Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/409

unit electric field, the charge transferred across unit area in unit time by the anions is

${\frac {v\nu c}{M}}\left(-{\frac {dV}{dx}}+{\frac {RT}{c\nu }}{\frac {dc}{dx}}\right)$.

We have therefore, if the total current be denoted by i,

$i-(u+v){\frac {\nu c}{M}}{\frac {dV}{dx}}-(u-v){\frac {RT}{M}}{\frac {dc}{dx}}$,

or

$-{\frac {dV}{dx}}dx={\frac {Mdx}{(u+v)\nu c}}i+{\frac {u-v}{u+v}}{\frac {RT}{\nu c}}{\frac {dc}{dx}}dx$.

The first term on the right evidently represents the product of the current into the ohmic resistance of the parallelepiped dx, while the second term represents the internal electromotive force of the parallelepiped. It follows that if r denote the specific resistance, we must have

$u+v=M/r\nu c$

in agreement with Kohlrausch's equation;^[1] while by integrating the expression for the internal electromotive force of the parallelepiped dx, we obtain for the electromotive force of a cell whose activity depends on the transference of electrolyte between the concentrations c₁ and c₂, the value

${\frac {u-v}{u+v}}{\frac {RT}{\nu }}\int {\frac {1}{c}}{\frac {dc}{dx}}dx$,

or

${\frac {u-v}{u+v}}{\frac {RT}{\nu }}\log {\frac {c_{2}}{c_{1}}}$,

in agreement with the result already obtained.

It may be remarked that although the current arising from a concentration cell which is kept at a constant temperature is capable of performing work, yet this work is provided, not by any diminution in the total internal energy of the cell, but by the abstraction of thermal energy from neighbouring bodies. This indeed (as may be seen by reference to W. Thomson's general

1. Cf. p. 374.