Page:Scientific Papers of Josiah Willard Gibbs.djvu/136

Now the value of the last of these determinants will be zero, when the composition of one of the three phases is such as can be produced by combining the other two. Hence, the pressure of three coexistent phases will in general be a maximum or minimum for constant temperature, and the temperature a maximum or minimum for constant pressure, when the above condition in regard to the composition of the coexistent phases is satisfied. The series of simultaneous values of $t$ and $p$ for which the condition is satisfied separates those simultaneous values of $t$ and $p$ for which three coexistent phases are not possible, from those for which there are two triads of coexistent phases. These propositions may be extended to higher values of $n$ and illustrated by the boiling temperatures and pressures of saturated solutions of $n-2$ different solids in solvents having two independently variable components.

Internal Stability of Homogeneous Fluids as indicated by Fundamental Equations.

We will now consider the stability of a fluid enclosed in a rigid envelop which is non-conducting to heat and impermeable to all the components of the fluid. The fluid is supposed initially homogeneous in the sense in which we have before used the word, i.e., uniform in every respect throughout its whole extent. Let $S_{1},~S_{2},~\ldots ~S_{n}$ be the ultimate components of the fluid; we may then consider every body which can be formed out of the fluid to be composed of $S_{1},~S_{2},~\ldots ~S_{n}$, and that in only one way. Let $m_{1},~m_{2},~\ldots ~m_{n}$ denote the quantities of these substances in any such body, and let $\epsilon ,~\eta ,~v$, denote its energy, entropy, and volume. The fundamental equation for compounds of $S_{1},~S_{2},~\ldots ~S_{n}$, if completely determined, will give us all possible sets of simultaneous values of these variables for homogeneous bodies.

Now, if it is possible to assign such values to the constants $T,~P,~M_{1},~M_{2},~\ldots ~M_{n}$ that the value of the expression

$\epsilon -T\eta +Pv-M_{1}m_{1}-M_{2}m_{2}\ldots -M_{n}m_{n}$

(133)

shall be zero for the given fluid, and shall be positive for every other phase of the same components, i.e., for every homogeneous body^[1] not identical in nature and state with the given fluid (but composed entirely of $S_{1},~S_{2},~\ldots ~S_{n}$), the condition of the given fluid will be stable. For, in any condition whatever of the given mass, whether or not homogeneous, or fluid, if the value of the expression (133) is not

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1. A vacuum is throughout this discussion to be regarded as a limiting case of an extremely rarified body. We may thus avoid the necessity of the specific mention of a vacuum in propositions of this kind.