BoltzmannStatistics热力学统计物理英文教学课件

§7.BoltzmannStatistics§7.1Thestatisticsexpressionofthermodynamicsquantities7.1.1ThestatisticsexpressionofinternalenergyIntroduceanewfunctionZ1calledpartitionfunctionThenWiththeformula(7.1.1)and(7.1.3)itgets

7.1.2GeneralizedworkGeneralizedforceperformedbyenvironmentfollowsthatForinstance,

Ininfinitesimalquasi-staticprocesstheworkperformedbyenvironmentfollowsthatThetotaldifferentialofisgivenbyRemark!(1)Thefirsttermdenotestheworkperformedbyenvironment;(2)Thesecondtermdenotestheheatabsorbingfromenvironment.

7.1.3ThestatisticsexpressionofentropyBasedonthefirstlawofthermodynamics,withintegralfactor1/TfordQ,itgetsthatWiththeformulas(7.1.4)and(7.1.6)itgetsThetotaldifferentialoflnZ1followsthatThereforedQgetsanotherintegralfactorβ

LetItwillprovethatkisBoltzmannconstant.Comparingtheformula(7.1.10)and(7.1.11)andwiththeformula(7.1.12)onegetsHeretheintegralconstantischosetobezero.

Withlogarithmcalculationfortheformula(7.1.3)itgetsBasedonBoltzmanndistribution,itgetsthereforeComparingwiththeformula(6.6.4)itgets

ThisiscalledBoltzmannrelation.

Remark!Themorethemicrostatenumberis,thebiggertheentropyis.(2)ForBoseandFermisystemssatisfyingtheclassicallimitedcondition,theentropyisgivenby

7.1.4Thestatisticsexpressionoffreeenergy

F=U-TSThisformula

is

appliedtolocalsystems.or

Thisformula

is

appliedtoBoseandFermisystem.

7.1.5TheclassicalstatisticsexpressionWhenΔωl

issmallenoughitfollowsTheinternalenergy,equationofstateandentropyarethesamewithaboveresultsaslongaspartitionfunctionexpression(7.1.18)isapplied.

§7.2The

equationofstateoftheidealgasTheenergyofmonatomicmoleculeInthescopeofdxdydzdpxdpydpz,theprobablemicrostatenumberisgivenbyThereforepartitionfunctionfollowsthat

Withlogarithmcalculationfortheformula(7.2.3)itgetshereisthevolumeoftheidealgas.Remark!(1)TheBoltzmannconstantisobtainedfromcomparingtheformulawithpV=nRT.(2)Forbiatomicmolecule,multi-atomicmolecule,althoughtheenergiesincludetranslationalenergy,rotationalenergyandvibrationalenergy,theformula(7.2.5)suitsforeverycase.

(3)Theresultsarethesamebyusingclassicalstatisticstheory.(4)TheotherexpressionofclassicallimitedconditionisthattheaveragespacebetweenmoleculesismorebiggerthendeBrogliewavelength.

Ifε=3kT/2,

thus,§7.3MaxwellVelocityDistributionLaw7.3.1MaxwellvelocitydistributionTheclassicalexpressionofBoltzmanndistributionNooutfield,Thestatenumberoftranslationofmoleculemasscenterfollowsthat

InthescopeofVanddpx

dpy

dpz

,themoleculenumberfollowsthat

TheparameterαisgivenbyInthescopeofVanddpx

dpy

dpz

,themoleculenumberfollowsthat

Let,denotingthemoleculenumberoftheunitofvolume,thusthemoleculenumberwithinthevelocityscopeofdvx

dvy

dvz

isgivenbyThisformulaisknownverywellandcalledMaxwellvelocitydistributionlaw.

