Page:Scientific Papers of Josiah Willard Gibbs.djvu/467

Rh Integrating from one point to another in the electrolyte, The evaluation of these integrals which denote the resistance and electromotive force for a finite part of the electrolyte depends on the distribution of the ions in the cell. For one salt with varying concentration, or, since $$c_{1}\gamma_{1} = c_{2}\gamma_{2}$$ and $$c_{1}d\gamma_{1} = c_{2}d\gamma_{2}$$,   The resistance depends on the concentration throughout the part of the cell considered, but the electromotive force depends only on the concentration at the terminal points ($$'$$ and $$''$$).

For $$c_{1}M_{1}$$ and $$c_{2}M_{2}$$ we may write $$\frac{v_{1}}{a_{\text{H}}}$$ and $$\frac{v_{2}}{a_{\text{H}}}$$, where $$v_{1}$$ and $$v_{2}$$ are the "valencies" of the molecules. This gives I think this is identical with your equation ($$V$$) when your ions have the same valency.

Planck's problem is less simple. We may regard it as relating to a tube connecting the two great reservoirs filled with different electrolytes of same concentration, i.e., $$\textstyle \sum_{0} \displaystyle c_{0} \gamma_{0}' = \textstyle \sum_{0} \displaystyle c_{0} \gamma_{0}''$$. I use (0) for an y ion, (1) for any cation, (2) for any anion. [The accents ($$'$$) and ($$''$$) refer to the two reservoirs.]

The tube is supposed to have reached a stationary state and dissociation is complete. The number of ions is immaterial, but they all must have the same valency $$v$$. Now by equations (3) and (4), since $$c_{0}M_{0} = \frac{v}{a_{\text{H}}}$$,