Page:Scientific Papers of Josiah Willard Gibbs.djvu/198

with reference to any solid or liquid bodies. But the common rule that the density of a gas necessary for equilibrium with a solid or liquid is not altered by the presence of a different gas which is not absorbed by the solid or liquid, if construed strictly, will involve consequences in regard to solids and liquids which are entirely inadmissible. To show this, we will assume, the correctness of the rule mentioned. Let $S_{1}$ denote the common component of the gaseous and liquid or solid masses, and $S_{2}$ the insoluble gas, and let quantities relating to the gaseous mass be distinguished when necessary by the index (g), and those relating to the liquid or solid by the index (l). Now while the gas is in equilibrium with the liquid or solid, let the quantity which it contains of $S_{2}$ receive the increment $dm_{2}$, its volume and the quantity which it contains of the other component, as well as the temperature, remaining constant. The potential for $S_{1}$ in the gaseous mass will receive the increment

$\left({\frac {d\mu _{1}}{dm_{2}}}\right)_{t,v,m}^{(G)}dm_{2}$

and the pressure will receive the increment

$\left({\frac {dp}{dm_{2}}}\right)_{t,v,m}^{(G)}dm_{2}.$

Now the liquid or solid remaining in equilibrium with the gas must experience the same variations in the values of $\mu _{1}$ and $p$. But by (272)

$\left({\frac {d\mu _{1}}{dp}}\right)_{t,m}^{(L)}=\left({\frac {dv}{dm_{1}}}\right)_{t,p,m}^{(L)}\cdot$

therefore⁠$\left({\frac {dv}{dm_{1}}}\right)_{t,p,m}^{(L)}={\frac {\left({\frac {d\mu _{1}}{dm_{2}}}\right)_{t,v,m}^{(G)}}{\left({\frac {dp}{dm_{2}}}\right)_{t,v,m}^{(G)}}}\cdot$

It will be observed that the first member of this equation relates solely to the liquid or solid, and the second member solely to the gas. Now we may suppose the same gaseous mass to be capable of equilibrium with several different liquids or solids, and the first member of this equation must therefore have the same value for all such liquids or solids; which is quite inadmissible. In the simplest case, in which the liquid or solid is identical in substance with the vapor which it yields, it is evident that the expression in question denotes the reciprocal of the density of the solid or liquid. Hence, when the gas is in equilibrium with one of its components both in the solid and liquid states (as when a moist gas is in equilibrium with ice and water), it would be necessary that the solid and liquid should have the same density.