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ION–ION INTERACTIONS 291

An obvious improvement of the theory consisted in removing the assumption of
point-charge ions and taking into account their finite size. With the use of an ion size
parameter a, the expression for the mean ionic-activity coefficient became

However, the value of the ion size parameter a could not be theoretically
evaluated. Hence, an experimentally calibrated value was used. With this calibrated
value for a, the values of at other concentrations [calculated from Eq. (3.119)] were
compared with experiment.
The finite-ion-size model yielded agreement with experiment at concentrations
up to 0.1 N. It also introduced through the value of a, the ion size parameter, a
specificity to the electrolyte (making NaCl different from KC1), whereas the point-
charge model yielded activity coefficients that depended only upon the valence type
of electrolyte. Thus, while the limiting law sees only the charges on the ions, it is blind
to the specific characteristics that an ionic species may have, and this defect is
overcome by the finite-ion-size model.
Unfortunately, the value ofa obtained from experiment by Eq. (3.119) varies with
concentration (as it would not if it represented simply the collisional diameters), and
–3
as the concentration increases beyond about 0.1 mol dm , a sometimes has to assume
physically impossible (e.g., negative) values. Evidently these changes demanded by
experiment not only reflect real changes in the sizes of ions but represent other effects
neglected in the simplifying Debye–Hückel model. Hence, the basic postulates of the
Debye–Hückel model must be scrutinized.
The basic postulates can be put down as follows: (1) The central ion sees the
surrounding ions in the form of a smoothed-out charge density and not as discrete
charges. (2) All the ions in the electrolytic solution are free to contribute to the charge
density and there is, for instance, no pairing up of positive and negative ions to form
any electrically neutral couples. (3) Only long-range Coulombic forces are relevant to
ion–ion interactions; short-range non-Coulombic forces, such as dispersion forces,
play a negligible role. (4) The solution is sufficiently dilute to make [which depends
on concentration through —cf. Eq. (3.20)] small enough to warrant the linearization
of the Boltzmann equation (3.10). (5) The only role of the solvent is to provide a
dielectric medium for the operation of interionic forces; i.e., the removal of a number
of ions from the solvent to cling more or less permanently to ions other than the central
ion is neglected.
It is because it is implicitly attempting to represent all these various aspects of the
real situation inside an ionic solution that the experimentally calibrated ion size
parameter varies with concentration. Of course, a certain amount of concentration
variation of the ion size parameter is understandable because the parameter depends
upon the radius of solvated ions and this time-averaged radius might be expected to