| 777 foot-pounds (Lat. 45°) | are equivalent to | 1 B.Th.U. (℔ deg. Fahr.) |
| 1399 foot-pounds ” | ” ” | 1 ℔ deg. C. |
| 426.3 kilogrammetres | ” ” | 1 kilogram-deg. C. or kilo-calorie. |
| 426.3 grammetres | ” ” | 1 gram-deg. C. or calorie. |
| 4.180 joules | ” ” | 1 gram-deg. C. or calorie. |
The water for the heat units is supposed to be taken at 20° C. or 68° F., and the degree of temperature is supposed to be measured by the hydrogen thermometer. The acceleration of gravity in latitude 45° is taken as 980.7 C.G.S. For details of more recent and accurate methods of determination, the reader should refer to the article [Calorimetry], where tables of the variation of the specific heat of water with temperature are also given.
The second law of thermodynamics is a title often used to denote Carnot’s principle or some equivalent mathematical expression. In some cases this title is not conferred on Carnot’s principle itself, but on some axiom from which the principle may be indirectly deduced. These axioms, however, cannot as a rule be directly applied, so that it would appear preferable to take Carnot’s principle itself as the second law. It may be observed that, as a matter of history, Carnot’s principle was established and generally admitted before the principle of the conservation of energy as applied to heat, and that from this point of view the titles, first and second laws, are not particularly appropriate.
20. Combination of Carnot’s Principle with the Mechanical Theory.—A very instructive paper, as showing the state of the science of heat about this time, is that of C. H. A. Holtzmann, “On the Heat and Elasticity of Gases and Vapours” (Mannheim, 1845; Taylor’s Scientific Memoirs, iv. 189). He points out that the theory of Laplace and Poisson does not agree with facts when applied to vapours, and that Clapeyron’s formulae, though probably correct, contain an undetermined function (Carnot’s F′t, Clapeyron’s 1/C) of the temperature. He determines the value of this function to be J/T by assuming, with Séguin and Mayer, that the work done in the isothermal expansion of a gas is a measure of the heat absorbed. From the then accepted value .078 of the difference of the specific heats of air, he finds the numerical value of J to be 374 kilogrammetres per kilo-calorie. Assuming the heat equivalent of the work to remain in the gas, he obtains expressions similar to Clapeyron’s for the total heat and the specific heats. In consequence of this assumption, the formulae he obtained for adiabatic expansion were necessarily wrong, but no data existed at that time for testing them. In applying his formulae to vapours, he obtained an expression for the saturation-pressure of steam, which agreed with the empirical formula of Roche, and satisfied other experimental data on the supposition that the coefficient of expansion of steam was .00423, and its specific heat 1.69—values which are now known to be impossible, but which appeared at the time to give a very satisfactory explanation of the phenomena.
The essay of Hermann Helmholtz, On the Conservation of Force (Berlin, 1847), discusses all the known cases of the transformation of energy, and is justly regarded as one of the chief landmarks in the establishment of the energy-principle. Helmholtz gives an admirable statement of the fundamental principle as applied to heat, but makes no attempt to formulate the correct equations of thermodynamics on the mechanical theory. He points out the fallacy of Holtzmann’s (and Mayer’s) calculation of the equivalent, but admits that it is supported by Joule’s experiments, though he does not seem to appreciate the true value of Joule’s work. He considers that Holtzmann’s formulae are well supported by experiment, and are much preferable to Clapeyron’s, because the value of the undetermined function F′t is found. But he fails to notice that Holtzmann’s equations are fundamentally inconsistent with the conservation of energy, because the heat equivalent of the external work done is supposed to remain in the gas.
That a quantity of heat equivalent to the work performed actually disappears when a gas does work in expansion, was first shown by Joule in the paper on condensation and rarefaction of air (1845) already referred to. At the conclusion of this paper he felt justified by direct experimental evidence in reasserting definitely the hypothesis of Séguin (loc. cit. p. 383) that “the steam while expanding in the cylinder loses heat in quantity exactly proportional to the mechanical force developed, and that on the condensation of the steam the heat thus converted into power is not given back.” He did not see his way to reconcile this conclusion with Clapeyron’s description of Carnot’s cycle. At a later date, in a letter to Professor W. Thomson (Lord Kelvin) (1848), he pointed out that, since, according to his own experiments, the work done in the expansion of a gas at constant temperature is equivalent to the heat absorbed, by equating Carnot’s expressions (given in § 17) for the work done and the heat absorbed, the value of Carnot’s function F′t must be equal to J/T, in order to reconcile his principle with the mechanical theory.
Professor W. Thomson gave an account of Carnot’s theory (Trans. Roy. Soc. Edin., Jan. 1849), in which he recognized the discrepancy between Clapeyron’s statement and Joule’s experiments, but did not see his way out of the difficulty. He therefore adopted Carnot’s principle provisionally, and proceeded to calculate a table of values of Carnot’s function F′t, from the values of the total-heat and vapour-pressure of steam-then recently determined by Regnault (Mémoires de l’Institut de Paris, 1847). In making the calculation, he assumed that the specific volume v of saturated steam at any temperature T and pressure p is that given by the gaseous laws, pv = RT. The results are otherwise correct so far as Regnault’s data are accurate, because the values of the efficiency per degree F′t are not affected by any assumption with regard to the nature of heat. He obtained the values of the efficiency F′t over a finite range from t to 0° C., by adding up the values of F′t for the separate degrees. This latter proceeding is inconsistent with the mechanical theory, but is the correct method on the assumption that the heat given up to the condenser is equal to that taken from the source. The values he obtained for F′t agreed very well with those previously given by Carnot and Clapeyron, and showed that this function diminishes with rise of temperature roughly in the inverse ratio of T, as suggested by Joule.
