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 16

Thomson: ‘‘On the absolute thermometric scale founded on Carnot’s theory of themotive power of heat, and calculated from Regnault’s observations.” PhilosophicalMagazine, Vol. 33 (1848) pp. 313–317.Rudolf Julius Emmanuel Clausius (1822–1888)59Compared to this daring proposition Kelvin’s previous introduction of theabsolute scale T (t ) (273 qtC )q K seems straightforward, and rather plain. As itwas, however, the logarithmic scale was never seriously considered, not even byKelvin.One might think that nobody really wanted the temperature scale on athermometer to look like a slide rule. Yet, in the meteorological range between– 30°C and +50°C the function IJ(t) is nearly linear. And also, for to – 273°C therescaled temperature W tends to f , which is not a bad value for the absoluteminimum of temperature. One could almost wish that Kelvin’s proposition hadbeen accepted.

That would make it easier to explain to students why the minimumtemperature cannot be reached.Insert 3.2Rudolf Julius Emmanuel Clausius (1822–1888)By 1850 the efforts of Rumford, Mayer, Joule and Helmholtz had finallysucceeded to create an overwhelming feeling that something was wrongwith the idea that heat passes from boiler to cooler unchanged in amount:Some of the heat, in the passage, ought to be converted to work.

But how toimplement that new knowledge? Kelvin despaired: 25 If we abandon[Carnot’s] principle we meet with innumerable other difficulties … and anentire reconstruction of the theory of heat [is needed].Clausius was less pessimistic: 26 I believe we should not be dauntedby these difficulties. … [and] then, too, I do not think the difficulties are soserious as Thomson [Kelvin] does. And indeed, it took Clausiussurprisingly slight touches in surprisingly few spots of Carnot’s andClapeyron’s works to come up with an expression for the Carnot functionFƍ(t) which determines the efficiency e of a Carnot cycle between t andt+dt.

We recall that Carnot had proved e=Fƍ(t)dt. And Clausius was thefirst person to argue convincingly that F c(t ) 273o1C t T1 holds, cf.Insert 3.3.25W. Thomson: ‘‘An account of Carnot’s theory of the motive power of heat.” Transactionsof the Royal Society of Edinburgh 16 (1849). pp. 5412–574.26 R. Clausius: ‘‘Über die bewegende Kraft der Wärme und die Gesetze, welche sich darausfür die Wärme selbst ableiten lassen.” Annalen der Physik und Chemie 155 (1850).pp.

368–397. Translation by W.F. Magie: ‘‘On the motive power of heat, and on the lawswhich can be deduced from it for the theory of heat.” Dover (1960). Loc.cit. pp. 109–152.603 EntropyClausius’s derivation of the internal energyand the calculation of the Carnot functionWhen a body absorbs the heat dQ it changes the temperature by dt and the volumeby dV, as dictated by the heat capacity Cv and the latent heat O 27 so that we havedQ = Cv(t,V) dt + Ȝ(t,V) dV.Truesdell, who had the knack of a pregnant expression, calls this equation thedoctrine of the latent and specific heat.28 Applied to an infinitesimal Carnot processabcd this reads, cf. Fig. 3.7:ab # (dV,dt=0)bc # (įƍV,dt)cd # (dƍV,dt=0)da # (įV,dt)d Qabd Qbcd Qcdd Qdadt + Ȝ(t,V)dVdt + Ȝ(t,V+dV) įƍVCv(t-dt,V+įV)dt - Ȝ(t-dt,V+įV)dƍVCv(t,V)dt - Ȝ(t,V)įVCv (t,V)== - Cv(t,V+dV)==All framed quantities are zero, since the process is composed of isotherms andadiabates.

Thus with a little calculation – expanding the coefficients – Clausiusarrived at formulae forheat exchanged: d Qab+dQcd = ( wwOV ww%88 )dtdVheat absorbed: d Qab = ȜdVwork done: dp dV =wpwtdV dt.Fig. 3.7. (p,V)-diagram of an infinitesimally small Carnot cycle in a gas.The work was calculated as the area of the parallelogram.By the first law the heat exchanged equals the work done: HencewpwO wCV=wtwVwtw ( O p ) wCV=0wtwVorwhich may be considered as the integrability condition of the differential formdU = Cv dt +(Ȝ – p) dVordU = d Q – p dV.Thus Clausius arrived at the notion of the state function internal energy U,generally a function of t and V.

Clausius assumed – correctly – that in an ideal gasU depends only on t. Therefore O = p holds and the efficiency e of the Carnotprocess is27In modern thermodynamics the term latent heat is reserved as a generic expression for theheat of a phase transition – like heat of melting, or heat of evaporation –, but this was notso in the 19th century.28C. Truesdell: ‘‘The tragicomical History of Thermodynamics 1822–1854”. SpringerVerlag New York (1980) [The specific heat is the heat capacity per mass.].Rudolf Julius Emmanuel Clausius (1822–1888)Gwork doneheat absorbedO M8 PO611FV273 q C VFVand the universal Carnot function Fƍ(t) is now calculated once and for all:F '(t )1273 ’C t1.T[It is true that Clausius in 1850 calculated the work done only for an ideal gas. Theabove generalization to an arbitrary fluid came in 1854.29]Insert 3.3Notation and mode of reasoning of Clausius is nearly identical to that ofClapeyron with the one difference, – an essential difference indeed – thatthe total heat exchange of an infinitesimal Carnot cycle is not zero; rather itis equal to the work.

