Page:Scientific Papers of Josiah Willard Gibbs.djvu/369

increment of the intrinsic energy of the ponderable matter, and the third the increment of the energy due to gravitation.^[1] But by (682)

$\delta m_{\text{a}}=a_{\text{a}}\delta \epsilon .$

It is therefore necessary for equilibrium that

$V''-V'+a_{\text{a}}(\mu _{\text{a}}''-\mu _{\text{a}}'-\Upsilon ''+\Upsilon ')=0.$

(684)

To extend this relation to all the electrodes we may write

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

(685)

For each of the other cations (specified by _{$_{\text{b}}$} etc.) there will be a similar condition, and for each of the anions a condition of the form

$V'-a_{\text{g}}(\mu _{\text{g}}'-\Upsilon ')=V''-a_{\text{g}}(\mu _{\text{g}}''-\Upsilon '')=V'''-a_{\text{g}}(\mu _{\text{g}}'''-\Upsilon ''')={\text{etc.}}$

(686)

When the effect of gravity may be neglected, and there are but two electrodes, as in a galvanic or electrolytic cell, we have for any cation

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

(687)

and for any anion

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

(688)

where $V''-V'$ denotes the electromotive force of the combination. That is:—

When all the conditions of equilibrium are fulfilled in a galvanic or electrolytic cell, the electromotive force is equal to the difference in the values of the potential for any ion or apparent ion at the surfaces of the electrodes multiplied by the electro-chemical equivalent of that ion, the greater potential of an anion being at the same electrode as the greater electrical potential, and the reverse being true of a cation.

Let us apply this principle to different cases.

(I.) If the ion is an independently variable component of an electrode, or by itself constitutes an electrode, the potential for the ion (in any case of equilibrium which does not depend upon passive resistances to change) will have the same value within the electrode as on its surface, and will be determined by the composition of the electrode with its temperature and pressure. This might be illustrated by a cell with electrodes of mercury containing certain quantities of zinc in solution (or with one such electrode and the other of pure zinc) and an electrolytic fluid containing a salt of zinc, but not capable of dissolving the mercury.^[2] We may regard

1. It is here supposed that the gravitational potential may be regarded as constant for each electrode. When this is not the case the expression may be applied to small parts of the electrodes taken separately.
2. If the electrolytic fluid dissolved the mercury as well as the zinc, equilibrium could only subsist when the electromotive force is zero, and the composition of the electrodes identical. For when the electrodes are formed of the two metals in different proportions, that which has the greater potential for zinc will have the less potential for mercury. (See equation (98).) This is inconsistent with equilibrium, according to the principle mentioned above, if both metals can act as cations.