Page:Scientific Papers of Josiah Willard Gibbs.djvu/186

the condition is not satisfied, if equilibrium subsists, it will be at least practically unstable.

Hence, the phase at any point of a fluid mass, which is in stable equilibrium under the influence of gravity (whether this force is due to external bodies or to the mass itself), and which has throughout the same independently variable components, is completely determined by the phase at any other point and the difference of the values of the gravitational potential for the two points.

Fundamental Equations of Ideal Gases and Gas-Mixtures.

For a constant quantity of a perfect or ideal gas, the product of the volume and pressure is proportional to the temperature, and the variations of energy are proportional to the variations of temperature. For a unit of such a gas we may write

$pv=at$

$d\epsilon =cdt,$

$a$ and $c$ denoting constants. By integration, we obtain the equation

$\epsilon =ct+E,$

in which $E$ also denotes a constant. If by these equations we eliminate $t$ and $p$ from (11) we obtain

$d\epsilon ={\frac {\epsilon -E}{c}}d\eta -{\frac {a}{v}}{\frac {\epsilon -E}{c}}dv,$

or $c{\frac {d\epsilon }{\epsilon -E}}=d\eta -a{\frac {dv}{v}}\cdot$

The integral of this equation may be written in the form

$c\log {\frac {\epsilon -E}{c}}=\eta -a\log {v}-H,$

where $H$ denotes a fourth constant. We may regard $E$ as denoting the energy of a unit of the gas for $t=1$; $H$ its entropy for $t=1$ and $v=1$; $a$ its pressure in the latter state, or its volume for $t=1$ and $p=1$; $c$ its specific heat at constant volume. We may extend the application of the equation to any quantity of the gas, without altering the values of the constants, if we substitute ${\frac {\epsilon }{m}},{\frac {\eta }{m}},{\frac {v}{m}}$ for $\epsilon ,\eta ,v$, respectively. This will give

$c\log {\frac {\epsilon -Em}{cm}}={\frac {\eta }{m}}-H+a\log {\frac {m}{v}}\cdot$

(255)

This is a fundamental equation (see pages 85–89) for an ideal gas of invariable composition. It will be observed that if we do not have to consider the properties of the matter which forms the gas as