Thevolumeelementofsphericalpolarcoordinatessubstitutesdvx

dvy

dvz

,andintegralforvariablesθandφ,thusthemoleculenumberinunitofvolumeandthevelocityscopeofdvisgivenby

7.3.2ThreecharacteristicvelocityThemostprobablevelocity（vm)μismolemass.(2)Meanvelocity()

(3)Squaremeanroot(vs)7.3.3ApplicationofMaxwellvelocitydistributionlawCalculatethemoleculenumberofcollisioninunitoftimeatunitarea.SolutiondAisaareaelement,

dΓdAdtdenotesthemoleculenumberofcollisionindtatdA.

dΓdAdt=namely

§7.4Energy

EquipartitionTheorem7.4.1Energy

equipartitiontheoremForaclassicalsystemwhichisinequilibriumstatewithtemperatureTtheaveragevalueofeverysquaretermofaparticleenergyequals.

εp

and

εqdenotetheparticlekineticenergyandpotentialenergyrespactivily.herepiismomentum,aiisapositivecoefficient.

Thefirsttermequalszero,itfollowsthat

Potentialenergycanbedenotedassquaretermsbiisapositivecoefficient.Similarlyitgetsthat7.4.2Applicationofenergyequipartitiontheorem(1)MonatomicmoleculegasAccordingtoenergyequipartitiontheoremthemeanenergyis

TheinternalenergyofmonatomicmoleculeoftheidealgasTheheatcapacityasconstantvolumewithTheheatcapacityasconstantpressurewith

(2)BiatomicmoleculegasThefirstterm:translationalenergy;M=m1+m2Thesecondterm:rotationalenergyencirclingcenterofmass,

I=μr2momentofinertia,Thethirdterm:relativemovementenergyoftwoatoms,relativemovementkineticenergy,

u(r)istheinteractionenergyoftwoatoms.ForrigiditybiatomicmoleculeTheinternalenergyandheatcapacityquantitiesfollowsas,,

gasTemperature(k)He2911.660931.673H22891.4071971.453921.597(3)ThesolidTheatomicvibrationinthesolidisconsideredasharmonicoscillationofindependenceeachother.TheenergyofonedegreeoffreedomisTheinternalenergyofthesolidis

U=3NkTTheheatcapacityasconstantvolumewithThisresultisagreementwiththeexperimentalresultofDulong-Petit.

§7.5TheInternalEnergyAndHeatCapacityoftheIdealGas7.5.1Thebasicexpressionofinternalenergyandheatcapacityet,ev

ander

denotetranslationalenergy,vibrationalenergyandrotationalenergyofbiatomicmoleculeidealgas.Thetotalpartitionfunctioncanbewrittenastheproductoftranslationalpartitionfunction,vibrationalpartitionfunction,rotationalpartitionfunction.

TheinternalenergyofbiatomicmoleculeidealgasisTheheatcapacityasconstantvolumewithThetranslationalpartitionfunctionhasbeengivenby

7.5.2Whyisthecontributionofvibrationdegreeoffreedomtoheatcapacitynearlyzerointhecaseofnormaltemperature.Therelativevibrationcanbeconsideredaslinearharmonicoscillation

vibrationalpartitionfunction

Basedonitgets

IntroducevibrationcharacteristictemperatureθvThe

formulas(7.5.8)and(7.5.9)followsas

θv~103K,normaltemperatureT<<θv,thereforeUvand

canbe

approximatelyTheformula(7.5.9’)indicatesthatthecontributionofvibrationdegreeoffreedomtoheatcapacityisnearlyzerointhecaseofnormaltemperature.Energylevelinterval,transitionenergyisverybig,oscillatorcannotbeexcitatedtohighenergylevelandfreezeingroundstate.

