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ION–ION INTERACTIONS 289

The result of equating these two expressions for the excess charge density is the
fundamental partial differential equation of the Debye–Hückel model, the linearized
P–B equation (Fig. 3.36),

where

By assuming that ions can be regarded as point charges, the solution of the
linearized P–B equation turns out to be (Fig. 3.37)

Such a variation of potential with distance from a typical (central or reference) ion
corresponds to a charge distribution that can be expressed as a function of distance r
from the central ion by

This variation of the excess charge density with distance around the central or
typical ion yielded a simple physical picture. A reference positive ion can be thought
of as being surrounded by a cloud of negative charge of radius . The charge density
in this ionic atmosphere, or ionic cloud, decays in the manner indicated by Eq. (3.35).
Thus, the interactions between a reference ion and the surrounding ions of the solution
are equivalent to the interactions between the reference ion and the ionic cloud, which
in the point-charge model set up at the central ion a potential given by

The magnitude of central ion–ionic-cloud interactions is given by introducing the
expression for into the expression (3.3) for the work of creating the ionic cloud,
i.e., setting up the ionic interaction situation. Thus, one obtains for the energy of such
interactions