Page:Scientific Papers of Josiah Willard Gibbs.djvu/462

We may avoid 'hedging' in regard to $B$ by using the differential equation (2). We may simply say that this equation holds for changes produced by varying the quantity of ($D$), when $\gamma _{\text{D}}$ is small. It is not limited to changes in which t is constant, for the change in $\mu _{\text{D}}$ due to $t$ appearing in (1) (both explicitly, and implicitly in $B$) becomes negligible when multiplied by the small quantity $\gamma _{\text{D}}$.

The formula contains the molecular weight $M_{\text{D}}$, and if all the solutum has not the same molecular formula, the $\gamma _{\text{D}}$ must be understood as relating only to a single kind of molecule.

Thus if a salt (₁₂) is partly dissociated into the ions (₁) and (₂), we will have the three equations

$\mu _{1}=B_{1}+{\frac {At}{M_{1}}}\log \gamma _{1},$ $\mu _{2}=B_{2}+{\frac {At}{M_{2}}}\log \gamma _{2},$ $\mu _{12}=B_{12}+{\frac {At}{M_{12}}}\log \gamma _{12}.$

The three potentials are also connected by the relation

$M_{1}\mu _{1}+M_{2}\mu _{2}=M_{12}\mu _{12},$

which determines the amount of dissociation. We have, namely,

$M_{1}B_{1}+M_{2}B_{2}-M_{12}B_{12}+At\log {\frac {\gamma _{1}\gamma _{2}}{\gamma _{12}}}=0,$

which makes constant, for constant temperature and solvent.

I may observe in passing that this relation, eq. (1) or (2), which is so fundamental in the modern theory of solutions, is somewhat vaguely indicated in my "Equilib. Het. Subs." (See [this volume] pp. 135–138, 156, and 164–165.) I say vaguely, because the coefficient of the logarithm is only given (in the general case) as constant for a given solvent and temperature. The generalization that this coefficient is in all cases of exactly the same form as for gases, even to the details which arise in cases of dissociation, is due to van't Hoff in connection with Arrhenius, who suggested that the "discords" are but "harmonies not understood," and that exceptions vanish when we use the true molecular weights. At all events, eq. (2) with (98) (E.H.S.) gives for a solvent ($S$) with one dissolved substance ($D$),

$dp={\frac {\eta }{v}}dt+{\frac {m_{\text{S}}}{v}}d\mu _{\text{S}}+{\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}}.$

If we integrate, keeping $t$ constant and also $\mu _{\text{S}}$ (by connection with the pure solvent through a semi-permeable diaphragm), we have van't HofTs Law,

$p-p'={\frac {At}{M_{\text{D}}}}\gamma _{\text{D}}.$