[物理化学英语课件]统计热力学Elementary_statistical_thermodynamics.ppt

[物理化学英语课件]统计热力学Elementary_statistical_thermodynamics.ppt

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Chapter8Elementarystatisticalthermodynamics8.1IntroductionStatisticalthermodynamics,orstatisticalmechanics,isthestudyofthemicroscopicbehaviorsofthermodynamicsystemsusingstatisticalmethodsandprobabilitytheory.TheessentialprobleminstatisticalthermodynamicsistodeterminethedistributionofagivenamountofenergyEoverNparticlesinasystem.Themacroscopicproperties,suchasthermodynamicenergy,heatcapacity,etc.,canbecalculatedintermsofpartitionfunctions.Statisticalthermodynamicsisabridgeofconnectingbetweenmacroscopicandmicroscopicpropertiesofasystem.DefinitionofstatisticalthermodynamicsTherearetwokindsofsystemsInteractingsystem（相倚子系统）Non-interactingsystem（独立子系统，forinstance,idealgas）Onlythelatterwillbeintroducedinthischapter.TwokindsofparticlesIdenticalparticles,orindistinguishableparticles（suchasgaseousmolecules）,isalsocallednon-localizedparticles.Distinguishableparticles（Suchastheatomsincrystal）,isalsocalledlocalizedparticles.8.2Energylevelanditsdegeneracy012345Energylevelsaresaidtobedegenerate,ifthesameenergylevelisobtainedbymorethanonequantummechanicalstate.Theyarethencalleddegenerateenergylevels.Thenumberofquantumstatesatthesameenergyleveliscalledthedegreeofdegeneracy.Amolecularenergystateisthesumofanelectronic（e）,nuclear（n）,vibrational（v）,rotational（r）andtranslational（t）component,suchthat:

ThedegreeoffreedomofmovementTranslation:

x,y,zF=3RotationForlinearmolecules,F=2Fornon-linearmolecules,F=3VibrationApolyatomicmoleculecontainingnatomshas3ndegreesoffreedomtotally.Threeofthesedegreesoffreedomcanbeassignedtotranslationalmotionofthecenterofmass,twoorthreetorotationalmotion.3n-5foralinearmolecule;3n-6foranonlinearmoleculeCO2has33-5=4degreesoffreedomofvibration;nonlinearmoleculeofH2Ohas33-6=3degreesoffreedomofvibration.8.2.1TranslationalparticleTheexpressionfortheallowedtranslationalenergylevelsofaparticleofmassmconfinedwithina3-dimensionalboxwithsidesoflengtha,b,cisWhereh6.62610-34Js，nx,ny,nzareintegralscalledquantumnumbers.Thenumberofthemis1,2,.Ifa=b=c,equationbecomesallenergylevelsexceptgroundenergylevelaredegenerate.ExampleAt300K,101.325kPa,1molofH2wasaddedintoacubicbox.Calculatetheenergylevelt,0atgroundstate,andtheenergydifferencebetweenthefirstexcitedstateandgroundstate.SolutionTaketheH2attheconditionasanidealgas,thenthevolumeofitisThemassofhydrogenmoleculeistheenergydifferenceissosmallthatthetranslationalparticlesareexcitedeasilytopopulateondifferentexcitedstates,andthattheenergychangesofdifferentenergylevelscanbethinkofasacontinuouschangeapproximately.8.2.2Rigidrotator（diatomic）Theequationforrotationalenergylevelofdiatomicmoleculesis：

whereJisrotationalquantumnumber,Iisthemomentofinertia（转动惯量）isthereducedmass（折合质量）,Thedegreeofdegeneracyis8.2.3One-dimensionalharmonicoscillatorWherevquantumnumber，whenv=0，theenergyiscalledzeropointenergy.Onedimensionalharmonicvibrationisnon-degenerate.8.2.4ElectronandatomicnucleusThedifferencesbetweenenergylevelsofelectronmotionandnucleusmotionarebigenoughtokeeptheelectronsandnucleistayattheirgroundstates.Bothdegreeofdegeneracy,ge,0,forelectronmotionatgroundstateanddegreeofdegeneracy,gn,0,fornucleusmotionatgroundstatearedifferentfordifferentsubstances,buttheyareconstantforagivensubstance.8.3Distributionandmicrostate8.3.1DistributionofenergylevelsWecalltheoccupationnumbernithenumberofdistributioninenergyleveli.Forexample,adistributionof6identicalparticlesamong9unitsofenergymustsatisfywiththeconditionsThetotalnumberofwaysofdistributionis26.8.3.2DistributionofstatesThenumberofparticlesoccupiedinamicroscopicquantumstateiscalledthenumberofdistributionofstates.OnedistributionDofenergylevelshasacertainnumberofmicrostatesWD,thesumofallWDisthetotalnumberofmicrostatesofasystem.Thatis.Thereare16waysofdistributionoffourdistinguishableparticlesintwoidenticalboxes.8.3.3DistinguishableparticlesConsiderNdistinguishableorlocalizedparticlesdistributeintoNnondegenerateenergylevels.Nowconsideranotherkindofdistributionthatthenumbersofparticlesoccupiedindifferentenergylevelsaredenotedasn1,n2,ni.Alltheenergylevelsarestillnondegenerate.threedifferentdistributionsofsixparticles.Theexchangesofparticlesinthesameenergyleveldonotcreatenewmicrostatebecauseeveryenergylevelhasonlyonequantumstate.ThenumbersofmicrostatesforthreedistributionsareWenowconsiderthatthedegreeofdegeneracyofenergylevelsisg1,g2,gi.Supposenumberofquantumstatesisunconstrained.Considerniparticlesoccupyenergyleveli,everyparticlecanchoseonefromallquantumstatesintheenergylevel.HencethewaysofselectionforniparticlesareForallenergylevels,thenumberofmicrostatescausedbythedegeneracyoflevelsisthenumberofmicrostatesforacertaindistributionDcanbewrittenas8.3.4Identicalparticlesassumethatthereisnorestrictiononthenumberofparticleswhichcanoccupyagivenenergylevelandthatenergylevelsisnondegenerate.thereisonlyonewayforniparticlestooccupytheenergyleveli.Therefore,thenumberofmicrostatesofadistributionDforasystemisWD=1.Ifenergylevelisdegenerate,Itiseasytoseethatthereis（2+1）waysofdistributing2particlesintwoquantumstateswhichcanbewrittenasSuppose8identicalparticlespopulatein4quantumstatesinanenergylevel.Thisisequivalenttothepermutationofthesumof8personsand（4-1）dividingwalls,bothpersonsanddividingwallsareindistinguishable.thenumberofmicrostatesforniparticlesdistributingingiquantumstatesinanenergylevelisonekindofdistributionistheproductsofthenumberofmicrostatesforeverylevelmultipliedbyoneanother.ThatisIfnigi,thisequationcanbesimplifiedintoComparethisequationwithequation,wecanseethatunderthesameconditionsofN,ni,andgi,thenumberofmicrostatesofadistinguishable-particlesystemisN!

