Page:Scientific Papers of Josiah Willard Gibbs.djvu/110

consistent with eqs. (43), that the condition (45) shall hold true for any change in the system irrespective of the equations of condition (39), (40), (41).

For this it is necessary and sufficient that

$t=T,\,\,\,p=P,$

(50)

$\mu _{1}\delta m_{1}\geqq M_{1}\delta m_{1},\,\,\,\mu _{2}\delta m_{2}\geqq M_{2}\delta m_{2},\,...\,\,\mu _{n}\delta m_{n}\geqq M_{n}\delta m_{n}$

(51)

for each of the original parts as previously defined, and that

$D\epsilon -TD\eta +PDv-M_{1}Dm_{1}-M_{2}Dm_{2}...-M_{n}Dm_{n}\geqq 0,$

(52)

for each of the new parts as previously defined. If to these conditions we add equations (43), we may treat $T,P,M_{1},M_{2},...M_{n}$ simply as unknown quantities to be eliminated.

In regard to conditions (51), it will be observed that if a substance $S_{1}$, is an actual component of the part of the given mass distinguished by a single accent, $\delta m'_{1}$ may be either positive or negative, and we shall have $\mu _{1}'=M_{1}$; but if $S_{1}$ is only a possible component of that part, $\delta m'_{1}$ will be incapable of a negative value, and we will have $\mu _{1}'\geqq M_{1}$.

The formulæ (50), (51), and (43) express the same particular conditions of equilibrium which we have before obtained by a less general process. It remains to discuss (52). This formula must hold true of any infinitesimal mass in the system in its varied state which is not approximately homogeneous with any of the surrounding masses, the expressions $D\epsilon ,D\eta ,Dv,Dm_{1},Dm_{2},...Dm_{n}$ denoting the energy, entropy, and volume of this infinitesimal mass, and the quantities of the substances $S_{1},S_{2},...S_{n}$ which we regard as comppsing it (not necessarily as independently variable components). If there is more than one way in which this mass may be considered as composed of these substances, we may choose whichever is most convenient. Indeed it follows directly from the relations existing between $M_{1},M_{2},...$ and $M_{n}$ that the result would be the same in any case. Now, if we assume that the values of $D\epsilon ,D\eta ,Dv,Dm_{1},Dm_{2},...Dm_{n}$ are proportional to the values of $\epsilon ,\eta ,v,m_{1},m_{2},...m_{n}$ for any large homogeneous mass of similar composition, and of the same temperature and pressure, the condition is equivalent to this, that

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

(53)

for any large homogeneous body which can be formed out of the substances $S_{1},S_{2},...S_{n}$.

But the validity of this last transformation cannot be admitted without considerable limitation. It is assumed that the relation between the energy, entropy, volume, and the quantities of the different components of a very small mass surrounded by substances of different composition and state is the same as if the mass in