Page:Scientific Papers of Josiah Willard Gibbs.djvu/299

Again, if more than one kind of surface of discontinuity is possible between $A$ and $B$, for any given values of the temperature and potentials, it will be impossible for that having the greater tension to displace the other, at the temperature and with the potentials considered. Hence, when $p_{C}$ has the value determined by equation (571), and consequently $\sigma _{AC}+\sigma _{BC}$ one value of the tension for the surface between $A$ and $B$, it is impossible that the ordinary tension of the surface $\sigma _{AB}$ should be greater than this. If $\sigma _{AB}=\sigma _{AC}+\sigma _{BC}$, when equation (571) is satisfied, we may presume that a thin film of the phase $C$ actually exists at the surface between $A$ and $B$, and that a variation of the phases such as would make p greater than the second member of (571) cannot be brought about at that surface, as it would be prevented by the formation of a larger mass of the phase $C$. But if $\sigma _{AB}<\sigma _{AC}+\sigma _{BC}$ wnen equation (571) is satisfied, this equation does not mark the limit of the stability of the surface between $A$ and $B$, for the temperature or potentials must receive a finite change before the film of phase $C$, or (as we shall see in the following paragraph) a lentiform mass of that phase, can be formed.

The work which must be expended in order to form on the surface between indefinitely large masses of phases $A$ and $B$ a lentiform mass of phase $C$ in equilibrium, may evidently be represented by the formula

${\begin{aligned}W&=\sigma _{AC}S_{AC}+\sigma _{BC}S_{BC}-\sigma _{AB}S_{AB}\\&-p_{C}V_{C}+p_{A}V_{A}+p_{B}V_{B},\\\end{aligned}}$

(573)

where $S_{AC},S_{BC}$ denote the areas of the surfaces formed between $A$ and $C$, and $B$ and $C$; and $S_{AB}V_{A},V_{B}$ the diminution of the area of the surface between $A$ and $B$; $V_{C}$ the volume formed of the phase $C$; and $V_{A},V_{B}$ the diminution of the volumes of the phases $A$ and $B$. Let us now suppose $\sigma _{AC},\sigma _{BC},\sigma _{AB},p_{A},p_{B}$ remain constant and the external boundary of the surface between $A$ and $B$ to remain fixed, while $p_{C}$ increases and the surfaces of tension receive such alterations as are necessary for equilibrium. It is not necessary that this should be physically possible in the actual system; we may suppose the changes to take place, for the sake of argument, although involving changes in the fundamental equations of the masses and surfaces considered. Then, regarding $W$ simply as an abbreviation for the second member of the preceding equation, we have

${\begin{aligned}dW&=\sigma _{AC}ds_{AC}+\sigma _{BC}-\sigma _{AB}S_{AB}\\&-p_{C}dV_{c}+p_{A}dV_{A}+p_{B}dV_{B}-V_{C}dp_{c}.\\\end{aligned}}$

(574)

But the conditions of equilibrium require that

${\begin{aligned}\sigma _{AC}dS_{AC}&+\sigma _{BC}-\sigma _{AB}S_{AB}\\&-p_{C}dV_{c}+p_{A}dV_{A}+p_{B}dV_{B}=0.\\\end{aligned}}$

(575)