9
EP271 Kinetic Theory Notes M.P. Bradley Kinetic Theory of Gases Supplementary Lecture Notes for EP271 Michael Patrick Bradley (Please note that the calculation of pressure in terms of atomic/molecular mean‐ square velocity closely follows the treatment given by I.N. Levine in “Physical Chemistry” 3 rd . ed. The discussion of molecular energy distributions follows Levine and also the notes of Prof. Adam Bourassa) Statistical Mechanics We have learned from our textbook about entropy via Clausius’ definition of the minimum entropy change dS associated with an amount of heat δQ absorbed by a system at temperature T: dS = δQ T This is a definition of entropy in terms of macroscopically measurable heat transfer and temperature, which “tames” the process‐dependent heat transfer δQ through the use of an “integrating factor”, which turns out to be just 1/ T , where T is our previously defined “thermodynamic temperature”, normally measured in Kelvin. While this definition is perfectly serviceable and useful, and enables us to put Carnot’s results for heat engine efficiency on a nice theoretical basis, it does not give insight into the physical meaning of the entropy S. For that, we have to go to the atomic picture of matter. The statistical behavior of large collections of atoms or molecules gives a way of calculating the entropy and related quantities. This approach to understanding thermodynamics from the bottom up is referred to as “statistical mechanics”. Gases A gas is a non‐condensed (i.e. “volume‐filling”) collection of atoms or molecules (particles). Except for collisions between particles, they don’t really interact with one another. We have talked extensively about gases in this course but up till now we have not really considered their microscopic behaviour in detail. However since we know that gases are composed on many small freely moving atoms and/or molecules, we should be able to use this information to help us get a better understanding of gas behaviour. So, what is a gas ? A gas is a collection of N atoms or molecules in a non‐ condensed (i.e. volume‐filling) state where they are moving relatively quickly (typical speeds in excess of 300 m/s at room temperature), but in random directions. The gas particles (i.e. atoms or molecules) are typically separated by distances which are much larger than the particle size. Particles in a gas are typically not strongly interacting and are moving randomly in all directions.

Kinetic Theory of Gases - WordPress.com · 2015. 4. 15. · EP271 Kinetic Theory Notes M.P. Bradley Kinetic Theory of Gases Supplementary Lecture Notes for EP271 Michael Patrick Bradley

  • Upload
    others

  • View
    6

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Kinetic Theory of Gases - WordPress.com · 2015. 4. 15. · EP271 Kinetic Theory Notes M.P. Bradley Kinetic Theory of Gases Supplementary Lecture Notes for EP271 Michael Patrick Bradley

EP271KineticTheoryNotesM.P.Bradley

KineticTheoryofGasesSupplementaryLectureNotesforEP271MichaelPatrickBradley(Pleasenotethatthecalculationofpressureintermsofatomic/molecularmean‐squarevelocitycloselyfollowsthetreatmentgivenbyI.N.Levinein“PhysicalChemistry”3rd.ed.ThediscussionofmolecularenergydistributionsfollowsLevineandalsothenotesofProf.AdamBourassa)StatisticalMechanicsWehavelearnedfromourtextbookaboutentropyviaClausius’definitionoftheminimumentropychangedSassociatedwithanamountofheat

δQabsorbedbya

systemattemperatureT:

dS =δQT

Thisisadefinitionofentropyintermsofmacroscopicallymeasurableheattransferandtemperature,which“tames”theprocess‐dependentheattransfer

