Müller I. A history of thermodynamics. The doctrine of energy and entropy (Müller I. A history of thermodynamics. The doctrine of energy and entropy.pdf), страница 15

p. 72.17 S. Carnot: ibidem pp. 73–78.15Benoît Pierre Émile Clapeyron (1799–1864)55subject. Both Clapeyron and Kelvin recognized the need to know thevalues of that function, but were frustrated in their attempts to eithermeasure or calculate it, cf. Inserts 3.1, 3.2. The problem was left open forClausius to solve – twenty five years after Carnot.It seems likely that Carnot, before publication of his work, did havesecond thoughts about the validity of his ‘‘Reflections”, particularly aboutthe caloric theory. E.

Mendoza who has had access to Carnot’s manuscript,quotes Carnot’s original summary, where Carnot concludes: The fundamental law that we have proposed … seems to us to have been placedbeyond doubt.18 In the published version this triumphant sentence isreplaced by the more thoughtful one: The fundamental law that we haveproposed seems to us to require … new verification. It is based upon thetheory of heat as it is understood today … [whose] foundation does notappear to be of unquestionable solidity.Carnot died in 1832 at the age of 36 years in a cholera epidemic. He leftbehind unpublished notes, in which he shows himself sceptical of thecaloric theory,19 and where he speculates Ɣ on the conversion of heat intowork, Ɣ on the conservation of motion, and Ɣ on the impossibility toproduce work by cooling a heat bath without transmitting heat to a reservoirof lower temperature.

Had he lived longer, it seems likely that he mighthave anticipated Clausius,s work by nearly 30 years.As it was, however, his book which he tried to sell for 3 francs, found noreaders and Carnot would have been entirely forgotten, perhaps, were it notfor Clapeyron, like Carnot a former student of the École Polytechnique.Benoît Pierre Émile Clapeyron (1799–1864)Mendoza20 writes that the list of men associated with the early period of theÉcole Polytechnique in Paris – founded in 1794 as a school for armyengineers – reads like the author index of a book on mathematical physics.Among the first instructors were Lagrange, Fourier, Laplace, Berthollet,Ampère, Malus, and Dulong; among former students who stayed on asinstructors were Cauchy, Arago, Désormes, Coriolis, Poisson, Gay-Lussac,Petit, and Lamé; other students included Fresnel, Biot, Sadi Carnot, andClapeyron.

Maybe Clapeyron is not the most eminent one among thesemen, but his contribution was competent and he is remembered for it.18E. Mendoza: Footnote to Carnot’s ‘‘Reflections”: Dover (1960) loc.cit p. 46.Mendoza (ed.): Appendix to Carnot’s ‘‘Reflections”: Selection from the posthumousmanuscripts of Carnot: Dover (1960) loc.cit. p. 60.20 E.

Mendoza : Introduction to Carnot’s ‘‘Reflections” Dover loc.cit. p. ix.19 E.563 EntropyClapeyron’s work21 is a big step forward from Carnot in clarity, but itmarks time with respect to the caloric theory. An interesting feature of thework is the introduction of the graphical representation of reversiblethermodynamic processes in a (pressure,volume)-diagram, such that thework of the process equals the area below its graph. This is a method ofvisualization which is still used today, see Fig. 3.4 above. Apart from that,Clapeyron’s analysis is also perfect. I believe that his paper could havebecome a classic, if only the physics had not been below par.However, even so, for some arguments involving heat, it does not matterwhether heat is caloric or motion. Thus Clapeyron was able to establish avalid relationship between the slope of the vapour pressure curve p(t) andthe latent heat, or heat of evaporation R(t), see Insert 3.1.

That relationcontains the Carnot function Fƍ(t) and it makes it possible to find the valuesof that function, if only R(t) and p(t) are measured. Extensive measurementsof that type were published by Regnault22 in 1847, but that did not helpClapeyron in 1834, of course. His results remain indeterminate, because ashe says …unfortunately there are no experiments which allow us todetermine the values of that function [the Carnot function] at all values ofthe temperature.The Clausius-Clapeyron equationClapeyron considered a Carnot process of wet steam.

That process consists ofhorizontal isobars and steep adiabates. The isobars are also isotherms, since thevapour pressure depends only on temperature: p = p(t). If the process isinfinitesimal – with the temperature difference dt, and the evaporation of the massfraction dx of liquid water on the isothermal branch – we haveR dx – for the heat absorbed, anddpdtdt [(VƎ – Vƍ)dx ] – for the work done.Fig. 3.6. (p,V)-diagram of infinitesimal Carnot process in wet steamR is the latent heat of evaporation and dp is the height of the small cycle, cf.Fig.

