4. Blackbody Radiation, Boltzmann Statistics, Temperature, and Thermodynamic Equilibrium

Transcription

1 4. Blackbody Radaton, Boltzmann Statstcs, Temperature, and Thermodynamc Equlbrum Blackbody radaton, temperature, and thermodynamc equlbrum gve a tghtly coupled descrpton of systems (atmospheres, volumes, surfaces) that obey Boltzmann statstcs at the macroscopc level. They are mportant because of the compact descrptons of systems that they gve when Boltzmann statstcs apply, ether approxmately or nearly exactly. Fortunately, ths s most of the tme n the Earth s stratosphere and troposphere, and n other planetary atmospheres as long as the densty s suffcent that collsons among atmospherc molecules, rather than photochemcal and photophyscal propertes, determne the energy populatons of the ensemble of molecules. The descrpton s compact because the defnton of temperature mples thermodynamc equlbrum, Boltzmann statstcs, and blackbody radaton as a lmtng case of the emsson of radaton. 4.1 Thermodynamc Equlbrum The exstence of thermodynamc equlbrum n a closed volume of gas (well approxmated by a small atmospherc ar parcel, of cm-scale sze or larger at the tropopause or below, based on collsonal mean free paths of ar molecules) means that the total energy of the gas volume and ts parttonng among dfferent energy levels (.e., the nternal rotatonal, vbratonal, and electronc levels and the translatonal energy) do not change macroscopcally over tme. The mcroscopc state remans n constant flux at equlbrum due to the changes n nternal and translatonal energes caused by collsons and the absorpton and emsson of radaton; these changes average to zero macroscopcally. 4.2 Boltzmann statstcs Consder a smplfed set of energy states an atom or molecule may occupy (Fgure 4.1). There are a number of dscrete energy states and, above the dssocaton lmt, where the molecule s no longer chemcally bound, a contnuum descrbed by the densty of states ρ(e). At thermal equlbrum, the Boltzmann factor, whch gves the relatve populaton for a gven bound state,, wth dscrete energy E, s e, where k s Boltzmann s constant and T s the temperature. Fgure 4.2. Trply degenerate p- orbtals of an atom. Fgure 4.1. A smplfed energy level dagram for an atom or molecule, ncludng dscrete bound states and a densty functon of contnuum states. The concept of degeneracy of states s also requred for the Boltzmann descrpton. Ths s nothng more than the realzaton that there are, n many or most cases, more than one dstnct quantum state at a gven energy. As an example, n the famlar case of a set of p-orbtals of an atom (shown schematcally n Fgure 4.2), there are three dstnct states, where the lobes of the orbtals (each of whch can house two electrons) are orented along ether the

2 x- y- or z-axs. These orbtals are degenerate n energy unless ther spatal symmetry s broken by an external nteracton. In the soon-to-be-famlar case of the rotatonal states of a datomc molecule, a state wth rotatonal angular momentum J has a degeneracy of 2J+1. Ths degeneracy may be broken by an electrc feld E or magnetc feld m, f the molecule has ether an electrc or magnetc dpole moment, to separate the states n energy. The degeneracy g = the number of energy levels wth energy E. Fgure 4.3 shows an example approprate to rotatonal levels of a lnear molecule (cf. Chapter 6.2). For statstcal mechancs purposes, degeneracy thus smply means havng more than one state at a partcular energy. Boltzmann statstcs may then be descrbed startng from the Boltzmann factors that gve relatve populatons. For the dscrete states, the populaton n each state P e. (4.1) g =5 g =3 g =1 Fgure 4.3. Degenerate energy levels approprate to rotatonal levels of a lnear molecule are shown on the left. The rght sde demonstrates how the degeneracy may be broken, e.g., by a magnetc feld m or an electrc feld E, f the molecule has a magnetc or an electrc dpole moment, respectvely. The populaton at each energy P ge. (4.2) The partton functon Q descrbes how relatve populaton numbers are apportoned among the dfferent energes and states: Q ge. (4.3) The extenson to nclude contnuum states s straghtforward: b E Q ge + ρ E e de = 1 b ( ). (4.4) The contnuum term wll be omtted from further dscussons. It s not normally needed n atmospherc spectroscopy snce the dssocaton energes are so hgh that contnuum states do not normally contrbute to the populatons of planetary atmospheres n nearequlbrum condtons. It can readly be rentroduced as needed, substtutng the sum over states wth the ntegral over densty of states as n Eq. (4.4). The partton functon provdes the normalzaton factor so that fractonal populatons for states are gven as: E P = e / Q. (4.5) The fractonal populaton at energy E, P = ge / Q. (4.6) The normalzaton by Q also makes the arbtrary choce of the zero of energy (the orgn of the energy levels) cancel out of the populaton statstcs (problem 4.1).

