Page:Scientific Papers of Josiah Willard Gibbs.djvu/185

Condition (244) may be reduced to the form

$\Delta D\epsilon -T\Delta D\eta -(\Upsilon +M_{1})\Delta Dm_{1}...-(\Upsilon +M_{n})\Delta Dm_{n}\geqq 0;$

(248)

and by (246) and (247) to

$\Delta D\epsilon -t\Delta D\eta -\mu _{1}\Delta Dm_{1}...-\mu _{n}\Delta Dm_{n}\geqq 0.$

(249)

If values determined subsequently to the change of phase are distinguished by accents, this condition may be written

$D\epsilon '-tD\eta '-\mu _{1}Dm'_{1}...-\mu _{n}Dm'_{n}-D\epsilon +tD\eta +\mu _{1}Dm_{1}...-\mu _{n}Dm_{n}\geqq 0,$

(250)

which may be reduced by (93) to

$D\epsilon '-tD\eta '-\mu _{1}Dm'_{1}...-\mu _{n}Dm'_{n}+pDv\geqq 0.$

(251)

Now if the element of volume $Dv$ is adjacent to a surface of discontinuity, let us suppose $D\epsilon ',D\eta ',Dm'_{1},...Dm'_{n}$ to be determined (for the same element of volume) by the phase existing on the other side of the surface of discontinuity. As $t,\mu _{1},...\mu _{n}$ have the same values on both sides of this surface, the condition may be reduced by (93) to

$-p'Dv+pDv\geqq 0.$

(252)

That is, the pressure must not be greater on one side of a surface of

discontinuity than on the other.

Applied more generally, (251) expresses the condition of equilibrium with respect to the possibility of discontinuous changes of phases at any point. As $Dv'=Dv$, the condition may also be written

$D\epsilon '-tD\eta '+pDv'-\mu _{1}Dm'_{1}...-Dm'_{n}\geqq 0,$

(253)

which must hold true when $t,p,\mu _{1},...\mu _{n}$ have values determined by any point in the mass, and $D\epsilon ',D\eta ',Dm'_{1},...Dm'_{n}$ have values determined by any possible phase of the substances of which the mass is composed. The application of the condition is, however, subject to the limitations considered on pages 74–79. It may easily be shown (see page 104) that for constant values of $t,p,\mu _{1},...\mu _{n}$, and of $Dv'$, the first member of (253) will have the least possible value when $D\epsilon ',D\eta ',Dm'_{1},...Dm'_{n}$ are determined by a phase for which the temperature has the value $t$, and the potentials the values $\mu _{1},...\mu _{n}$. It will be sufficient, therefore, to consider the condition as applied to such phases, in which case it may be reduced by (93) to

$p-p'\geqq 0.$

(254)

That is, the pressure at any point must be as great as that of any phase of the same components, for which the temperature and the potentials have the same values as at that point. We may also express this condition by saying that the pressure must be as great as is consistent with equations (246), (247). This condition with the equations mentioned will always be sufficient for equilibrium; when