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 17

Lindsay: ‘‘On different forms of the fundamentalequations of the mechanical theory of heat and their convenience for application”. In:‘‘The Second Law of Thermodynamics.” J. Kestin (ed.), Stroudsburgh (Pa), DowdenHutchinson and Ross (1976).643 Entropyx away from infinitesimal Carnot cycles x away from ideal gasesx away from Carnot cycles altogether, x away from cycles of whatevertype, and x away from reversible processes.36 In the end he came up with theconcept of entropy and the properties of entropy, and that is his greatestachievement.

We shall presently review his progress.Among the people, whom we are discussing in this book, Clausius wasthe first one who lived and worked entirely in the place that was to becomethe natural habitat of the scientist: The autonomous university with tenuredprofessors,37 often as public or civil servants. With Clausius the time ofdoctor-brewer-soldier-spy had come to an end, at least in thermodynamics.General and compulsory education had begun and universities sprang up tosatisfy the need for higher education and they had to be staffed.

Thus onekilled two birds with one stone: When a professor was no good as ascientist, he could at least teach and thus earn part of his keep. On the otherhand, if he was good, the teaching duties left him enough time to doresearch.38 Clausius belonged to the latter category. He was a professor inZürich and Bonn, and his achievements are considerable: He helped tocreate the kinetic theory of ideal and real gases and, of course, he was thediscoverer of entropy and the second law.

His work on the kinetic theorywas largely eclipsed by the progress made in that field by Maxwell inEngland and Boltzmann in Vienna. And in his work on thermodynamics hehad to fight off numerous objections and claims of priority by other people,who had thought, or said, or written something similar at about the sametime.

By and large Clausius was successful in those disputes. Brush callsClausius one of the outstanding physicists of the nineteenth century.3936Reversible processes are those – in the present context of single fluids – in whichtemperature and pressure are always homogeneous, i.e. spatially constant, throughout theprocess, and therefore equal to temperature and pressure at the boundary. If that processruns backwards in time, the heat absorbed is reversed (sic) into heat emitted, or vice versa.A hallmark of the reversible process is the expression -pdV for the work dW. Thatexpression for dW is not valid for an irreversible process, which may exhibit turbulence,shear stresses and temperature gradients inside the cylinder of an engine (say) duringexpansion or compression.

Irreversibility usually results from rapid heating and working.37 Tenure was intended to protect freedom of thought as much as to guarantee financialsecurity.38 The system worked fairly well for one hundred years before it was undermined by jobseekers or frustrated managers, who failed in their industrial career. They are withoutscientific ability or interest, and spend their time attending committee meetings,reformulating curricula, and tending their gardens.39 Stephen G. Brush: ‘‘Kinetic Theory” Vol I. Pergamon Press, Oxford (1965).Second Law of Thermodynamics65Second Law of ThermodynamicsClausius keeps his criticism of Carnot mild when he says that … Carnothas formed a peculiar opinion [of the transformation of heat in a cycle]. Hesets out to correct that opinion, starting from an axiom which has becomeknown as the second law of thermodynamics:Heat cannot pass by itself from a colder to a warmer body.This statement, suggestive though it is, has often been criticized asvague.

And indeed, Clausius himself did not feel entirely satisfied with it.Or else he would not have tried to make the sentence more rigorous in apage-long comment, which, however, only succeeds in removing whateversuggestiveness the original statement may have had. 40 We need not godeeper into this because, after all, in the end there will be an unequivocalmathematical statement of the second law.The technique of exploitation of the axiom makes use of Carnot’s idea ofletting two reversible Carnot machines compete, – one a heat engine and theother one a heat pump, or refrigerator, cf. Fig. 3.9; the pump becomes anengine when it is reversed and vice versa; and the heats exchanged arechanging sign upon reversal. Both machines work in the temperature rangebetween TLow and THigh and one produces the work which the other oneconsumes, cf.

Fig. 3.9. Thus Clausius concludes that both machines mustexchange the same amounts of heat at both temperatures, lest heat flowfrom cold to hot, which is forbidden by the axiom. So the efficiencies ofboth machines are equal, – if they work as heat engines. And, since nothingis said about the working agents in them, the efficiency must be universal.So far this is all much like Carnot’s argument.Fig. 3.9. Clausius’s competing reversible Carnot engines40E.g.

see R. Clausius: ‘‘Die mechanische Wärmetheorie” [The mechanical theory of heat](3.ed.) Vieweg Verlag, Braunschweig (1887) p. 34.663 EntropyBut then, unlike Carnot, Clausius knew that the work WO of the heatengine is the difference between Qboiler and |Qcooler| so that the efficiency ofany engine, – not necessarily a reversible Carnot engine – is given byeWOQ boiler1QcoolerQboiler.Qcooler could conceivably be zero; at least, if it were, that would not contradictthe first law, which only forbids WO to be bigger than Qboiler.

