Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 43
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6.7. Inversion curves and critical isochor and isotherm Also: Mini-table of critical data“Thirteen ways of looking at the van der Waals equation”.23 And I believethat in a recent book 24 I have presented a fourteenth way.Students of thermodynamics are often mystified by the non-monotoneisotherms exhibited in Fig. 6.6 and, in particular, by the branch with apositive slope, which suggests instability.
These features are reflections ofthe non-convex character of the function ij(r), but we shall not go into that,although at present – while I write this – there is great interest in similarphenomena occurring in phase transitions in solids, like shape memoryalloys. An instructive mechanical model for non-monotone stress-straincurves has been proposed and investigated by the author.25Van der Waals equationAll N atoms of a monatomic gas in a volume V with the surface V and outernormal n move according to Newton’s law of motionµ xDKD(Į = 1,2,…).If that equation is multiplied by xĮ, and then averaged over a long time IJ, andsummed over all Į, one obtainsN¦ µ xD 12324252DN¦KxD D(angular brackets denote averages).D 1M.M.
Abbott: Chemical Engineering Progress, February (1989).I. Müller, W. Weiss: “Entropy and energy,...” loc.cit. (2001).I. Müller, P. Villaggio: “A model for an elastic-plastic body” Archive for RationalMechanics and Analysis. 65 (1977).Johannes Diderik Van Der Waals (1837–1923)1811The left hand side is equal to –3NkT, since each atom has an average value /2 kTof kinetic energy. The right hand side was called virial by Clausius. The virial hastwo parts WS and Wi due to forces on atoms from the surface and from other atomsrespectively. Therefore we write-3 NkT = WS + Wi .Assuming that only atoms in the immediate neighbourhood of the surface elementdA of V feel the effect of the surface, and that the sum of forces from the surfaceon those atoms is equal to –pndA on average, we obtain WS = -3 pV.
Hence follows1pV = N k T + /3 Wi..Without the inner virial Wi we thus have regained the ideal gas law.The force on atom Į from atom ȕ may be written K αβ K xα xβxβ xαxβ xα. Itfollows for WiWiWiN ÇNÇK xα xβα 1β 1x x xβ xααβxαN N1 Ç Ç K x x x xα β α2 α 1β 1βNN Ç K x x x xα β α2α 1βfor any β.The last step requires that on average each atom is surrounded by others in thesame manner. We set Ňxȕ-xĮŇ= r and convert the sum in an integral by definingthe particle density n(r) .Wi N2 V³ K ( r ) r n ( r ) dV or, by isotropyf2 SN ³ K ( r ) n ( r ) r 3dr.0The force K(r) and the potential ij(r) of Fig.
6.5 are related by K(r) = particle density n(r) may approximately be given byNVexp(dMdrand theM kT ) , so that an atomon average is surrounded by a cloud of other atoms which is densest, where ij(r)has its minimum. Insertion providesfWi2SN2VkT 3³ (1 exp( kTM )) r 2 dr .0We set ij = for r < d andMkT 1 for r > d as indicated in Fig. 6.5 and obtain1 N 4π d 3ËÛNN Ô ϕ (r ) 4π r 2 dr Ü3 Ì NkT 2 32dVVÌÍÜÝor, with a and b from Fig. 6.5,Wi1826 Third Law of ThermodynamicsWibaÛË3 Ì NkT V 2 Ü .vv ÝÍElimination of Wi between this and the equation for pV provides the van der Waalsequation, provided we assume that b << v holds which is reasonable.Insert 6.1ThrottlingAdiabatic throttling is an isenthalpic process of lowering the pressure by ǻp < 0.The temperature changes accordingly so that ǻh =0 can be satisfied. Therefore wehave'T'p( wwhp ) T( wwTh ) p.The denominator is the specific heat cp, and the numerator may be rewritten – byuse of the Gibbs equation – in the form ( wwRJ )6v 6 ( ww6v ) R .
Hence follows∆T∆pvT È 1 È v Ø1ØÉ ÉÊ ÙÚ Ùc p Ê v T p T Úand we conclude that cooling occurs, if the thermal expansion coefficientD 1v ( ww6v ) R is greater than 1/T . For ideal gases we have Į = 1/T.Insert 6.2HeliumHelium deserves its own section, although it was liquefied in the samemanner as hydrogen, by adiabatic expansion and throttling. It just tookmore time, because the boiling point was lower: 4.2K. It was HeikeKammerlingh-Onnes (1853–1926) who succeeded in 1908 and whoeventually reached 0.8K by adiabatic evaporation of the liquid.Kammerlingh-Onnes received the Nobel prize for his efforts in 1912.He did not succeed, however, to freeze helium, and later it turned out thatit cannot be done, no matter how far the temperature is lowered, at least notunder ordinary pressures.
