Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 55
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59It seems then, that despite his partisanship for Einstein’s theory ofrelativity, Einstein’s light quanta and Compton’s photons did not impressEddington – at least not at the time when he published the book.Another peculiarity about Eddington is that he still believed in theelement coronium – a hypothetical element of relative molecular mass ofabout 0.4 – which had been postulated by Dimitrij Iwanowitch Mendelejew(1934 – 1907) 60 in order to fill a perceived gap in the periodic table. Surelyby 1926 atomic physicists did not give credence to this fictitious element,although Mendelejew’s reputation was so great that many scientists clung tothe coronium.
So also the eminent geophysicist Alfred Lothar Wegener(1880 – 1930) – author of the continental drift theory – who says 61,...because of Mendelejews lucky shot with the prediction of germaniumit seems to me that the hypothesis [about coronium] deserves our attention58Ibidem, p. 31.Ibidem, p. 29.60 D.I. Mendelejew: Chemisches Centralblatt (1904) Vol. I p. 137.,,61 A.L. Wegener: Thermodynamik der Atmosphä re [ Thermodynmics of the atmosphere]Verlag J.A. Barth, Leipzig (1911).59,,8 Thermodynamics of Irreversible ProcessesLong before there was a thermodynamic theory of irreversible processes,there were phenomenological equations, i.e.
equations governing the fluxesof momentum, energy and partial masses. They were read off from theobserved phenomena of thermal conduction, internal friction and diffusion.Even the appropriate field equation for temperature was formulatedcorrectly, – for special cases – before the first law of thermodynamics waspronounced and accepted. Thus it was that complex problems of heatconduction were being solved routinely in the 19th century before anybodyknew what heat was.It took more than a century after phenomenological equations had beenformulated – and proved their reliability for engineering applications –before transport processes were incorporated into a consistent thermodynamic scheme.
And the first theories of irreversible processes clung soclosely to the laws of equilibrium – or near-equilibrium – that they achievedno more than confirmation of the 19th century formulae, and proof of theirconsistency with the doctrines of energy and entropy.It is only most recently that non-equilibrium thermodynamics has beenrephrased and given a formal mathematical structure with symmetrichyperbolic field equations. That structure is motivated by the classical laws,of course, but not in any obvious manner; no specific assumptions arecarried over from equilibrium thermodynamics into the new theory ofextended thermodynamics.
It has thus been possible to modify the classicallaws in an unprejudiced manner, and to extrapolate them into the range ofrarefied gases and of non-Newtonian fluids. The kinetic theory of gases hasprovided a trustworthy heuristic tool for this extension of thermodynamicswhich, at this time, has only just begun.Phenomenological EquationsJean Baptiste Joseph Baron de Fourier (1768–1830)Fourier came from poor parents and, besides, he became an orphan at theage of eight. So his ambitions to be a mathematician and artillery manseemed to be stymied and they would doubtless not have led him anywhere,2348 Thermodynamics of Irreversible Processeswere it not for the French revolution and Napoléon Bonaparte. As it was,the revolution happened in 1789 and Fourier could enter a military school –the later École Polytechnique of early 19th century fame, cf. Chap. 3 – andafter graduation he stayed on as an instructor.Napoléon took Fourier along on his disastrous Egyptian campaign andmade him a baron in recognition of his great mathematical discoverieswhich were related to heat conduction and the calculation of temperaturefields.
Those discoveries were first published in the Bulletin des Sciences(Société Philomatique, année 1808). After that first work, Fourier continueda lively scientific production and eventually he summarized his life’s workin the book “Théorie analytique de la chaleur” in 1824. This book is notavailable to me; therefore I refer to a German edition, published in 1884.1The translator claims that his work is identical to the original except that hecorrected numerous misprints.The work is essentially a book on analysis.
