Page:Scientific Papers of Josiah Willard Gibbs.djvu/368

332 It will be observed that the choice of the substances which we regard as the components of the fluid is to some extent arbitrary, and that the same physical relations may be expressed by different equations of the form (682), in which the fluxes are expressed with reference to different sets of components. If the components chosen are such as represent what we believe to be the actual molecular constitution of the fluid, those of which the fluxes appear in the equation of the form (682) are called the ions, and the constants of the equation are called their electro-chemical equivalents. For our present purpose, which has nothing to do with any theories of molecular constitution, we may choose such a set of components as may be convenient, and call those ions, of which the fluxes appear in the equation of the form (682), without farther limitation.

Now, since the fluxes of the independently variable components of an electrolytic fluid do not necessitate any electrical currents, all the conditions of equilibrium which relate to the movements of these components will be the same as if the fluid were incapable of the electrolytic process. Therefore all the conditions of equilibrium which we have found without reference to electrical considerations, will apply to an electrolytic fluid and its independently variable components. But we have still to seek the remaining conditions of equilibrium, which relate to the possibility of electrolytic conduction.

For simplicity, we shall suppose that the fluid is without internal surfaces of discontinuity (and therefore homogeneous except so far as it may be slightly affected by gravity), and that it meets metallic conductors (electrodes) in different parts of its surface, being otherwise bounded by non-conductors. The only electrical currents which it is necessary to consider are those which enter the electrolyte at one electrode and leave it at another.

If all the conditions of equilibrium are fulfilled in a given state of the system, except those which relate to changes involving a flux of electricity, and we imagine the state of the system to be varied by the passage from one electrode to another of the quantity of electricity $$\delta e$$ accompanied by the quantity $$\delta m_{\text{a}}$$ of the component specified, without any flux of the other components or any variation in the total entropy, the total variation of energy in the system will be represented by the expression in which $$V', V$$ denote the electrical potentials in pieces of the same kind of metal connected with the two electrodes, $$\Upsilon ', \Upsilon $$, the gravitational potentials at the two electrodes, and $$\mu_{\text{a}}, \mu_{\text{a}}$$, the intrinsic potentials for the substance specified. The first term represents the increment of the potential energy of electricity, the second the