Page:Scientific Papers of Josiah Willard Gibbs.djvu/113

The condition (52) understood in either of these ways (or in others which will suggest themselves to the reader) will have a perfectly definite meaning, and will be valid as the necessary and sufficient condition of equilibrium in regard to the formation of new parts, when the conditions of equilibrium in regard to the original parts, (50), (51), and (43), are satisfied.

In regard to the condition (53), it may be shown that with (50), (51), and (43) it is always sufficient for equilibrium. To prove this, it is only necessary to show that when (50), (51), and (43) are satisfied, and (52) is not, (53) will also not be satisfied. We will first observe that an expression of the form

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

(54)

denotes the work obtainable by the formation (by a reversible process) of a body of which $\epsilon ,\eta ,v,m_{1},m_{2},...m_{n}$ are the energy, entropy, volume, and the quantities of the components, within a medium having the pressure $P$, the temperature $T$, and the potentials $M_{1},M_{2},...M_{n}$. (The medium is supposed so large that its properties are not sensibly altered in any part by the formation of the body.) For $\epsilon$ is the energy of the body formed, and the remaining terms represent (as may be seen by applying equation (12) to the medium) the decrease of the energy of the medium, if, after the formation of the body, the joint entropy of the medium and the body, their joint volumes and joint quantities of matter, were the same as the entropy, etc., of the medium before the formation of the body. This consideration may convince us that for any given finite values of $v$ and of $T,P,M_{1}$ , etc., this expression cannot be infinite when $\epsilon ,\eta ,m_{1}$, etc., are determined by any real body, whether homogeneous or not (but of the given volume), even when $T,P,M_{1}$, etc., do not represent the values of the temperature, pressure, and potentials of any real substance. (If the substances $S_{1},S_{2},...S_{n}$ are all actual components of any homogeneous part of the system of which the equilibrium is discussed, that part will afford an example of a body having the temperature, pressure, and potentials of the medium supposed.)

Now by integrating equation (12) on the supposition that the nature and state of the mass considered remain unchanged, we obtain the equation

$\epsilon =t\eta -pv+\mu _{1}m_{1}+\mu _{2}m_{2}...+\mu _{n}m_{n},$

(55)

which will hold true of any homogeneous mass whatever. Therefore for any one of the original parts, by (50) and (51),

$\epsilon -T\eta +Pv-M_{1}m_{1}-M_{1}m_{2}...+M_{n}m_{n}=0.$

(56)

If the condition (52) is not satisfied in regard to all possible new parts, let $N$ be a new part occurring in an original part $O$, for which