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ION–ION INTERACTIONS 239

Notice that one has obtained two expressions for the charge density in the
volume element dV at a distance r from the central ion. One has the Poisson equation
[Eq. (3.4)]

and one also has the “linearized” Boltzmann distribution

where refers to the summation over all species of ions typified by i.
If one equates these two expressions, one can obtain the linearized Poisson–
Boltzmann (P–B) expression

The constants in the right-hand parentheses can all be lumped together and called
a new constant i.e.,

At this point, the symbol is used only to reduce the tedium of writing. It turns
out later, however, that is not only a shorthand symbol; it contains information
concerning several fundamental aspects of the distribution of ions around an ion in
solution. In Chapter 6 it will be shown that it also contains information concerning the
distribution of charges near a metal surface in contact with an ionic solution. In terms
of the linearized P–B expression (3.19) is

3.3.7. Solution of the Linearized P–B Equation
The rather messy-looking linearized P–B equation (3.21) can be tidied up by a
mathematical trick. Introducing a new variable defined by