Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 51
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Einstein had come close to doing that in his paper onstimulated emission, see above. His expression hcQ for the recoil of anemitting atom is in fact the magnitude of the momentum. This can easily beconfirmed, since light – being electro-magnetic radiation – exerts a pressurep = 1/3 e on a wall, where e is the energy density, cf. Chap. 2. From thisresult it follows that the momentum p of the light quanta is in fact equal tohQc n , where n is the direction of their motion, see Insert 7.3.Arthur Holly Compton (1892–1962) proved this expression for themomentum directly when he observed collisions of light quanta withelectrons, in which – naturally – momentum and energy had to beconserved.
The observed Compton effect settled the matter. Thus the lightquantum now had energy and momentum and could be considered a particleCompton proposed the name photon and that was generally accepted aftersome time.Radiation pressure and momentum of light quanta1As in Insert 4.1 we consider that /6 of the photons with the energy hȞ and the(unknown) momentum pȞ move in the six spatial directions perpendicular to thesides of a cube. The walls reflect them elastically.
In this manner the photons withmomentum pȞ exert a pressure2 RQ EPQ6on a wall, where nȞ is the number density.The energy density is obviously hȞ·nȞ and since – by Maxwell’s equations – theenergy density equals three times the pressure, the momentum pȞ of a quantumequals hcQ in magnitude.Insert 7.3Photons, A New Name for Light Quanta215Bose’s derivation of the Planck distributionLet V be a volume, homogeneously filled with NȞ photons with frequenciesbetween Ȟ and Ȟ + dȞ. Accordingly the spectral energy is EȞ = NȞhȞ.
The photonsoccupy a spherical shell of volume2§ hQ · hdQV 4 S¨ ¸© c ¹ cin the phase space spanned by space and momentum coordinates. The phase spacehas cells of size h3 which can accommodate only two photons, – one each for the2two possible polarizations. Therefore there are AQ V 4 S Qc3 dQ cells in thespherical shell.Bose introduced the idea that the distribution of photons is characterized by prȞ ,the number of cells occupied by r photons in the range dȞ.
Their spectral entropy isthereforeSQk ln WQWQwhereMaximizing this under the constraints¦pAQQrAQ !.f p Q!r 0 r¦ rpand N QrSνÈQrwe obtainrØN È A Ø È A Ø k É ln ν É1 ν Ù ln É1 ν Ù Ù Nν .Ê Aν Ê Nν Ú Ê Nν Ú ÚWith NȞ = EȞ/hȞ we getSν1EνTNνAνlnhence Nνhν 1 NνAνkAν kT exp hν 1and with the above value for AȞ and EȞ = eȞ(Ȟ,T)VdȞeν (ν , T )8πν2hνc exp hν 13 kT which is the Planck distribution once again, but now derived without any referenceto classical thinking and classical formulae and, of course, without anyinterpolation between empirical functions.Insert. 7.4216 7 Radiation ThermodynamicsPhoton GasNow that photons may be considered as particles, endowed with momentumand energy, we may write an equation of transport for a photon gas. Letf(x,p,t)dp be the number density of photons with momenta betweenp = JQE n and p + dp. Since all photons have the speed c, the density functionsatisfies the photon transport equationwfwf EP MwZ MwV5 ( f ),which represents an equation of balance for the number of photons with xand t and with momentum p.
The equation is a little like the Boltzmannequation, cf. Chap. 4, except that the right hand side, which represents thesource density of photons, is not specific yet. Since the photons do notinteract among themselves – at least not normally – the right hand side isdue exclusively to interaction of the photons with matter. S(f) is zero, whenthe radiation is in equilibrium with matter, and, of course, when there is nomatter, there is no production either.Multiplication of the photon transport equation by a generic functionȥ(x,p, t) and integration leads to the equation of transport for radiativequantities Ô ψ fd pt Ô ψ cnk fd pxkÈ ψψ Øcnfd p Ô ψ S ( f )d p.kÔ ÉÊ txk ÙÚThe right hand side represents the production density of photons.
1/y is thevolume of a cell of (x,p)-space, and it is equal to h3 according to Bose.For ȥ = 1, R L JEQ P L , cp = hȞ, and – M (ln [f [f (1 [f ) ln(1 [f )) weobtain equations of balance for the number of photons and for momentum,energy and entropy with densities, fluxes and source densities as indicatedin Table 7.1.The entropic terms in the table are those appropriate for a Bose gas, forwhich the photon gas is the prototype, see above and Chap.