7.5.3Whyisnottheheatcapacityofhydrogenagreementwithexperiment?(1)Heteronuclear(CO,NO,HCl)RotationalenergylevelandrotationalpartitionfunctionareIntroducevibrationcharacteristictemperatureθr

Inthecaseofnormaltemperature,,canbeconsideredtobeacontinuousvariable.Thusintegralsubstitutescalculationsum.LetThereforeitgets

(2)ThequestionaboutH2Ortho-hydrogenstate:spinparallel,

oddnumberforl,probabilityis.Parahydrogenstate:spinreverseparallel,evennumberforl,probabilityis.denotetherotationpartitionfunctionsofOrtho-hydrogenandparahydrogenrespectively.H2

isat

thestate

ofhighl.SimilarlyitgetsBecausethemomentofinertiaIofhydrogenissmall,sothevibrationcharacteristictemperatureθrisbig.Inthecaseoflowtemperature(92K),energy

equipartitiontheoremisnotapplicable.7.5.4Whydoesnotthecontributionofelectrontotheheatcapacityofgasbetakenintoaccount.Thedifferencebetweenexcitationstateenergyandgroundstateenergyforaelectronis1~10eV,namely10-19~10-18J,correspondingtemperature104~105K.Itistoohightoexcitingaelectrontoexcitationstate.

7.5.5Calculationthermodynamicsquantitiesbyusingclassicalpartitionfunction.Theenergyofdifferentcorediatomicmoleculeis

WeobtainthatWiththeformulas(7.5.21)-(7.5.23),wehave

§7.6TheEntropyoftheIdealGas7.6.1TheentropyoftheidealgaswithFormonatomicmoleculeidealgaswehave

7.6.2ThechemicalpotentialofthemonatomicmoleculeidealgasAccordingtotheformulas(7.1.16’)and(7.6.4),itgetsFortheidealgas,μ<0

§7.7TheEinsteinTheoryofSolidHeatCapacity3Noscillators:oscillatorenergylevelisIntroducevibrationcharacteristictemperatureθEDotsdenoteexperimentalresult;SolidlinedenotesEinsteintheoryresult.θE=1320KDiscussion:(1)WhenT>>θE,

CV=3Nk(7.7.7)Thisformula

isagreementwithenergyequipartitiontheorem.TheeffectofquantumisneglectedandTheclassicalstatisticsisapplied.(2)WhenT<<θE,Thedifferencebetweenexcitationstateenergyandgroundstateenergyismuchbigsothat3Noscillatorsareingroundstate.

§7.8ParamagnetismSolid

Anespeciallyinterestingapplicationofclassicalstatistics(Boltzmannstatistics)istheparamagneticbehaviorofsubstances.Ifahomogeneousfieldpointsinz-direction,thetotalangularmomentumofamagneticionis1/2.ThemagneticmomentumisTwopossibleenergyare–μBandμB.Thepartitionfunction

ofthissystemis

GeneralizedforceperformedbyenvironmentMagnetizationMis

Discussion:HighTorweakfieldTherelation(7.8.4)isknownasCurie’slaw.(2)LowTorstrongfieldM=Nμ(7.8.5)

TheinternalenergyofthissystemisThisispotentialenergyinoutsidefield.TheentropyofthissystemisDiscussion:HighTorweakfield

ThenThemicrostatenumberis(2)LowTorstrongfieldThenThemicrostatenumberis1,namelyallmagneticmomentumpointsinthedirectionofH.§7.9ThestateofNegativeTemperature

IfSdecreaseswithincreasingU,

thusTis

negative.Theexampleofaparamagneticsystemwithj=1/2(two-levelsystem,nuclearspinsystem)allowsustodiscussapossibleextensionofthenotionoftemperature.EachoftheNparticlesofthesystemshallbeabletoassumetwopossibleenergies,.Letthenumberofnuclearmagneticmomentuminthelevel+εbeN+,andthatin-εbeN-.Ofcoursewehave

N=N++N-(7.9.2)ThetotalenergyofsystemisEquations(7.9.2)and(7.9.3)canbesolvedforN+andN-,Equation(7.9.4)immediatelyallowsforthecalculationoftheentropyofthesystem.Wheretheformulalnm!=m(lnm-1)isusedforN+,N->>1.

Equation(7.9.6)yieldsforthetemperatureDiscussion:(1)AslongasE<0,wehaveT>0,asusual.(2)WhenE>0,wehaveT<0.(3)WhenE=-Nε,allmagneticmomentumspointthedirectionofB.Ω=1,S=0.

TheEandSincreasewiththeincreasingofT.WhenN+