Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 39
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In particular,the present strategy diagram lacks the lateral regions, denoted by a and b inFig. 5.6. This is due to the fact that we have not accounted for an entropy ofmixing in the present case. For socio-thermodynamics in full – includingthe entropy of mixing – I refer to my recent article “Socio-thermodynamics– integration and segregation in a population.” 45 In that paper the analogy isfully developed, including first and second laws of socio-thermodynamics,and with the proper interpretations of working and heating etc.464546I.
Müller: Continuum Mechanics and Thermodynamics 14 (2002) pp. 389--404.The simplified presentation given above follows a paper by J. Kalisch, I. Müller:“Strategic and evolutionary equilibria in a population of hawks and doves.” Rendiconti delCircolo Matematico in Palermo, Serie II, Supplemento 78 (2006), pp. 163–171.1645 Chemical PotentialsThe upshot of the present investigation is that, if integration of species –or, perhaps, ethnic groups – is desired and segregation is to be avoided,political leaders should provide for low prices, if they can. In good timesintegration is no problem, but in bad times segregation is likely to occur.We all know that. But here is a mathematical representation of the factwith – conceivably – the possibility for a quantification of parameters.The analogy of segregation in a population and the miscibility gap insolutions and alloys has been noticed before by Jürgen Mimkes, a metallurgist.47 His approach is more phenomenological than mine, without amodel from game theory.
Mimkes has studied the integration and segregation of protestants and catholics in Northern Ireland, and he came tointeresting conclusions about mixed marriages.It is interesting to note that socio-thermodynamics is only accessible tochemical engineers and metallurgists.
These are the only people who knowphase diagrams and their usefulness. It cannot be expected, in our society,that sociologists will appreciate the potential of these ideas. They havenever seen a phase diagram in their lives.That paper also includes evolutionary processes, which make the hawk fraction change sothat the population may eventually reach the evolutionarily stable strategy appropriate tothe price level IJ.47 J. Mimkes: “Binary alloys as a model for a multicultural society.” Journal of ThermalAnalysis 43 (1995).6 Third Law of ThermodynamicsIn cold bodies the atoms find potential energy barriers difficult to surmount,because the thermal motion is weak.
That is the reason for liquefaction andsolidification when the intermolecular van der Waals forces overwhelm thefree-flying gas atoms. If the temperature tends to zero, no barriers –however small – can be overcome so that a body must assume the state oflowest energy. No other state can be realized and therefore the entropy mustbe zero. That is what the third law of thermodynamics says.On the other hand cold bodies have slow atoms and slow atoms havelarge de Broglie wave lengths so that the quantum mechanical wavecharacter may create macroscopic effects. This is the reason for gasdegeneracy which is, however, often disguised by the van der Waals forces.In particular, in cold mixtures even the smallest malus for the formationof unequal next neighbours prevents the existence of such unequal pairs andshould lead to un-mixing. This is in fact observed in a cold mixture ofliquid He3 and He4.
In the process of un-mixing the mixture sheds itsentropy of mixing. Obviously it must do so, if the entropy is to vanish.Let us consider low-temperature phenomena in this chapter and let usrecord the history of low-temperature thermodynamics and, in particular, ofthe science of cryogenics, whose objective it is to reach low temperatures.The field is currently an active field of research and lower and lowertemperatures are being reached.Capitulation of EntropyIt may happen – actually it happens more often than not – that a chemicalreaction is constrained. This means that, at a given pressure p, the reactantspersist at temperatures where, according to the law of mass action, theyshould long have been converted into resultants; the Gibbs free energy g islower for the resultants than for the reactants, and yet the resultants do norform.
We may say that the mixture of reactants is under-cooled, or overheated depending on the case. As we have understood on the occasion ofthe ammonia synthesis, the phenomenon is due to energetic barriers whichmust be overcome – or bypassed – before the reaction can occur. Thebypass may be achieved by an appropriate catalyst.1666 Third Law of ThermodynamicsAn analogous behaviour occurs in phase transitions,1 mostly in solids: Itmay happen that there exist different crystalline lattice structures in thesame substance, one stable and one meta-stable, i.e. as good as stable or,anyway, persisting nearly indefinitely.
Hermann Walter Nernst (1864–1941) studied such cases, particularly for low and lowest temperatures.Take tin for example. Tin, or pewter, as white tin is a perfectly goodmetal at room temperature – with a tetragonal lattice structure – popular fortin plates, pewter cups, organ pipes, or toy soldiers.2 Kept at 13.2°C and1atm, white tin crumbles into the unattractive cubic grey tin in a few hours.However, if it is not given the time, white tin is meta-stable below 13.2°Cand may persist virtually forever.3It is for a pressure of 1atm that the phase equilibrium occurs at 13.2°C.At other pressures that temperature is different and we denote it by Twļg(p);its value is known for all p.
