Page:Scientific Papers of Josiah Willard Gibbs.djvu/101

for all variations which do not conflict with the equations of condition. These equations must express that the entropy of the whole given mass does not vary, nor its volume, nor the total quantities of any of the substances $S_{1},S_{2},...,S_{n}$. We will suppose that there are no other equations of condition. It will then be necessary for equilibrium that

$t'\delta \eta '-p'\delta v'+\mu _{1}\delta m'_{1}+\mu _{2}\delta m'_{2}...+\mu _{n}\delta m'_{n}+t''\delta \eta ''-p''\delta v''+\mu _{1}\delta m''_{1}+\mu _{2}\delta m''_{2}...+\mu _{n}\delta m''_{n}+etc.\geqq 0$

(15)

for any values of the variations for which

$\delta \eta '+\delta \eta ''+\delta \eta '''+{\text{etc.}}=0,$

(16)

$\delta v'+\delta v''+\delta v'''+{\text{etc.}}=0,$

(17)

$\delta m'_{1}+\delta m''_{1}+\delta m'''_{1}+{\text{etc.}}=0,$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$ (18)
$\delta m'_{2}+\delta m''_{2}+\delta m'''_{2}+{\text{etc.}}=0,$
$.............................................$
$\delta m'_{n}+\delta m''_{n}+\delta m'''_{n}+{\text{etc.}}=0.$

For this it is evidently necessary and sufficient that

$t'=t''=t'''={\text{etc.}}$

(19)

$p'=p''=p'''={\text{etc.}}$

(20)

$\mu _{1}'=\mu _{1}''=\mu _{1}'''={\text{etc.}}$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$ (21)
$\mu _{2}'=\mu _{2}''=\mu _{2}'''={\text{etc.}}$
$................................$
$\mu _{n}'=\mu _{n}''=\mu _{n}'''={\text{etc.}}$

Equations (19) and (20) express the conditions of thermal and mechanical equilibrium, viz., that the temperature and the pressure must be constant throughout the whole mass. In equations (21) we have the conditions characteristic of chemical equilibrium. If we call a quantity $\mu _{x}$ as defined by such an equation as (12), the potential for the substance $S_{x}$ in the homogeneous mass considered, these conditions may be expressed as follows:—

The potential for each component substance must be constant throughout the whole mass.

It will be remembered that we have supposed that there is no restriction upon the freedom of motion or combination of the component substances, and that each is an actual component of all parts of the given mass.

The state of the whole mass will be completely determined (if we regard as immaterial the position and form of the various homogeneous parts of which it is composed), when the values are determined of the quantities of which the variations occur in (15). The number