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decrease with an increase of concentration. One must try to isolate that part of the
changes in the ion size parameter that does not reflect real changes in the sizes of ions
but represents the impact of, for instance, ionic solvation upon activity coefficients.
This question of the influence of ion–solvent interactions (Chapter 2) upon the ion–ion
interactions will be considered in Section 3.6.

3.5.7. Parentage of the Theory of Ion–Ion Interactions
Stress has been laid on the contribution of Debye and Hückel (1923) to the
development of the theory of ion–ion interactions. It was Debye and Hückel who
ushered in the electrostatic theory of ionic solutions and worked out predictions that
precisely fitted experiments for sufficiently low concentrations of ions. It is not often
realized, however, that the credit due to Debye and Hückel as the parents of the theory
of ionic solutions is the credit that is quite justifiably accorded to foster parents. The
true parents were Milner and Gouy. These authors made important contributions very
early in the growth of the theory of ion–ion interactions.
Milner’s contribution (1912) was direct. He attempted to find out the virial 18
equation for a mixture of ions. However. Milner’s statistical mechanical approach
lacked the mathematical simplicity of the ionic-cloud model of Debye and Hückel and
proved too unwieldy to yield a general solution testable by experiment. Nevertheless,
his contribution was a seminal one in that for the first time the behavior of an ionic
solution had been linked mathematically to the interionic forces.
19
The contribution of Gouy (1910) was indirect. Milner’s treatment was not
sufficiently fruitful because he did not formulate a mathematically treatable model.
Gouy developed such a model in his treatment of the distribution of the excess charge
density in the solution near an electrode. Whereas Milner sought to describe the
interactions between series of discrete ions, it was Gouy who suggested the smoothing
out of the ionic charges into a continuous distribution of charge and took the vital step
of using Poisson’s equation to relate the electrostatic potential and the charge density
in the continuum. Thus, Gouy was the first to evolve the ionic-atmosphere model.
It was with an awareness of the work of Milner and Gouy that Debye and Hückel
attacked the problem. Their contributions, however, were vital ones. By choosing one
ion out of the ionic solution and making an analogy between this charged reference
ion and the charged electrode of Gouy, by using the Gouy type of approach to obtain
the variation of charge density and potential with distance from the central ion and

18
Virial is derived from the Latin word for force, and the virial equation of state is a relationship between
pressure, volume, and temperature of the form

where .... the virial coefficients, represent interactions between constituent particles.
19
Chapman made an independent contribution in 1913 on the same lines as that of Gouy.