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ION–ION INTERACTIONS 241

Hence, if is an exponential function of r, one will obtain a differential equation of
the form of Eq. (3.25). In other words, the “primitive” or “origin” of the differential
equation must have had an exponential in
Two possible exponential functions, however, would lead to the same final
differential equation; one of them would have a positive exponent and the other a
negative one [Eq. (3.26)]. The general solution of the linearized P–B equation can
therefore be written as

where A and B are constants to be evaluated. Or, from Eq. (3.22),

The constant B is evaluated by using the boundary condition that far enough from
a central ion situated at r = 0, the thermal forces completely dominate the Coulombic
forces, which decrease as and there is electroneutrality (i.e., the electrostatic
potential vanishes at distances sufficiently far from such an ion, as
This condition would be satisfied only if B = 0. Thus, if B had a finite value,
Eq. (3.28) shows that the electrostatic potential would shoot up to infinity (i.e.,
a physically unreasonable proposition. Hence,

To evaluate the integration constant A, a hypothetical condition will be considered
in which the solution is so dilute and on the average the ions are so far apart that there
is a negligible interionic field. Further, the central ion is assumed to be a point charge,
i.e., to have a radius negligible compared with the distances otherwise to be considered.
Hence, the potential near the central ion is, in this special case, simply that due to an
isolated point charge of value
This is given directly from Coulomb’s law as

At the same time, for this hypothetical solution in which the concentration tends
to zero, i.e., it is seen from Eq. (3.20) that Thus, in Eq. (3.29),
and one has