δQthroughtheuseofan“integratingfactor”,whichturnsouttobejust

1/T ,whereTisourpreviouslydefined“thermodynamictemperature”,normallymeasuredinKelvin.Whilethisdefinitionisperfectlyserviceableanduseful,andenablesustoputCarnot’sresultsforheatengineefficiencyonanicetheoreticalbasis,itdoesnotgiveinsightintothephysicalmeaningoftheentropyS.Forthat,wehavetogototheatomicpictureofmatter.Thestatisticalbehavioroflargecollectionsofatomsormoleculesgivesawayofcalculatingtheentropyandrelatedquantities.Thisapproachtounderstandingthermodynamicsfromthebottomupisreferredtoas“statisticalmechanics”.GasesAgasisanon‐condensed(i.e.“volume‐filling”)collectionofatomsormolecules(particles).Exceptforcollisionsbetweenparticles,theydon’treallyinteractwithoneanother.Wehavetalkedextensivelyaboutgasesinthiscoursebutuptillnowwehavenotreallyconsideredtheirmicroscopicbehaviourindetail.Howeversinceweknowthatgasesarecomposedonmanysmallfreelymovingatomsand/ormolecules,weshouldbeabletousethisinformationtohelpusgetabetterunderstandingofgasbehaviour.So,whatisagas?AgasisacollectionofNatomsormoleculesinanon‐condensed(i.e.volume‐filling)statewheretheyaremovingrelativelyquickly(typicalspeedsinexcessof300m/satroomtemperature),butinrandomdirections.Thegasparticles(i.e.atomsormolecules)aretypicallyseparatedbydistanceswhicharemuchlargerthantheparticlesize.Particlesinagasaretypicallynotstronglyinteractingandaremovingrandomlyinalldirections.

Page 2: Kinetic Theory of Gases - WordPress.com · 2015. 4. 15. · EP271 Kinetic Theory Notes M.P. Bradley Kinetic Theory of Gases Supplementary Lecture Notes for EP271 Michael Patrick Bradley

EP271KineticTheoryNotesM.P.Bradley

Afewbasicphysicspoints:Eachgasmoleculehasaposition(x,y,z)andavectorvelocitywithcomponents(vx,vy,vz)Eachgasmolecule’spositionevolves“ballistically”accordingtothesimplekinematicequations:

x(t)=x0+vxty(t)=y0+vytz(t)=z0+vzt

(Notethattheinitialposition(x0,y0,z0)oftheatom/moleculeattimet=0israndom.)Becausetherearenosignificantforcesbetweengasmolecules,normallytheatomic/molecularvelocitiesremainconstantuntiltheatomormoleculescollideswithawallofthechamberoranotheratomormolecule.EquipartitionofEnergy Toquantifythekineticenergyofanatomormoleculeweneedaveryimportantresultfromstatisticalmechanics/kinetictheoryofgasescalledthe“equipartitiontheorem”orthe“equipartitionofenergy”.Theequipartitiontheoremstatesthateach“degreeoffreedom”ofanatomormoleculehasanassociatedmeanenergyof

12 kBT wherekB=1.3806503x10‐23J/KisBoltzmann’sconstant.

NowBoltzmann’sconstantisrelatedtoouroldfriendthemolargasconstant

R =8.314J/mol.Kviatherelation

R = NA kB whereNA=6.0221415x1023particles/mole.Sonowwehaveadirect,quantitativeconnectionbetweentemperatureTandmeanmolecularenergy—thisisaveryimportantresult.DegreesofFreedom(DOF)So—whatisamolecular“degreeoffreedom”?Wellinformally,amoleculardegreeoffreedom(DOF)isanyofthe“onedimensional”waysamoleculecanstoreenergy,e.g.becauseofavelocitycomponentsvx,vy,vzORarotationalmotionaboutthex,y,orzaxes,orinvibrationalmodesassociatedwithchemicalbonds.Formally,eachDOFcorrespondstoaquadraticterminthetotalatomicormolecularenergyexpression(theatomicormolecular“Hamiltonian”).Forasimplediatomicmoleculethisis

E = 12mCM vx

2 + vy2 + vz

2( ) + 12 Ixxωx

2 + 12 Iyyωy

2

Page 3: Kinetic Theory of Gases - WordPress.com · 2015. 4. 15. · EP271 Kinetic Theory Notes M.P. Bradley Kinetic Theory of Gases Supplementary Lecture Notes for EP271 Michael Patrick Bradley

EP271KineticTheoryNotesM.P.Bradley

wherewehaveignoredvibrationalmotionbecausethisnormallynotexcitedexceptathightemperatures.Sincethekineticenergyofanatomormoleculeisdueto3velocitycomponentswehave3degreesoffreedomassociatedwiththis,whichgivesusthefollowingresult:

