Page:Scientific Papers of Josiah Willard Gibbs.djvu/273

make $p'-p'',t,\mu _{3},\mu _{4}$, etc. constant is in this case equivalent to making $t,\mu _{1},\mu _{3},\mu _{4}$, etc. constant.

Substantially the same is true of the superficial density of entropy or of energy, when either of these has the same density in the two homogeneous masses.^[1]

Concerning the Stability of Surfaces of Discontinuity between Fluid Masses.

We shall first consider the stability of a film separating homogeneous masses with respect to changes in its nature, while its position and the nature of the homogeneous masses are not altered. For this purpose, it will be convenient to suppose that the homogeneous masses are very large, and thoroughly stable with respect to the possible formation of any different homogeneous masses out of their components, and that the surface of discontinuity is plane and uniform.

Let us distinguish the quantities which relate to the actual components of one or both of the homogeneous masses by the suffixes $_{a},_{b}$, etc., and those which relate to components which are found only at the surface of discontinuity by the suffixes $_{g},_{h}$ etc., and consider the variation of the energy of the whole system in consequence of a given change in the nature of a small part of the surface of discontinuity, while the entropy of the whole system and the total quantities of the several components remain constant, as well as the volume of each of the homogeneous masses, as determined by the surface of tension. This small part of the surface of discontinuity in its changed state is supposed to be still uniform in nature, and such as may subsist in equilibrium between the given homogeneous masses, which will evidently not be sensibly altered in nature or thermodynamic state.

The remainder of the surface of discontinuity is also supposed to

1. With respect to questions which concern only the form of surfaces of discontinuity, such precision as we have employed in regard to the position of the dividing surface is evidently quite unnecessary. This precision has not been used for the sake of the mechanical part of the problem, which does not require the surface to be defined with greater nicety than we can employ in our observations, but in order to give determinate values to the superficial densities of energy, entropy, and the component substances, which quantities, as has been seen, play an important part in the relations between the tension of a surface of discontinuity, and the composition of the masses which it separates.
   The product $\sigma s$ of the superficial tension and the area of the surface, may be regarded as the available energy due to the surface in a system in which the temperature and the potentials $\mu _{1},\mu _{2}$, etc.—or the differences of these potentials and the gravitational potential (see page 148) when the system is subject to gravity—are maintained sensibly constant. The value of $\sigma$, as well as that of $s$, is sensibly independent of the precise position which we may assign to the dividing surface (so long as this is sensibly coincident with the surface of discontinuity), but $\epsilon _{S}$, the superficial density of energy, as the term is used in this paper, like the superficial densities of entropy and of the component substances, requires a more precise localization of the dividing surface.