Thus the heat Q is not a state function anymore, i.e. afunction of t and V (say). To be sure, there is a state function, but it is not Q.Clausius denotes it by U, cf. Insert 3.3, and he calls U the sum of the freeheat and of the heat consumed in doing internal work, meaning the sum ofthe kinetic energies of all molecules and of the potential energy of theintermolecular forces. 30 Nowadays we say that U is the internal energy inorder to distinguish it from the kinetic energy of the flow of a fluid andfrom the potential energy of the fluid in a gravitational field.A change of U is either due to heat exchanged or work done, or both:dU = dQ – pdV.With this relation the first law of thermodynamics finally left thecompass of verbiage – like heat is motion or heat is equivalent to work, orimpossibility of the perpetuum mobile, etc. – and was cast into a29R. Clausius: ‘‘Über eine veränderte Form des zweiten Hauptsatzes der mechanischenWärmetheorie”.

Annalen der Physik und Chemie 169 (1854). English translation: ‘‘On amodified form of the second fundamental theorem in the mechanical theory of heat.”Philosophical Magazine (4) 12, (1856).30 It was Kelvin who, in 1851, has proposed the name energy for U: W. Thomson: ‘‘On thedynamical theory of heat, with numerical results deduced from Mr. Joule’s equivalent of athermal unit, and M. Regnault’s observations on steam.” Transactions of the RoyalSociety of Edinburgh 20 (1851). p. 475.Clausius concurred: … in the sequel I shall call U the energy.

It is quite surprising thatClausius let himself be preceded by Kelvin in this matter, because Clausius himself was aninveterate name-fixer. He invented the virial for something or other in his theory of realgases, see Chap. 6, and he proposed the ergal as a word for the potential energy, whichseemed too long for his taste. And, of course, he invented the word entropy, see below.623 Entropymathematical equation, albeit for the special case of reversible processesand for a closed system, i.e. a body of fixed mass.Clausius reasonably – and correctly – assumes that U is independent of Vin an ideal gas and a linear function of t, so that the specific heats areconstant.

Because, he says: …we are naturally led to take the view that themutual attraction of the particles… no longer acts in gases, so that U doesnot feel how far apart the particles are, or how big the volume is. For anideal gas we may write31U(T,V) = U(TR) + m z Pk (T TR ) ,where TR is a reference temperature, usually chosen as 298K. The factor zhas the value 3/2, 5/2, and 3 for one-, two-, or more-atomic gases respectively.Actually Clausius could have proved his view – at least as far as it relatesto the V-independence of U – from Gay-Lussac’s experiment, mentioned inChap.

2, on the adiabatic expansion of an ideal gas into an empty volume,where U must be unchanged after the process, and the temperature isobserved to be unchanged, although the density does change, of course. Asit is, Clausius mentions the (p,V,t)-relation of Mariotte and Gay-Lussac onevery second page, but he seems to be unaware of Gay-Lussac’s expansionexperiment, or he does not recognize its significance.In his paper of 1850, which we are discussing, Clausius deals with idealgases and saturated vapour. Having determined the universal Carnot function, he is able to write the Clausius-Clapeyron equation, cf.

Insert 3.1. Alsohe can obtain the adiabatic (p,V,t)-relation in an ideal gas, whose prototypeis pV Ȗ = const, – well-known to all students of thermodynamics – whereJ = Cp/Cv is the ratio of specific heats. Later, in 1854,32 Clausius applies thisknowledge to calculate the efficiency e of a Carnot cycle of an ideal gas inany range of temperature, no matter how big; certainly not infinitesimal.

Heobtains, cf. Insert 3.4e 1TLow,THighso that even the maximal efficiency is smaller than one, unless TLow = 0holds of course, which, however, is clearly impractical.31This is a modern version which, once again, is somewhat anachronistic. Clausius wasconcerned with air and he used the poor value of the specific heat – given by Delarocheand Bérard – which had already haunted the works of Carnot and Mayer.To do full justice to the specific heats, even of ideal gases, one could write a book all byitself. But that would be a different book from the present one.32 R.

Clausius: (1854) loc.cit.Rudolf Julius Emmanuel Clausius (1822–1888)63Efficiency of a Carnot cycle of a monatomic ideal gasWe refer to Fig. 3.8 which shows a graphical representation of a Carnot cyclebetween temperatures THigh and TLow. For a monatomic ideal gas we have for thework and the heat exchanged on the four branchesFig. 3.8 Graph of a Carnot processVkW12 m TH ln 2 ,V1µQ12Vkm TH ln 2V1µ3kW23 m(T T ) ,2µ L HQ230VkW34 m TL ln 4 ,V3µVkQ12 m TL ln 4V3µ3kW41 m(T T ) ,2µ H LQ41 0Therefore the efficiency comes out asem Pk TH ln VV12 m Pk TL ln VV43km P TH lnV2V1The last equation results from the observation that1TL.THV2V1V3V4holds.Insert 3.4With all this – by Clausius’s work of 1850 – thermodynamics acquired adistinctly modern appearance.

His assumptions were quickly confirmed byexperimenters,33 or by reference to older experiments, which Clausius hadeither not known, or not used. Nowadays a large part of a modern course onthermodynamics is based on that paper by Clausius: the part that deals withideal gases, and a large portion of the part on wet steam.For Clausius, however, that was only the beginning. He proceededwith two more papers 34,35 in which he took five important steps forward:33W. Thomson, J.P.

Joule: ‘‘On the thermal effects of fluids in motion.” PhilosophicalTransactions of the Royal Society of London 143 (1853).34R. Clausius: (1854) loc.cit.35 R. Clausius: ‘‘Über verschiedene für die Anwendungen bequeme Formen derHauptgleichungen der mechanischen Wärmetheorie”. Poggendorff’s Annalen der Physik125 (1865). English translation by R.B.