εtr = 12mv

2 = 12m v 2 =

32kBT

RelationbetweenAverageAtomic/MolecularKineticEnergy&GasPressure:Let’simagineourgasatomsormoleculestobeconfinedinsidearectangularboxofdimensions(lx,ly,lz).Supposewelookatonewallofthebox(sayoneofthelxxlzwalls).Whatpressureisexertedonthatwall?Well,pressureisthemacroscopicmanifestationofthefactthatthewallisbeingbombardedbillionsandbillionsoftimespersecondbyfast‐movinggasparticles.Eachtimeagasparticlecollideswiththewallitchangesthey‐componentofitsmomentumfrompyto–py.Iftheparticle’smomentumbeforethecollisionwaspy=mvy,thenafterthecollisionitwillbe–mvy.

NowthegeneralformofNewton’s2ndlawis

F = d p

dt,i.e.forceisrelatedtothetime

rateofchangeofmomentum.Nowinthecaseofourgasmolecule,whenitcollideswiththewallandreversesitsy‐componentofmomentum,itdoessobecauseduringthedurationtimeofthecollisionΔt(whichmaybeveryshort,lessthanananosecond)thereisanaverageforceFyexertedonitbythewall.Sinceaction=reaction,byNewton’s3rdlaw,duringthecollisionthemoleculeexertsthesameforcebackonthewall,butactingintheoppositedirection.So,wecanfindthetotalforceonthewallbyaddinguptheforcesexertedbyallthemoleculeswhentheycollidewithit.The“translational”kineticenergyofagasatomormolecule(i.e.thepartassociatedwiththemolecule’smotion(=translation)isgivenby:

εtr = 12mv

2 = 12m(vx

2 + vy2 + vz

2)

Fy,i = may,i = mdpy,idt

=ddt(mvy,i)

2mvy,i =< Fy,i > Δt

Nowthecleverobservationinvolvedinthisderivationistorealizethattheaveragetimebetweencollisionscanberelatedtotheboxsizeasfollows:

Page 4: Kinetic Theory of Gases - WordPress.com · 2015. 4. 15. · EP271 Kinetic Theory Notes M.P. Bradley Kinetic Theory of Gases Supplementary Lecture Notes for EP271 Michael Patrick Bradley

EP271KineticTheoryNotesM.P.Bradley

Δt =2lyvyi.Thisthenenablesustowritetheaverageforceas

< Fy >= < Fy,i >i=1

N

∑ =mvy,i

2

ly=mly

vy,i2

i=1

N

∑i=1

N

Nowtheatoms/moleculesofthegasDONOTallhavethesamespeed,soallthevyiaredifferent.However,wecancalculateandtalkaboutanaverageormeanvalueofthesquareofthemolecularspeed

< vy2 >≡

1N

vy,i2

i=1

N

Withthisdefinitionthey‐componentoftheforceonthewallweselectedis

< Fy >=mNly

< vy2 > Now,thepressureexertedonthewallisjusttheforceonthe

walldividedbyitscross‐sectionalarea(recallthat1Pascal=1Newton/m2)sowe

gettheresultthat

P =mN < vy

2 >

lxlylz=mN < vy

2 >

V

Nowthereisnothingspecialaboutthey‐direction,wejusthadtopicksomethingtogetstarted,soonaverageitmustbetruethat

<vx2>=< vy

2 >=< vz2 >

Nowsince

< v 2 >=< vx2 + vy

2 + vz2 >=

1N

(vx,i2 + vy,i

2 + vz,i2 )

i=1

N

∑ =1N

vx,i2

i=1

N

∑ +1N

vy,i2

1

N

∑ +1N

vz,i2

1

N

Wehavetheresultthat

< v 2 >=< vx2 > + < vy

2 > + < vz2 >

Since

< vx2 >=< vy

2 >=< vz2 > wecanwrite

< vx2 >=< vy

2 >=< vz2 >=

13

< v 2 >

Sowegetfinallytheresultthat

P =mN < v 2 >

3Vforanidealgas(i.e.oneinwhichthe

moleculestravelballisticallybetweencollisionsandtherearenointermolecularforces).Wecangoabitfurtherhere.Theaveragetranslationalkineticenergyofagasatom

ormoleculeis

< εtr >= 12m < v 2 >⇒< v 2 >=

2 < εtr >m

Thereforewecanre‐writeour

expressionforthepressureofaclassicalidealgasintermsofthemeanatomicormolecularkineticenergy:

PV =23N < εtr >

Thisisourfirstreallyimportantkinetictheoryresult.YoucanseehowwehavebeenabletorelateamacroscopicallymeasurableparameterlikepressuretotheaveragebehaviourofalargenumberNofatomsormolecules.

Page 5: Kinetic Theory of Gases - WordPress.com · 2015. 4. 15. · EP271 Kinetic Theory Notes M.P. Bradley Kinetic Theory of Gases Supplementary Lecture Notes for EP271 Michael Patrick Bradley

EP271KineticTheoryNotesM.P.Bradley

KINETICTHEORYANDDISTRIBUTIONS OK,upuntilnowwehavejustsampledkinetictheoryinthemostbasicway:weacceptthatagasiscomposedonalargenumberNoratomsofmolecules,andwecouldevencalculateanicerelationshipbetweenpressureandthemean‐squarevelocityofthosemolecules(whichinturnwasrelatedtotheirmeankineticenergy):

P =mN < v 2 >

3V butsofarthatwasaboutasfaraswewereabletogo.Now,wecangofarther.TheMaxwell(orMaxwellBoltzmannMB)distribution:Sinceweknowthattheatomsormoleculesinagasdonotallhavethesame

velocities,weneedawayfordescribingthis.TheMaxwellBoltzmanndistributiongivestheanswer.TheprobabilityPthatagivenatomormoleculehasanxcomponentofvelocitybetweenvxandvx+dvxisgivenbyP=f(vx)dvx,where

f (vx ) =m

2πkBT

1/ 2

e−mvx

2

2kBT

Noticethatthishastheformofa“Gaussian”curveordistribution

P ~ e−vx2 / 2σ 2

wherethe“standarddeviation”

σ 2 = kBT /m SincethedefiniteintegralofaGaussianovertheentiredomainisgivenby

e−x 2

2σ 2 dx−∞

∫ = 2πσ ,wecanseethatthevelocitydistributionisNormalized,i.e.

thedefiniteintegralfrom

vx = −∞ to

vx = +∞ isequalto1:

f (vx )dvx =1−∞

∫ Thisisimportantbecausef(vx)isaprobabilitydistribution.Probabilitiesare

measuredfrom0to1,with0meaningnochancewhatsoever,and1meaningcompletecertainty.Sinceeverygasmoleculehassomevalueofthexcomponentofitsvelocitylyingbetween‐∞and+∞,theintegraloftheprobabilitydistributionovertheentirerangeofvelocitiesmustaddto1,andthenormalizationconditionguaranteesthat.Thereisnothingspecialaboutthex‐componentofvelocity(itisjustalabelwe

pickedforconvenience)sothesamedistributionappliestovyandvzMaxwellSpeedDistribution:Thevelocityofagivenmoleculeisavectorgivenbyitscomponents(vx,vy,vz),

eachofwhichisdistributedaccordingtotheabovedistributionlaw.Butinmanycasewearemoreinterestedinthespeedvofthemolecule,whichisgivenby

Page 6: Kinetic Theory of Gases - WordPress.com · 2015. 4. 15. · EP271 Kinetic Theory Notes M.P. Bradley Kinetic Theory of Gases Supplementary Lecture Notes for EP271 Michael Patrick Bradley

EP271KineticTheoryNotesM.P.Bradley

v = vx2 + vy

2 + vz2 .Thespeedisparticularimportantbecausethekineticenergyis

givenby

12mv

2 Nowitisimportanttorealizethattheprobabilitydistributionformolecularspeed

isdifferentfromtheprobabilitydistributionformolecularvelocitycomponents,eventhoughthespeedisafunctionofthosesamevelocitycomponents.Whyisthis?Well,theanswerissimple,onceyoulookattheproblemcorrectly.Basicallytheimportantthingisthattherearemanycombinationsofvelocitycomponents(vx,vy,vz)whichwillgivethesamefinalvalueofthespeed

v = vx2 + vy

2 + vz2 .

