Page:Scientific Papers of Josiah Willard Gibbs.djvu/258

If we substitute $\geqq$ for $=$ in this equation, the formula will hold true of all variations whether reversible or not;^[1] for if the variation of energy could have a value less than that of the second member of the equation, there must be variation in the condition of $M$ in which its energy is diminished without change of its entropy or of the quantities of its various components.

It is important, however, to observe that for any given values of $\delta \eta ,\delta m_{1},\delta m_{2}$, etc., while there may be possible variations of the nature and state of $M$ for which the value of $\delta \epsilon$ is greater than that of the second member of (477), there must always be possible variations for which the value of $\delta \epsilon$ is equal to that of the second member. It will be convenient to have a notation which will enable us to express this by an equation. Let ${\mathfrak {d}}\epsilon$ denote the smallest value (i.e., the value nearest to $\infty$) of $\delta \epsilon$ consistent with given values of the other variations, then

${\mathfrak {d}}\epsilon =t\delta \eta +\mu _{1}\delta m_{1}+\mu _{2}m_{2}+{\text{etc.}}$

(478)

For the internal equilibrium of the whole mass which consists of the parts $M,M',M''$, it is necessary that

$\delta \epsilon +\delta \epsilon '+\delta \epsilon ''\geqq 0$

(479)

for all variations which do not affect the enclosing surface or the total entropy or the total quantity of any of the various components. If we also regard the surfaces separating $M,M'$, and $M''$ as invariable, we may derive from this condition, by equations (478) and (12), the following as a necessary condition of equilibrium:—

$t\delta \eta +\mu _{1}\delta m_{1}+\mu _{2}\delta m_{2}+{\text{etc.}}$

$+t'\delta \eta '+\mu _{1}'\delta m_{1}'+\mu _{2}'\delta m_{2}'+{\text{etc.}}$

$+t''\delta \eta ''+\mu _{1}''\delta m_{1}''+\mu _{2}''\delta m_{2}''+{\text{etc.}}\geqq 0,$

(480)

1. To illustrate the difference between variations which are reversible, and those which are not, we may conceive of two entirely different substances meeting in equilibrium at a mathematical surface without being at all mixed. We may also conceive of them as mixed in a thin film about the surface where they meet, and then the amount of mixture is capable of variation both by increase and by diminution. But when they are absolutely unmixed, the amount of mixture can be increased, but is incapable of diminution, and it is then consistent with equilibrium that the value of $\delta \epsilon$ (for a variation of the system in which the substances commence to mix) should be greater than the second member of (477). It is not necessary to determine whether precisely such cases actually occur; but it would not be legitimate to overlook the possible occurrence of cases in which variations may be possible while the opposite variations are not.
   It will be observed that the sense in which the term reversible is here used is entirely different from that in which it is frequently used in treatises on thermodynamics, where a process by which a system is brought from a state A to a state B is called reversible, to signify that the system may also be brought from the state B to the state A through the same series of intermediate states taken in the reverse order by means of external agencies of the opposite character. The variation of a system from a state A to a state B (supposed to differ infinitely little from the first) is here called reversible when the system is capable of another state B' which bears the same relation to the state A that A bears to B.