Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 18
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Now, it is easy – atleast in principle – to determine the thermal equation, because p, T, and Vare all measurable quantities and they need only be put down in tables, ordiagrams, or – in modern times – on CD’s. But that is not so with thecaloric equation, because U is not measurable. U(T,V) must be calculatedfrom caloric measurements of the heat capacities CV (T,V) and Cp(T,V). Suchmeasurements are difficult and time-consuming, – hence expensive – andthey are unreliable to boot. And this is where the Gibbs equation helps. Ithelps to reduce – drastically – the number of caloric measurements needed,cf.
Insert 3.6 and Insert 3.7.Calculating U(T,V) from measurements of heat capacitiesThe heat capacities CV and Cp are defined by the equation dQ = CdT. Thus theydetermine the temperature change of a mass for a given application of heat dQ ateither constant V or p. In this way CV and Cp can be measured.
By dQ = dU + p dV ,and since we do know that U is a function of T and V – we just do not know theform of that function – we may writeCVÈ U ØÉÊÙT ÚVÈ U ØÉÊÙT ÚVCVand CpÈ È U ØØ È V ØÈ U ØÉÊÙ É ÉÊÙ p Ù ÉÊÙT ÚVÊ V Ú TÚ T ÚVÈ U Øand ÉÊ V ÙÚ TÈ VÉÊTCpCVØÙÚ p.or.pHaving measured CV(T,V) and Cp(T,V) and p(T,V) we may thus calculate U(T,V) byintegration to within an additive constant.The integrability condition implied by the Gibbs equation provideswp .wUpTwTwVHence follows that the V-dependence of U , hence Cp, need not be measured: Itmay be calculated from the thermal equation of state.
Moreover, differentiationwith respect to T provides the equation42Actually the equation was first written and exploited by Clausius, but Gibbs extended it tomixtures, see Chap. 5; the extension became known as Gibbs’s fundamental equation and,as time went by, that name was also used for the special case of a single body.703 EntropyÈ 2 p Ø,TÉÊ T 2 ÚÙ Vso that the V-dependence of CV is also determined by p(T,V). Therefore the onlycaloric measurements needed are those of CV as a function of T for one volume, V0(say).
The number of caloric measurements is therefore considerably reduced, andthat is a direct result of the Gibbs equation and the second law.È CV ØÉÊ V ÙÚTInsert 3.6Once we know the thermal and caloric equations of state we maycalculate the entropy S(T,V), or S(T,p) – by integration of the Gibbsequation – to within an additive constant.
Thus for an ideal gas of mass mwe obtainS (T , p )ÈkTkp ØS (TR , p R ) m É ( z 1) ln lnµ TR µp R ÙÚ .ÊTherefore the entropy of an ideal gas grows with lnT and lnV: Theisothermal expansion of a gas increases its entropy.Clausius-Clapeyron equation revisitedIf the Gibbs equation is applied to the reversible evaporation of a liquid underconstant pressure – and temperature – it may be written in the form(U-TS+pV)Ǝ = (U-TS+pV)ƍ ,where, once again, ƍ and Ǝcharacterize liquid and vapour. Thus the combinationU-TS+pV, called free enthalpy or Gibbs free energy, is continuous across theinterface between liquid and vapour, along with T and p.
Therefore the vapourpressure must be a function of temperature only. We have p=p(T) and thederivative of that function is given by the Clausius-Clapeyron equation, cf.Insert 3.1. When we realize that the heat of evaporation equals R=T(SƎ-Sƍ), wemay write the Clausius-Clapeyron equation in the formU cc U cdp ,pTcccV VdTwhich is clearly – for steam – the analogue to the integrability condition ofInsert. 3.6. The relation permits us to dispense with measurements of the latent heatof steam and to replace them with the much easier (p,T)-measurements.Insert 3.7There is a school of thermodynamicists – the axiomatists – who thrive on formalarguments, and who would never let considerations of measurability enter theirthoughts.43 One can hear members of that school say, that the temperature T is43Such an attitude is not uncommon in other branches of physics as well. Thus in mechanicsthere is a school of thought that considers Newton’s law F = m a as the definition of theforce rather than a physical law between measurable quantities.Exploitation of the Second Law71defined as ( wwUS )V .
That interpretation of the Gibbs equation ignores the fact that weshould never know anything about either U or S, let alone U=U(S,V), unless we haddetermined them first by measurements of p,V,T, and CV(T,V0) in the mannerdescribed above.Actually, the measurability of T is a consequence of its continuity at a diathermicwall, i.e. a wall permeable for heat.
That continuity is the real defining property oftemperature, and it gives temperature its central role in thermodynamics.The chief witness of the formal interpretation of temperature is Gibbs,unfortunately, the illustrious pioneer of thermodynamics of mixtures. He, however,for all his acumen, was an inveterate theoretician, and I believe that he never madea single thermodynamic measurement in his whole life. We shall come back to thisdiscussion in the context of chemical potentials, cf. Chap.