TheMaxwellBoltzmanndistributionformolecularspeedsisgivenby

g(v) =m

2πkBT

3 / 2

4πv 2e−mv 2

2kBT .Noticethatishassomesimilaritiestothevelocity

componentdistributionsbutitisnotthesame!ApplicationsoftheMaxwellDistributions:Calculationoftheaverage/meanmolecularspeed:

v = v = vg(v)dv−∞

∫ .Whydoesthisformulawork?Well,ifwethinkaboutitsimply,werememberthattheprobabilityofgettingspeedvisgivenby

P(v) ~ g(v)dv .Thentheweightedaveragespeed(ormathematicalexpectation)isgivenbythesum

vP(v)∑ (whereIhavebeenratherrelaxedaboutthediscretization,butthebasicideaiscorrect).SinceinthelimitofverymanyparticlesNwecantakethesumoverintoanintegral,wethengettheformulaquotedabove.Nowletusapplyit:

v = 4π m2πkBT

3 / 2

e−mv2 / 2kBTv 3dv

0

Noticethattheintegralextendsfrom0to∞withnonegativevaluesinthe

integrationdomainbecausethespeedisapositivedefinitequantity.Wecanevaluatethisintegralusingastandardresult:

x 2n+1e−ax2

dx =n!2an+10

∫ whichisvalidfora>0andn=0,1,2,3….

ApplyingthisstandardGaussianintegraltoourcalculationof<v>weget(usingx

v,a=m/2kBT,n=1)

Page 7: Kinetic Theory of Gases - WordPress.com · 2015. 4. 15. · EP271 Kinetic Theory Notes M.P. Bradley Kinetic Theory of Gases Supplementary Lecture Notes for EP271 Michael Patrick Bradley

EP271KineticTheoryNotesM.P.Bradley

v = 4π m2πkBT

3 / 21!

2 m2kBT( )

2

= 4π m2πkBT

3 / 24kB

2T 2

2m2

=2 2π

kBTm

1/ 2

Thusweendupfinallywith

v =8πkBTm

Wecanseethattheaveragemolecularspeedinagasincreasesasthe

temperatureTgoesup(as

T1/ 2 ,anddecreasesforheavieratomsormoleculesas

m−1/ 2).Thisseemsintuitivelyreasonable.Mean‐SquareSpeed:Wecouldcalculatethemeanofthesquareofthespeed(normallycalledthe

“meansquarespeed”)inthesamemannerasabove:

v 2 = v 2g(v)dv0

∫ .Asbeforethisisjustaweightedaverageormathematicalexpectationvalueforthesquaredspeedv2,wheretheprobabilityistheweightfactor.Thiscalculationisnottoodifficultbutitisalittlebitinvolved(infactIaskedyoutocarryoutasoneofthehomeworkproblems).Howeverthereisanotherwaytoapproachthisproblemwhichismuchsimpler,usingourearlierresultontheEquipartitionofEnergytheorem.Recalltheequipartitionofenergytheorem:foreverymolecularoratomicdegree

offreedom(basicallycorrespondingtoeveryquadratictermintheenergy),thereisonaverage1/2kBTofassociatedenergy.Fortranslationalkineticenergyofanatomormolecule,thekineticenergyis

εtr = 12mv

2 = 12mvx

2 + 12mvy

2 + 12mvz

2 .Sincetherearethreequadratictermsintheenergyexpression(correspondingtothe3translationaldirectionsofmotionx,y,z),theaverageormeankineticenergymustbe3/2kBT,i.e.

