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286 CHAPTER 3

3.5.6. The Debye–Hückel Theory of Ionic Solutions: An Assessment
It is appropriate at this stage to recount the achievements in the theory of ionic
solutions described thus far. Starting with the point of view that ion–ion interactions
are bound to operate in an electrolytic solution, in going from a hypothetical state of
noninteracting ions to a state in which the ions of species i interact with the ionic
solution, the chemical-potential change was considered a quantitative measure
of these interactions. As a first approximation, the ion–ion interactions were assumed
to be purely Coulombic in origin. Hence, the chemical-potential change arising from
the interactions of species i with the electrolytic solution is given by the Avogadro
number times the electrostatic work W resulting from taking a discharged reference
ion and charging it up in the solution to its final charge In other words, the charging
work is given by the same formula as that used in the Born theory of solvation, i.e.,

where is the electrostatic potential at the surface of the reference ion that is
contributed by the other ions in the ionic solution. The problem therefore was to obtain
a theoretical expression for the potential This involved an understanding of the
distribution of ions around a given reference ion.
It was in tackling this apparently complicated task that appeal was made to the
Debye–Hückel simplifying model for the distribution of ions in an ionic solution. This
model treats only one ion—the central ion—as a discrete charge, the charge of the
other ions being smoothed out to give a continuous charge density. Because of the
tendency of negative charge to accumulate near a positive ion, and vice versa, the
smoothed-out positive and negative charge densities do not cancel out; rather, their
imbalance gives rise to an excess local charge density which of course dies away
toward zero as the distance from the central ion is increased. Thus, the calculation of
the distribution of ions in an electrolytic solution reduces to the calculation of the
variation of excess charge density with distance r from the central ion.
The excess charge density was taken to be given, on the one hand, by Poisson’s
equation of electrostatics

and on the other, by the linearized Boltzmann distribution law