Page:Scientific Papers of Josiah Willard Gibbs.djvu/144

and that when $t'=t''$

$\psi ''+p'v''-\mu _{1}'m_{1}''...+\mu _{n}'m_{n}''-\psi '-p'v'+\mu _{1}'m'_{1}...+\mu _{n}'m'_{n}>0.$

(152)

These conditions may be written in the form

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

(153)

$[\Delta \psi +p\Delta v-\mu _{1}\Delta m_{1}...-\mu _{n}\Delta m_{n}]_{t}>0,$

(154)

in which the subscript letters indicate the quantities which are to be regarded as constant, m standing for all the quantities $m_{1},...m_{n}$. If these conditions hold true within any given limits, (150) will also hold true of any two infinitesimally differing phases within the same limits. To prove this, we will consider a third phase, determined by the equations

$t'''=t',$

(155)

and $v'''=v'',\,\,m'''_{1}=m''_{1},...m'''_{n}=m''_{n}.$

(156)

Now by (153), $\psi '''-\psi ''+(t'''-t'')\eta ''<0;$

(157)

and by (154), $\psi '''+p'v'''-\mu _{1}'m'''_{1}...-\mu _{n}'m'''_{n}-\psi '-p'v'+\mu _{1}'m'_{1}...+\mu _{1}'m'_{1}>0.$

(158)

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

(159)

which by (155) and (156) is equivalent to (150). Therefore, the conditions (153) and (154) in respect to the phases within any given limits are necessary and sufficient for the stability of all the phases within those limits. It will be observed that in (153) we have the condition of thermal stability of a body considered as unchangeable in composition and in volume, and in (154), the condition of mechanical and chemical stability of the body considered as maintained at a constant temperature. Comparing equation (88), we see that the condition (153) will be satisfied, if ${\frac {d^{2}\psi }{dt^{2}}}<0,$, i.e., if ${\frac {d\eta }{dt}}$ or $t{\frac {d\eta }{dt}}$ (the specific heat for constant volume) is positive. When $n=1,$ i.e.,

when the composition of the body is invariable, the condition (154) will evidently not be altered, if we regard $m$ as constant, by which the condition will be reduced to

$[\Delta \psi +p\Delta v]_{t,m}>0.$

(160)

This condition will evidently be satisfied if ${\frac {d^{2}\psi }{dv^{2}}}>0,$, i.e., if $-{\frac {dp}{dv}}$ or $-v{\frac {dp}{dv}}$ (the elasticity for constant temperature) is positive. But when $n>1$, (154) may be abbreviated more symmetrically by making $v$ constant.