISSON–BOLTZMANN EQUATION

One approach to understanding the discrepancies between the experimental
values of the activity coefficient and the predictions of the Debye–Hückel model has
just been described (Section 3.6); it involved a consideration of the influence of
solvation.
An alternative approach is based on the view that the failure of the Debye–Hückel
theory at high concentrations stems from the fact that the development of the theory
involved the linearization of the Boltzmann equation (see Section 3.3.5). If such a view
is taken, there is an obvious solution to the problem: instead of linearizing the