At that temperature ǻg = gw – gg vanishes, andbelow we have gw > gg, so that grey tin is the stable phase. ǻg may beconsidered as the frustrated driving force for the transition and it issometimes called the affinity of the transition. It depends on T and p and hastwo partsǻg(T,p) = ǻh(T,p) – T·ǻs(T,p),an energetic and an entropic one.ǻh(T,p) is the latent heat of the transition and ǻs(T,p) is the entropychange.4 For any given p the latent heat ǻh(T,p) can be measured as afunction of T by encouraging the transition catalytically, e.g. by dopingwhite tin with a small amount of grey tin. And ǻs(T,p) may be calculated byintegration of cp(T,p)/T of both variants, white and grey, between T = 0, – oras low as possible – and the extant T.
Thus we havegw½T c p ( W, p)T c p ( W, p) °°dW s g (0, p) ³'g (T , p) 'h(T , p) T ®sw (0, p) ³dW¾WW00°°¿¯1234From the point of view of thermodynamics phase transitions are much like chemicalreactions, although the phenomena differ in appearance. One might go so far as to say thatphase transitions are chemical reactions of a particularly simple type.In ancient times tin was much in demand because, alloyed to copper, it provided bronze,the relatively hard material used for weapons, tools, and beads and baubles in the bronzeage (sic).Not so, however, when it coexists with previously formed traces of grey tin. If that is thecase, tin appliances are affected by the tin disease at low temperature.
A church may loseits organ pipes in a short time, and that loss did in fact occur during a cold winter night inSt. Petersburg in the 19th century.Note that the heat and entropy of transition depend on T and p, if the transition occurs inthe under-cooled range. If it occurs at the equilibrium point, both quantities depend onlyon one variable, since T = Twļg(p) holds at that point.Inaccessibility of Absolute Zero167Inspection shows that for T ĺ 0 the affinity tends to the latent heat.
Thiswould even be true, if the specific heats cp(T,p) were constant for T ĺ 0. Inreality, in Nernst’s time – between the 19th and the 20th century – there wasalready ample evidence that all specific heats tend to zero polynomially,with Tĺ0, e.g. as (a·T 3) for non-conductors, or as (a·T 3+b·T) forconductors. Given this observation, the integrals in ǻs(T,p) themselves tendto zero, and the curly bracket reduces to sw(0,p) – sg(0,p).
This differencemay be related to the heat of transition ǻh(Twļg(p)) at the equilibrium point,because in phase equilibrium we have ǻg(Twļg(p)) = 0, ors w (Twl g ( p )) s g (Twl g ( p ))s w (0, p ) s g (0, p )'h(Twl g ( p ))Twl g ( p )'h(Twl g ( p ))Tw l g ( p )orTwl g ( p )³0wgc p (W , p ) c p (W , p )WdW .From some measurements Nernst convinced himself that this expression – which after all is equal to ǻs(T,p) for T ĺ 0 – is zero, irrespective ofthe pressure p, and for all transitions.5 So he came to pronounce his law ortheorem which we may express by saying that the entropies of differentphases of a crystalline body become equal for T ĺ 0, irrespective of thelattice structure.
Moreover, they are independent of the pressure p.This became known as the third law of thermodynamics.We recall Berthelot, who had assumed the affinity to be given by the heatof transition. And we recall Helmholtz, who had insisted that thecontribution of the entropy of the transition must not be neglected.Helmholtz was right, of course, but the third law provides a lowtemperature niche for Berthelot: Not only does T·ǻs(T,p) go to zero, ǻs(T,p)itself goes to zero. The entropy capitulates to low temperature and gives upits efficacy to influence reactions and transitions.Inaccessibility of Absolute ZeroIn 1912 Nernst pointed out that absolute zero could not be reached becauseof the third law.6 Indeed, since s(T,p) tends to the same value for T ĺ 0irrespective of pressure, the graphs for different p’s must look qualitatively56W.
Nernst: “Über die Berechnung chemischer Gleichgewichte aus thermodynamischenMessungen” [On calculations of chemical equilibria from thermodynamic measurements]Königliche Gesellschaft der Wissenschaften Göttingen 1, (1906).W. Nernst: “Thermodynamik und spezifische Wärme” [Thermodynamics and specificheat]. Berichte der königlichen preußischen Akademie der Wissenschaften (1912).1686 Third Law of Thermodynamicslike those of Fig. 6.1.a.
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