Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 27
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To be sure, eminent people do not take kindly to criticism,and they become addicted to praise and may need it every hour of everyday; but Boltzmann did get that kind of attention: He was a celebrity withan exceptional salary for the time and full recognition by all the people whocounted. Even the Zermelo controversy seems to have rankled in his mindonly slightly: In his essay “The Journey of a German Professor to Eldorado”Boltzmann reports good-humouredly that Felix Klein tried to push him intowriting a review article on statistical mechanics by threatening to askZermelo to do it, if Boltzmann continued to delay.So, no! The neurasthenic condition which darkened Boltzmann’s life,seems more like the depressing mood that afflicts a certain percentage of thehuman population normally and which is nowadays treated effectively withcertain psycho-pharmaca, vulgarly known as happiness pills.It is true though that Boltzmann did not reign supreme in the scientificcircles in Vienna; there was also Ernst Mach (1838–1916), a physicist ofsome note in gas dynamics.
Mach was a thorn in Boltzmann’s flesh,because he insisted that physics should be restricted to what we can see,hear, feel, and smell, or taste, and that excluded atoms. As late as 189760W. Thomson: (1874) loc.cit.S.G. Brush: (1976) loc.cit. p. 597.62 M.J. Klein: American Science 58 (1970).61Boltzmann and Philosophy109Mach maintained that atoms did not exist, 63 and it is therefore clear that hehad no appreciation for the kinetic theory of gases. Mach also taughtphilosophy and his classes were full of students eager to imbibe his tastyintellectual philosophical concoction. Boltzmann taught hard science andinsisted that his students master a good deal of mathematics; consequentlythere were few students. That situation irritated Boltzmann, and he decidedto teach philosophy himself.He brought to the task a healthy contempt of philosophers.
After Machhad retired, Boltzmann taught Naturphilosophie in Vienna. And in hisinaugural lecture64 he gave an account of the failure of his efforts to learnsomething about philosophy:So as to go to the deepest depths I picked up Hegel; but what an unclear,senseless torrent of words I was to find there! My bad luck conducted mefrom Hegel to Schopenhauer … and even in Kant there were many thingsthat I could grasp so little that, judging by the sharpness of his mind inother respects, I almost suspected that he was pulling the reader’s leg, oreven deceiving him.For a lecture to the philosophical society of Vienna he proposed the title:Proof that Schopenhauer is a stupid, ignorant philosophaster, scribblingnonsense and dispensing hollow verbiage that fundamentally and foreverrots people’s brains.When the organizers objected, he pointed out – to no avail – that he wasmerely quoting Schopenhauer, who had written these exact same wordsabout Hegel.
Boltzmann had to change the title to a tame one: On a Thesisof Schopenhauer,65 but he got his own back by explaining the controversy indetail to the audience: Apparently Schopenhauer wrote that sentence aboutHegel in a fit of pique, when Hegel had failed to support him for anappointment to an academic position. In contrast to that Boltzmann’sintended title had been chosen out of an objective evaluation ofSchopenhauer’s work, – or so he says.It is thus clear that Boltzmann was not an optimal choice for a teacher ofconventional philosophy.
His disdain for philosophy, that doctrine of claptrap and idle whim was expressed frequently, with or without provocation.It is a good thing, perhaps, that Boltzmann did not also apply himself to the63I recommend an excellent account of Boltzmann’s professional work and psyche by D.Lindley: “Boltzmann’s atom.” The Free Press, New York, London (2001).
Lindley startshis Introduction with the apodictic quotation from Mach: “I don´t believe that atomsexist.”64 L. Boltzmann: “Eine Antrittsvorlesung zur Naturphilosophie” [Inaugural lecture on naturalphilosophy] Reprinted in the journal “Zeit” December 11, 1903 See also: E. Broda: loc.cit.65 L. Boltzmann: “Über eine These von Schopenhauer” Lecture to the Philosophical Societyof Vienna, given on January 21, 1905. See also: E.
Broda: loc.cit.110 4 Entropie as S = k ln Wteaching of theology. Because indeed, his ideas in that field are again quiteunconventional as the following paragraph shows.66…..only a madman denies the existence of God. However, it is true that allour mental images of God are only inadequate anthropomorphisms, so thatthe God whom we imagine does not exist in the shape in which weimagine him. Therefore, if someone says that he is convinced of God’sexistence and someone else says that he does not believe in God, maybeboth think exactly the same…Boltzmann sincerely admired Darwin’s discoveries, however.
Indeed, thereis not a single public lecture in which he did not advertise Darwin’s work.That work represents the type of natural philosophy that appealed toBoltzmann. And it is true that there is some congeniality between the twoscientists in their emphasis upon the underlying randomness of eitherthermodynamic processes or biological evolution: The vast majority of allmutations are detrimental to the progeny, just as the vast majority ofcollisions in a gas lead to more disorder.
In contrast to a gas, however, thesmall minority of advantageous mutative events is assisted by naturalselection so that nature can create order in living organisms. Naturalselection in this view plays the role of the infamous Maxwell demon, seeabove.67Despite his partisanship for Darwin’s ideas Boltzmann professes to seenothing in his convictions that runs counter to religion.68In the last ten years of his life Boltzmann did not really do any originalresearch, nor did he follow the research of others. Planck’s radiation theoryof 1900, and Einstein’s works on photons, on E = mc2, and on the Brownianmovement – all in 1905 – passed by him. In the end his neurastheniacaught up with him in a summer vacation.
