Page:Scientific Papers of Josiah Willard Gibbs.djvu/291

may regard $\sigma$ as positive (for if $\sigma$ is not positive when $p'=p''$, the surface when plane would not be stable in regard to position, as it certainly is, in every actual case, when the proper conditions are fulfilled with respect to its perimeter), we see by (550) that the pressure in the interior mass must be the greater; i.e., we may regard $\sigma ,p'-p''$, and $r$ as all positive. By (555), the value of $W$ will also be positive. But it is evident from equation (552), which defines $W$, that the value of this quantity is necessarily real, in any possible case of equilibrium, and can only become infinite when $r$ becomes infinite and $p'=p''$. Hence, by (556) and (558), as $p'-p''$ increases from very small values, $W,r$, and $\sigma$ have single, real, and positive values until they simultaneously reach the value zero. Within this limit, our method is evidently applicable; beyond this limit, if such exist, it will hardly be profitable to seek to interpret the equations. But it must be remembered that the vanishing of the radius of the somewhat arbitrarily determined dividing surface may not necessarily involve the vanishing of the physical heterogeneity. It is evident, however (see pp. 225–227), that the globule must become insensible in magnitude before $r$ can vanish.

It may easily be shown that the quantity denoted by $W$ is the work which would be required to form (by a reversible process) the heterogeneous globule in the interior of a very large mass having initially the uniform phase of the exterior mass. For this work is equal to the increment of energy of the system when the globule is formed without change of the entropy or volume of the whole system or of the quantities of the several components. Now $[\eta ],[m_{1}],[m_{2}]$, etc. denote the increments of entropy and of the components in the space where the globule is formed. Hence these quantities with the negative sign will be equal to the increments of entropy and of the components in the rest of the system. And hence, by equation (86),

$-t[\eta ]-\mu _{1}[m_{1}]-\mu _{2}[m_{2}]-{\text{etc.}}$

will denote the increment of energy in all the system except where the globule is formed. But $[\epsilon ]$ denotes the increment of energy in that part of the system. Therefore, by (552), $W$ denotes the total increment of energy in the circumstances supposed, or the work required for the formation of the globule.

The conclusions which may be drawn from these considerations with respect to the stability of the homogeneous mass of the pressure $p''$ (supposed less than $p'$, the pressure belonging to a different phase of the same temperature and potentials) are very obvious. Within those limits within which the method used has been justified, the mass in question must be regarded as in strictness stable with respect to the growth of a globule of the kind considered, since $W$, the work