3 Some defntons may now be stated more frmly: A system at equlbrum s one where the populatons of energy levels are descrbed by Boltzmann statstcs. A system at equlbrum may be descrbed by a temperature and, conversely; Temperature s a characterstc of an equlbrum system. A system that s not at equlbrum does not have a defned temperature. (In the early days of laser development, the achevement of a populaton nverson, necessary for lasng or masng, has been descrbed as achevng a negatve temperature,.e., e E upper /kt > e E lower /kt T < 0 ). At equlbrum, the radaton s n equlbrum wth the molecules, at the same temperature. Ther energy dstrbutons are descrbed by the blackbody radaton law, whch wll be ntroduced n the next secton. Local thermodynamc equlbrum The expresson local thermodynamc equlbrum (LTE) s frequently encountered, partcularly n astrophyscs and atmospherc scence. Ths s smply a way of expressng that local behavor (say at a certan alttude n the atmosphere) s reasonably well descrbed as beng n equlbrum and characterzed by a temperature, whereas on larger scales, where the atmospherc structure vares, ths cannot be the case, snce the temperature vares. As a very rough rule, LTE n a regon of an atmosphere s establshed when there are 10 collsons per photochemcal or reacton event. In the Earth s atmosphere, non-lte condtons are normally encountered n the mesosphere and above, wth mesospherc CO 2 as a common example. An atmosphere may be stable even though t s not overall n equlbrum, dependng on the boundary condtons of gravty and heatng. Most atmospheres are not completely n equlbrum (no planetary atmospheres are): Snce the Earth s troposphere s heated from the bottom, and warmer ar s more buoyant, there s a natural mxng as a counter effect to the thermodynamc temperature lapse (tropos s Greek for to turn ). The stratosphere, on the other hand, s heated nternally, manly through absorpton of ultravolet radaton by ozone. Ths heatng ncreases wth heght relatve to atmospherc densty. The stratosphere s more stratfed, and stable. Stuatons where a system s descrbed by more than one temperature are frequently encountered. For example, n an astrophyscal plasma, one may hear of a radaton temperature and a dfferent knetc temperature. In ths case the separate phases (radaton and matter) are reasonably well descrbed by temperatures, but are not strongly enough coupled together through absorpton and emsson to establsh equlbrum. Analogously, n laboratory spectroscopy, one frequently hears of separate rotatonal and vbratonal temperatures produced by certan sample preparaton technques (e.g., supersonc expansons n molecular beams).

4 4.3 Blackbody radaton A blackbody s an dealzed object that absorbs and emts radaton at all wavelengths wth 100% effcency. Although a blackbody s an deal stuaton, t s a fundamentally mportant lmt for emsson and absorpton spectroscopy, as wll be descrbed n Chapter 5. Someone who has encountered a buffalo or a moose on a road at nght may perceve t as a pretty good blackbody! A blackbody s completely characterzed by a temperature T and as such obeys several convenent laws as descrbed below. Blackbody emsson s Lambertan. Relaton of ntensty wth wavelength and temperature (Planck s law) The Planck s law blackbody flux densty for emsson from a surface per wavenumber s: 2 3 2πhc σ Rσ =, hcσ e 1 (e.g., n W m-1 or erg s -1 cm -1 ), (4.7) where σ s the wavenumber, (e.g. m -1 or cm -1 ), h s the Planck constant, c the speed of lght, and k the Boltzmann constant. Includng the ncrement n wavenumber dσ gves Rd σ σ, flux densty n power per unt area, 2 3 2πhc σ dσ Rd σ σ =. (4.8) hcσ e 1 The blackbody radaton densty, defned as the radaton densty nsde a hohlraum, a cavty whose walls are at a constant temperature and n thermodynamc equlbrum wth the nsde, s 3 8πhcσ dσ ρσ ( ) dσ=. (4.9) hc e σ 1 Radaton constants The frst radaton constant c 2 1 2π hc = W m 2. The second radaton constant c2 hc/ k= m K. The emtted flux densty s thus: 3 c1σ dσ Rd σ σ =. (4.10) c2σ / T e 1 Often, exctaton levels are gven n K rather than n wavenumbers, for convenence n relatng atomc and molecular physcs to a partcular temperature regme. c2 provdes the relatonshp between them. Fgure 4.4 Blackbody emsson per unt area s greater at every wavenumber when the temperature s hgher. The Sun and Earth are approxmated here as blackbodes.

