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ION–ION INTERACTIONS 243

Further, one has the linearized P–B equation

From these two equations, one has the linear relation between excess charge density
and potential, i.e.,

and by inserting the solution (3.33) for the linearized P–B equation, the result is

Here then is the desired expression for the spatial distribution of the charge density
with distance r from the central ion (Fig. 3.10). Since the excess charge density results
from an unequal distribution of positive and negative ions, Eq. (3.35) also describes
the distribution of ions around a reference or sample ion.
To understand this distribution of ions, however, one must be sufficiently attuned
to mathematical language to read the physical significance of Eq. (3.35). The physical
ideas implicit in the distribution will therefore be stated in pictorial terms. One can say
that the central reference ion is surrounded by a cloud, or atmosphere, of excess charge
(Fig. 3.11). This ionic cloud extends into the solution (i.e., r increases), and the excess

Fig. 3.10. The variation of
the excess charge density
as a function of distance from
the central ion.