Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 31
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His masterpiece “On the equilibrium ofheterogeneous substances” was published in the “Transactions of theConnecticut Academy of Sciences”2 by reluctant editors, who knew nothingof thermodynamics and who may have been put off by the size of themanuscript – 316 pages! The paper carries Clausius’s triumphant sloganabout the energy and entropy of the universe as a motto in the heading, seeChap. 3, but it extends Clausius’s work quite considerably.The publication was not entirely ignored. In fact, in 1880 the AmericanAcademy of Arts and Sciences in Boston awarded Gibbs the Rumfordmedal – a legacy of the long-dead Graf Rumford. However, Gibbs remainedlargely unknown where it mattered at the time, in Europe.Friedrich Wilhelm Ostwald (1853–1932), one of the founders of physicalchemistry, explains the initial neglect of Gibbs’s work: Only partly, he says,is this due to the small circulation of the Connecticut Transactions; indeed,he has identified what he calls an intrinsic handicap of the work: … theform of the paper by its abstract style and its difficult representationdemands a higher than usual attentiveness of the reader.
And it is true thatGibbs wrote overlong sentences, because he strove for maximal generalityand total un-ambiguity, and that effort proved to be counterproductive toclarity of style. However, it is also true that the concepts in the theory ofmixtures, with which Gibbs had to deal, are somewhat further removedfrom everyday experience – and bred-in perspicuity – than those occurringin single liquids and gases.Ostwald translated Gibbs’s work into German in 1892, and in 1899le Chatelier translated it into French.
Then it turned out that Gibbs hadanticipated much of the work of European researchers of the previousdecades, and that he had in fact gone far beyond their results in some cases.Ostwald encourages researchers to study Gibbs’s work because … apartfrom the vast number of fruitful results which the work has alreadyprovided, there are still hidden treasures.
Gibbs revised Ostwald’stranslation but … lacked the time to make annotations, whereas thetranslator [Ostwald] lacked the courage.31I. Asimov: “Biographies …” loc.cit.J.W. Gibbs: Vol III, part 1 (1876), part 2 (1878).3 So Ostwald in the foreword of his translation: “Thermodynamische Studien von J. WillardGibbs” [Thermodynamic studies by J. Willard Gibbs] Verlag W. Engelmann, Leipzig(1892).2Entropy of Mixing. Gibbs Paradox129Those translations made Gibbs known. His work came to be universallyrecognized, and in 1901 he received the Copley medal of the Royal Societyof London. In 1950 – nearly fifty years after his death – he was elected amember of the Hall of Fame for Great Americans.The greatest achievement, perhaps, of Gibbs is the discovery of thechemical potentials of the constituents of a mixture.
The chemical potentialof a constituent is representative for the presence of that constituent in themixture in much the same way as temperature is representative for thepresence of heat. I shall explain as we go along.While evolution has provided us, the human race, with a good sensitivityfor temperature, it has done less well with chemical potentials. To be sure,our senses of smell and taste can discern foreign admixtures to air or water,but such observations are at a low level of distinctness.
Therefore thethermodynamic laws of mixtures have to be learned intellectually – ratherthan intuitively – and Gibbs taught us how this is best done.Because of that it seems impossible to explain Gibbs’s work – and to doit justice – without going into some technicalities. Nor is it possible torelegate all the more technical points into Inserts. Therefore I am afraid thatparts of this chapter may read more like pages out of a textbook than Ishould have liked.Entropy of Mixing. Gibbs ParadoxChemical thermodynamics deals with mixtures – or solutions, or alloys –and the first person in modern times who laid down the laws of mixing, wasJohn Dalton again, the re-discoverer of the atom, see Chap.
4. Dalton’s law,as we now understand it, has two parts.The first one is valid for all mixtures, or solutions, and it states that, inequilibrium, the pressure p of the mixture and the densities of mass, energyand entropy of the mixture are sums of the respective partial quantitiesappropriate for the constituents. If we have Ȟ constituents, indexed by Į =1,2,…Ȟ, we may thus writeQQRQQU D (T , p E ) , U u ¦ U D uD (T , p E ) , U s ¦ U D s D ( T , p E ) .¦ RD , U ¦DDD111D 1The second part of Dalton’s law refers to ideal gases: If we are looking ata mixture of ideal gases, the partial quantities ȡĮ, uĮ, and sĮ depend on T andon only their own pĮ, and, moreover, the dependence is the same as in asingle gas, i.e.
cf. Chap. 31305 Chemical PotentialsRD UDM6,PDuDsDk(T TR ) , andµDkTkp .s D ( T R , p R ) ( z D 1)lnlnµDTRµDpRuD (TR ) z DA typical mixing process is indicated in Fig. 5.1, where Ȟ singleconstituents under the pressure p and at temperature T are allowed to mixafter the opening of the connecting valves. When the mixing is complete,the volume, internal energy and entropy of the mixture may be differentfrom their values before mixing. We writeQV¦VD VMix ,D 1QU¦UD U Mix ,QSD 1SD S¦DMix1and thus we identify the volume, internal energy and entropy of mixing.Fig.
