Page:Scientific Papers of Josiah Willard Gibbs.djvu/135

Concerning Cases in which the Number of Coexistent Phases is less than n + 1.

When $n>1$, if the quantities of all the components $S_{1},S_{2},...S_{n}$ are proportional in two coexistent phases, the two equations of the form of (127) and (128) relating to these phases will be sufficient for the elimination of the variations of all the potentials. In fact, the condition of the coexistence of the two phases together with the condition of the equality of the $n-1$ ratios of $m'_{1},m'_{2},...m'_{n}$ with the $n-1$ ratios of $m''_{1},m''_{2},...m''_{n}$ is sufficient to determine $p$ as a function of $t$ if the fundamental equation is known for each of the phases. The differential equation in this case may be expressed in the form of (130), $m'$ and $m''$ denoting either the quantities of any one of the components or the total quantities of matter in the bodies to which they relate. Equation (131) will also hold true in this case if the total quantity of matter in each of the bodies is unity. But this case differs from the preceding in that the matter which absorbs the heat $Q$ in passing from one state to another, and to which the other letters in the formula relate, although the same in quantity, is not in general the same in kind at different temperatures and pressures. Yet the case will often occur that one of the phases is essentially invariable in composition, especially when it is a crystalline body, and in this case the matter to which the letters in (131) relate will not vary with the temperature and pressure.

When $n=2$, two coexistent phases are capable, when the temperature is constant, of a single variation in phase. But as (130) will hold true in this case when $m'_{1}:m'_{2}::m''_{1}::m''_{2}$, it follows that for constant temperature the pressure is in general a maximum or a minimum when the composition of the two phases is identical. In like manner, the temperature of the two coexistent phases is in general a maximum or a minimum, for constant pressure, when the composition of the two phases is identical. Hence, 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 pair of coexistent phases. This may be applied to a liquid having two independently variable components in connection with the vapor which it yields, or in connection with any solid which may be formed in it.

When $n=3$, we have for three coexistent phases three equations of the form of (127), from which we may obtain the following,

${\begin{vmatrix}v'&m'_{1}&m'_{2}\\v''&m''_{1}&m''_{2}\\v'''&m'''_{1}&m'''_{2}\end{vmatrix}}dp={\begin{vmatrix}\eta '&m'_{1}&m'_{2}\\\eta ''&m''_{1}&m''_{2}\\\eta '''&m'''_{1}&m'''_{2}\end{vmatrix}}dt+{\begin{vmatrix}m'_{1}&m'_{2}&m'_{3}\\m''_{1}&m''_{2}&m''_{3}\\m'''_{1}&m'''_{2}&m'''_{3}\end{vmatrix}}d\mu _{3}$

(132)