“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 dwdt + 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 ƒδH/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 ƒdH/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 ƒdH/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 ƒdH/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
