Page:Scientific Papers of Josiah Willard Gibbs.djvu/312

276 surface. On the same principle, we may use $$\Gamma_{1}$$ and $$\Gamma_{2}$$ to denote the values of $$m_{1}^S$$ and $$m_{2}^S$$ per unit of surface, and $$m_{1}', m_{2}, \gamma_{1}', \gamma_{2}$$ to denote the quantities of the substance and its densities in the two homogeneous masses.

With such a notation, which may be extended to cases in which the film is impermeable to any number of components, the equations relating to the surface and the contiguous masses will evidently have the same form as if the substances specified by the different suffixes were all really different. The superficial tension will be a function of $$\mu_{1}$$ and $$\mu_{2}$$, with the temperature and the potentials for the other components, and $$-\Gamma_{1}, -\Gamma_{2}$$ will be equal to its differential coefficients with respect to $$\mu_{1}$$ and $$\mu_{2}$$. In a word, all the general relations which have been demonstrated may be applied to this case, if we remember always to treat the component as a different substance according as it is found on one side or the other of the impermeable film.

When there is free passage for the component specified by the suffixes $$_{1}$$ and $$_{2}$$ through other parts of the system (or through any flaws in the film), we shall have in case of equilibrium $$\mu_{1} = \mu_{2}$$. If we wish to obtain the fundamental equation for the surface when satisfying this condition, without reference to other possible states of the surface, we may set a single symbol for $$\mu_{1}$$ and $$\mu_{2}$$ in the more general form of the fundamental equation. Cases may occur of an impermeability which is not absolute, but which renders the transmission of some of the components exceedingly slow. In such cases, it may be necessary to distinguish at least two different fundamental equations, one relating to a state of approximate equilibrium which may be quickly established, and another relating to the ultimate state of complete equilibrium. The latter may be derived from the former by such substitutions as that just indicated.

Let us now seek the complete value of the variation of the energy of a system of heterogeneous fluid masses, in which the influence of gravity and of the surfaces of discontinuity shall be included, and deduce from it the conditions of internal equilibrium for such a system. In accordance with the method which has been developed, the intrinsic energy (i.e. the part of the energy which is independent of gravity), the entropy, and the quantities of the several components must each be divided into two parts, one of which we regard as belonging to the surfaces which divide approximately homogeneous