Page:Scientific Papers of Josiah Willard Gibbs.djvu/114

the condition is not satisfied. It is evident that the value of the expression

$\epsilon -T\eta +Pv-M_{1}m_{1}-M_{2}m_{2}...-M_{n}m_{n}$

(57)

applied to a mass like $O$ including some very small masses like $N$, will be negative, and will decrease if the number of these masses like $N$ is increased, until there remains within the whole mass no portion of any sensible size without these masses like $N$, which, it will be remembered, have no sensible size. But it cannot decrease without limit, as the value of (54) cannot become infinite. Now we need not inquire whether the least value of (57) (for constant values of $T,P,M_{1},M_{2},...M_{n}$) would be obtained by excluding entirely the mass like $O$, and filling the whole space considered with masses like $N$, or whether a certain mixture would give a smaller value,—it is certain that the least possible value of (57) per unit of volume, and that a negative value, will be realized by a mass having a certain homogeneity. If the new part $N$ for which the condition (52) is not satisfied occurs between two different original parts $O'$ and $O''$, the argument need not be essentially varied. We may consider the value of (57) for a body consisting of masses like $O'$ and $O''$ separated by a lamina $N$. This value may be decreased by increasing the extent of this lamina, which may be done within a given volume by giving it a convoluted form; and it will be evident, as before, that the least possible value of (57) will be for a homogeneous mass, and that the value will be negative. And such a mass will be not merely an ideal combination, but a body capable of existing, for as the expression (57) has for this mass in the state considered its least possible value per unit of volume, the energy of the mass included in a unit of volume is the least possible for the same matter with the same entropy and volume, hence, if confined in a non-conducting vessel, it will be in a state of not unstable equilibrium. Therefore when (50), (51), and (43) are satisfied, if the condition (52) is not satisfied in regard to all possible new parts, there will be some homogeneous body which can be formed out of the substances $S_{1},S_{2},...S_{n}$ which will not satisfy condition (53).

Therefore, if the initially existing masses satisfy the conditions (50), (51), and (43), and condition (53) is satisfied by every homogeneous body which can be formed out of the given matter, there will be equilibrium.

On the other hand, (53) is not a necessary condition of equilibrium. For we may easily conceive that the condition (52) shall hold true (for any very small formations within or between any of the given masses), while the condition (53) is not satisfied (for all large masses formed of the given matter), and experience shows that this is very often the case. Supersaturated solutions, superheated water, etc.