Page:Scientific Papers of Josiah Willard Gibbs.djvu/457

where $n_{\text{D}}$ denotes the number of molecules of the form ($D$). Hence we have for the solution

$dp={\frac {\eta }{v}}dt+\gamma _{\text{S}}d\mu _{\text{S}}+At\,d{\frac {n_{\text{D}}}{v}}\cdot$

[6]

If $t$ is constant, and also $\mu _{\text{S}}$,—a condition realized in equilibrium, when the solution is separated from the pure solvent by a diaphragm permeable to the solvent but not to the solutum, the equation reduces to

$dp=At\,d{\frac {n{\text{D}}}{v}}\cdot$

Whence⁠$p-p'={\frac {At}{M_{\text{D}}}}\gamma _{\text{D}}=At{\frac {n_{\text{D}}}{v}},$

[7]

$p'$ being the pressure where $\gamma _{\text{D}}=0$, i.e., in the pure solvent. Here $p-p'$ is the so-called osmotic pressure, and ${\frac {At}{M_{\text{D}}}}\gamma _{\text{D}}$ is the pressure as calculated^[1] by the laws of Boyle, Charles, and Avogadro for the solutum in the space occupied by the solution. The equation manifestly expresses van't HofF's law.

For a coexistent solid phase of the solvent, with constant pressure, the general equation gives

$0=\eta dt+m_{\text{S}}d\mu _{\text{S}}+v\,At\,d\gamma _{\text{D}}$

for the solution, and

$0=\eta 'dt+m_{1}d\mu _{\text{S}}$

for the solid coexistent phase. Here $t$ and $\mu _{\text{S}}$ have necessarily the same values in the two equations, and we may suppose the quantity of one of the phases to be so chosen as to make the values of $m_{\text{S}}$ equal in the two equations. This gives

$(\eta '-\eta )dt=v{\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}}.$

[8]

In integrating from $\gamma _{\text{D}}=0$ to any small value of $\gamma _{\text{D}}$, we may treat the coefficients of $dt$ and $d\gamma _{\text{D}}$ as having the same constant values as when $\gamma _{\text{D}}=0$. This gives

$(\eta '-\eta )\Delta t=v{\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}}={\frac {At}{M_{\text{D}}}}m_{\text{D}}.$

If we write $Q_{\text{S}}$ for ${\frac {t(\eta '-\eta )}{m_{\text{S}}}}$ (the latent heat of melting for the unit of weight of the solvent), we get

$-\Delta t={\frac {At}{M_{\text{D}}}}m_{\text{D}}{\frac {t}{Q_{\text{S}}m_{\text{S}}}},$

or⁠$-\Delta t={\frac {\frac {m_{\text{D}}}{M_{\text{D}}}}{\frac {m_{\text{S}}}{M_{\text{S}}}}}{\frac {At^{2}}{Q_{\text{S}}M_{\text{S}}}}={\frac {m_{\text{D}}}{M_{\text{D}}}}{\frac {At^{2}}{Q_{\text{S}}m_{\text{S}}}}\cdot$

[9]

1. Not experimentally found.