R. J. E. Clausius (Pogg. Ann., 1850, 79, p. 369) and W. J. M. Rankine (Trans. Roy. Soc. Edin., 1850) were the first to develop the correct equations of thermodynamics on the mechanical theory. When heat was supplied to a body to change its temperature or state, part remained in the body as intrinsic heat energy E, but part was converted into external work of expansion W and ceased to exist as heat. The part remaining in the body was always the same for the same change of state, however performed, as required by Carnot’s fundamental axiom, but the part corresponding to the external work was necessarily different for different values of the work done. Thus in any cycle in which the body was exactly restored to its initial state, the heat remaining in the body would always be the same, or as Carnot puts it, the quantities of heat absorbed and given out in its diverse transformations are exactly “compensated,” so far as the body is concerned. But the quantities of heat absorbed and given out are not necessarily equal. On the contrary, they differ by the equivalent of the external work done in the cycle. Applying this principle to the case of steam, Clausius deduced a fact previously unknown, that the specific heat of steam maintained in a state of saturation is negative, which was also deduced by Rankine (loc. cit.) about the same time. In applying the principle to gases Clausius assumes (with Mayer and Holtzmann) that the heat absorbed by a gas in isothermal expansion is equivalent to the work done, but he does not appear to be acquainted with Joule’s experiment, and the reasons he adduces in support of this assumption are not conclusive. This being admitted, he deduces from the energy principle alone the propositions already given by Carnot with reference to gases, and shows in addition that the specific heat of a perfect gas must be independent of the density. In the second part of his paper he introduces Carnot’s principle, which he quotes as follows: “The performance of work is equivalent to a transference of heat from a hot to a cold body without the quantity of heat being thereby diminished.” This is not Carnot’s way of stating his principle (see § 15), but has the effect of exaggerating the importance of Clapeyron’s unnecessary assumption. By equating the expressions given by Carnot for the work done and the heat absorbed in the expansion of a gas, he deduces (following Holtzmann) the value J/T for Carnot’s function F′t (which Clapeyron denotes by 1/C). He shows that this assumption gives values of Carnot’s function which agree fairly well with those calculated by Clapeyron and Thomson, and that it leads to values of the mechanical equivalent not differing greatly from those of Joule. Substituting the value J/T for C in the analytical expressions given by Clapeyron for the latent heat of expansion and vaporization, these relations are immediately reduced to their modern form (see [Thermodynamics], § 4). Being unacquainted with Carnot’s original work, but recognizing the invalidity of Clapeyron’s description of Carnot’s cycle, Clausius substituted a proof consistent with the mechanical theory, which he based on the axiom that “heat cannot of itself pass from cold to hot.” The proof on this basis involves the application of the energy principle, which does not appear to be necessary, and the axiom to which final appeal is made does not appear more convincing than Carnot’s. Strange to say, Clausius did not in this paper give the expression for the efficiency in a Carnot cycle of finite range (Carnot’s Ft) which follows immediately from the value J/T assumed for the efficiency F′t of a cycle of infinitesimal range at the temperature t C or T Abs.
Rankine did not make the same assumption as Clausius explicitly, but applied the mechanical theory of heat to the development of his hypothesis of molecular vortices, and deduced from it a number of results similar to those obtained by Clausius. Unfortunately the paper (loc. cit.) was not published till some time later, but in a summary given in the Phil. Mag. (July 1851) the principal results were detailed. Assuming the value of Joule’s equivalent, Rankine deduced the value 0.2404 for the specific heat of air at constant pressure, in place of 0.267 as found by Delaroche and Bérard. The subsequent verification of this value by Regnault (Comptes rendus, 1853) afforded strong confirmation of the accuracy of Joule’s work. In a note appended to the abstract in the Phil. Mag. Rankine states that he has succeeded in proving that the maximum efficiency of an engine working in a Carnot cycle of finite range t1 to t0 is of the form (t1 − t0) / (t1 − k), where k is a constant, the same for all substances. This is correct if t represents temperature Centigrade, and k = −273.
Professor W. Thomson (Lord Kelvin) in a paper “On the Dynamical Theory of Heat” (Trans. Roy. Soc. Edin., 1851, first published in the Phil. Mag., 1852) gave a very clear statement of the position of the theory at that time. He showed that the value F′t = J/T, assumed for Carnot’s function by Clausius without any experimental justification, rested solely on the evidence of Joule’s experiment, and might possibly not be true at all temperatures. Assuming the value J/T with this reservation, he gave as the expression for the efficiency over a finite range t1 to t0 C., or T1 to T0 Abs., the result,