The caloric theory of heat as developed by Carnot in his famous "Reflexions on the Motive Power of Heat" (Paris, 1824) leads immediately to the correct solution of the relations between heat and motive power (energy or work) in all reversible processes, and appears to be in some respects preferable to the mechanical theory as a method of expression, because it emphasises more clearly the distinction first clearly stated by Carnot, between reversible and irreversible transformations, and because it directly provides the natural measure of a quantity of heat as distinct from a quantity of thermal energy.
Carnot first introduced the method of the cyclical process in discussing the action of a heat engine, and showed that, in the ideal case, if there were no direct transference of heat between bodies at different temperatures, the transformations of heat and motive power in such a cycle were reversible. Assuming that it was impossible to imagine a heat engine capable of producing motive power perpetually without taking any heat from the boiler, he concluded that the quantity of motive power, W, produced from a given quantity of heat, Q, by means of a reversible engine, working between given temperature limits in a cyclical process, was the maximum obtainable; or that the efficiency must be independent of the agents employed, and must be a function of the temperature limits alone. He expressed this by the equation W/Q=F(t), between finite limits 0° and t°C., or by the equivalent equation dW/dt=QF'(t) for a cycle of infinitesimal range, dt, at a temperature, t, where F'(t) (generally known as Carnot's function) is the derived function of F(t), and must be the same for all substances at the same temperature.
Applying the equation in this form to a gas obeying the law pv=RT, he showed that the heat absorbed in isothermal expansion from ν0 to ν was given by the expression Q=R loge (ν/ν0) /F'(t), and that the difference of the specific heats at constant pressure and volume, given by the expression Sp - Sν = R/FT'(t), must be independent of the pressure, and the same for equal volumes of all gases. These results were new, but were confirmed experimentally by Dulong five years later. Carnot showed, further, that if the ratio Sp / Sν was constant (as found by Gay Lussac and Welter, and assumed by Laplace and Poisson), both Sp and Sν must be independent of the pressure.
The results so far obtained by Carnot, including the description of his reversible cycle and the deduction of his fundamental principle, were independent of any assumption as to the nature of heat. Applying the assumption of the caloric theory, that the quantity of caloric required to change the state of a substance from (ν0, t0) to (ν, t) was the same by any reversible process, Carnot deduced that, if Sν was independent of the pressure, the function F'(t) must be constant,=A. This assumes that heat is measured as caloric, and that temperature is measured on the scale of a gas, obeying the law pv=RT, and having Sν independent of the pressure, which is equivalent to the modern definition of a perfect gas. Putting F'(t)=A, he obtains for the work W produced from a quantity of caloric, Q, supplied at a temperature, T, in a cycle of finite range T to T0, an expression equivalent to the following:- W=AQ(T - T0).
Carnot was unable to reconcile this solution with the imperfect experimental data available in his day, and particularly with the observation of Delaroche and Bérard, supported by Laplace's theory, that the specific heat of air, Sp, diminished with increase of pressure, which we know now, from the experiments of Regnault, to have been incorrect. He therefore made no serious attempt to apply the solution, and subsequent writers have apparently failed to observe that it is the correct final solution of the problem on the caloric theory. With our present knowledge, it is easy to see that this solution of Carnot's is also consistent with the mechanical theory, and contains implicitly all the relations of heat and work so far as they relate to reversible processes. The quantity, Q, of caloric remains constant in reversible expansion such as is postulated by Carnot, when no heat is supplied. The work done is directly proportional to the temperature range T - T0. The absolute motive power or equivalent work-value of a quantity of caloric, Q, supplied at a temperature, T, is the maximum work obtainable from a perfect gas (and therefore from any other substance whatever) when T0=0, namely, AQT. The efficiency of the cycle with range T to T0 is W/AQT=(T - T0)/T. The external work done in the cycle is the difference of the work-values of the caloric supplied and rejected, a result which is readily extended to cycles of any form.
To complete Carnot's solution, it is necessary to enquire what happens to caloric in irreversible processes, such as friction, or the direct passage of heat from a hotter to a colder body. Carnot, as we see from his posthumous notes, had already, before his early death in 1832, arrived at the general conception of the conservation of motive power, and had planned experiments in which the motive power consumed in friction should be measured at the same time as the caloric generated. According to his theory, it would have been natural to assume that the motive power of the caloric generated at any temperature, namely AQT, should be equal to the motive power consumed in friction. But he realised that further experimental evidence was necessary, which was first supplied by Joule.
