Page:Scientific Papers of Josiah Willard Gibbs.djvu/464

We may regard ${\frac {v}{v'-v}}$ as constant in integrating (for small $\gamma _{\text{D}}$), which gives

$P-p={\frac {v}{v'-v}}At{\frac {\gamma _{\text{D}}}{M_{\text{D}}}}\cdot$

Raoult found values about 5 per cent, larger than this, which agrees very well with the fact that ${\frac {At}{v'-v}}$ is somewhat larger than ${\frac {PM_{\text{S}}}{m_{\text{S}}}}\cdot$ It is also to be observed that $M_{\text{D}}$ relates to the molecules in the solution, but $M_{\text{S}}$ to the molecules in the vapor. Or, with a coexistent vapor phase of the solutum (alone or mixed with other vapors or gases), we have

${\begin{aligned}\mu &=B+{\frac {At}{M_{\text{D}}}}\log \gamma _{\text{D}},\\\mu &=B'+{\frac {At}{M_{\text{D}}}}\log \gamma _{\text{D}}',\\{\frac {B'-B}{At}}M_{\text{D}}&=\log {\frac {\gamma _{\text{D}}}{\gamma _{\text{D}}'}},\end{aligned}}$

which makes ${\frac {\gamma _{\text{D}}}{\gamma _{\text{D}}'}}$ constant for the same solvent, solutum, and temperature, according to Henry's Law.

So for the galvanic cell which you first consider, I should write

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

$\gamma _{a}',\gamma _{a}''$ being the densities, supposed small, of the cation (a) in the two electrodes, which are supposed identical except for the dissolved (a). Here $a_{a}$ has reference to the solution and $M_{a}$ to the electrodes. It may be more convenient to divide a a into the factors $E_{a},a_{\text{H}}$, where $a_{\text{H}}$ is the weight of hydrogen which carries the unit of electricity, and $E_{a}$ the weight of (a) which carries the same quantity of electricity as the unit of weight of hydrogen. In other words $E_{a}$ is Faraday's "electrochemical equivalent" and $a_{a}$ is Maxwell's "electrochemical equivalent." This gives

$V''-V'=a_{\text{H}}At{\frac {E_{a}}{M_{a}}}\log {\frac {\gamma _{a}'}{\gamma _{a}''}},$

where $a_{\text{H}}A$ is your $R$ and ${\frac {M_{a}}{E_{a}}}$ your $v,v'.$^[1]

1. [The valence of the ion].