Page:Scientific Papers of Josiah Willard Gibbs.djvu/300

Hence,

$dW=-V_{C}dp_{C}.$

(576)

Now it is evident that $V_{C}$ will diminish as $p_{C}$ increases. Let us integrate the last equation supposing $p_{C}$ to increase from its original value until $V_{C}$ vanishes. This will give

$W''-W'={\text{a negative quantity,}}$

(577)

where $W'$ and $W''$ denote the initial and final values of $W$. But $W''=0$. Hence $W$ is positive. But this is the value of $W$ in the original system containing the lentiform mass, and expresses the work necessary to form the mass between the phases $A$ and $B$. It is therefore impossible that such a mass should form on a surface between these phases. We must however observe the same limitation as in the less general case already discussed,—that $p_{C}-p_{A},p_{C}-p_{B}$ must not be so great that the dimensions of the lentiform mass are of insensible magnitude. It may also be observed that the value of these differences may be so small that there will not be room on the surface between the masses of phases $A$ and $B$ for a mass of phase $C$ sufficiently large for equilibrium. In this case we may consider a mass of phase $C$ which is in equilibrium upon the surface between $A$ and $B$ in virtue of a constraint applied to the line in which the three surfaces of discontinuity intersect, which will not allow this line to become longer, although not preventing it from becoming shorter. We may prove that the value of $W$ is positive by such an integration as we have used before.

Substitution of Pressures for Potentials in Fundamental Equations for Surfaces.

The fundamental equation of a surface which gives the value of the tension in terms of the temperature and potentials seems best adapted to the purposes of theoretical discussion, especially when the number of components is large or undetermined. But the experimental determination of the fundamental equations, or the application of any result indicated by theory to actual cases, will be facilitated by the use of other quantities in place of the potentials, which shall be capable of more direct measurement, and of which the numerical expression (when the necessary measurements have been made) shall depend upon less complex considerations. The numerical value of a potential depends not only upon the system of units employed, but also upon the arbitrary constants involved in the definition of the energy and entropy of the substance to which the potential relates, or, it may be, of the elementary substances of which that substance is formed. (See page 96.) This fact and the want of means of direct measurement may give a certain vagueness to the idea of the