Page:Scientific Papers of Josiah Willard Gibbs.djvu/120

These conditions of equilibrium do not of course depend in any way upon the supposition that the volume of each fluid mass is kept constant, if the diaphragm is in any case supposed immovable. In fact, we may easily obtain the same conditions of equilibrium, if we suppose the volumes variable. In this case, as the equilibrium must be preserved by forces acting upon the external surfaces of the fluids, the variation of the energy of the sources of these forces must appear in the general condition of equilibrium, which will be

$\delta \epsilon '+\delta \epsilon ''+P'\delta v'+P''\delta v''\geqq 0,$

(79)

$P$ and $P''$ denoting the external forces per unit of area. (Compare (14).) From this condition we may evidently derive the same internal conditions of equilibrium as before, and in addition the external conditions

$p'=P',\,\,\,p''=P''.$

(80)

In the preceding paragraphs it is assumed that the permeability of the diaphragm is perfect, and its impermeability absolute, i.e., that it offers no resistance to the passage of the components of the fluids in certain proportions, except such as vanishes with the velocity, and that in other proportions the components cannot pass at all. How far these conditions are satisfied in any particular case is of course to be determined by experiment.

If the diaphragm is permeable to all the $n$ components without restriction, the temperature and the potentials for all the components must be the same on both sides. Now, as one may easily convince himself, a mass having $n$ components is capable of only $n+1$ independent variations in nature and state. Hence, if the fluid on one side of the diaphragm remains without change, that on the other side cannot (in general) vary in nature or state. Yet the pressure will not necessarily be the same on both sides. For, although the pressure is a function of the temperature and the n potentials, it may be a many-valued function (or any one of several functions) of these variables. But when the pressures are different on the two sides, the fluid which has the less pressure will be practically unstable, in the sense in which the term has been used on page 79. For

$\epsilon ''-t''\eta ''+p''v''-\mu _{1}''m''_{1}-\mu _{2}''m''_{2}...-\mu _{n}''m''_{n}=0,$

(81)

as appears from equation (12) if integrated on the supposition that the nature and state of the mass remain unchanged. Therefore, if $p'<p''$ while $t'=t'',\,\mu _{1}'=\mu _{1}''$, etc.,

$\epsilon ''-t''\eta ''+p''v''-\mu _{1}''m''_{1}-\mu _{2}''m''_{2}...-\mu _{n}''m''_{n}<0.$

(82)

This relation indicates the instability of the fluid to which the single accents refer. (See page 79.)