Page 196 - Modern physical chemistry

ISS Equilibria in Condensed Phases

8.8 Variation of Electric Potential in an Ionic Atmosphere

Equation (8.63) applies to any spherically symmetric distribution of charge. On the
other hand, condition (8.49) describes the effect of thermal agitation on the distribution
of ions in a potential field. Combining these equations gives us

[8.64J

where we set
2
2000N Ae d _ b2 [8.65J
ekT J.l-.
Let us rearrange the overall equation (8.64),
2
:2 (riP )-b (riP) = 0, [8.66J

and factor the operator to get
[8.67J
or
[8.68J
where
d
D=-. [8.69J
dr
Equation (8.67) is satisfied when
(D+bXriP) = 0, [8.70J
while equation (8.68) is satisfied when

[8.71 J

The solution of (8.70) is
[8.72J

while the solution of (8.71) is
[8.73J
So equation (8.66) is satisfied by both (8.72) and (8.73); we have the general solution

[8.74J
or
e- br
ebr
iP=A-+B-. [8.75J
r r
If B were different from zero, the last term, and iP, would increase without limit as r
increases. But this is not allowed. For the ionic atmosphere, we have
-br
e
iP=A--. [8.76J
r
Integration constant A is determined by the potential at the surface of the ion.