Page:Scientific Papers of Josiah Willard Gibbs.djvu/121

But independently of any assumption in regard to the permeability of the diaphragm, the following relation will hold true in any case in which each of the two fluid masses may be regarded as uniform throughout in nature and state. Let the character ${\scriptstyle {\text{D}}}$ be used with the variables which express the nature, state, and quantity of the fluids to denote the increments of the values of these quantities actually occurring in a time either finite or infinitesimal. Then, as the heat received by the two masses cannot exceed $t'{\scriptstyle {\text{D}}}\eta '+t''{\scriptstyle {\text{D}}}\eta ''$, and as the increase of their energy is equal to the difference of the heat they receive and the work they do,

${\scriptstyle {\text{D}}}\epsilon '+{\scriptstyle {\text{D}}}\epsilon ''\leqq t'{\scriptstyle {\text{D}}}\eta '+t''{\scriptstyle {\text{D}}}\eta ''-p'{\scriptstyle {\text{D}}}v'-p''{\scriptstyle {\text{D}}}v''$

(83)

i.e., by (12),

$\mu _{1}'{\scriptstyle {\text{D}}}m'_{1}+\mu _{1}''{\scriptstyle {\text{D}}}m''_{1}+\mu _{2}'{\scriptstyle {\text{D}}}m'_{2}+\mu _{2}''{\scriptstyle {\text{D}}}m''_{2}+\,{\text{etc.}}\,\leqq 0,$

(84)

or

$(\mu _{1}''-\mu _{1}'){\scriptstyle {\text{D}}}m''_{1}+(\mu _{2}''-\mu _{2}'){\scriptstyle {\text{D}}}m''_{2}+\,{\text{etc.}}\,\leqq 0.$

(85)

It is evident that the sign $=$ holds true only in the limiting case in which no motion takes place.

Definition and Properties of Fundamental Equations.

The solution of the problems of equilibrium which we have been considering has been made to depend upon the equations which express the relations between the energy, entropy, volume, and the quantities of the various components, for homogeneous combinations of the substances which are found in the given mass. The nature of such equations must be determined by experiment. As, however, it is only differences of energy and of entropy that can be measured, or indeed, that have a physical meaning, the values of these quantities are so far arbitrary, that we may choose independently for each simple substance the state in which its energy and its entropy are both zero. The values of the energy and the entropy of any compound body in any particular state will then be fixed. Its energy will be the sum of the work and heat expended in bringing its components from the states in which their energies and their entropies are zero into combination and to the state in question; and its entropy is the value of the integral $\int {\frac {dQ}{T}}$ for any reversible process by which that change is effected ($dQ$ denoting an element of the heat communicated to the matter thus treated, and $t$ the temperature of the matter receiving it). In the determination both of the energy and of the entropy, it ia understood that at the close of the process, all bodies which have been used, other than those to which the determinations relate, have been restored to their original state, with the