Page:Scientific Papers of Josiah Willard Gibbs.djvu/465

The meagreness of the results obtained in my E.H.S. in the matter of electrolysis has a deeper reason than the difficulty of the evaluation of the potentials.

In the first place, cases of true equilibrium (even for open circuit) are quite exceptional. Thus the single case of unequal concentration of the electrolyte cannot be one of equilibrium since the process of diffusion cannot be stopped. Cases in which equilibrium does not subsist were formally excluded by my subject, and indeed could not be satisfactorily treated without the introduction of new ideas quite foreign to those necessary for the treatment of equilibrium.

Again, the consideration of the electrical potential in the electrolyte, and especially the consideration of the difference of potential in electrolyte and electrode, involves the consideration of quantities of which we have no apparent means of physical measurement, while the difference of potential in "pieces of metal of the same kind attached to the electrodes" is exactly one of the things which we can and do measure.

Nevertheless, with some hedging in regard to the definition of the electrical potential, we may apply

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

to points in electrolyte ($'$) and electrode ($''$).

This gives

$V''-V'=a_{a}\left(B+{\frac {At}{M_{a}}}\log \gamma _{a}-\mu _{a}''\right),$

The $G$ like the $P$ of your formula seems to depend on the solvent, presumably varies with the temperature, but as Nernst remarks does not depend on the other ion associated with ($a$), so long as the solution is dilute.

The case of unequal concentration, or, in general, cases in which the electrolyte is not homogeneous, I should treat as follows; Let us suppose for convenience that the cell is in form of a rectangular parallelepiped with edge parallel to axis of x and cross section of unit area. The electrolyte is supposed homogeneous in planes parallel to the ends, which are formed by the electrodes.

Of course we should have equilibrium if proper forces could be applied to prevent the migration of the ions and also of the part of the solutum which is not dissociated. What would these forces be? For the molecules (₁₂) which are not dissociated, the force per unit of mass would be ${\frac {d\mu _{12}}{dx}}\cdot$ (The problem is practically the same as that discussed in E.H.S. [this volume], pp. 144 ff.) If the unit of mass of