Thermodynamics and Statistical Mechanics

Thermodynamics and Statistical MechanicsBlackbody Radiation

Thermo & Stat Mech - Spring 2006 Class 21

Classical Energy Equipartition


Quantum Harmonic Oscillator


Energy of EM Waves in a CavityCalculate the number of standing wave states in the frequency range between w and w + dw. Then Bose-Einstein statistics is used to determine the number of photons in each state. With that information the energy can be calculated.


Density of States (Lec 17)


Density of States for Photons


Photon Energy


Graph


Chart1

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5000

6000

7000

Photon Energy (eV)

Energy Density

Black Body Radiation

Sheet1


k =8.63E-05eV/K1.31E+19

h =4.14E-15eV-s

c =3.00E+08m/s

2.00E-02eV

500060007000

0000

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Sheet1

5000

6000

7000

Photon Energy (eV)

Energy Density


Sheet2

Sheet3

Total Energy Density


Energy Flux


Human Eye and the SunIt is often stated in textbooks that the peak sensitivity of the human eye is at the wavelength at which the energy radiated by the sun is maximum.


Human EyePeak at 555 nm


A Problem?The graph of the energy density as a function of frequency, for a blackbody at a temperature of 5800 K, has a maximum at 1.41 eV. Photons of that energy have a wavelength of 879 nm, which is out of the visible spectrum, in the infrared. What is wrong?


SolutionThe energy density is plotted as a function of energy, while the eyes sensitivity is plotted as a function of wavelength. If we plot the energy density as a function of wavelength, its peak does not occur at 879 nm. How can this be?


Change to Wavelength


Wavelength Spectrum


Graph


Chart1

000

000

000

000

000

000

000

000

000.0000000022

000.0000003642

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296480.623192683454395.664755643731028.552656445

5000

5800

7000

Wavelength (m)

Energy Density


Sheet1


k =8.63E-05eV/K1.31E+19

h =4.14E-15eV-s

c =3.00E+08m/s

1.00E-02eV1.41E+00

500058007000

0000

1.00E-02559517626236304650082356598012785930949689264

2.00E-022212138056024210257435504932809031177044972028300

3.00E-024919418535671000573424956939776069566173828242400

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9.00E-024123467030244130048547812860096800595273208954782000

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1.30E-0181974804837565000971814228058742001200338364670940000

1.40E-01939194191294908001115371609105350001380208590980240000

1.50E-011065045468345350001267064167650660001570843629321840000

1.60E-011196995670669840001426573830157540001771912605126140000

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Sheet1

5000

5800

7000

Photon Energy (eV)

Energy Density


Sheet2


k =1.38E-23J/K5.00E-24

h =6.63E-34J-s

c =3.00E+08m/s

5.00E-09m5.00E-07

500058007000

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4.65E-07467985.6812364451103506.943821752778324.060158980

