Page:Scientific Papers of Josiah Willard Gibbs.djvu/295

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EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES.

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of $A$ and $B$ shall become less than the pressure of $A$ and $B$ . A system consisting of the phases $A$ and $B$ will be entirely stable with respect to the formation of any phase like $C$ . (This case is not quite identical with that considered on page 104, since the system in question contains two different phases, but the principles involved are entirely the same.)

With respect to variations of the phases $A$ and $B$ in the opposite direction we must consider two cases separately. It will be convenient to denote the pressures of the three phases by $p_{A},p_{B},p_{C}$ , and to regard these quantities as functions of the temperature and potentials.

If $\sigma _{AB}=\sigma _{AC}+\sigma _{BC}$ for values of the temperature and potentials which make $p_{A}=p_{B}=p_{C}$ , it will not be possible to alter the temperature and potentials at the surface of contact of the phases $A$ and $B$ so that $p_{A}=p_{b}$ , and $p_{C}>p_{A}$ , for the relation of the temperature and potentials necessary for the equality of the three pressures will be preserved by the increase of the mass of the phase $C$ . Such variations of the phases $A$ and $B$ might be brought about in separate masses, but if these were brought into contact, there would be an immediate formation of a mass of the phase $C$ , with reduction of the phases of the adjacent masses to such as satisfy the conditions of equilibrium with that phase.

But if $\sigma _{AB}<\sigma _{AC}+\sigma _{BC}$ , we can vary the temperature and potentials so that $p_{A}=p_{B}$ , and $p_{C}>p_{A}$ , and it will not be possible for a sheet of the phase of $C$ to form immediately, i.e., while the pressure of $C$ is sensibly equal to that of $A$ and $B$ ; for mechanical work equal to $\sigma _{AC}+\sigma _{BC}-\sigma _{AB}$ per unit of surface might be obtained by bringing the system into its original condition, and therefore produced without any external expenditure, unless it be that of heat at the temperature of the system, which is evidently incapable of producing the work. The stability of the system in respect to such a change must therefore extend beyond the point where the pressure of $C$ commences to be greater than that of $A$ and $B$ . We arrive at the same result if we use the expression (520) as a test of stability. Since this expression has a finite positive value when the pressures of the phases are all equal, the ordinary surface of discontinuity must be stable, and it must require a finite change in the circumstances of the case to make it become unstable.^[1]

↑ It is true that such a case as we are now considering is formally excluded in the discussion referred to, which relates to a plane surface, and in which the system is supposed thoroughly stable with respect to the possible formation of any different homogeneous masses. Yet the reader will easily convince himself that the criterion (520) is perfectly valid in this case with respect to the possible formation of a thin sheet of the phase $C$ , which, as we have seen, may be treated simply as a different kind of surface of discontinuity.

[1] It is true that such a case as we are now considering is formally excluded in the discussion referred to, which relates to a plane surface, and in which the system is supposed thoroughly stable with respect to the possible formation of any different homogeneous masses. Yet the reader will easily convince himself that the criterion (520) is perfectly valid in this case with respect to the possible formation of a thin sheet of the phase $C$ , which, as we have seen, may be treated simply as a different kind of surface of discontinuity.

[1]