98 



NATURE 



TMarch i6, 1911 



iinat»iiU' a heat fnj{iiU' capable of producing motive power 

 perpitualiy wilhom takinf» any h<at from the boik-r, he 

 concluded that the quantity of motive power, W, produced 

 from a given quantity of heat, Q, by means of u reversible 

 engine, working between given temperature limits in a 

 tyclical process, was the maximum obtainable, or that the 

 efficiency must be indep«'ndent of the agents employed, and 

 must be a function of the temperature limits alone. He 

 expressed this by the equation \V7y=F{n. between finite 

 limits 0° and t° C, or by the equivalent equation 

 d\V/<i< = QF'(/) for a cycle of infinitesimal range, dt, at 

 a temperature, <, where F'(<) (generally known as Carnot's 

 function) is the derived function of F(/), and must be the 

 same for all substances at the same temperature. 



Applying the equation in this form to a gas obeying the 

 law /)f = KT, he showed that the heat absorbed in iso- 

 thermal expansion from r, to v was given by the ex- 

 pression Q = K log,(t'/t',)/F'(<), and that the difference of 

 the specific heats at constant pressure and volume, given 

 by the expression S^, - S, = R/TF"'(/), must be independent 

 of the pressure, and the same for equal volumes of all 

 gases. These results were new, but were confirmed ex- 

 perimentally by Di.long five years later. Carnot showed, 

 further, that if the ratio S^/S„ was constant (as found 

 by Gay Lussac and Welter, and assumed by Laplace and 

 Poisson), both Sp and S, 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 assump- 

 tion 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 (f„, f„) to (v, i) 

 was the same by any reversible process, Carnot deduced 

 that, if S, was independent of the pressure, the function 

 F'(/) 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 /)t' = RT, and having 

 Sr independent of the pressure, which is equivalent to the 

 modern definition of a perfect gas. Putting F'(() = 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 T„, an expression equivalent to the 

 following : — 



W = AQ<T-T.). 



Carnot was unable to reconcile this solution with the 

 imperfect experimental data available in his day, and 

 particularly with the observation of Delaroche and B^rard, 

 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 know- 

 ledge, it is easy to see that this solution of Carnot's is 

 also consistent with the mechanical theory, and contains 

 implicitly aH 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 — T„. 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 what- 

 ever) when T„ = o, namely, AQT." The efficiencv of the 

 cycle with range T to T„ is W/AQT = (T-T„)/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 inquire 

 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 

 NO. 2:59, VOL. 861 



motive power of the calcine y< m rauci at any temperature, 

 namely, A^l, should be equal to the motive p<jwer con- 

 sumed in friction. But he realised that further experi- 

 mental 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 b> 

 called the carnot, is that quaniuy which is capable «■• 

 doing one joule of work per degree fall. The mechanicai 

 equivalent of Q carnots at T abs. is QT joules. From 

 Carnot's data, the work done in a cycle per gram of 

 steam vaporised at 100° C. per degree fall is o-bii kilo 

 grammetres, or nearly 6 joules. The caloric of vaporisa 

 tion is 6 carnots. Similarly, from Kelvin's data for tlv 

 pressure required to lower the freezing point 1° C, th- 

 caloric of fusion of ice is 12 carnots. Since this definitio; 

 is independent of calorimetric measurements, it may b' 

 employed in a calorimetric test, in which steam is cor. 

 densed at 100° C. on one side of a conducting partition . 

 while ice is melted at 0° C. on the other, to determine h. 

 direct experiment what happens when caloric fall- 

 irreversibly by conduction from 100° C. to 0° C. \\ > 

 know that for each gram of steam condensed, or for each 

 6 carnots supplied at 100° C, 540/79-5 grams of ii • 

 approximately would be melted, or 8-17 carnots of calorii 

 would appear at 0° C. The quantity of caloric is in- 

 creased 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 — T) had been developed 

 and converted into heat by friction at the lower tempera- 

 ture. 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 ol(! 

 principle of the universal conservation of caloric, which i- 

 true only for reversible processes, must therefore b< 

 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 calori- 

 metric 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. Evolu- 

 tion 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, d\\ !dt = QF'{t), was adopted 

 without material modification into the mechanical theory, 

 and QF'(0 remained simply a quantity of Carnot's caloric 

 (though Q was measured in energy units and F'(t) re- 

 ceived the appropriate value J/T required to reduce energ\ 

 units^ to caloric), it was inevitable that Carnot's calor'n 

 should make its reappearance sooner or later in th- 

 mechanical theory. It first reappears, disguised as a trip!- 

 integral, in Kelvin's solution {Phil. Mag., iv., 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 temp>era- 

 ture IS effected reversibly. Caloric reappeared next as the 