5.1. Pure constituents at T, p before mixing (top). Homogeneous mixture at T, p(bottom). Note that the volume may have changed during the mixing processFor ideal gas mixtures VMix and UMix are both zero and SMix comes out asQS Mix k ¦ N D lnD 1ND,Nwhere NĮ is the number of atoms of gas Į and 0Q¦D10 D . ByAvogadro’s law – and, of course, by the thermal equation of statepD ρD kµα T – the numbers NĮ are independent of the nature of the gases.Therefore the entropy of mixing is the same, irrespective of the gases thatare being mixed. This is an observation due to Gibbs and the Gibbsparadox4 is closely related to it: If the same gas fills all volumes at thebeginning, the situation before and after opening of the valves is the sameone, and yet the entropies should differ, since the entropy of mixing does4J.W. Gibbs: loc.cit.
pp. 227–229.Homogeneity of Gibbs Free Energy for a Single Body131not depend on the nature of the gases, but only on their number of atoms ormolecules.The Gibbs paradox persists to this day. The simplicity of the argumentmakes it mind-boggling. Most physicists think that the paradox is resolvedby quantum thermodynamics, but it is not! Not, that is, as it has beendescribed above, namely as a proposition on the equations of state of amixture and its constituents as formulated by Dalton’s law.5Gibbs himself attempted to resolve the paradox by discussing thepossibility of un-mixing different gases, and the impossibility of such anun-mixing process in the case of a single gas. It is in this context that Gibbspronounced his often-quoted dictum: … the impossibility of an uncompensated decrease of entropy seems to be reduced to an improbability, seeFig.
4.6. Gibbs also suggested to imagine mixing of different gases whichare more and more alike and declared it noteworthy that the entropy ofmixing was independent of the degree of similarity of the gases. None ofthis really helps with the paradox, as far as I can see, although it providedlater scientists with a specious argument. Thus Arnold Alfred Sommerfeld(1868–1951) 6 pointed out that gases are inherently distinct and that there isno way to make them gradually more and more similar.
Then Sommerfeldquickly left the subject, giving the impression that he had said somethingrelevant to the Gibbs paradox which, however, is not so, – or not in any waythat I can see.Homogeneity of Gibbs Free Energy for a Single BodySo far, when we have discussed the trend toward equilibrium, or theincrease of disorder, or the impending heat death, we might have imaginedthat equilibrium is a homogeneous state in all variables. The truth is,however, that indeed, temperature T and pressure p7 are homogeneous inequilibrium, but the mass density is not, or not necessarily.
What ishomogeneous are the fields of temperature, pressure and specific Gibbs free5The easiest way to deal with a paradox is to maintain that it does not exist, or does not existanymore. The Gibbs paradox is particularly prone to that kind of solution, because it sohappens that a superficially similar phenomenon occurs in statistical thermodynamics. Thatstatistical paradox was based on an incorrect way of counting realizations of a distribution,and it has indeed been resolved by quantum statistics of an ideal gas, cf.
Chap. 6. It is easyto confuse the two phenomena.6 A. Sommerfeld: „Vorlesungen über theoretische Physik, Bd. V, Thermodynamik undStatistik“ [Lectures on theoretical physics, Vol. V. Thermodynamics and Statistics]Dietrich’sche Verlagsbuchhandlung, Wiesbaden, 1952 p. 76.7 Pressure is only homogeneous in equilibrium in the absence of gravitation.1325 Chemical PotentialsXenergy u – Ts + pv.8 The specific Gibbs free energy is usually abbreviatedby the letter g and it is also known as the chemical potential,9 although thatname is perhaps not quite appropriate in a single body.We proceed to show briefly how, and why, this unlikely combination – atfirst sight – of u,s,v with T and p comes to play a central role inthermodynamics: We know that the entropy S of a closed body with animpermeable and adiabatic surface at rest tends to a maximum, which isreached in equilibrium. The interior of the body may at first be in anarbitrary state of non-equilibrium with turbulent flow (say) and largegradients of temperature and pressure.
While the body approaches equilibrium, its mass m and energy U + Ekin are constant, because of theproperties of the surface. In order to find necessary conditions for equilibrium we must therefore maximize S under the constraints of constant mand U + Ekin. If we take care of the constraints by Lagrange multipliers Ȝmand ȜE , we have to find the conditions for a maximum of³8 UU d8 OO ³ U d8 O' ³ U (W 8122) d8 .8The specific values s and u of entropy and internal energy are assumed tosatisfy the Gibbs equation locally:10T ds du p dvor, equivalentlyT d(ρ s )d(ρu ) − gdρ .Since u is a function of T and ȡ, the variables in the expression to bemaximized are the values of the fields T(x), Xl(x), and ȡ(x) at each point x.By differentiation we obtain the necessary conditions for thermodynamicequilibrium in the formXl = 0,wU WwUU0 O'w6w6wUWwUU O O O'wUwUandhence with the0Gibbs equation:1TgOE.TOmTherefore in thermodynamic equilibrium the body is at rest throughout V,and T and g = u – Ts + pv are homogeneous.
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