Page 369 - MODERN ELECTROCHEMISTRY

ION–ION INTERACTIONS 305

Fig. 3.42. The probability of finding
an ion of charge in a dr-thick
spherical shell of radius r around a
reference ion of charge

proportional, first, to the ratio of the volume of the shell to the total volume V
of the solution; second, to the total number of negative ions present; and third, to
the Boltzmann factor exp(–U/kT), where U is the potential energy of a negative ion
at a distance r from a cation, i.e.,

Since is the concentration of negative ions in the solution and

it is clear that

or, writing

one has

A similar equation is valid for the probability of finding a positive ion in a dr-thick
shell at a radius r from a reference negative ion. Hence, in general, one may write for