Page:Scientific Papers of Josiah Willard Gibbs.djvu/99

position and state of this part. (By homogeneous is meant that the part in question is uniform throughout, not only in chemical composition, but also in physical state.) If we consider the amount and kind of matter in this homogeneous mass as fixed, its energy $\epsilon$ is a function of its entropy $\eta$, and its volume $v$, and the differentials of these quantities are subject to the relation

$d\epsilon =td\eta -pdv,$

(11)

$t$ denoting the (absolute) temperature of the mass, and $p$ its pressure. For $td\eta$ is the heat received, and $pdv$ the work done, by the mass during its change of state. But if we consider the matter in the mass as variable, and write $m_{1},m_{2},...,m_{n}$ for the quantities of the various substances $S_{1},S_{2},...,S_{n}$ of which the mass is composed, $\epsilon$ will evidently be a function of $\eta ,v,m_{1},m_{2},...,m_{n}$, and we shall have for the complete value of the differential of $\epsilon$

$d\epsilon =td\eta -pdv+\mu _{1}dm_{1}+\mu _{2}dm_{2}+...+\mu _{n}dm_{n},$

(12)

$\mu _{1},\mu _{2},...,\mu _{n}$ denoting the differential coefficients of $\epsilon$ taken with respect to $m_{1},m_{2},...,m_{n}$.

The substances $S_{1},S_{2},...,S_{n}$, of which we consider the mass composed, must of course be such that the values of the differentials $dm_{1},dm_{2},...,dm_{n}$ shall be independent, and shall express every possible variation in the composition of the homogeneous mass considered, including those produced by the absorption of substances different from any initially present. It may therefore be necessary to have terms in the equation relating to component substances which do not initially occur in the homogeneous mass considered, provided, of course, that these substances, or their components, are to be found in some part of the whole given mass.

If the conditions mentioned are satisfied, the choice of the substances which we are to regard as the components of the mass considered, may be determined entirely by convenience, and independently of any theory in regard to the internal constitution of the mass. The number of components will sometimes be greater, and sometimes less, than the number of chemical elements present. For example, in considering the equilibrium in a vessel containing water and free hydrogen and oxygen, we should be obliged to recognize three components in the gaseous part. But in considering the equilibrium of dilute sulphuric acid with the vapor which it yields, we should have only two components to consider in the liquid mass, sulphuric acid (anhydrous, or of any particular degree of concentration) and (additional) water. If, however, we are considering sulphuric acid in a state of maximum concentration in connection with substances which might possibly afford water to the acid, it must be noticed that the condition of the independence of the differentials will require that we