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 “the intrinsic energy,” the energy taken in during an isothermal transformation is represented by e, of which H is taken in as heat, while the remainder, the change of free (or mechanical or available) energy of the system is the unnamed quantity denoted by the symbol w, which is “the work done by the applied forces” at uniform temperature. It is pointed out that it is w and not e that is the potential energy-function for isothermal change, of which the form can be determined directly by dynamical and physical experiment, and from which alone the criteria of equilibrium and stress are to be derived—simply for the reason that for all reversible paths at constant temperature between the same terminal configurations, there must, by Carnot’s principle, be the same gain or loss of heat. And a system of formulae are given (5) to (11)—Ex. gr. e = w − t dw/dt + J ∫ sdt for finding the total energy e for any temperature t when w and the thermal capacity s of the system, in a standard state, have thus been ascertained, and another for establishing connexion between the form of w for one temperature and its form for adjacent temperatures—which are identical with those developed by Helmholtz long afterwards, in 1882, except that the entropy appears only as an unnamed integral. The progress of physical science is formally identified with the exploration of this function w for physical systems, with continually increasing exactness and range—except where pure kinetic considerations prevail, in which cases the wider Hamiltonian dynamical formulation is fundamental. Another aspect of the matter will be developed below.
A somewhat different procedure, in terms of entropy as fundamental, has been adopted and developed by Planck. In an isolated system the trend of change must be in the direction which increases the entropy φ, by Clausius’ form of the principle. But in experiment it is a system at constant temperature rather than an adiabatic one that usually is involved; this can be attained formally by including in the isolated system (cf. infra) a source of heat at that temperature and of unlimited capacity, when the energy of the original system increases by δE this source must give up heat of amount δE, and its entropy therefore diminishes δE/T. Thus for the original system maintained at constant temperature T it is δφ − δE/T that must always be positive in spontaneous change, which is the same criterion as was reached above. Reference may also be made to H.A. Lorentz’s Collected Scientific Papers, part i.
A striking anticipation, almost contemporaneous, of Gibbs’s thermodynamic potential theory (infra) was made by Clerk Maxwell in connexion with the discussion of Andrews’s experiments on the critical temperature of mixed gases, in a letter published in Sir G.G. Stokes’s Scientific Correspondence (vol. ii. p. 34).
Available Energy.—The same quantity φ, which Clausius named the entropy, arose in various ways in the early development of the subject, in the train of ideas of Rankine and Kelvin relating to the expression of the available energy A of the material system. Suppose there were accessible an auxiliary system containing an unlimited quantity of heat at absolute temperature T0, forming a condenser into which heat can be discharged from the working system, or from which it may be recovered at that temperature: we proceed to find how much of the heat of our system is available for transformation into mechanical work, in a process which reduces the whole system to the temperature of this condenser. Provided the process of reduction is performed reversibly, it is immaterial, by Carnot’s principle, in what manner it is effected: thus in following it out in detail we can consider each elementary quantity of heat δH removed from the system as set aside at its actual temperature between T and T + δT for the production of mechanical work δW and the residue of it δH0 as directly discharged into the condenser at T0. The principle of Carnot gives δH/T = δH0/T0, so that the portion of the heat δH that is not available for work is δH0, equal to T0δH/T. In the whole process the part not available in connexion with the condenser at T0 is therefore T0 ∫ δH/T. This quantity must be the same whatever reversible process is employed: thus, for example, we may first transform the system reversibly from the state C to the state D, and then from the state D to the final state of uniform temperature T0. It follows that the value of T0 ∫ dH/T, representing the heat degraded, is the same along all reversible paths of transformation from the state C to the state D; so that the function ∫ dH/T is the excess of a definite quantity φ connected with the system in the former state as compared with the latter.
