Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 52
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Schrödinger seems to advocate that possibility whenhe declares37 radiation to be the cause, when plants decrease their entropyduring growth in the process of photosynthesis of glucose. We shall reviewthat proposition in Chap. 11.Dissipative and radiative entropy sourcesWe consider a black stone plate of thickness L = 0.1m exposed to solar radiationperpendicular to the plate. The plate absorbs the radiation in a thin surface layer oftemperature T1. That layer reemits part of the absorbed energy and the rest istransmitted through the plate by heat conduction. On the dark side – away from thesun – the plate emits radiation according to its temperature T2 and according to theStefan-Boltzmann law.
The emitted radiation on the dark side again comes from athin surface layer. We look at stationary conditions. The heat flux is governed byFourier’s law, cf. Chap. 8 so that we haveqκT1 T2Land T ( x)T2 T1 T2Lx (0 x L) .First we determine T1 and T2. We balance the in-and effluxes in the whole plate andin the surface layer on the dark side and obtain respectively36W. Weiss: “The balance of entropy on earth.” Thermodynamics and ContinuumMechanics 8, (1996).37 In his booklet: E. Schrödinger: “What is Life ?” Cambridge: At the University Press.
NewYork: The Macmillan Company (1945).Photons, A New Name for Light Quantac4Q¤ aT14With Nc4aT24andc4aT24κT1 T2L221.1341 W2 , the solar constant,mW , appropriate for stone, and Q0.74 mK¤the only relevant solution isT1 = 355K andT2 = 296K.The area density of entropy sources has four terms in principle which we denote byȈrr – due to photon-photon interaction.
Here absent.Ȉrm – source of radiative entropy due to matter.Ȉmr – source of material entropy due to radiation.Ȉmm – dissipative entropy source due to heat conduction.Ȉrm may be calculated from the entries of Table 7. 3 as the balance of in- andeffluxes of radiative entropy2Ȉ rmc 4 3 r¤ a T¤4 3RE2c 4 3 c 4 3a T a T24 3 14 35.032Ȉmr may be calculated according to Clausius, cf. Chap.
3, asWm2 KQT., i.e. as heatabsorbed or emitted divided by the appropriate temperature. ThusȈ mr1 Èc 4Ø 1 caT 4ÉÊ Q¤ aT1 ÙÚ 4T1T2 4 20.243Wm2 K.And Ȉmm+Ȉmr must together be zero, because outside the plate there is no materialentropy flux. Therefore we have6 mm0.243Wm2 K.We conclude that, whatever entropy is produced by heat conduction is balanced bya decrease of the entropy of matter due to absorption and emission of radiation.
Wealso see that the radiative entropy source is about 20 times bigger than thedissipative material one. Absorption, emission and scattering of radiation seems tobe the prevalent mechanism of entropy production in the plate.Insert 7.5The most interesting – and most important – application of radiationthermodynamics is the physics of stars. And yet, the physicists of the 19thcentury, who raised their eyes to the stars, as it were, were unaware of thedecisive role of radiation for stellar structure.
They thought, perhaps, thatthe only role of radiation in a star was to carry the energy away from it.222 7 Radiation ThermodynamicsAlthough they were mistaken in this assumption, their work laid a foundation and – by good luck – it could be used later as a basis for Eddington’smore informed work. Let us review this preliminary work first, before wediscuss the radiation thermodynamics of stars.Convective EquilibriumIn the 19th century the only conceivable source of solar energy – or stellarenergy – was the contraction of the stars under their gravitational pull asfirst envisaged by Helmholtz, cf.
Chap. 2 and Insert 2.2. According to thecontraction hypothesis, the heating occurs everywhere in the star while, ofcourse, the cooling by radiation occurs near the surface. Thus it makessense to think of a star as hot inside and cool – relatively cool – near thesurface. And it was known that heat conduction could not account for thetransfer of heat from the inner regions of a star to the surface, because thethermal conductivity is much too small.