3.6.21E. Clapeyron: ‘‘Mémoire sur la puissance motrice de la chaleur” Journal de l´ÉcolePolytechnique. Vol XIV (1834) pp. 153–190. Translations: (English) ‘‘Memoir on themotive power of heat.” Scientific Memoirs Vol. 1 (1837) pp. 347–376. (German) ‘‘Überdie bewegende Kraft der Wärme.” Annalen der Physik und Chemie Vol 135 (1843).22 H.V. Regnault: ‘‘Relations des expériences …pour déterminer les principales lois et lesdonnées numériques qui entrent dans le calcul des machines à vapeur.” Mémoires del’Académie des Sciences de l´Institut de France , Paris, Vol.

21 (1847) pp. 1–748.William Thomson (1824–1907), Lord Kelvin since 189257Vƍ and VƎ are the volumes of the boiling water and of the saturated vapour of whichthe wet steam is composed.The ratio of the two quantities is the efficiency e. And by Carnot’s results it isequal to e = Fƍ(t)dt, see above. Therefore we havedpdtF ' (t )RV '' V '.Thus Carnot’s universal function could be calculated from measurements of R, ofdpdt, – the slope of the vapour pressure curve – and of the vapour volume VƎ, all attemperature t.

Kelvin attempted such calculations, see below.Clausius found later – in 1850, cf. Insert 3.3 – that Fƍ(t) equals T1 , where T is theabsolute temperature, and therefore the relationdpdTRT (V c ' V ' )is called the Clausius-Clapeyron relation. Nowadays it is used to calculate the latentheat R of a new refrigerant (say) from the vapour pressure curve; the latter is easierto measure than R.Insert 3.1William Thomson (1824–1907), Lord Kelvin since 1892Kelvin has accompanied the development of thermodynamics for more thanhalf a century, starting from his graduation in 1845.

He went to Paris aftergraduation to work and study under Regnault, the careful and influentialexperimenter, whom we have already mentioned. Later Kelvin encouragedand supported Joule, and together the two men discovered the JouleThomson effect in real gases, see Chaps. 2 and 6. Kelvin suggested theabsolute temperature scale that bears his name, and he was a forerunner ofthe second law with the idea that there is a continuous degradation, ordissipation of energy into heat. However, Kelvin missed out himself on theparadigmatic changes23 in thermodynamics. To be sure, when theyoccurred, he was often the first, or one of the first, to interpret and rephrasethem, and apply them.

Therefore a history of thermodynamics is incompletewithout a prominent place for Kelvin. Perhaps his greatest achievement isthat he suggested the possibility of convective equilibrium, see Chap. 7,23This term has been made popular by Thomas S. Kuhn in his book: ‘‘The structure ofscientific revolutions.” The University of Chicago Press, Chicago and London. Thirdedition (1996).583 Entropywhich goes a long way to determine the structure of stars and the conditionsin the lower atmosphere of the earth. Another original result of his is theThomson formula for super-saturation in the processes of boiling andcondensation on account of surface energy.

However, here I choose tohighlight Kelvin’s capacity for original thought by a proposition he madefor an absolute temperature scale, – an alternative to the Kelvin scale whichwe all know; see Insert 3.2. The proposition is intimately linked to theCarnot function Fƍ(t) which Kelvin attempted to calculate from Regnault’sdata. The new scale would have been logarithmic, and absolute zero wouldhave been pushed to -’, a fact that gives the proposition its charm.Kelvin’s alternative absolute temperature scaleWe recall the Carnot function Fƍ(t), a universal function of the temperature t, whichneither Carnot nor Clapeyron had been able to determine.

After Regnault’s datawere published, Kelvin used them to calculate Fƍ(t) for 230 values of t between 0°Cand 230°C.24 He proposed to rescale the temperature, and to introduce IJ(t) such thatthe Carnot efficiency Fƍ(t)dt for a small fall dt of caloric would be equal to cd W ,where c is a constant, independent of t or IJ. Kelvin found that feature appealing. Hesays: This [scale] may justly be termed an absolute scale.

By integration IJ(t) resultsasW t W ( 0 ) 1ct³ F ' ( x ) dx .0Had Kelvin been able to fit an analytic function to Regnault’s data, and to hiscalculations of Fƍ(t), he would have found a hyperbolaF ' (t )1273 qC tand his new scale would have been logarithmic:W (t ) W (0) 1 273 qC tln273 qCc.IJ(0) and c need to be determined by assigning IJ-values to two fix-points, e.g.melting ice and boiling water.However, not even the 230 values, which Kelvin possessed, were good enoughto suggest the hyperbola in a convincing manner.Therefore Kelvin had to wait for Clausius to determine Fƍ(t) in 1850, cf.Insert 3.3. When Kelvin’s papers were reprinted in 1882, he added a note in whichindeed he proposes the logarithmic temperature scale.24W.