However, if theengine is a reversible Carnot engine with its universal efficiency, thatefficiency is equal to that of an ideal gas – see above – so that we must haveQboilerTHighQcoolerTLow.It is clear from this equation that it is not the heat that passes through aCarnot engine unchanged in amount; rather it is Q/T , the entropy.Clausius sees two types of transformations going on in the heat engine:The conversion of heat into work, and the passage of heat of hightemperature to that of low temperature.

Therefore in 186541 he proposes tocall QT the entropy, … after the Greek word IJȡȠʌȒ = transformation, orchange and he denotes it by S. He says that he has intentionally chosen theword to be similar to energy, because he feels that the two quantities … areclosely related in their physical meaning. Well, maybe they appeared so toClausius. However, it seems very much the question, in what way twoquantities with different dimensions can be close.The last equation shows that |Qcooler| cannot be zero, except for theimpractical case TLow = 0.

Thus even for the optimal engine – the Carnotengine – there must be a cooler. Far from getting more work than the heatsupplied to the boiler, we now see that we cannot even get that much: Theboiler heat cannot all be converted into work. Therefore we cannot gainwork by just cooling a single heat reservoir, like the sea. Students ofthermodynamics like to express the situation by saying, rather flippantly:1st law: You cannot win.2nd law: You cannot even break even.All of this still refers to cycles, or actually Carnot cycles. In Insert 3.5 weshow in the shortest possible manner, how Clausius extrapolated theseresults to arbitrary cycles, and how he was able to consolidate the notion ofentropy as a state function S(T,V), whose significance is not restricted tocycles. The final result is the mathematical expression of the second law41R.

Clausius: (1865) loc.cit.Second Law of Thermodynamics67and it is an inequality: For a process from (TB,VB) to (TE, VE) the entropygrowth cannot be smaller than the sum of heats exchanged divided by thetemperature, where they are exchanged:ES(TE,VE) – S(TB,VB) tdQ[equality holds for reversible processes].TB³Clausius’s derivation of the second lawSince Qcooler< 0, the relationQb o ile rTH ig hQc o o le rTL owQb o ile r TH ig hmay be writ te n asQc o o le rTL ow0.In order to extrapolate this relation away from Carnot cycles to arbitrary cycles,Clausius decomposed such an arbitrary cycle into Carnot cycles with infinitesimalisothermal steps, cf.

Fig.3.10. On those steps the heat dQ is exchanged such thatdS=dQ/T is passing from the warm side to the cold one. Summation – or integration– thus leads to the equationvÔ dSvÔdQT0Hence follows for an open reversible process – not a cycle – between the points Band EES (TE ,VE ) S (TB , VB )dQ,TBÔwhere S(TE,VE) – S(TB,VB) is independent of the path from B to E, so that the entropyfunction S(T,V) is a state function. After the internal energy U(T,V) this is thesecond state function discovered by Clausius.Fig.

3.10. Smooth cycle decomposed into narrow Carnot cyclesIt remains to learn how this relation is affected by irreversibility. For thatpurpose Clausius reverted to the two competing Carnot engines, – one driving theother one. But now, one of them, the heat engine, was supposed to workirreversibly. In that case the process in the heat engine cannot be represented by a683 Entropygraph in a (p,V)-diagram, and therefore we show it schematically in Fig.

3.11. Itturns out that the system of two engines contradicts Clausius’s axiom, if the heatpump absorbs more heat at the low temperature than the heat engine delivers there.And now the reverse case cannot be excluded, because the engine changes its heatexchanges when it is made to work as a pump. Therefore for the irreversible heatengine we haveQb o ile r TH ig hQc o o le r  0 ,TL owIt follows that the efficiency of the irreversible engine is lower than that of thereversible engine, and a fortiori – by the same sequence of arguments as before –that in an arbitrary irreversible process between points B and E we haveS (TE , VE ) S (TB , VB )E dQ.ÔB TThe two relations for the change of entropy – one for the reversible and the otherfor the irreversible process – may be combined in a single •alternative, as we havedone in the main text..Fig.

3.11. Two competing Carnot engines with an irreversible heat engineInsert 3.5Exploitation of the Second LawAn important corollary of the second law concerns a reversible processbetween B and E, when those two point are infinitesimally close. In thatcase we havedSdQTand when we eliminate dQ between that relation and the first law in theform dQ = (dU + pdV), we obtaindS =1(dU + pdV).TExploitation of the Second Law69This equation is called the Gibbs equation.42 Its importance can hardly beoverestimated; it saves time and money and it is literally worth billions tothe chemical industry, because it reduces drastically the number ofmeasurements, which must be made in order to determine the internalenergy U = U(T,V) as a function of T and V.Let us consider this:Both the thermal equation of statep=p(T,V) and the caloric equation ofstate U = U(T,V) are needed explicitly for the calculation of nearly allthermodynamic processes, and they must be measured.