Page 197 - Modern physical chemistry

8.9 Free Energy of the Ionic Atmosphere 189

8.9 Free Energy of the Ionic Atmosphere

If only ion j were present, it would impose the potential
z·e
A. _ J [8.77]
'I'ion - 4ner
on its surroundings. But when the ionic strength J1 is zero, b is zero and formula (8.76)
reduces to
A
tPion =-. [8.78]
r
These expressions are the same when

A= zje. [8.79]
4ne
So equation (8.76) becomes
A. = Z je -br = Z je (1- b ) [8.80]
'I' e r+ ....
4ner 4ner
For dilute solutions, parameter b is small. Higher terms in expansion (8.80) can then
be neglected. Subtracting out tPion from the result yields

tP - _ zjeb [8.81]
attn - 4ne'
the contribution to potential ¢ from the atmosphere of the ion.
To obtain the work done in setting up the atmosphere, one begins with the central
ion discharged, Electric charge is then brought up from infinity continuously. At a given
stage, the fraction of final charge on ionj is! Then zJe in equation (8.81) is replaced with
zjej and the next charge brought up to the central ion is zJed!
The total work done against the potential due to the atmosphere is

2
)
(
W= f l tPatmzjeq[ =- Ilz.2e2b z.2e b . [8.82]
_J __ jdj= __ J __
/=0 0 4ne 8ne
Since this is net work done reversibly, it is the Gibbs energy associated with setting up
the ionic atmosphere:
2 2
ilG atm =_ Zj e b. [8.83]
8ne
The approximation applied to expression (8.80) may be improved as follows.
In the argument, expression e-Or - 1 was replaced with -br. But a better average approx-
imation to

[8.84]

is achieved by constructing

[8.85]