4.70E-07473924.7275233731107549.293777512762291.201781410

4.75E-07479547.5620490741110916.455729922745223.718718620

4.80E-07484856.5467641821113630.730784572727194.028793310

4.85E-07489854.6372376121115714.565593352708271.729425680

4.90E-07494545.3304171431117190.4643742688523.610714090

4.95E-07498932.6145405871118080.909295362668013.680280620

5.00E-07503020.9212725741118408.28874462646803.198259260

5.05E-07506815.0801153281118194.83299652624950.720957130.0000005

5.10E-07510320.2751185591117462.556812942602512.151861140.000000505

5.15E-07513542.0038934041116233.208511452579540.798794490.00000051

5.20E-07516486.0389179661114528.22505552556087.436149440.000000515

5.25E-07519158.3911071751112368.692734522532200.371235420.00000052

5.30E-07521565.2756072271109775.313018732507925.513884980.000000525

5.35E-07523713.0797643721106768.373191982483306.448555130.00000053

5.40E-07525608.3332093391103367.721384332458384.508248430.000000535

5.45E-07527257.6799917081099592.745645622433198.849657450.00000054

5.50E-07528667.8526931981095462.356720222407786.529008260.000000545

5.55E-07529845.6484446571090994.974202882382182.578144330.00000055

5.60E-07530797.9067686071086208.515774032356420.080451220.000000555

5.65E-07531531.4891672031081120.389232192330530.246276290.00000056

5.70E-07532053.2603743691075747.487058852304542.487545960.000000565

5.75E-07532370.07119051070106.183268932278484.491326480.00000057

5.80E-07532488.7428184271064212.332317022252382.292113260.000000575

5.85E-07532416.0526201441058081.269845592226260.342668690.00000058

5.90E-07532158.7212150761051727.815077212200141.583259610.000000585

5.95E-07531723.4008423741045166.274667342174047.509173520.00000059

6.00E-07531116.6649116311038410.447848542147998.236416910.000000595

6.05E-07530344.9986686871031473.632709942122012.565521410.0000006

6.10E-07529414.7909056021024368.633468722096108.04340250.000000605

6.15E-07528332.326646421017107.768601752070301.023232320.00000061

6.20E-07527103.7807430711009702.879716852044606.722303540.000000615

6.25E-07525735.2123184551002165.341053562019039.277873480.00000062

6.30E-07524232.559996612994506.0695126911993611.800989680.000000625

6.35E-07522601.637862615986735.5351233081968336.428307180.00000063

6.40E-07520848.132097697978863.7718640631943224.371916690.000000635

6.45E-07518977.598237849970900.3887635871918285.96720960.00000064

6.50E-07516995.459006884962854.5812120111893530.718811970.000000645

6.55E-07514907.002677635954735.1424223331868967.344624530.00000065

6.60E-07512717.381917568946550.4749866191844603.818010140.000000655

6.65E-07510431.613077628938308.6024777261820447.408173340.00000066

6.70E-07508054.575885597930017.1810525551796504.718779360.000000665

6.75E-07505591.013507601921683.5110176961772781.724862210.00000067

6.80E-07503045.532943705913314.5483228031749283.808072860.000000675

6.85E-07500422.6057257904916.9159511561726015.790319730.00000068

6.90E-07497726.568887275896496.9151805771702981.965854350.000000685

6.95E-07494961.626178803888060.5366913821680186.131855510.00000069

7.00E-07492131.849500815879613.4715010991657631.6175650.000000695

7.05E-07489241.180532094871161.1217086171635321.312028130.0000007

7.10E-07486293.432530006862708.6110329651613257.690491490.000000705

7.15E-07483292.292282331854260.7951343331591442.839509820.00000071

7.20E-07480241.322191375845822.2717070531569878.480813170.000000715

7.25E-07477143.962472615837397.3903362031548565.993984380.00000072

7.30E-07474003.533451479828990.2621112381527506.437995970.000000725

7.35E-07470823.23794316820604.7689916241506700.571654290.00000073

7.40E-07467606.163701583812244.5729208791486148.872997550.000000735

7.45E-07464355.285924747803913.1246866451465851.557693140.00000074

7.50E-07461073.469804772795613.6725255891445808.596478170.000000745

7.55E-07457763.473111953787349.2704729171426019.731685880.00000075

7.60E-07454427.948803048779122.7864571641406484.492899170.000000755

7.65E-07451069.447644935770936.9101417391387202.211771120.00000076

7.70E-07447690.42084553762794.1605153471368172.036050850.000000765

7.75E-07444293.222684676754696.8932340931349392.942851870.00000077

7.80E-07440880.113138373746647.307718511330863.751198520.000000775

7.85E-07437453.260490385738647.4540092731312583.133884820.00000078

7.90E-07434014.743925892730699.2393857391294549.628678680.000000785

7.95E-07430566.556102376722804.4347517511276761.64890310.00000079

8.00E-07427110.605693493714964.6807934651259217.493424680.000000795

8.05E-07423648.719902141707181.4939141661241915.356078610.0000008

8.10E-07420182.646939397699456.2719512291224853.334557880.000000805

8.15E-07416714.058466389691790.2996805191208029.438793370.00000081

8.20E-07413244.551996566684184.7541136681191441.598850540.000000815

8.25E-07409775.653256179676640.7095936921175087.672366680.00000082

8.30E-07406308.818501083669159.1426945341158965.451552510.000000825

8.35E-07402845.436788303661740.9369300741143072.669779930.00000083

8.40E-07399386.832201039654386.8872782051127407.007777410.000000835

8.45E-07395934.266026052647097.7045255451111966.099453260.00000084

8.50E-07392488.938882616639874.0194383031096747.537365880.000000845

8.55E-07389051.992802365632716.3867647941081748.87785970.00000085

8.60E-07385624.513259638625625.2890750381066967.645884150.000000855

8.65E-07382207.531152013618601.1404427811052401.339512370.00000086

8.70E-07378802.024730926611644.2899752231038047.434175650.000000865

8.75E-07375408.921482398604755.025195631023903.386628750.00000087

8.80E-07372029.099957994597933.5752839131009966.638660360.000000875

8.85E-07368663.3915563591180.114180164996234.6205625580.00000088

8.90E-07365312.582255239584494.763556007982704.7543721990.000000885

8.95E-07361977.414295702577877.595658531969374.4568966880.00000089

9.00E-07358658.587817006571328.636031455956241.1425358270.000000895

9.05E-07355356.762444772564847.866118038943302.2259109860.0000009

9.10E-07352072.558831882558435.225750148930555.124312180.000000905

9.15E-07348806.560153239552090.615527769917997.2599731540.00000091

9.20E-07345559.313555064545813.899093119905626.0621840450.000000915

9.25E-07342331.331559546539604.905303408893438.9692507160.00000092

9.30E-07339123.093425673533463.430306166881433.4303093690.000000925

9.35E-07335935.046467101527389.239520933869606.9070046460.00000093

9.40E-07332767.60732794521382.069531001857956.8750389520.000000935

9.45E-07329621.163217375515441.629888762846480.8256003790.00000094

9.50E-07326496.073104023509567.604838116835176.2666761980.000000945

9.55E-07323392.668870958503759.654957253824040.7242585410.00000095

9.60E-07320311.256432346498017.41872506813071.7434485370.000000955

9.65E-07317252.116812622492340.514014226802266.8894648480.00000096

9.70E-07314215.507189148486728.539514069791623.7485622250.000000965

9.75E-07311201.661899298481181.076085966781139.9288654270.00000097

9.80E-07308210.793412901475697.688054188770813.0611235420.000000975

9.85E-07305243.093270977470277.924434824760640.7993894830.00000098

9.90E-07302298.732991675464921.320105381750620.8216292180.000000985

9.95E-07299377.864944356459627.396917563740750.8302649620.00000099

1.00E-06296480.623192683454395.664755643731028.5526564450.000000995

Sheet2

5000

5800

7000

Wavelength (m)

Energy Density


Sheet3

AgreementThe maximum of the graph for 5800 K, the temperature of the suns surface has a maximum at 500 nm, in reasonable agreement with the peak sensitivity of the human eye.


Wien Displacement Law


Photon Gas


Helmholtz Function


Pressure


4/07/06

Thermodynamics and Statistical Mechanics

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Transcript of Thermodynamics and Statistical Mechanics