6. Forequilibrium the entropy has to have a maximum and that occurs for thedensity functionfGSW ( R ,6 )[JQM6G 1,where T is the temperature of the matter with which the radiation is inequilibrium. The equilibrium density function is the Planck distribution, ofPhotons, A New Name for Light Quanta217course; it is homogeneous and isotropic. Insertion of fequ into the tableprovides the entries of Table 7.2, most of which are zero.Of some interest are beams emanating from a spherical source S intoempty space. Inside the source the radiation is supposed to be inequilibrium and the temperature is TS. Therefore in a point outside thesource the density function is given by, cf. Fig. 7.4fGSW ( x,V , p)®¯fGSW (Q ,65 ) for 0 d M d 2S , 0 d E d E 00else.arcsin 4TThe distribution is strongly non-homogeneous and non-isotropic andtherefore it is a non-equilibrium distribution, although within the sphericalcone of angle ȕo it is a Planck distribution appropriate for the temperatureTS.
We may calculate the entries of Table 7.1 for this distribution andobtain the results of Table 7.3.Table 7.1 Thermodynamic fields of radiation. [*] stands for –M (ln [f numberDensitynÔ fdpMomentum2LenergyGentropyJ³ JEQ PL³ JQ³ [*]Fluxf(1 [f ) ln(1 [f ))Source DensityÔ SdpfdpÔ cnkfdp2LM³ JQPL PM fdpfdp,M³ JQEPMfdpfdpMM³ [*]EPMfdpTable 7.2 Equilibrium values of radiative fields. C] ( 3)[8S 5 M 415 J 3E 3Ôhνcn j SdpÔ hν SdpÔ k ln(1 7.8·10-161.202DensitynumberMomentum15] ( 3) CS4M6Flux030013energyentropyC6 4343 C6Source Density04C6 G KL0000yf)SdpJm 3K 4,218 7 Radiation ThermodynamicsFig.
7. 4. Radiation from a spherical sourceTable 7.3. Radiative thermodynamic quantities of rays emanating from a spherical source S.r2R21stands forDensitynumbernFlux2 ª0º3T « »604 M S4 5 42 « »¬1¼E C 15] (3)1 a 15ζ (3) T 3 (2 2k4 S)π2LMMomentum 2LenergyG14E12C65C65 (1 ),MentropyJ1 4 3C 65 (1 2 312)MME40C65 4 ª1 3 1 13 0 0º2«6»« 1 3 1»12 3 0»«0 6«»1103 3«0»¬¼2 ª0º«0»24 «1 »¬ ¼4 T4Sourceª 0ºC 654T2 « »042 « »«¬1»¼ª 0ºE 4 3 T2 « »C 65 2 04 « »4 3¬«1¼»000Photons, A New Name for Light Quanta219Some of the entries in the table permit simple calculations for thetemperatures of the sun and the planets as follows.
The sun has the radius99r ¤ = 0.7·10 m and it is at the distance RE = 150·10 m from the earth. Alsowe know from measurements that the energy flux density reaching the earthfrom the sun equals 1341 W/m2, – the so-called solar constant. Therefore wehave2c4 raT¤ ¤ 24RE1341W.m2From this relation the surface temperature of the sun may be calculatedand it comes out as T¤ = 5700 K.If rP is the radius of a planet with the distance RP from the sun, thetemperature TP of the planet follows from the equation2c4 r2aT¤ ¤ 2 SrP4RPc42aTP 4SrP ,4since it absorbs solar radiation on the circle ʌrP2 exposed to the sun and emitsradiation on its whole surface 4ʌrP2.
Since we know the distances RP of allplanets, we may prepare a table of planetary temperatures as shown inTable 7.4. The value for the earth is a trifle low – the mean temperature ofthe earth is 288K – but this is due to the fact that all kinds of secondaryeffects have been ignored by the calculation, e.g. the albedo, or coefficientof reflection, and the cloud cover. The same is true for the values of otherplanets.Let us also be interested in incoming and outgoing entropy fluxes of abody under solar radiation. The entropy flux density from the sun to theearth reads, according to Table 7.32ϕpc 4 3 r¤a T¤24 3RE0.30W.m2 KTable 7.4 Planetary temperaturesMercuryRP [m]TP [K]50·104759Earth150·102759MarsJupiter9230·10222770·109122220 7 Radiation ThermodynamicsOn the other hand, a body with the temperature T= 298K, the leaf of aplant (say), emits entropy at the rateϕnc 4 3a T4 32.00W.m2 KThus between absorption and emission the leaf has produced radiativeentropy, because it emits more than it absorbs.A more detailed investigation of this phenomenon was recently presentedby Wolf Weiss as part of a memoir on the entropy sources of the earth’satmosphere.36 As a preliminary exercise Weiss considers radiative andmaterial entropy sources in a black stone plate exposed to the sun.
Thisexercise shows what can be done without using explicit expressions for thesource terms, if only conditions are stationary, cf. Insert 7.5. In that case thesources may be calculated from the balance of in- and effluxes of entropyand energy, and it turns out that the scattering of radiation provides thebiggest contribution to the entropy production; far bigger than the dissipation of matter.Therefore it is conceivable – at least from the entropic point of view –that the entropy source of matter is negative, if only it is accompanied byradiative scattering.
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