A quantity of caloric is defined in Carnot's equation as measured by work done in a Carnot cycle per degree fall. The absolute unit of caloric, which may appropriately be called the Carnot, is that quantity which is capable of doing one joule of work per degree fall. The mechanical equivalent of Q carnots at T Abs. is QT joules. From Carnot's data, the work done in a cycle per gramme of steam vaporised at 100°C. per degree fall is 0.611 kilogrammetres, or nearly 6 joules. The caloric of vaporisation is 6 carnots. Similarly, from Kelvin's data for the pressure required to lower the freezing point 1°C., the caloric of fusion of ice is 1.2 carnots. Since this definition is independent of calorimetric measurements, it may be employed in a calorimetric test, in which steam is condensed at 100°C. on one side of a conducting partition while ice is melted at 0°C. on the other, to determine by direct experiment what happens when caloric falls irreversibly by conduction from 100°C. to 0°C. We know that for each gramme of steam condensed, or for each 6 carnots supplied at 100°C., 540/79.5 grammes of ice approximately would be melted, or 8.17 carnots of caloric would appear at 0°C. The quantity of caloric is increased in the proportion 373/273. The motive power of the caloric remains constant if no useful work is done. The increase of the quantity of caloric is the same as if the available motive power AQ(T - T0) had been developed and converted into heat by friction at the lower temperature. Whenever motive power is wasted in friction, or "in the useless re-establishment of the equilibrium of caloric," a quantity of caloric equivalent to the wasted motive power is generated. The total quantity of caloric in an isolated system remains constant only if all the transformations are reversible, in which case the motive power developed exactly suffices to restore the initial state. In all other cases there is an increase of caloric. The old principle of the universal conservation of caloric, which is true only for reversible processes, must therefore be modified as follows:- "The total quantity of caloric in any system cannot be diminished except by taking heat from it."
This principle, with various modifications to suit special cases (such as conditions of constant temperature, pressure, or volume) is immediately recognised as one of the most fruitful in modern thermodynamics. But it appeals more forcibly to the imagination of the student, if established, as roughly sketched above, by a direct investigation of the properties of Carnot's caloric.
The caloric theory is seen to be perfectly consistent with Carnot's principle and with the mechanical theory for all reversible processes. Caloric is the natural measure of a quantity of heat in accordance with Carnot's equation, if we adopt the gas-scale of temperature. The only defect of the caloric theory lay in the tacit assumption, so easily rectified, that the ordinary calorimetric units were units of caloric. The quantity measured in an ordinary calorimetric experiment is the motive power or energy of the caloric, and not the caloric itself. If this had been realised in 1850, it would have been quite unnecessary to recast and revolutionise the entire theory of heat. Evolution might have proceeded along safer lines, with the retention of caloric, and the investigation of its properties, which are of such fundamental importance in all questions of equilibrium in physics.
Since Carnot's equation, dW /dt=QF'(t), was adopted without material modification into the mechanical theory, and QF'(t) remained simply a quantity of Carnot's caloric (though Q was measured in energy units and F'(t) received the appropriate value J/T required to reduce energy units to caloric) it was inevitable that Carnot's caloric should make its reappearance sooner or later in the mechanical theory. It first reappears, disguised as a triple integral, in Kelvin's solution ("Phil. Mag.," 4, p. 305, 1852) of the problem of finding the available work in an unequally heated body. The solution (as corrected later) is equivalent to the statement that the total quantity of caloric remains constant when the equalisation of temperature is effected reversibly. Caloric reappeared next as the "thermodynamic function" of Rankine, and the "equivalence-value of a transformation" (Clausius "Pogg. Ann.," 93, p. 497, 1854). Finally, in 1865, when its importance was more fully recognised, Clausius ("Pogg. Ann.," 125, p. 390) gave it the name of "entropy," and defined it as the integral of dQ/T. Such a definition appeals to the mathematician only. In justice to Carnot, it should be called caloric, and defined directly by his equation W=AQ(T - T0), which any schoolboy could understand. Even the mathematician would gain by thinking of caloric as a fluid, like electricity, capable of being generated by friction or other irreversible processes. Conduction of caloric is closely associated with the electrons, and the science of heat would gain, like the science of electricity, by attaching a more material conception to the true measure of a quantity of heat, as distinguished from a quantity of thermal energy.
A vote of thanks to Prof. Callendar for his Presidential Address moved by Dr. Chree and seconded by Dr. Russell, was carried unanimously.