Page:Scientific Papers of Josiah Willard Gibbs.djvu/145

Again, by (91) and (96), the condition (142) may be brought to the form

$\zeta ''+t''\eta ''-p''v''-\mu _{1}'m''_{1}...-\mu _{n}'m''_{n}-\zeta '+t'\eta ''-p'v''-\mu _{1}'m'_{1}...-\mu _{n}'m'_{n}>0.$

(161)

Therefore, for the stability of all phases within any given limits it is necessary and sufficient that within the same limits

$[\Delta \zeta +\eta \Delta t-v\Delta p]_{m}<0,$

(162)

and $[\Delta \zeta -\mu _{1}\Delta m_{1}...-\mu _{n}\Delta m_{n}]_{t,p}>0,$

(163)

as may easily be proved by the method used with (153) and (154). The first of these formulæ expresses the thermal and mechanical conditions of stability for a body considered as unchangeable in composition, and the second the conditions of chemical stability for a body considered as maintained at a constant temperature and pressure. If $n=1$, the second condition falls away, and as in this case $\zeta =m\mu$, condition (162) becomes identical with (148).

The foregoing discussion will serve to illustrate the relation of the general condition of stability in regard to continuous changes to some of the principal forms of fundamental equations. It is evident that each of the conditions (146), (149), (154), (162), (163) involves in general several particular conditions of stability. We will now give our attention to the latter. Let

$\Phi =\epsilon -t'\eta +p'v-\mu _{1}'m_{1}...-\mu _{n}'m_{n},$

(164)

the accented letters referring to one phase and the unaccented to another. It is by (142) the necessary and sufficient condition of the stability of the first phase that, for constant values of the quantities relating to that phase and of $v$, the value of $\Phi$ shall be a minimum when the second phase is identical with the first. Differentiating (164), we have by (86)

$d\Phi =(t-t')d\eta -(p-p')dv+(\mu _{1}-\mu _{1}')dm_{1}...+(\mu _{n}-\mu _{n}')dm_{n}.$

(165)

Therefore, the above condition requires that if we regard $v,m_{1},...m_{n}$ as having the constant values indicated by accenting these letters, $t$ shall be an increasing function of $\eta$, when the variable phase differs sufficiently little from the fixed. But as the fixed phase may be any one within the limits of stability, $t$ must be an increasing function of $\eta$ (within these limits) for any constant values of $v,m_{1},...m_{n}$. This condition may be written

$\left({\frac {\Delta t}{\Delta \eta }}\right)_{v,m_{1},...m_{n}}\cdot$

(166)

When this condition is satisfied, the value of $\Phi$, for any given values of $v,m_{1},...m_{n}$, will be a minimum when $t=t'$. And therefore, in applying the general condition of stability relating to the value of $\Phi$, we need only consider the phases for which $t=t'$.