Page:Scientific Papers of Josiah Willard Gibbs.djvu/116

of equilibrium, even in the case of compounds of variable proportions, i.e., even when bodies can exist which are compounded in proportions infinitesimally varied from those of the solids considered. (Those solids which are capable of absorbing fluids form of course an exception, so far as their fluid components are concerned.) It is true that a solid may be increased by the formation of new solid matter on the surface where it meets a fluid, which is not homogeneous with the previously existing solid, but such a deposit will properly be treated as a distinct part of the system (viz., as one of the parts which we have called new). Yet it is worthy of notice that if a homogeneous solid which is a compound of variable proportions is in contact and equilibrium with a fluid, and the actual components of the solid (considered as of variable composition) are also actual components of the fluid, and the condition (53) is satisfied in regard to all bodies which can be formed out of the actual components of the fluid (which will always be the case unless the fluid is practically unstable), all the conditions will hold true of the solid, which would be necessary for equilibrium if it were fluid.

This follows directly from the principles stated on the preceding pages. For in this case the value of (57) will be zero as determined either for the solid or for the fluid considered with reference to their ultimate components, and will not be negative for any body whatever which can be formed of these components; and these conditions are sufficient for equilibrium independently of the solidity of one of the masses. Yet the point is perhaps of sufficient importance to demand a more detailed consideration.

Let $S_{a},...S_{g}$ be the actual components of the solid, and $S_{h},...S_{k}$ its possible components (which occur as actual components in the fluid); then, considering the proportion of the components of the solid as variable, we shall have for this body by equation (12)

$d\epsilon '=t'd\eta '-p'dv'+\mu _{a}'dm'_{a}...+\mu _{g}'dm'_{g}+\mu _{h}'dm'_{h}...+\mu _{k}'dm'_{k}.$

(58)

By this equation the potentials $\mu _{a}',...\mu _{k}'$ are perfectly defined. But the differentials dm $dm'_{a},...dm'_{k}$, considered as independent, evidently express variations which are not possible in the sense required in the criterion of equilibrium. We might, however, introduce them into the general condition of equilibrium, if we should express the dependence between them by the proper equations of condition. But it will be more in accordance with our method hitherto, if we consider the solid to have only a single independently variable component $S_{x}$ of which the nature is represented by the solid itself. We may then write

$\delta \epsilon '=t'\delta \eta '-p'\delta v'+\mu '\delta m'_{x}.$

(59)