Page:Scientific Papers of Josiah Willard Gibbs.djvu/103

the form of the particular conditions of equilibrium as expressed by (19), (20), (22); but the number of single conditions contained in (22) is of course less than if all the component substances were components of all the parts. Whenever, therefore, each of the different homogeneous parts of the given mass may be regarded as composed of some or of all of the same set of substances, no one of which can be formed out of the others, the condition which (with equality of temperature and pressure) is necessary and sufficient for equilibrium between the different parts of the given mass may be expressed as follows:—

The potential for each of the component substances must have a constant value in all parts of the given mass of which that substance is an actual component, and have a value not less than this in all parts of which it is a possible component.

The number of equations afforded by these conditions, after elimination of $M_{1},M_{2},...M_{n}$, will be less than $(n+2)(\nu -1)$ by the number of terms in (15) in which the variation of the form $\delta m$ is either necessarily nothing or incapable of a negative value. The number of variables to be determined is diminished by the same number, or, if we choose, we may write an equation of the form $m=0$ for each of these terms. But when the substance is a possible component of the part concerned, there will also be a condition (expressed by $\geqq$) to show whether the supposition that the substance is not an actual component is consistent with equilibrium.

We will now suppose that the substances $S_{1},S_{2},...S_{n}$ are not all independent of each other, i.e., that some of them can be formed out of others. We will first consider a very simple case. Let $S_{3}$ be composed of $S_{1}$ and $S_{2}$ combined in the ratio of $a$ to $b$, $S_{1}$ and $S_{2}$ occurring as actual components in some parts of the given mass, and $S_{3}$ in other parts, which do not contain $S_{1}$ and $S_{2}$ as separately variable components. The general condition of equilibrium will still have the form of (15) with certain of the terms of the form $\mu \delta m$ omitted. It may be written more briefly

$\sum (t\delta \eta )-\sum (p\delta v)+\sum (\mu _{1}\delta m_{1})+\sum (\mu _{2}\delta m_{2})...+\sum (\mu _{n}\delta m_{n})\geqq 0$

(23)

the sign $\sum$ denoting summation in regard to the different parts of the given mass. But instead of the three equations of condition,

$\sum (\mu _{1}\delta m_{1})=0,\,\,\,\sum (\mu _{2}\delta m_{2})=0,\,\,\,\sum (\mu _{3}\delta m_{3})=0$

(24)

we shall have the two,

$\sum \delta m{1}+{\frac {a}{a+b}}\sum \delta m_{3}=0,$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$ (25)
$\sum \delta m{2}+{\frac {b}{a+b}}\sum \delta m_{3}=0.$

The other equations of condition,

$\sum \delta \eta =0,\,\,\,\sum \delta v=0,\,\,\,\sum \delta m_{4}=0$, etc.,

(26)