On the other hand, the importantrole of radiation inside the star was not recognized at the time. Thereforethe transfer had to happen by convection, the same mechanism thatdistributes the heat from the stove throughout the living room. Let usconsider this.The situation of hot below and cool above is akin to the state of ouratmosphere on a nice summer day, when the sun heats up the ground in themorning, and the ground heats up the air-layer next to it, which thusbecomes warmer than the air on top, and lighter than it should be for equilibrium.38 If that situation is only slightly disturbed, it causes thermalconvection, i.e. a vertical rise of the warm air.
Since the rising air enterszones of lower pressure, it expands and, since heat conduction is negligible,it cools adiabatically. When this goes on for some hours the air reaches aconvective equilibrium by mid-day. In that equilibrium the pressure P, andthe density ȡ within the lower layers – as far up as the convection reaches –obey the adiabatic equation of stateP = ț ȡȖ .The specific entropy is homogeneous in convective equilibrium. Ȗ is theratio of specific heats, equal to 7/5 in air and accordingly the air temperaturedrops by 1K for every 100 meters of height. In the atmosphere the convection stops at night and convective equilibrium breaks down.In a star there is no night, of course, and therefore the convection may besupposed to persist until the whole star is in convective equilibrium with38Not lighter than the air on top, however, as scientific folklore sometimes has it.Convective EquilibriumdPGȡdr223M r as mechanical equilibrium conditionor momentum balance andr2Pc γρ as (pressure, density) relation .γρcThe index c refers to the centre of the star and Mr is the mass inside thesphere of radius r.POf course, this cannot have been acceptable for all, because the adiabatic equationgof state refers to ideal gases and the sun has a mean density of 1.4 cm3 , larger thanthe density of water and a thousand times denser than air.
Could that matterpossibly behave like an ideal gas? Well it does, at least approximately, but thephysicists in the 19th century – without any knowledge of the atomic structure –could not begin to understand that. They put the problem on the shelf andproceeded anyway to calculate the potential energy of a gas sphere with radius Rand mass Mr, cf.
Insert 7.6:' RQV3(J 1) / 4).5J 6 4 2Potential energy of a starAccording to Insert 2.2 the potential energy of a spherical mass is equal toR M2 G G Ô 2r dr ,22 0 rR1E pot2MR1where the second term depends on the mass distribution in the star. That term maybe rewritten, in terms of Epot itself, for a star in convective equilibrium by thefollowing string of equations involving partial integrations and the repeated use ofthe mechanical equilibrium condition, the adiabatic equation of state and the2identity dMr = ȡ 4ʌr dr.R M2G Ô 2r dr2 0 r11R1 dPdrÔ Mrȡ dr20R24ʌ Ô P r dr2 Ȗ 1 0Ȗ11Ȗ6 Ȗ 1RdPȡMdrÔ rdr2 Ȗ 1 01ȖR 3 dP4ʌ Ô rdrdr6 Ȗ 1 01ȖȖ Mr PdM rÔ2 Ȗ 1 0 ȡ1MRMrG ÔdM r6 Ȗ 10 r1ȖE pot .Insertion into the original equation for Epot provides the equation given in the maintext.Insert 7.6224 7 Radiation ThermodynamicsThe pioneer of convective equilibrium was W. Thomson (Lord Kelvin)who conceived of the idea and suggested it for the atmosphere of the earthand for the sun.39 He says:The essence of convective equilibrium is that the density and thetemperature are so distributed throughout the whole fluid mass that thesurfaces of equal density and equal temperature remain unchanged whencurrents are produced in it by any disturbing influence gentle enough thatchanges in pressure due to inertial motions are negligible.And about stars he says that…the natural stirring produced in a great free fluid mass like the Sun’s bythe cooling of the surface, must, I believe, maintain a somewhat closeapproximation to convective equilibrium throughout the whole mass.J.
Homer Lane investigated the problem thoroughly. The long title of hispaper reveals his main assumption that the stellar material be considered asan ideal gas: “On the theoretical temperature of the sun under thehypothesis of a gaseous mass maintaining its volume by its internal energyand depending on the laws of gases known to terrestrial experiment.”40Lane obtained a fairly simple, albeit non-linear second order differentialequations for P(r), or ȡ(r), or 2U (( TT )) by differentiating the momentumbalance in convective equilibrium.
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