It is usual to change the law of sign of δH so that gain of heat by the system is reckoned positive; then, relative to a condenser of unlimited capacity at T0, the state C contains more mechanically available energy than the state D by the amount EC − ED + T0 ∫ dH/T, that is, by EC − ED − T0(φC − φD). In this way the existence of an entropy function with a definite value for each state of the system is again seen to be the direct analytical equivalent of Carnot’s axiom that no process can be more efficient than a reversible process between the same initial and final states. The name motivity of a system was proposed by Lord Kelvin in 1879 for this conception of available energy. It is here specified as relative to a condenser of unlimited capacity at an assigned temperature T0: some such specification is necessary to the definition; in fact, if T0 were the absolute zero, all the energy would be mechanically available.
But we can obtain an intrinsically different and self-contained comparison of the available energies in a system in two different states at different temperatures, by ascertaining how much energy would be dissipated in each in a reduction to the same standard state of the system itself, at a standard temperature T0. We have only to reverse the operation, and change back this standard state to each of the others in turn. This will involve abstractions of heat δH0 from the various portions of the system in the standard state, and returns of δH to the state at T0; if this return were δH0T/T0 instead of δH, there would be no loss of availability in the direct process; hence there is actual dissipation δH − δH0T/T0, that is T(δφ − δφ0). On passing from state 1 to state 2 through this standard state 0 the difference of these dissipations will represent the energy of the system that has become unavailable. Thus in this sense E − Tφ + Tφ0 + const. represents for each state the amount of energy that is available; but instead of implying an unlimited source of heat at the standard temperature T0, it implies that there is no extraneous source. The available energy thus defined differs from E − Tφ, the free energy of Helmholtz, or the work function of the applied forces of Kelvin, which involves no reference to any standard state, by a simple linear function of the temperature alone which is immaterial as regards its applications.
The determination of the available mechanical energy arising from differences of temperature between the parts of the same system is a more complex problem, because it involves a determination of the common temperature to which reversible processes will ultimately reduce them; for the simple case in which no changes of state occur the solution was given by Lord Kelvin in 1853, in connexion with the above train of ideas (cf. Tait’s Thermodynamics, §179). In the present exposition the system is sensibly in equilibrium at each stage, so that its temperature T is always uniform throughout; isolated portions at different temperatures would be treated as different systems.
Thermodynamic Potentials.—We have now to develop the relations involved in the general equation (1) of thermodynamics. Suppose the material system includes two coexistent states or phases, with opportunity for free interchange of constituents—for example, a salt solution and the aqueous vapour in equilibrium with it. Then in equilibrium a slight transfer δm of the water-substance of mass mr constituting the vapour, into the water-substance of mass ms, existing in the solution, should not produce any alteration of the first order in δE − Tδφ; therefore μr must be equal to μs. The quantity μr is called by Willard Gibbs the potential of the corresponding substance of mass mr; it may be defined as its marginal available energy per unit mass at the given temperature. If then a system involves in this way coexistent phases which remain permanently separate, the potentials of any constituent must be the same in all of them in which that constituent exists, for otherwise it would tend to pass from the phases in which its potential is higher to those in which it is lower. If the constituent is non-existent in any phase, its potential when in that phase would have to be higher than in the others in which it is actually present; but as the potential increases logarithmically when the density of the constituent is indefinitely diminished, this condition is automatically satisfied—or, more strictly, the constitutent cannot be entirely absent, but the presence of the merest trace will suffice to satisfy the condition of equality of potential. When the action of the force of gravity is taken into account, the potential of each constituent must include the gravitational potential gh; in the equilibrium state the total potential of each constituent, including this part, must be the same throughout all parts of the system into which it is freely mobile. An example is Dalton’s law of the independent distributions of the gases in the atmosphere, if it were in a state of rest. A similar statement applies to other forms of mechanical potential energy arising from actions at a distance.
When a slight constitutive change occurs in a galvanic element at given temperature, producing available energy of electric current, in a reversible manner and isothermally, at the expense of chemical energy, it is the free energy of the system E − Tφ, not its total intrinsic energy, whose value must be conserved during the process. Thus the electromotive force is equal to the change of this free energy per electrochemical equivalent of reaction in the cell. This proposition, developed by Gibbs and later by Helmholtz, modifies the earlier one of Kelvin—which tacitly assumed all the energy of reaction to be available—except in the cases such as that of a Daniell’s cell, in which the magnitude of the electromotive force does not depend sensibly on the temperature.