Page:Scientific Papers of Josiah Willard Gibbs.djvu/100

consider the acid in the state of maximum concentration as one of the components. The quantity of this component will then be capable of variation both in the positive and in the negative sense, while the quantity of the other component can increase but cannot decrease below the value 0.

For brevity's sake, we may call a substance S a an actual component of any homogeneous mass, to denote that the quantity $S_{a}$ of that substance in the given mass may be either increased or diminished (although we may have so chosen the other component substances that $m_{a}=0$); and we may call a substance $S_{b}$ a possible component to denote that it may be combined with, but cannot be subtracted from the homogeneous mass in question. In this case, as we have seen in the above example, we must so choose the component substances that $m_{b}=0$.

The units by which we measure the substances of which we regard the given mass as composed may each be chosen independently. To fix our ideas for the purpose of a general discussion, we may suppose all substances measured by weight or mass. Yet in special cases, it may be more convenient to adopt chemical equivalents as the units of the component substances.

It may be observed that it is not necessary for the validity of equation (12) that the variations of nature and state of the mass to which the equation refers should be such as do not disturb its homogeneity, provided that in all parts of the mass the variations of nature and state are infinitely small. For, if this last condition be not violated, an equation like (12) is certainly valid for all the infinitesimal parts of the (initially) homogeneous mass ; i.e., if we write $D\epsilon ,D\eta ,$ etc., for the energy, entropy, etc., of any infinitesimal part,

$dD\epsilon =tdD\eta -pdDv+\mu _{1}dDm_{1}++\mu _{2}dDm_{2}+...+\mu _{n}dDm_{n}$

(13)

whence we may derive equation (12) by integrating for the whole initially homogeneous mass.

�We will now suppose that the whole mass is divided into parts so that each part is homogeneous, and consider such variations in the energy of the system as are due to variations in the composition and state of the several parts remaining (at least approximately) homogeneous, and together occupying the whole space within the envelop. We will at first suppose the case to be such that the component substances are the same for each of the parts, each of the substances $S_{1},S_{2},...,S_{n}$ being an actual component of each part. If we distinguish the letters referring to the different parts by accents, the variation in the energy of the system may be expressed by $\delta \epsilon '+\delta \epsilon ''+etc.$, and the general condition of equilibrium requires that

$\delta \epsilon '+\delta \epsilon ''+etc.\geqq 0$

(14)