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

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

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  $$p'$$ 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 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 for the solution, and  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

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 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