Page:Scientific Papers of Josiah Willard Gibbs.djvu/458

$m_{\text{S}}Q_{\text{S}}$ is the latent heat of so much of the solvent as occurs in the solution. (Or make $m_{\text{S}}=1$.)

Raoult makes $\Delta t\propto {\frac {m_{\text{D}}}{M_{\text{D}}}}$, with exceptions.

With a coexistent gaseous phase of the solvent (the solutum being not volatile), we have for the solution

$dp=\gamma _{\text{S}}d\mu _{\text{S}}+{\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}},$

and for the gaseous phase

$dp=\gamma _{\text{S}}'d\mu _{\text{S}}.$

Here, on account of the coexistence of the phases, $p$ and $\mu _{\text{S}}$ and $dp$ and $d\mu _{\text{S}}$ have the same values. Hence

$dp={\frac {\gamma _{\text{S}}}{\gamma _{\text{S}}'}}dp+{\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}}.$ $-dp{\frac {(\gamma _{\text{S}}-\gamma _{\text{S}})}{\gamma _{\text{S}}}}={\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}}.$

Say$-dp={\frac {\gamma _{\text{S}}'}{\gamma _{\text{S}}}}{\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}},$

$p-P={\frac {\gamma _{\text{S}}'}{\gamma _{\text{S}}}}{\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}}.$[1]

[10]

or[2]$-{\frac {dp}{p}}={\frac {M_{\text{S}}}{\gamma _{\text{S}}}}{\frac {d\gamma _{\text{D}}}{M_{\text{D}}}},$

${\frac {p-P}{p}}={\frac {M_{\text{S}}}{\gamma _{\text{S}}}}{\frac {\gamma _{\text{D}}}{M_{\text{D}}}}\cdot$

[11]

$M_{\text{D}}$ is the molecular weight [of solutum] in solution;

$M_{\text{S}}$ is the molecular weight [of solvent] in vapor.

But the foregoing equation suggests a generalization which is not confined to cases in which the law of Henry has been proved. The letter $M$ in the equation has been defined as the molecular weight of the substance in the form of gas. Now the molecular weight which figures in the relation between the potential and the density of a substance in a liquid would naturally be the molecular weight of the substance as it exists in the liquid. It is therefore a natural supposition suggested by the equation that, in the case where Henry's law holds good, and consequently eq. [4], the molecular weight of the solutum is the same in the liquid and in the gaseous phase; that in

1. [$p$ is the vapor pressure of the pure solvent, $P$ that of the solution.]
2. Assuming that the vapor behaves like an ideal gas, we have $\gamma _{\text{S}}'={\frac {pM_{\text{S}}}{At}}\cdot$