22 



NATURE 



[September 5, 1912 



solely by the temperatures between which in the 

 limit the transfer of heat takes place." 



Objection is commonly taken to Carnot's proof, on 

 the ground that the combination which he imagines 

 might produce a balance of useful work without in- 

 fringing the principle of conservation of energy, or 

 constituting what we now understand as perpetual 

 motion of the ordinary kind in mechanics. It has 

 become the fashion to introduce the conservation 

 of energy in the course of the proof, and to make a 

 final appeal to some additional axiom. Any proof of 

 this kind must always be to some extent a matter of 

 taste; but since Carnot's principle cannot be deduced 

 from the conservation of energy alone, it seems a 

 pity to complicate the proof by appealing to it. For 

 the particular object in view, the absurdity of a heat- 

 engine working without fuel appears to afford the 

 most appropriate improbability which could be in- 

 voked. The final appeal must be to experiment in 

 any case. At the present time the experimental 

 verification of Carnot's principle in its widest applica- 

 tion so far outweighs the validity of any deductive 

 proof, that we might well rest content with the logic 

 that satisfied Carnot instead of confusing the issue 

 by disputing his reasoning. 



Carnot himself proceeded to test his principle in 

 every possible way by comparison with experiment so 

 far as the scanty data available in his time would 

 permit. He also made several important deductions 

 from it, which were contrary to received opinion at 

 the time, but have since been accurately verified. He 

 appears to have worked out these results analytically 

 in the first instance, as indicated bv his footnotes, 

 and to have translated his equations into words in 

 the text for the benefit of his non-mathematical 

 readers. In consequence of this, some of his most 

 important conclusions appear to have been overlooked 

 or attributed to others. Owing to want of exact 

 knowledge of the properties of substances over ex- 

 tended ranges of temperature, he was unable to apply 

 his principle directly in the general form for any tem- 

 perature limits. We still labour to a less extent' under 

 the same disability at the present day. He showed, 

 however, that a great simplification was effected in its 

 application by considering a cycle of infinitesimal 

 range at any temperature f. In this simple case the 

 principle is equivalent to the assertion that the work 

 obtainable from a unit of heat per degree fall (or per 

 degree range of the cycle) at a temperature t, is some 

 function F'f of the temperature (generally known as 

 Carnot's function), which must be the same for all 

 substances at the same temperature. From the rough 

 data then available for the properties of steam, 

 alcohol, and air, he was able to calculate the numerical 

 values of this function in kilogrammetres of work per 

 kilocalorie of heat at various temperatures between o° 

 and 100° C., and to show that it was probably the 

 same for dift'erent substances at the same temperature 

 within the limits of experimental error. For the 

 vapour of alcohol at its boiling-point, 787° C, he found 

 the value F'/ = r23o kilogrammetres per kilocalorie per 

 degree fall. For steam at the same temperature he 

 found nearly the same value, namely, F't= r2i2. Thus 

 no advantage in point of efficiency could be gained by 

 employing the vapour of alcohol in place of steam. 

 He was also able to show that the work obtainable 

 from a kilocalorie per degree fall probably diminished 

 with _ rise of temperature, but his data were not 

 sufficiently exact to indicate the law of the variation. 



The equation which Carnot emploved in deducing 

 the numerical values of his function from the experi- 

 mental data for steam and alcohol is simplv the direct 

 expression of his principle as applied to a saturated 

 vapour. It is now generallv known as Clapevron's 

 equation, because Carnot did not happen to give the 

 NO. 2236, VOL. 90] 



equation itself in algebraic form, although the prin- 

 ciple and details of the calculation were most miriutely 

 and accurately described. In calculating the vaiue of 

 his function for air, Carnot made use of the known 

 value of the difference of the specific heats at constant 

 pressure and volume. He showed that this difference 

 must be the same for equal volumes of all gases 

 measured under the same temperature and pressure, 

 whereas it had always previously been assumed that 

 the ratio (not the difference) of the specific heats was 

 the same for different gases. He also gave a general 

 expression for the heat absorbed by a gas in expand- 

 ing at constant temperature, and showed that it must 

 bear a constant ratio to the work of expansion. 

 These results were verified experimentally some years 

 later, in part by Dulong, and more completely by 

 Joule, but Carnot's theoretical prediction has generally 

 been overlooked, although it was of the greatest 

 interest and importance. The reason of this neglect 

 is probably to be found in the fact that Carnot's ex- 

 pressions contained the unknown function F'f of the 

 temperature, the form of which could not be deduced 

 without making some assumptions with regard to the 

 nature of heat and the scale on which temperature 

 should be measured. 



It was my privilege to discover a few years ago that 

 Carnot himself had actually given the correct solu- 

 tion of this fundamental problem in one of his most 

 important footnotes, where it had lain buried and 

 unnoticed for more than eighty years. He showed by 

 a most direct application of the caloric theory that if 

 temperature was measured on the scale of a perfect 

 gas (which is now universally adopted) the value of 

 his function F'i on the caloric theory would be the 

 same at all temperatures, and might be represented 

 simply by a numerical constant A (our " mechanical 

 equivalent ") depending on the units adopted for 

 work and heat. In other words, the work W done 

 by a quantity of caloric Q in a Carnot cycle of range 

 T to T„ on the gas scale would be represented by the 

 simple equation : 



W = .\Q(T-T„). 



It is at once obvious that this solution, obtained 

 by Carnot from the caloric theory, so far from being 

 inconsistent with the mechanical theory of heat, is a 

 direct statement of the law of conservation of energy 

 as applied to the Carnot cycle. If the lower limit T„ 

 of the cycle is taken at the absolute zero of the gas- 

 thermometer, we observe that the maximum quantity 

 of work obtainable from a quantity of caloric Q at a 

 temperature T is simply AQT, which represents the 

 absolute value of the energy carried by the caloric 

 taken from the source at the temperature T. The 

 energy of the caloric rejected at the temperature T„ 

 is AQT„. The external work done is equal to the 

 difference between the quantities of heat energy sup- 

 plied and rejected in the cycle. 



The analogy which Carnot himself employed in the 

 interpretation of this equation was the oft-quoted 

 analogy of the waterfall. Caloric might be regarded 

 as possessing motive-power or energy in virtue of 

 elevation of temperature just as water may be said 

 to possess motive-power in virtue of its head or pres- 

 sure. The limit of motive-power obtainable by a 

 reversible motor in either case would be directly pro- 

 portional to the head or fall measured on a suitable 

 scale. Caloric itself was not motive-power, but must 

 be regarded simply as the vehicle or carrier of energy, 

 the production of motive-power from caloric depending 

 essentially (as Carnot puts it) not on the actual con- 

 sumption of caloric, but on the fall of temperature 

 available. The measure of a quantity of caloric is 

 the work done per degree fall, which corresponds with 

 the measure of a quantity of water by weight, i.e. in 

 kilogrammetres per metre fall. 