timesthatofanidentical-particlesystem.8.4Themostprobabledistribution,equilibrium8.4Themostprobabledistribution,equilibriumdistribution,andBoltzmanndistributiondistribution,andBoltzmanndistribution8.4.1TheprincipleofequalaprioriprobabilitiesStatisticalthermodynamicsisbasedonthefundamentalassumptionthatallpossibleconfigurationsofagivensystem,whichsatisfythegivenboundaryconditionssuchastemperature,volumeandnumberofparticles,areequallylikelytooccur.ExampleConsidertheorientationsofthreeunconstrainedanddistinguishablespin-1/2particles.Whatistheprobabilitythattwoarespinupandonespindownatanyinstant?

SolutionOftheeightpossiblespinconfigurationsforthesystem,Thesecond,third,andfourthcomprisethesubsettwoupandonedown.Therefore,theprobabilityforthisparticularconfigurationisP=3/88.4.2ThemostprobabledistributionTheprobabilityfordistributionDisthemicrostatesofthreeharmonicoscillatorswhicharedistinguishableparticleswithtotalvibrationalenergyof=WI+WII+WIII=3+6+1=10Whichdistributionisthemostprobabledistribution?

WDiscalledthermodynamicprobabilityofdistributionD8.4.3EquilibriumdistributionInasystemwithlargenumberofN,themostprobabledistributionmayrepresentalldistributions.StirlingsapproximationamoreaccurateformConsiderasystemconsistingofNlocalizedparticleswhichdistributeovertwodegeneratequantumstates,AandB.MdenotesforthenumberofparticlesinstateAand（N-M）instateB.thenumberofmicrostatesforthisdistributioncanbeexpressedasWhenM=N/2,WDhasamaximumvalue.（）（）!

2/!

2/!

NNNWB=Everyparticlehastwopossibilitiestopopulateonthequantumstates,stateAorstateB.Thetotalnumberofmicrostatesforthesystemwouldbe2N.TheprobabilityforthemostprobabledistributionisWhenthenumberofparticlesinasystemisabout1024,theprobabilityisthenWeconsideranotherdistributionthathasadistributionnumberdeviatingmfromN/2,itsprobabilitywouldbeWhenmN,intermsofStirlingsapproximationthisequationcanbeconvertedintoTheprobabilityforalldistributionsrangingfromisthesummationoftheirprobabilities.Byusingerrorfunctionweobtainthedistributionatequilibriumismostcertainlygoingtobethemostprobabledistribution,orattheveryleast,withthesekindofnumbers,somethingveryclosetoit.8.4.4BoltzmanndistributionForalargenumberofnoninteractingparticles=1.3810-23JK-1,Boltzmannconstant.isproportionalcoefficient.ThepopulationcanalsobeexpressedintheformofenergyleveldistributionThetotalnumberisthenDefinetheparticlepartitionfunctionthenThedistributionthatobeystheseequationsiscalledtheBoltzmanndistribution.TheequationsarealsoknownasBoltzmanndistributionlaw.foranytwolevels：

Theratiotototalnumber：

Boltzmanndistributionisthemostprobabledistribution.ThemaximumvaluecanbederivedbyusingLagrangesmethodofundeterminedmultipliers8.5Computationsofthepartitionfunction8.5.1Somefeaturesofpartitionfunctions

（1）atT=0,thepartitionfunctionisequaltothedegeneracyofthegroundstate.

（2）WhenTissohighthatforeachtermi/kT=0,（3）factorizationpropertyIftheenergyisasumofthosefromindependentmodesofmotion,thenThepartitionfunctionsfor5modemotionsareexpressedas8.5.2Zero-pointenergyzero-pointenergyistheenergyatgroundstateortheenergyasthetemperatureisloweredtoabsolutezero.Supposesomeenergylevelofgroun