Page 343 - MODERN ELECTROCHEMISTRY

ION–ION INTERACTIONS 279

This individual ionic-activity coefficient can be transformed into a mean ionic-
activity coefficient by the same procedure as for the Debye–Hückel limiting law (see
Section 3.4.4). On going through the algebra, one finds that the expression for log
in the finite-ion-size model is

It will be recalled, however, that the thickness of the ionic cloud can be written
as [Eq. (3.85)]

Using this notation, one ends up with the final expression

If one compares Eq. (3.119) of the finite-ion-size model with Eq. (3.90) of the
point-charge approximation, it is clear that the only difference between the two
expressions is that the former contains a term in the denominator. Now,
one of the tests of a more general version of a theory is the correspondence principle;
i.e., the general version of a theory must reduce to the approximate version under the
conditions of applicability of the latter. Does Eq. (3.119) from the finite-ion-size model
reduce to Eq. (3.90) from the point-charge model?
Rewrite Eq. (3.119) in the form

and consider the term As the solution becomes increasingly dilute, the radius
of the ionic cloud becomes increasingly large compared with the ion size, and
simultaneously becomes increasingly small compared with unity, or

Thus, when the solution is sufficiently dilute to make i.e., to make the ion
size insignificant in comparison with the radius of the ion atmosphere,the finite-ion-
size model Eq (3.119) reduces to the corresponding Eq. (3.90) of the point-charge
model because the extra term tends to unity