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236 CHAPTER 3

Fig. 3.8. At a distance r from the
reference ion, the excess charge
density and electrostatic potential in
an infinitesimal volume element dV
are and respectively.

3.3.4. Excess Charge Density near the Central Ion Is Given by a
Classical Law for the Distribution of Point Charges in a
Coulombic Field

The excess charge density in the volume element dV is equal to the total ion density
(total number of ions per unit volume) times the charge on these ions. Let there be, per
unit volume, ions of type 1, each bearing charge of type 2 with charge
and of type i with charge where is the valence of the ion and is the
electronic charge. Then, the excess charge density in the volume element dV is given
by

In order to proceed further, one must link up the unknown quantities
to known quantities. The link is made on the basis of the Boltzmann
distribution law of classical statistical mechanics. Thus, one writes

where U can be described either as the change in potential energy of the i particles
when their concentration in the volume element dV is changed from the bulk value
to or as the work that must be done by a hypothetical external agency against the
time average of the electrical and other forces between ions in producing the above