Page 193 - Modern physical chemistry

8.6 An Integral Form of Coulomb sLaw 785

8.5 Average Electric Atmosphere about an Ion

Each ion in an electrolytic solution attracts oppositely charged ions and repels like
charged ions. This organizing influence is opposed by the random motion associated with
the internal energy of the system. In the resulting compromise, each ion is surrounded
by a neighborhood of considerable size containing, on the average, an excess of oppo-
sitely charged ions. A positive ion thus sports a negative atmosphere; a negative ion, a
positive atmosphere. Since it is more difficult to remove an ion from its charged atmos-
phere than from an environment of neutral solvent molecules, the activity of the ion is
reduced below what its concentration would indicate.
Consider a representative ion j about which the average electric potential is ¢>(r) ,
where r is the distance from the center of the ion and the ion is taken to be spherically
symmetric. The potential energy of an atmosphere ion carrying charge q.i is qi¢> .
The average charge density at a given point in the atmosphere equals the charge qj
multiplied by the number density of ions NAc; summed over the ionic species. With Cj the
moles of i per 1000 cm 3 , we multiply by 1000, the number of 1000 cm in a cubic meter.
3
In the potential field, average concentration c i is given by the Boltzmann distribution law
(3.81). Thus, we construct the formula

[8.45]
As long as ¢>/T is small enough, the exponential may be expanded and the higher terms
neglected:
[8.46]

The first term vanishes because of the electrical neutrality condition.
If z; is the number of units of charge on ion i while e is the charge on a proton, we have

[8.47]
Also from (8.24), we have
[8.48]

where mj is the molality of ion i in the solution and d is the density of the solution. Sub-
stituting these into the final form of (8.46) leads to the average charge density

[8.49]

where
[8.50]

Parameter J.l is called the ionic strength of the solution.
Formula (8.49) describes how the electric potential ¢> entails a nonzero charge density
p. Ions are distributed by thermal agitation up and down the potential slopes, following
the Boltzmann equation (3.81).
8.6 An Integral Form of Coulomb 's Law

The potential ¢> arises from the electric forces acting between the charges. These are
described by Coulomb's law.