Page:Scientific Papers of Josiah Willard Gibbs.djvu/393

fluids will depend upon the rigidity of the diaphragm. Even when the diaphragm is permeable to all the components without restriction, equality of pressure in the two fluids is not always necessary for equilibrium.

Effect of gravity.—In a system subject to the action of gravity, the potential for each substance, instead of having a uniform value throughout the system, so far as the substance actually occurs as an independently variable component, will decrease uniformly with increasing height, the difference of its values at different levels being equal to the difference of level multiplied by the force of gravity.

Fundamental equations.—Let $\epsilon ,\eta ,v,t$ and $p$ denote respectively the energy, entropy, volume, (absolute) temperature, and pressure of a homogeneous mass, which may be either fluid or solid, provided that it is subject only to hydrostatic pressures, and let $m_{1},m_{2},...m_{n}$ denote the quantities of its independently variable components, and $\mu _{1},\mu _{2},...\mu _{n}$ the potentials for these components. It is easily shown that $\epsilon$ is a function of $\eta ,v,m_{1},m_{2},...m_{n}$, and that the complete value of $d\epsilon$ is given by the equation

$d\epsilon =td\eta -pdv+\mu _{1}dm_{1}+\mu _{2}dm_{2}...+\mu _{n}dm_{n}.$

(5)

Now if $\epsilon$ is known in terms of $\eta ,v,m_{1},m_{2},...m_{n}$, we can obtain by differentiation $t,p,\mu _{1},\mu _{2},...\mu _{n}$ in terms of the same variables. This will make $n+3$ independent known relations between the $2n+5$ variables, $\epsilon ,\eta ,v,m_{1},m_{2},...m_{n},t,p,\mu _{1},\mu _{2},...\mu _{n}$. These are all that exist, for of these variables, $n+2$ are evidently independent. Now upon these relations depend a very large class of the properties of the compound considered,—we may say in general, all its thermal, mechanical, and chemical properties, so far as active tendencies are concerned, in cases in which the form of the mass does not require consideration. A single equation from which all these relations may be deduced may be called a fundamental equation. An equation between $\epsilon ,\eta ,v,m_{1},m_{2},...m_{n},t,p,\mu _{1},\mu _{2},...\mu _{n}$ is a fundamental equation. But there are other equations which possess the same property.

If we suppose the quantity $\psi$ to be determined for such a mass as we are considering by equation (3), we may obtain by differentiation and comparison with (5)

$d\psi =-\eta dt-pdv+\mu _{1}dm_{1}+\mu _{2}dm_{2}...+\mu _{n}dm_{n}.$

(6)

If, then, $\psi$ is known as a function of $t,v,m_{1},m_{2},...m_{n}$, we can find $\eta ,p,\mu _{1},\mu _{2},...\mu _{n}$ in terms of the same variables. If we then substitute for $\psi$ in our original equation its value taken from equation (3) we shall have again $n+3$ independent relations between the same $2n+5$ variables as before. Let

$\zeta =\epsilon -t\eta +pv,$

(7)