Page 296 - MODERN ELECTROCHEMISTRY

232 CHAPTER 3

Then as the work of charging an electrical conductor is proved to be

W = 1/2 [(charge on the conductor) × (conductor’s electrostatic potential)] (3.2)

one obtains

where is the electrostatic potential of the ion due to the influence on it by the
electrostatic interactions of the surrounding field.
The essence of the task therefore in computing the chemical-potential change due
to the interactions of the ionic species i with the ionic solution is the calculation of the
electrostatic potential produced at a reference ion by the rest of the ions in solution.
Theory must aim at this quantity.
If one knew the time-averaged spatial distribution of the ions, then one could find
out how all the other charges are distributed as a function of distance from the reference
ion. At that stage, one of the fundamental laws of electrostatics could be used, namely,
the law of the superposition of potentials, according to which the potential at a point
due to an assembly of charges is the sum of the potentials due to each of the charges
in the assembly.
Thus, the problem of calculating the chemical-potential change due to the
interactions between one ionic species and the assembly of all the other ions has been
reduced to the following problem: On a time average, how are the ions distributed
around any specified ion? If that distribution became known, it would then be easy to
calculate the electrostatic potential of the specified ion due to the other ions and then,
by Eq. (3.3), the energy of that interaction. Thus, the task is to develop a model that
describes the equilibrium spatial distribution of ions inside an electrolytic solution and
then to describe that model mathematically.

3.3.2. A Prelude to the Ionic-Cloud Theory
A spectacular advance in the understanding of the distribution of charges around
an ion in solution was achieved in 1923 by Debye and Hückel. It is as significant in
the understanding of ionic solutions as the Maxwell theory of the distribution of
velocities is in the understanding of gases.
Before going into the details of their theory, a moment’s reflection on the
magnitude of the problem will promote appreciation of their achievement. Consider,
for example, a aqueous solution of sodium chloride. There will be
sodium ions per cubic centimeter of solution and the same number
of chloride ions, together, of course, with the corresponding number of water mole-
cules. Nature takes these ions and arranges them so that there