Page:Scientific Papers of Josiah Willard Gibbs.djvu/132

substance. For we may imagine the substance brought from the state in which $\psi =0$ and the temperature is the same as that of the given mass, first to any specified state of the same temperature, and then into combination with the given mass. In the first part of the process the work expended is evidently represented by the value of $\psi$ for the unit of the substance in the state specified. Let this be denoted by $\psi '$, and let $\mu$ denote the potential in question, and $W$ the work expended in bringing a unit of the substance from the specified state into combination with the given mass as aforesaid; then

$\mu =\psi '+W.$

(123)

Now as the state of the substance for which $\epsilon =0$ and $\eta =0$ is arbitrary, we may simultaneously increase the energies of the unit of the substance in all possible states by any constant $C$, and the entropies of the substance in all possible states by any constant $K$. The value of $\psi$, or $\epsilon -t\eta$, for any state would then be increased by $C-tK$, $t$ denoting the temperature of the state. Applying this to $\psi '$ in (123) and observing that the last term in this equation is independent of the values of these constants, we see that the potential would be increased by the same quantity $C-tK$, $t$ being the temperature of the mass in which the potential is to be determined.

On Coexistent Phases of Matter.

In considering the different homogeneous bodies which can be formed out of any set of component substances, it will be convenient to have a term which shall refer solely to the composition and thermodynamic state of any such body without regard to its quantity or form. We may call such bodies as differ in composition or state different phases of the matter considered, regarding all bodies which differ only in quantity and form 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, we shall call coexistent.

If a homogeneous body has $n$ independently variable components, the phase of the body is evidently capable of $n+1$ independent variations. A system of $r$ coexistent phases, each of which has the same $n$ independently variable components is capable of $n+2-r$ variations of phase. For the temperature, the pressure, and the potentials for the actual components have the same values in the different phases, and the variations of these quantities are by (97) subject to as many conditions as there are different phases. Therefore, the number of independent variations in the values of these quantities, i.e., the number of independent variations of phase of the system, will be $n+2-r.$