Page:Scientific Papers of Josiah Willard Gibbs.djvu/395

and thermodynamic state of any such body without regard to its size or form. The word phase has been chosen for this purpose. Such bodies as differ in composition or state are called different phases of the matter considered, all bodies which differ only in size and form being regarded as different examples of the same phase. Phases which can exist together, the dividing surfaces being plane, in an equilibrium which does not depend upon passive resistances to change, are called coexistent.

The number of independent variations of which a system of coexistent phases is capable is $n+2-r$, where $r$ denotes the number of phases, and $n$ the number of independently variable components in the whole system. For the system of phases is completely specified by the temperature, the pressure, and the $n$ potentials, and between these $n+2$ quantities there are $r$ independent relations (one for each phase), which characterize the system of phases.

When the number of phases exceeds the number of components by unity, the system is capable of a single variation of phase. The pressure and all the potentials may be regarded as functions of the temperature. The determination of these functions depends upon the elimination of the proper quantities from the fundamental equations in $p,t,\mu _{1},\mu _{2}$, etc. for the several members of the system. But without a knowledge of these fundamental equations, the values of the differential coefficients such as - may be expressed in terms of the entropies and volumes of the different bodies and the quantities of their several components. For this end we have only to eliminate the differentials of the potentials from the different equations of the form (12) relating to the different bodies. In the simplest case, when there is but one component, we obtain the well-known formula

${\frac {dp}{dt}}={\frac {\eta '-\eta ''}{v'-v''}}={\frac {Q}{t(v''-v')}},$

in which $v',v'',\eta ',\eta ''$ denote the volumes and entropies of a given quantity of the substance in the two phases, and $Q$ the heat which it absorbs in passing from one phase to the other. It is easily shown that if the temperature of two coexistent phases of two components is maintained constant, the pressure is in general a maximum or minimum when the composition of the phases is identical. In like manner, if the pressure of the phases is maintained constant, the temperature is in general a maximum or minimum when the composition of the phases is identical. The series of simultaneous values of $t$ and $p$ for which the composition of two coexistent phases is identical separates those simultaneous values of $t$ and $p$ for which no coexistent phases are possible from those for which there are two pairs of coexistent phases.