The effects produced on electromotive forces by difference of concentrations in dilute solutions can thus be accounted for and traced out, from the knowledge of the form of the free energy for such cases; as also the effects of pressure in the case of gas batteries. The free energy does not sensibly depend on whether the substance is solid or fused—for the two states are in equilibrium at the temperature of fusion—though the total energy differs in these two cases by the heat of fusion; for this reason, as Gibbs pointed out, voltaic potential-differences are the same for the fused as for the solid state of the substances concerned.
Relations involving Constitution only.—The potential of a component in a given solution can depend only on the temperature and pressure of the solution, and the densities of the various components, including itself; as no distance-actions are usually involved in chemical physics, it will not depend on the aggregate masses present. The example above mentioned, of two coexistent phases liquid and vapour, indicates that there may thus be relations between the constitutions of the phases present in a chemical system which do not involve their total masses. These are developed in a very direct manner in Willard Gibbs’s original procedure. In so far as attractions at a distance (a uniform force such as gravity being excepted) and capillary actions at the interfaces between the phases are inoperative, the fundamental equation (1) can be integrated. Increasing the volume k times, and all the masses to the same extent—in fact, placing alongside each other k identical systems at the same temperature and pressure—will increase φ and E in the same ratio k; thus E must be a homogeneous function of the first degree of the independent variables φ, v, m1, ..., mn, and therefore by Euler’s theorem relating to such functions
E = Tφ − pv + μ1m1 + ... + μnmn.
This integral equation merely expresses the additive character of the energies and entropies of adjacent portions of the system at uniform temperature, and thus depends only on the absence of sensible physical action directly across finite distances. If we form from it the expression for the complete differential δE, and subtract (1), there remains the relation
0 = φδT − vδp + m1δμ1 + ... + mnδμn.
(2)
This implies that in each phase the change of pressure depends on and is determined by the changes in T, μ1, ... μn alone; as we know beforehand that a physical property like pressure is an analytical function of the state of the system, it is therefore a function of these n + 1 quantities. When they are all independently variable, the densities of the various constituents and of the entropy in the phase are expressed by the partial fluxions of p with respect to them: thus
| φ | = | dp | , | mr | = | dp | . |
| v | dT | v | dμr |
But when, as in the case above referred to of chloride of ammonium gas existing partially dissociated along with its constituents, the masses are not independent, necessary linear relations, furnished by the laws of definite combining proportions, subsist between the partial fluxions, and the form of the function which expresses p is thus restricted, in a manner which is easily expressible in each special case.
This proposition that the pressure in any phase is a function of the temperature and of the potentials of the independent constituents, thus appears as a consequence of Carnot’s axiom combined with the energy principle and the absence of effective actions at a distance. It shows that at a given temperature and pressure the potentials are not all independent, that there is a necessary relation connecting them. This is the equation of state or constitution of the phase, whose existence forms one mode of expression of Carnot’s principle, and in which all the properties of the phase are involved and can thence be derived by simple differentiation.