It is completely unaffected byany speculations about the nature of heat, or whether heat is the weightlesscaloric or a form of motion. Fourier says:One can only form hypotheses on the inner nature of heat, but theknowledge of the mathematical laws that govern its effects is independentof all hypotheses.2It is true that Fourier’s pronouncements are couched in long and oldfashioned sentences like this one:If two corpuscles of a body lie infinitely close and have differenttemperatures, the warmer corpuscle transmits a certain amount of its heatto the other one; and this heat – given from the warmer corpuscle to thecolder one at a given time and during a given moment – is proportional tothe temperature difference, if that difference has a small value.3However, Fourier also summarizes this cumbersome statement in the simplevectorial expressionqiNwT,wxiwhich is Fourier’s law for the heat flux q; ț is the thermal conductivity.Fourier calls it the internal conductivity.
He proceeds from there byassuming that the rate of change of temperature of a corpuscle is proportional to the difference of the heat fluxes on opposite sides and thus hecomes to formulate the differential equation of heat conduction, viz.1M. Fourier: “Analytische Theorie der Wärme.” Translated by Dr. B. Weinstein. Springer,Berlin (1884).2 Ibidem: Introduction, p. 11.3 Ibidem.
p. 451/2.Phenomenological EquationswTwtO235w 2T,wxi wxiwhere Ȝ is Fourier’s external conductivity, in modern terms it is the ratio ofț and the density of the heat capacity. This equation is the prototype of allparabolic equations and Fourier presented solutions for a large variety ofboundary and initial values in his book.Among many other problems solved, there is the one – a particularly ingenious one – by which the yearly periodic change of temperature on thesurface of the earth propagates as a damped wave into the interior, so that atcertain depths the earth is colder in summer than in winter.As a tool for the solution of heat conduction problems Fourier developedwhat we now call harmonic analysis – or Fourier analysis – by which anyfunction can be decomposed into a series of harmonic functions, and heexpresses his amazement about the discovery by saying:It is remarkable that the graphs of quite arbitrary lines and areas can berepresented by convergent series [of harmonic functions] … Thus thereare functions which are represented by curves, … which exhibit anosculation on finite intervals, while in other points they differ.4The harmonic analysis has found numerous applications in mathematics,physics and engineering.
It transcends the narrow field of heat conductionand proves its usefulness everywhere. Let me quote Fourier on the subject:The main property [of mathematical analysis] is clarity; [the theory]possesses no symbol for the expression of confused ideas. It combines themost diverse phenomena and discovers hidden analogies.5His lifelong preoccupation with heat conductionhad left Fourier with an idée fixe:He believed heat to be essential to health so healways kept his dwelling place overheated andswathed himself in layer upon layer of clothes. Hedied of a fall down the stairs.6Fig.
8.1. Jean Baptiste Joseph Baron de Fourier4Ibidem. p. 160.Ibidem. Forword, p. XIV.6 I. Asimov: “Biographies…” loc.cit. p. 234.52368 Thermodynamics of Irreversible ProcessesFourier’s book has a distinctly modern appearance.7 This is all the moresurprising, if the book is compared with contemporary ones, like Carnot’s,which appeared in he same year. Maybe that shows that physics is moredifficult than mathematics, but the fact remains that every line of Fourier’sbook can be read and understood, while large parts of Carnot’s book mustbe read, thought over and then discarded.One of the eager readers of Fourier’s book was the young W.
Thomson(later Lord Kelvin). Fourier’s results troubled him and in 1862 he wrote:For 18 years I have been worried by the thought that essential results ofthermodynamics have been overlooked by geologists.8Kelvin praises … the admirable analysis which led Fourier to solutions andhe uses its results to determine the age of the consistentior status – thesolid state – of the earth. That expression goes back to Leibniz. Theprevailing idea was that, at some time in the past, the earth was liquid.Obviously it had to cool off to a solid of at most 7000°F before thegeological history could begin. And Kelvin sets out to determine when thatwas.Fourier had given the temperature field in two half spaces initially attemperatures To ± ǻT asT ( x, t ) To 2'T 2S ³0xOte z 2 dz .Kelvin took ǻT = 7000°F and in effect fitted Fourier’s solution tox a constant surface temperature To of the earth,x the known value of Fourier’s external conductivity,x the known value of the present temperature gradient near the earth’ssurface,and calculated the corresponding value for t as 100 million years.
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