Page:Scientific Papers of Josiah Willard Gibbs.djvu/372

electrode. But whether the electrodes are liquid or solid, if the external electromotive force $V'-V''$ applied to an electrolytic combination is varied, when it is too weak to produce a lasting current, and the electrodes are thereby brought into a new state of polarization in which they make equilibrium with the altered value of the electromotive force, without change in the nature of the electrodes or of the electrolytic fluid, then by (508) or (675)

$d\sigma '=-\Gamma _{\text{a}}'d\mu _{\text{a}}',$ $d\sigma ''=-\Gamma _{\text{a}}''d\mu _{\text{a}}'';$

and by (687),

$d(V'-V'')=-a_{\text{a}}(d\mu _{\text{a}}'-d\mu _{\text{a}}'').$

Hence

$d(V'-V'')={\frac {a_{\text{a}}}{\Gamma _{\text{a}}'}}d\sigma '-{\frac {a_{\text{a}}}{\Gamma _{\text{a}}''}}d\sigma ''.$

(689)

If we suppose that the state of polarization of only one of the electrodes is affected (as will be the case when its surface is very small compared with that of the other), we have

$d\sigma '={\frac {\Gamma _{\text{a}}'}{a_{\text{a}}}}d(V'-V'').$

(690)

The superficial tension of one of the electrodes is then a function of the electromotive force.

This principle has been applied by M. Lippmann to the construction of the electrometer which bears his name.^[1] In applying equations (689) and (690) to dilute sulphuric acid between electrodes of mercury, as in a Lippmann's electrometer, we may suppose that the suffix refers to hydrogen. It will be most convenient to suppose the dividing surface to be so placed as to make the surface-density of mercury zero. (See page 234.) The matter which exists in excess or deficiency at the surface may then be expressed by the surface-densities of sulphuric acid, of water, and of hydrogen. The value of the last may be determined from equation (690). According to M. Lippmann's determinations, it is negative when the surface is in its natural state (i.e., the state to which it tends when no external electromotive force is applied), since $\sigma '$ increases with $V''-V'$. When $V''-V'$ is equal to nine-tenths of the electromotive force of a Daniell's cell, the electrode to which $V''$ relates remaining in its natural state, the tension $\sigma '$ of the surface of the other electrode has a maximum value, and there is no excess or deficiency of hydrogen at that surface. This is the condition toward which a surface tends when it is extended while no flux of electricity takes place. The flux of electricity per unit of new surface formed, which will maintain a surface in a

1. See his memoir: "Relations entre les phenomenes electriques et capillaires," Annales de Chimie et de Physique, 5° série, t. v, p. 494.