Page:Scientific Papers of Josiah Willard Gibbs.djvu/129

$m_{s}$, are not the same, nor those of $m_{W}$ and $m_{w}$, and hence it might seem that the potential for water in the given liquid considered as composed of the hydrate and water, viz.,

$\left({\frac {d\epsilon }{dm_{W}}}\right)_{\eta ,v,m_{S}},$

would be different from the potential for water in the same liquid considered as composed of anhydrous salt and water, viz.,

$\left({\frac {d\epsilon }{dm_{w}}}\right)_{\eta ,v,m_{s}}\cdot$

The value of the two expressions is, however, the same, for, although $m_{W}$ is not equal to $m_{w}$, we may of course suppose $dm_{W}$ to be equal to $dm_{w}$, and then the numerators in the two fractions will also be equal, as they each denote the increase of energy of the liquid, when the quantity $dm_{W}$ or $dm_{w}$ of water is added without altering the entropy and volume of the liquid. Precisely the same considerations will apply to any other case.

In fact, we may give a definition of a potential which shall not presuppose any choice of a particular set of substances as the components of the homogeneous mass considered.

Definition.—If to any homogeneous mass we suppose an infinitesimal quantity of any substance to be added, the mass remaining homogeneous and its entropy and volume remaining unchanged, the increase of the energy of the mass divided by the quantity of the substance added is the potential for that substance in the mass considered. (For the purposes of this definition, any chemical element or combination of elements in given proportions may be considered a substance, whether capable or not of existing by itself as a homogeneous body.)

In the above definition we may evidently substitute for entropy, volume, and energy, respectively, either temperature, volume, and the function $\psi$; or entropy, pressure, and the function $\chi$; or temperature, pressure, and the function $\zeta$. (Compare equation (104).)

In the same homogeneous mass, therefore, we may distinguish the potentials for an indefinite number of substances, each of which has a perfectly determined value.

Between the potentials for different substances in the same homogeneous mass the same equations will subsist as between the units of these substances. That is, if the substances, $S{a},S_{b}$, etc., $S_{k},S_{l}$, etc., are components of any given homogeneous mass, and are such that

$\alpha {\mathfrak {S}}_{a}+\beta {\mathfrak {S}}_{b}+{\text{etc.}}=\kappa {\mathfrak {S}}_{k}+\lambda {\mathfrak {S}}_{l}+{\text{etc.,}}$

(120)

${\mathfrak {S}}_{a},{\mathfrak {S}}_{b}$, etc., ${\mathfrak {S}}_{k},{\mathfrak {S}}_{l}$, etc., denoting the units of the several substances, and $\alpha ,\beta$, etc., $\kappa ,\lambda$, etc., denoting numbers, then if $\mu _{a},\mu _{b}$, etc., $\mu _{k},\mu _{l}$,