It took pressures of 20 atm, or so, to make heliumsolid.The reason for the persistence of the liquid phase is supposed to bequantum mechanical. According to quantum mechanics a particle withmomentum p and energyp22Pmay be considered as a de Broglie wave withHeliuma wave length Ohpand a frequency Q21 p 26h 2P .183Such a particle has anequal probability to be anywhere in space, so that it cannot be localized. Aparticle, however, which we know to be boxed in, in a range of lineardimension ǻx, is represented in quantum mechanics by a packet of deBroglie waves, i.e.
a superposition of such waves with momenta in therange ǻp. Between ǻx and ǻp there is the relation ǻx ǻp = h, which is calledHeisenberg’s uncertainty relation. Thus either x or p may be fixed, but notboth.The above is a subject of single-particle quantum mechanics, governedby the Schrödinger equation. The uncertainty relation is extrapolated tothermodynamics by the assumption that ǻp may be interpreted as themomentum of a particle of a liquid (say) during its thermal motion. Thus wemay write'p2 PEor withE12kT : 'pPkT , 'xh.PkTǻx is therefore a typical de Broglie wave length of a particle of a body oftemperature T, such that ǻx3 represents the smallest volume element inwhich such a particle can be localized.
For an atom of liquid helium atT = 1K the uncertainty ǻx of position comes out as ǻx = 2·10-9m. This isconsiderably more than if the particle were confined to an elementary cell ina solid lattice structure. Therefore the solid lattice cannot form, and that iswhy helium remains liquid, or so they say.Once liquid helium was available, it could be used to cool othersubstances down to the neighbourhood of absolute zero. And it turned outthat some metals, like mercury and lead develop a very strange behaviourindeed. They lose their electrical resistance at some characteristictemperature.
We say that they become super-conductors, materials withzero resistivity in which a current, once induced, moves round and roundforever.Actually helium itself, below 2.19K – the so-called Ȝ-point – exhibits asomewhat similar unique phenomenon of its own: It behaves like a mixtureof a normal fluid with a small viscosity and a super-fluid, which has noviscosity at all. That liquid mixture is called He II as opposed to He I, liquidhelium above the Ȝ-point. The lower the temperature is, the higher is theproportion of the super-fluid.Strange phenomena occur in He II, or they should occur and do not. Thesubstance dumbfounded eminent scientists like Lev Davidovich Landau(1908–1968) and Evgenii Michailovich Lifshitz (1915–1985), Landau’scolleague, collaborator and frequent co-author.
It is, perhaps, worthwhile todescribe two of the more illustrative snares in which those scientists found26h = 6.626·10-34Js is the Planck constant.1846 Third Law of Thermodynamicsthemselves entangled for years. If nothing else, that will be a consolationfor those of us – less eminent than Landau and Lifshitz – who find itdifficult to adjust their minds to the evidence of the new and unusual. Let usconsider:Sound first: Sound in air – essentially a two-constituent mixture ofnitrogen and oxygen – permits two wave modes, both longitudinal. Oneconsists of the joint oscillation of both constituents with no relativevelocity, while for the other one the two constituents move relative to eachother with no motion of the mixture as a whole. Those modes may be calledthe first and second sound respectively. Both propagate with differentspeeds and both are usually coupled so that, if the first sound is stimulated,the second sound follows, and vice versa.
We never actually hear thesecond sound in air, because it is damped away within the distance of lessthan 1 mm from our vibrating vocal cords; this may be a good thing,because it saves us from hearing everything twice. Also temperatureoscillations are associated with both sounds, although in air they cannot bedetected, at least not by our coarse human senses.Sound in helium below the Ȝ-point is qualitatively similar, since itbehaves like a mixture. But quantitatively it differs, chiefly on account ofthe fact that one constituent, the super-fluid, is free of friction so thatdamping is absent.
The theory of first and second sound was first workedout by Lazlo Tisza (1907- ).27 A little later Landau developed essentiallythe same theory 28 and therefore the governing equations are very oftencalled the Landau equations. According to those equations the secondsound should be detectable, but it was not, or not for years. The first soundcame through helium loud and clear at one side when it was excited by avibrating membrane at the other side, but no second sound could bedetected.
At the end, after many vain attempts, a frustrated Lifshitz satdown and did a simple calculation, a calculation that should have been donebeforehand: He calculated the amplitudes.29 Then it turned out that,according to the Landau equations, the first and second sound were nowuncoupled, so that the second sound could not be stimulated by a vibratingmembrane, and that the first sound was not accompanied by a temperatureoscillation, but the second sound was. So Lifshitz suggested to use anelectric coil with an alternating current instead of a membrane. The Jouleheating of the coil produced temperature oscillations, and there was thesecond sound immediately, – as a thermal wave, in a manner of speaking.
Itpropagated with the speed predicted by the Landau equations.27L. Tisza: “Transport phenomena in He II.” Nature 141 (1938).Landau: “The theory of superfluidity of Helium II.” Journal of Physics (USSR) 5(1941).29 E.M. Lifshitz: “Radiation of sound in Helium II.” Journal of Physics (USSR) 8 (1944).28 L.D.Adiabatic Demagnetisation185For the Landau equations – and other achievements – Landau obtainedthe Nobel prize in 1962.
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