5 In photons ( E = hν = hcσ ), πσ c dσ σdσ Rn ( σ) dσ = =. (4.11) c2σ/ T c2σ/ T e 1 e 1 The blackbody emsson per unt of emttng area ncreases wth ncreasng temperature at all wavenumbers (Fgure 4.4). It s also greater per unt sold angle of observed area. Problem 4.2 determnes the flux densty per unt sold angle. In spectroscopy of the Earth s atmosphere, the balance of radaton from the Earth and the Sun n the nfrared s due to the much larger angular sze of the Earth (Problem 4.3 and Fgure 5.6). Raylegh-Jeans lmt For hν kt (hcσ kt ), R σ s approxmately lnear wth temperature: 2πhc 2 σ 3 dσ e hcσ /kt 1 2πkTcσ 2 dσ. (4.12) Ths s n common use n radofrequency and mcrowave work, especally n rado astronomy. Ths law was frst dscovered emprcally, and then led to the predcted ultravolet catastrophe. That s, emsson versus wavenumber and total emsson become nfnte! It was known to be wrong but t was classcally requred: Quantum theory was requred for the dervaton of Planck s law (Davdson, Chapter 6). Antenna temperature, nose temperature, system temperature Rado astronomers and aeronomers measurng at frequences where the Raylegh-Jeans lmt apples gve power sources as temperatures. Ths ncludes target sgnals as well as aspects of the nstrumentaton (detector nose and other nose sources) and the entre nstrument (the system) n ths way. Ths smply means that each contrbuton s equal to that whch would arse from a blackbody at that temperature n sgnal power or nose power. Ths s dscussed n more detal under nose sources n Secton x.y. Emssvty, reflecton coeffcent, Krchoff s law A blackbody s the most a surface at temperature T can emt. It can emt less: Emssvty: ε 1. The emssvtyε of a medum (gas, lqud, or sold) at wavelength equals ts absorptvty A. Thus, for a blackbody, ε = A = 1. For a non-blackbody, ε = A < 1. The reflectvty R s 0 for a blackbody: ε + R = 1. In general, blackbody or not, the emsson and reflecton are exactly balanced: Reflecton coeffcent (reflectvty): R 1. Krchoff s law: ε + R = 1 for opaque surfaces (the extenson to nclude transmsson s straghtforward). Relaton between flux densty and temperature (Stefan-Boltzmann constant) The total flux densty F of a blackbody can be calculated by ntegratng a blackbody over wavenumber:

6 0 F = R σ dσ = erg cm -2 s -1 o K -4 = W m -2 o K -4. Relaton between maxmum ntensty and temperature (Wen s law) The wavelength of maxmum ntensty of blackbody radaton s proportonal to temperature as llustrated n Fgure 4.4. The maxmum power per wavenumber occurs at σ max (cm -1 ) = T ( o K) number of photons per wavenumber occurs at σ (cm -1 ) = T ( o K) power per wavelength occurs at (µm) = / T ( o K) number of photons per wavelength occurs at (µm) = / T ( o K) References Davdson, N.R., Statstcal Mechancs, McGraw-Hll, New York, A good source for further nformaton especally on statstcal aspects of the development of the concepts of statstcal mechancs. Penner, S.S., Quanttatve Molecular Spectroscopy and Gas Emssvtes, Addson- Wesley, Readng, MA, ISBN: , Great detal on blackbody radaton. Problems (assgned February 11, due February 20) 4.1 Demonstrate that the determnaton of fractonal populatons of states and energy levels usng Boltzmann factors and the partton functon s ndependent of the choce of the zero of energy (the orgn of the energy levels). 4.2 Determne the blackbody radance (emtted flux densty per steradan) from Equaton 4.10 by nvokng Lambertan emsson and ntegratng over sold angle. 4.3 The Sun may be approxmated by a blackbody at 5900 o K. The average temperature of the Earth s near 288 o K. For these temperatures and for a satellte orbtng the Earth at 800 km, on the sunlt part of the orbt, at what wavenumber does the radaton receved from the Earth equal that receved from the Sun? 4.4 Calculate the fracton of radaton emtted at vsble wavelengths for a typcal ncandescent lght bulb (flament T = 3000 o K). What s the wavelength of maxmum emsson?