Page 365 - MODERN ELECTROCHEMISTRY

ION–ION INTERACTIONS 301

Boltzmann equation, one can take the higher terms. Thus, one obtains the unlinearized
P–B equation

In the special case of a symmetrical electrolyte with equal concentrations
of positive and negative ions, i e., one gets

But

and therefore

or

By utilizing a suitable software, one could obtain from Eq. (3.134) so-called rigorous
solutions.
Before proceeding further, however, it is appropriate to stress a logical inconsis-
tency in working with the unlinearized P–B equation (3.131). The unlinearized
Boltzmann equation (3.10) implies a nonlinear relationship between charge density
and potential. In contrast, the linearized Boltzmann equation (3.16) implies a linear
relationship of to
Now, a linear charge density–potential relation is consistent with the law of
superposition of potentials, which states that the electrostatic potential at a point due
to an assembly of charges is the sum of the potentials due to the individual charges.
Thus, when one uses an unlinearized P–B equation, one is assuming the validity of
the law of superposition of potentials in the Poisson equation and its invalidity in the
Boltzmann equation. This is a basic logical inconsistency which must reveal itself in
the predictions that emerge from the so-called rigorous solutions. This is indeed the
case, as will be shown below.