Page:Scientific Papers of Josiah Willard Gibbs.djvu/180

We shall hereafter, in the discussion of the fundamental equations of gases, have an example of the derivation of the fundamental equation for phases of dissipated energy (with respect to the molecular changes on which the proximate composition of the body depends) from the more general form of the fundamental equation.

The Conditions of Equilibrium for Heterogeneous Masses under the Influence of Gravity.

Let us now seek the conditions of equilibrium for a mass of various kinds of matter subject to the influence of gravity. It will be convenient to suppose the mass enclosed in an immovable envelop which is impermeable to matter and to heat, and in other respects, except in regard to gravity, to make the same suppositions as on page 62. The energy of the mass will now consist of two parts, one of which depends upon its intrinsic nature and state, and the other upon its position in space. Let $Dm$ denote an element of the mass, $D\epsilon$ the intrinsic energy of this element, $h$ its height above a fixed horizontal plane, and $g$ the force of gravity; then the total energy of the mass (when without sensible motions) will be expressed by the formula

$\int {D\epsilon }+\int {ghDm},$

(219)

in which the integrations include all the elements of the mass; and the general condition of equilibrium will be

$\delta \int {D\epsilon }+\delta \int {ghDm}\geqq 0,$

(220)

the variations being subject to certain equations of condition. These must express that the entropy of the whole mass is constant, that the surface bounding the whole mass is fixed, and that the total quantity of each of the component substances is constant. We shall suppose that there are no other equations of condition, and that the independently variable components are the same throughout the whole mass; and we shall at first limit ourselves to the consideration of the conditions of equilibrium with respect to the changes which may be expressed by infinitesimal variations of the quantities which define the initial state of the mass, without regarding the possibility of the formation at any place of infinitesimal masses entirely different from any initially existing in the same vicinity.

Let $D\eta ,Dv,Dm_{1},...Dm_{n}$ denote the entropy of the element $Dm$, its volume, and the quantities which it contains of the various components. Then

$Dm=Dm_{1}...+Dm_{n},$

(221)

and $\delta Dm=\delta Dm_{1}...+\delta Dm_{n}.$

(222)

Also, by equation (12),

$\delta D\epsilon =t\delta D\eta -p\delta Dv+\mu _{1}\delta Dm_{1}...\mu _{n}\delta Dm_{n}.$

(223)