temperature, pressure and other independent data specifying its
constitution, must form the variables of an analytical exposition.
We have, therefore, to substitute Tδφ for δH; also the change of
internal energy is determined by the change of constitution,
involving a differential relation of type
when the system consists of an intimate mixture (solution) of masses m1, m2, ... mn of given constituents, which differ physically or chemically but may be partially transformable into each other by chemical or physical action during the changes under consideration, the whole being of volume v and under extraneous pressure p, while W is potential energy arising from physical forces such as those of gravity, capillarity, &c. The variables m1, m2, ... mn may not be all independent; for example, if the system were chloride of ammonium gas existing along with its gaseous products of dissociation, hydrochloric acid and ammonia, only one of the three masses would be independently variable. The sufficient number of these variables (independent components) together with two other variables, which may be v and T, or v and φ, specifies and determines the state of the system, considered as matter in bulk, at each instant. It is usual to include δW in μ1δm1 + ...; in all cases where this is possible the single equation
thus expresses the complete variation of the energy-function E arising from change of state; and when the part involving the n constitutive differentials has been expressed in terms of the number of them that are really independent, this equation by itself becomes the unique expression of all the thermodynamic relations of the system. These are in fact the various relations ensuring that the right-hand side is an exact differential, and are of the type of reciprocal relations such as dμr/dφ = dT/dmr.
The condition that the state of the system be one of stable equilibrium is that δφ, the variation of entropy, be negative for all formally imaginable infinitesimal transformations which make δE vanish; for as δφ cannot actually be negative for any spontaneous variation, none of these transformations can then occur. From the form of the equation, this condition is the same as that δE − Tδφ must be positive for all possible variations of state of the system as above defined in terms of co-ordinates representing its constitution in bulk, without restriction.
We can change one of the independent variables expressing the state of the system from φ to T by subtracting δ(φT) from both sides of the equation of variation: then
It follows that for isothermal changes, i.e. those for which δT is maintained null by an environment at constant temperature, the condition of stable equilibrium is that the function E − Tφ shall be a minimum. If the system is subject to an external pressure p, which as well as the temperature is imposed constant from without and thus incapable of variation through internal changes, the condition of stable equilibrium is similarly that E − Tφ + pv shall be a minimum.
A chemical system maintained at constant temperature by communication of heat from its environment may thus have several states of stable equilibrium corresponding to different minima of the function here considered, just as there may be several minima of elevation on a landscape, one at the bottom of each depression; in fact, this analogy, when extended to space of n dimensions, exactly fits the case. If the system is sufficiently disturbed, for example, by electric shock, it may pass over (explosively) from a higher to a lower minimum, but never (without compensation from outside) in the opposite direction. The former passage, moreover, is often effected by introducing a new substance into the system; sometimes that substance is recovered unaltered at the end of the process, and then its action is said to be purely catalytic; its presence modifies the form of the function E − Tφ so as to obliterate the ridge between the two equilibrium states in the graphical representation.
There are systems in which the equilibrium states are but very slightly dependent on temperature and pressure within wide limits, outside which reaction takes place. Thus while there are cases in which a state of mobile dissociation exists in the system which changes continuously as a function of these variables, there are others in which change does not sensibly occur at all until a certain temperature of reaction is attained, after which it proceeds very rapidly owing to the heat developed, and the system soon becomes sensibly permanent in a transformed phase by completion of the reaction. In some cases of this latter type the cause of the delay in starting lies possibly in passive resistance to change, of the nature of viscosity or friction, which is competent to convert an unstable mechanical equilibrium into a moderately stable one; but in most such reactions there seems to be no exact equilibrium at any temperature, short of the ultimate state of dissipated energy in which the reaction is completed, although the velocity of reaction is found to diminish exponentially with change of temperature, and thus becomes insignificant at a small interval from the temperature of pronounced activity.
Free Energy.—The quantity E − Tφ thus plays the same fundamental part in the thermal statics of general chemical systems at uniform temperature that the potential energy plays in the statics of mechanical systems of unchanging constitution. It is a function of the geometrical co-ordinates, the physical and chemical constitution, and the temperature of the system, which determines the conditions of stable equilibrium at each temperature; it is, in fact, the potential energy generalized so as to include temperature, and thus be a single function relating to each temperature but at the same time affording a basis of connexion between the properties of the system at different temperatures. It has been called the free energy of the system by Helmholtz, for it is the part of the energy whose variation is connected with changes in the bodily structure of the system represented by the variables m1, m2, ... mn, and not with the irregular molecular motions represented by heat, so that it can take part freely in physical transformations. Yet this holds good only subject to the condition that the temperature is not varied; it has been seen above that for the more general variation neither δH nor δU is an exact differential, and no line of separation can be drawn between thermal and mechanical energies.
The study of the evolution of ideas in this, the most abstract branch of modern mathematical physics, is rendered difficult in the manner of most purely philosophical subjects by the variety of terminology, much of it only partially appropriate, that has been employed to express the fundamental principles by different investigators and at different stages of the development. Attentive examination will show, what is indeed hardly surprising, that the principles of the theory of free energy of Gibbs and Helmholtz had been already grasped and exemplified by Lord Kelvin in the very early days of the subject (see the paper “On the Thermoelastic and Thermomagnetic Properties of Matter, Part I.” Quarterly Journal of Mathematics, No. 1, April 1855; reprinted in Phil. Mag., January 1878, and in Math. and Phys. Papers, vol. i. pp. 291, seq.). Thus the striking new advance contained in the more modern work of J. Willard Gibbs (1875–1877) and of Helmholtz (1882) was rather the sustained general application of these ideas to chemical systems, such as the galvanic cell and dissociating gaseous systems, and in general fashion to heterogeneous concomitant phases. The fundamental paper of Kelvin connecting the electromotive force of the cell with the energy of chemical transformation is of date 1851, some years before the distinction between free energy and total energy had definitely crystallized out; and, possibly satisfied with the approximate exactness of his imperfect formula when applied to a Daniell’s cell (infra), and deterred by absence of experimental data, he did not return to the subject. In 1852 he briefly announced (Proc. Roy. Soc. Edin.) the principle of the dissipation of mechanical (or available) energy, including the necessity of compensation elsewhere when restoration occurs, in the form that “any restoration of mechanical energy, without more than an equivalent of dissipation, is impossible”—probably even in vital activity; but a sufficient specification of available energy (cf. infra) was not then developed. In the paper above referred to, where this was done, and illustrated by full application to solid elastic systems, the total energy is represented by c and is named
