Page:Scientific Papers of Josiah Willard Gibbs.djvu/403

where $\eta _{\text{S}},\Gamma _{1},\Gamma _{2}$, etc. are written for ${\frac {\eta _{\text{S}}}{s}},{\frac {m_{1}^{\text{S}}}{s}},{\frac {m_{2}^{\text{S}}}{s}}$ etc., and denote the superficial densities of entropy and of the various substances. We may regard $\sigma$ as a function of $t,\mu _{1},\mu _{2}$, etc., from which if known $\eta _{\text{S}},\Gamma _{1},\Gamma _{2}$, etc. may be determined in terms of the same variables. An equation between $\sigma ,t,\mu _{1},\mu _{2}$, etc. may therefore be called a fundamental equation for the surface of discontinuity. The same may be said of an equation between $\epsilon ^{\text{S}},\eta ^{\text{S}},m_{1}^{\text{S}},m_{2}^{\text{S}}$, etc.

It is necessary for the stability of a surface of discontinuity that its tension shall be as small as that of any other surface which can exist between the same homogeneous masses with the same temperature and potentials. Besides this condition, which relates to the nature of the surface of discontinuity, there are other conditions of stability, which relate to the possible motion of such surfaces. One of these is that the tension shall be positive. The others are of a less simple nature, depending upon the extent and form of the surface of discontinuity, and in general upon the whole system of which it is a part. The most simple case of a system with a surface of discontinuity is that of two coexistent phases separated by a spherical surface, the outer mass being of indefinite extent. When the interior mass and the surface of discontinuity are formed entirely of substances which are components of the surrounding mass, the equilibrium is always unstable; in other cases, the equilibrium may be stable. Thus, the equilibrium of a drop of water in an atmosphere of vapor is unstable, but may be made stable by the addition of a little salt. The analytical conditions which determine the stability or instability of the system are easily found, when the temperature and potentials of the system are regarded as known, as well as the fundamental equations for the interior mass and the surface of discontinuity.

The study of surfaces of discontinuity throws considerable light upon the subject of the stability of such phases of fluids as have a less pressure than other phases of the same components with the same temperature and potentials. Let the pressure of the phase of which the stability is in question be denoted by $p'$, and that of the other phase of the same temperature and potentials by $p''$. A spherical mass of the second phase and of a radius determined by the equation

$2\sigma =(p''-p')r,$

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would be in equilibrium with a surrounding mass of the first phase. This equilibrium, as we have just seen, is unstable, when the surrounding mass is indefinitely extended. A spherical mass a little larger would tend to increase indefinitely. The work required to form such a spherical mass, by a reversible process, in the interior of an infinite mass of the other phase, is given by the equation

$W=\sigma s-(p''-p')v''.$

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