5, which have a lot incommon with temperature.Continuing our discussion of the consequences of the second law, wenow come to another important corollary, namely that the entropy in anadiabatic process, – where dQ = 0 holds –, cannot decrease. It grows untilit reaches a maximum. We know from experience that, when we leavean adiabatic system alone, it tends to a state of homogeneity – theequilibrium, – in which all driving forces for heat conduction and expansionhave run down.44 That is the state of maximum entropy.And so Clausius could summarize his work in the triumphant slogan: 45Die Energie der Welt ist constant.Die Entropie der Welt strebt einem Maximum zu.Die Welt [the universe] was chosen in this statement as being the ultimatethermodynamic system, which presumably is not subject to heating andworking, so that dU = 0 holds, as well as dS > 0.So the world has a purpose, or a destination, the heat death, see Fig.
3.12,not an attractive end!It is often said that the world goes in a circle …suchthat the same states are always reproduced. Thereforethe world could exist forever. The second lawcontradicts this idea most resolutely … The entropytends to a maximum. The more closely that maximumis approached, the less cause for change exists. Andwhen the maximum is reached, no further changes canoccur; the world is then in a dead stagnant state.Fig.
3.12. Rudolf Clausius and his contemplation of the heat death4445See Chap. 5 for a formal proof and for an explanation of what exactly homogeneity means.R. Clausius: (1865) loc.cit. p. 400.723 EntropyTerroristic Nimbus of Entropy and Second LawConcerning the heat death modern science does not seem to have made upits mind entirely. Asimov46 writes:Though the laws of thermodynamics stand as firmly as ever, cosmologists…[show] a certain willingness to suspend judgement on the matter of heatdeath.At his time, however, Clausius’s predictions were much discussed.
Theteleological character of the entropy aroused quite some interest, not onlyamong physicists, but also among philosophers, historians, sociologists andeconomists. The gamut of reactions ranged from uneasiness about the bleakprospect to pessimism confirmed. Let us hear about three of the morecolourful opinions:The physicist Josef Loschmidt (1821–1895)47 deplored… the terroristic nimbus of the second law …, which lets it appear as adestructive principle of all life in the universe. 48Oswald Spengler (1880–1936), the historian and philosopher of historydevotes a paragraph of his book ‘‘The Decline of the West” 49 to entropy.He thinks that … the entropy firmly belongs to the multifarious symbols ofdecline, and in the growth of entropy toward the heat death he sees thescientific equivalent of the twilight of the gods of Germanic mythology:The end of the world as the completion of an inevitable evolution – that isthe twilight of the gods.
Thus the doctrine of entropy is the last, irreligiousversion of the myth.And the historian Henry Adams (1838–1918) – an apostle of humandegeneracy, and the author of a meta-thermodynamics of history – commented on entropy for the benefit of the ordinary, non-educated historian.He says:…. this merely means that the ash-heap becomes ever bigger.46I. Asimov: ‘‘Biographies” loc.cit. p.
364.J. Loschmidt: ‘‘Über den Zustand des Wärmegleichgewichts eines Systems von Körpernmit Rücksicht auf die Schwerkraft.” [On the state of the equilibrium of heat of a system ofbodies in regard to gravitation.] Sitzungsberichte der Akademie der Wissenschaften inWien, Abteilung 2, 73: pp. 128–142, 366–372 (1876), 75: pp. 287–298, (1877), 76:pp.
205–209, (1878).48 If the author of this book had had his way in the discussion with the publisher, this citationof Loschmidt would have been either the title or the subtitle of the book. But, alas, we allhave to yield to the idiosyncrasies of our real-time terrorists, – and to the show of paranoiaby our opinionators.49 O. Spengler: ‘‘Der Untergang des Abendlandes: Kapitel VI. Faustische und ApollinischeNaturerkenntnis. § 14: Die Entropie und der Mythos der Götterdämmerung.” Beck’scheVerlagsbuchhandlung.
München (1919) pp. 601–607.47Modern Version of Zeroth , First and Second Laws73Well, maybe it does. But then, Adams was an inveterate pessimist, to theextent even that he looked upon optimism as a sure symptom of idiocy.50The entropy and its properties have not ceased to stimulate originalthought throughout science to this day:x biologists calculate the entropy increase in the diversification ofspecies,x economists use entropy for estimating the distribution of goods,51x ecologists talk about the dissipation of resources in terms of entropy,x sociologists ascribe an entropy of mixing to the integration of ethnicgroups and a heat of mixing to their tendency to segregate.52It is true that there is the danger of a lack of intellectual thoroughness insuch extrapolations. Each one ought to be examined properly for mereshallow analogies.Modern Version of Zeroth, First and Second LawsEven though the historical development of thermodynamics makes interesting reading, it does not provide a full understanding of some of thesubtleties in the field.
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