Page:Scientific Papers of Josiah Willard Gibbs.djvu/390

IV.

ON THE EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES.

Abstract of the Preceding Paper by the Author.

[American Journal of Science, 3 ser., vol. xvi., pp. 441–458, Dec., 1878.]

It is an inference naturally suggested by the general increase of entropy which accompanies the changes occurring in any isolated material system that when the entropy of the system has reached a maximum, the system will be in a state of equilibrium. Although this principle has by no means escaped the attention of physicists, its importance does not appear to have been duly appreciated. Little has been done to develop the principle as a foundation for the general theory of thermodynamic equilibrium.

The principle may be formulated as follows, constituting a criterion of equilibrium:—

I. For the equilibrium of any isolated system it is necessary and sufficient that in all possible variations of the state of the system which do not alter its energy, the variation of its entropy shall either vanish or be negative.

The following form, which is easily shown to be equivalent to the preceding, is often more convenient in application:—

II. For the equilibrium of any isolated system it is necessary and sufficient that in all possible variations of the state of the system which do not alter its entropy, the variation of its energy shall either vanish or be positive.

If we denote the energy and entropy of the system by $\epsilon$ and $\eta$ respectively, the criterion of equilibrium may be expressed by either of the formulæ

$(\delta \eta )_{\epsilon }\geqq 0,$

(1)

$(\delta \epsilon )_{\eta }\geqq 0.$

(2)

Again, if we assume that the temperature of the system is uniform, and denote its absolute temperature by $t$, and set

$\psi =\epsilon -t\eta ,$

(3)

the remaining conditions of equilibrium may be expressed by the formula

$(\delta \psi )_{t}\geqq 0,$

(4)