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

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

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

Now

${\displaystyle {\frac {At}{v'-v}}={\frac {At}{v'}}={\frac {PM_{\text{S}}}{m_{\text{S}}}}}$ nearly, which gives

${\displaystyle {\frac {P-p}{P}}={\frac {m_{\text{D}}}{m_{\text{S}}}}{\frac {M_{\text{S}}}{M_{\text{D}}}},}$ which is Raoult's Law.

Raoult found values about 5 per cent, larger than this, which agrees very well with the fact that ${\displaystyle {\frac {At}{v'-v}}}$ is somewhat larger than ${\displaystyle {\frac {PM_{\text{S}}}{m_{\text{S}}}}\cdot }$ It is also to be observed that ${\displaystyle M_{\text{D}}}$ relates to the molecules in the solution, but ${\displaystyle M_{\text{S}}}$ 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

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

which makes ${\displaystyle {\frac {\gamma _{\text{D}}}{\gamma _{\text{D}}'}}}$ 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

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

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

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

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