统计热力学基础

统计热力学基础

化学化工学院尹世伟2015-7-14Elementsofstatisticalthermodynamics

LeonardKNash

物理化学上册第七章

傅献彩

热力学统计物理

汪志诚OutlineStatisticalViewpointThepartitionfunctionEvaluationofpartitionfunctionApplicationStatisticalviewpointTwoimportantconvictionsatnineteencentury:Energyormass-energyremainconstantEntropyincreaseduringthespontaneousenergy+entropyThermodynamicsWhatcanhappen?YESWhyofthosehappenings?NOTThermodynamicparametersGUHAStatisticalviewpointThermodynamics:N2+H2+NH3

PositionofequilibriumbymacroscopicexperimentalmeasurementssuchasGYES

Whythatistheequilibriumcondition.NOTHowNH3determinesthemagnitudeofthefree-energyofthatcompound?YESStatisticalmechanicsThermodynamicSystemAssemblyofunitsStatisticalmechanicsProvideatoolstocalculatethethermodynamicparameters(UHSA…)basedonsomeparametersdescribingsuchconstituentunitsasatomsandmolecules.Whatarethosesomeparameters?Inboundedsystemformicroscopicunits,theenergiesare“quantized”.Theenergiesofamacroscopicsystemformavirtualcontinuumofpossibilities,whichassociatedwithintegralvaluesofsome“quantumnumber”.QuantizationexamplesElectronenergiesofHydrogen

atom

:Thepossiblestatesofhydrogenatomarerelatedwithquantumnumbern.Thetotalenergiesofsystem(manyhydrogenatoms)areintegralofquantumnumber.Unitsofmolecules,besidesquantizedelectronicenergies,alsohavevariousvibrationalmotions.Underharmonicoscillationapproximation,theonlypermissiblevibrationalenergyaregivenbytheequations:vvibrationalquantumnumber,characteristicfrequencyPossiblevibrationalstatesisshowedbydistinctiveintegralvalueofvibrationalquantumnumber

QuantizationexamplesRotationalandtranslationalmotionsbothatomandmoleculesarerelatedtosetsofdiscretequantumstates.Itseemscompletelyhopelesspretensionwhenviewingamacroscopicthermodynamicsystemasanassemblyofmyraidsubmicroscopicentitles(1023)inmyraidever-changingquantumstates.Actually,justbecauseofenormousnumberinvolved,theproblemprovesunexpectedlytractablewhenwegiveitastatisticalformation!MicrostatesandconfigurationsSystemswithidenticalunitsandquantumstateswithanevenlyspacedenergies,suchasone-demensionalharmonicoscillatorsfixedpositionsincrystallattice.MicrostatesandconfigurationsThreelocalizedoscillatorssharethreequantaofenergyMicrostatesandconfigurationsMicrostatesandconfigurationsThenumber(W)ofmicrostatesassociatedwithanyconfigurationsinvolvingNdistinguishableunits:a

meansthenumbersofunitsassignedthesamenumberofenergyquantaaCondition:Anyspeciesofdistinguishableunitwithanyenergyspacingbetweentheirquantumlevels.WhenNisveryhugenumber>10000(SterlingequationP460physicalchemistry)Fivelocalizedoscillatorssharefivequantaofenergy(5-5assembly)MicrostatesandconfigurationsMicrostatesandconfigurations10localizedoscillatorssharefivequantaofenergyMicrostatesandconfigurations10localizedoscillatorsshare5quantaofenergyMicrostatesandconfigurations10-5assemblyhistgraphMicrostatesandconfigurationsResults:1.Thenumberofmicrostates(Wtot)skyrocketstounimaginablemagnitudesasthenumberifunitsincreasesfurther.(微观状态数随着粒子数的增加火箭式的增加)Forexamples,1000localizedharmonicoscillatorssharing1000energyquantapossessesmorethan10600differentmicrostatesGalaxyatomssmallerthan1070,universityatomssmallerthan106002.Theemergenceofapredominantconfigurationischaracteristicofanyassemblywithalargenumber(N)ofunits.（粒子数大到一定程度时，总会有一种“最可几”微观状态出现）predominantconfigurationTossingawell-balancedcoinintwoways:head(H)ortail(T)Toss4times.Thepossibleresultsare:IHHHHHHTTHTTTIIHHHTHTHTIVTHTTHHTHIIITHHTTTHTHTHHHTTHTTTHTHHHTHTHTTHHVTTTTAIAIIAIIIAIVAV0.1660.6661.0000.6660.166PredominantConfigurationThelargerN,thesmallergroupofconfigurationscenteredonthepredominantconfiguration.Homework1:makeanA-ratiosandWx-XplotwhenN=10,100,1000,10000withexcelororigin.A-ratiopredominantconfiguration1000-1000assembly1000harmonicoscillatorssharing1000quantanumberWtot=10600Theareaunderthiscurveisoftheorderof10600,buttheimportantthingisthatalmost100percentofthisareafallsunderacentralpeak–thesharpnessofwhichbecomesevermoreextremewithassembliesoflargerandlargernumbersofunits.predominantconfigurationpredominantconfigurationlocatedat:ddenoteachangefromthepredominantconfigurationtoanotherconfigurationonly“infinitesimally”differentfromit.Whenthenumberofunitsinassembly(N)ishugenumberandcomparablewithNA,dandWareregardsascontinuousvariableandfunction.DifferentialcalculusandmaximumvalueconditionsTheBoltzmannDistributionlawAnisolatedmacroscopicassemblyofNharmonicoscillatorssharelargenumber(Q)energyquantaAssemblyrequirement:identicalbutdistinguishablebyspatiallocationCondition:NandQareconstant.Whichofenormousnumberofconfigurationsassociatedmicrostates(W)canberealizeitsmaximumvalue?Thenumberofmicrostates(W)associatedwithconfigurationa-a,b-b,l-l…z-zTheBoltzmannDistributionlawLetusmakeaminimumchangeintheinitialconfiguration.Thenewconfiguration:TheBoltzmannDistributionlawIfinitialConfiguration(W)havemaximumvalue,hereNandQarehugenumber,then:Therefore,infinitesimalchangeresultsinW=W’NandQishugenumberfarlargerthan1TheBoltzmannDistributionlawConditions:Anisolatedmacroscopicassemblyofharmonicoscillatorswithuniformenergyspacingbetweentheirquantumstates.Descriptionofthepredominantconfiguration:Whataboutquantums