He sent his family to the beachand hanged himself in the pension on the crossbar of a window.66L. Boltzmann: “Über die Frage nach der objektiven Existenz der Vorgänge in derunbelebten Natur” [On the question of the objective existence of events in the inanimatenature] Sitzungsberichte der kaiserlichen Akademie der Wissenschaften in Wien.Mathematisch-naturwissenschaftliche Klasse; Bd. CVI. Abt.
II (1897) p. 83 ff.67 Boltzmann does not seem to have argued like that. I read about this idea in one ofAsimov’s scientific essays. I. Asimov: “The modern demonology” in “Asimov onPhysics.” Avon Publishers of Bard, New York (1979).68 Some church leaders see this differently. So also Pope Benedikt XVI.
Says he in hisinaugural speech on April 24th, 2005: ... each being is a thought of God and not theproduct of a blind evolutionary process. The catholic church does not like randomevolution, nor does George W. Bush, 43rd president of the United States of America, whoadvocates that intelligent design be taught in the schools of his country.Kinetic Theory of Rubber111Kinetic Theory of RubberWe have already remarked that the formula S = klnW can be extrapolatedaway from monatomic gases, where it was discovered.
One such extrapolation – an important one, and a particularly plausible one – occurred inthe 1930’s when chemists started to understand polymers and to use theirunderstanding to develop a thermal equation of state for rubber. The kinetictheory of rubber is a masterpiece of thermodynamics and statisticalthermodynamics, and it laid the foundation for an important modern branchof physics and technology: Polymer science.Fig.
4.7. A rubber bar in the un-stretched and stretched configurationsAt the base of the theory is the Gibbs equation, see Chap. 3. In the aboveform the term –pdV represents the work done on the gas. That term must bereplaced by PdL for a rubber bar of length L under the uni-axial load P,which depends on L and T, cf.
Fig 4.7. Therefore the appropriate form ofthe Gibbs equation for a bar readsTdS = dU – PdL .The Gibbs equation obviously impliesPwUwST,wLwLso that we may say that the load has an energetic and entropic part.The integrability condition implied by the Gibbs equation readswUwLP TwPwTand hence followswSwLwP.wT112 4 Entropie as S = k ln WTherefore the entropic part of the load may be identified as the slope ofthe tangent of the easily measured (P,T)-curve of the bar for a fixed lengthL. The energetic part is determined from the ordinate intercept of thattangent, cf. Fig.
4.8.Fig. 4.8. Left: (load, temperature)-curve for a generic material Right: ditto for rubber.Tangent identifies entropic and energetic parts of forceWhen the (P,T)-curve is measured for rubber it turns out to be a straightline through the origin of the (P,T)-diagram. Therefore in rubber wwSL is independent of T, and U does not depend on L. We obtainPTwSwL(for rubber) ,a relation that is sometimes expressed by saying that rubber elasticity isentropic, or that the elastic force of rubber is entropy induced; energy playsno role in rubber elasticity.This was first noticed by Kurt H. Meyer and Cesare Ferri 69 and theydescribe their discovery by saying: L´origine de la contraction [ducaoutchouc] se trouve dans l´orientation par la traction des chaînespolypréniques. A cette orientation s´opposent les mouvements thermiquesqui provoquent finalement le retour des chaînes orientées à des positionsdésordinées (variation de l´entropie).7069K.H. Meyer & C.
Ferri: “Sur l´élasticité du caoutchouc”. Helvetica Chimica Acta 29,p. 570 (1935).70 The cause for the contraction of rubber lies in the orientation imparted to the polymerchains by the traction. The orientation is opposed by the thermal motion which eventuallycauses the return of the oriented chains to disordered positions (change of entropy).Kinetic Theory of Rubber113Apart from rubber, and some synthetic polymers, entropic elasticity occurs onlyin gases. Indeed, different as gases and rubber may be in appearance,thermodynamically those materials are virtually identical.
A joker with an originalturn of mind has once commented on this similarity by saying that rubbers are theideal gases among the solids. 71It is clear then that we need S as a function of L, if we wish to calculatethe thermal equation of state P(T,L) of rubber. We know that S=klnW holdsand for the calculation of W we need a model for the chaînes desordineés,the unordered polymer chains. Werner Kuhn (1899–1963)72 has providedsuch a model by imagining the rubber molecules as chains of Nindependently oriented links of length b with an end-to-end distance r.Fig. 4.9 shows such a molecule and also its one-dimensional caricature,where N± links point to the right or left. Obviously for that simplifiedmodel – which we use here – we must have0hence0 D 0 D0rTT1r(20D0(0 0.The pair of numbers ^N , N` is called the distribution of links, and thenumber of possible realizations of this distribution is°°r°°Fig.
4.9. Model for rubber molecule and its one-dimensional caricature71I. Müller, W. Weiss: “Entropy and energy – a universal competition” Springer,Heidelberg (2005). In Chap. 5 of that book the analogy between thermodynamic propertiesof rubber and gases is highlighted by a juxtaposition.72 W. Kuhn: “Über die Gestalt fadenförmiger Moleküle in Lösungen” [On the shape offiliform molecules in solution] Kolloidzeitschrift 68, p. 2 (1934).114 4 Entropie as S = k ln WWN!N!N !N !ËN Èr ØÛ Ë N Èr ØÛÌ 2 ÉÊ1 Nb ÙÚ Ü !Ì 2 ÉÊ1 Nb ÙÚ Ü !ÍÝÍÝ.Thus W and Smol = klnW, the entropy of a molecule, are functions of theend-to-end distance r. That function may be simplified by use of the Stirlingformula and by an expansion of the logarithm, viz.2lna! = alna-aandr Ør1È r ØÈln É1 ÉÙÙ .Ê Nb ÚNb 2 Ê Nb ÚThe former is true for large values of a, and we apply it to N as well as toN r .
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