εtr = 12mv

2 = 12m v 2 = 3

2 kBT .Wecanrearrangethistoimmediatelygetour

desiredresultfor<v2>:

v 2 = 3 kBTm

Wesometimesliketotalkaboutthe“root‐mean‐square”speedor“RMS”speed

vRMS .Thisissimplygivenby

vRMS = v 2 =3kBTm

.NotethatthisisNOTthesame

asthemeanspeed(infactitisalittlelarger),butitdoeshavethesameunits.MostProbableAtomicorMolecularSpeed:Nowwehavecalculated2differentcharacteristicspeeds(themeanspeed

v andtheRMSspeed

vRMS )forgasatomsormoleculesofmassmattemperatureT.But

Page 8: Kinetic Theory of Gases - WordPress.com · 2015. 4. 15. · EP271 Kinetic Theory Notes M.P. Bradley Kinetic Theory of Gases Supplementary Lecture Notes for EP271 Michael Patrick Bradley

EP271KineticTheoryNotesM.P.Bradley

thereisstillanothercharacteristicmolecularspeedwhichmaybeuseful—the“mostprobable”speed

vMP .Thisisthemostlikelyspeedtoobserveifwepickasinglemoleculeatrandomandmeasureitsspeed.Itisgivensimplybythevaluewhichmaximizesthespeedprobabilitydistribution

g v( ) ,whichwefindbysettingthefirst

derivativeto0,i.e.

dg(v)dv vMP

= 0 .ForthemomentIwillleavethiscalculationforyou

asanexercise.Theresultthatwegetis

vMP = 2 kBTm

.Youcanverifythatthisisa

maximumbyapplyingthesecondderivativetest,althoughtheshapeofthedistributiong(v)makesthismore‐or‐lessobvious.SummaryofCharacteristicMolecularSpeeds:OK,sowehavenowcalculated3differentcharacteristicspeedsforgasatomsor

moleculesofmassm,attemperatureT.Wecansummarizetheseresultsinatable:

MostProbableAtomic/MolecularSpeed

vMP = 2 kBTm

≈1.41 kBT /m

MeanAtomic/MolecularSpeed

vMP =8πkBTm

≈1.60 kBT /m

RMSSpeed

vMP = 3 kBTm

≈1.73 kBT /m

Nowwecanseethatthespeedshavethefollowinghierarchyofmagnitudes:

vMP < v < vRMS .Also,sincethespeeddistributionfunctiong(v)fallsoffforspeedshigherthanthemostprobablyspeedvMPwealsohavethefollowingresult:

g(vMP ) > g( v ) > g vRMS( )Soisthephysicalsignificanceofallofthis?Well,wecanseeprettyclearlythat

althoughaboxorchambercontaininganatomicormoleculargascontainsatomsormoleculewithawiderangeofspeeds,movingrandomlyinalldirectionsandcollidingofthewallsandoffeachother,wecannonethelessmakesomeprettydefinitestatementsabouttheaverageormeanvaluesofmanyatomic/molecularquantities,suchasthespeedandenergy,aswellasresultsforthepressureexertedbythegasonthewallsofthechamber.Thisispowerofthestatisticalmechanicsapproach(recallthatthekinetictheoryofgasesistheearlieststatisticalmechanics‐basedtheory).Ourdiscussionsofspeedsshowedusthatthereareseveralcharacteristicspeeds

wecancalculate,allofwhicharecloseinvalueandallofwhichincreaseasthetemperatureincreases.Thisisthebasisforthecommonstatementthatthetemperatureisameasureoftheaveragemolecularmotion,andthatinwarmerbodies,atomsandmoleculearemovingfaster.Thecharacteristicenergyassociated

Page 9: Kinetic Theory of Gases - WordPress.com · 2015. 4. 15. · EP271 Kinetic Theory Notes M.P. Bradley Kinetic Theory of Gases Supplementary Lecture Notes for EP271 Michael Patrick Bradley

EP271KineticTheoryNotesM.P.Bradley

witheachdegreeoffreedomiskBT,andthecharacteristicmolecularspeedsallhave

theform(numericalfactor)x

kBTm

.

Thismakessenseonthebasisofunitsandisalsoconsistentwiththeformulafor

thespeedofsoundinagas:

csound = γkBTm

where

γ =cpcvisthespecificheatratio

(Moran,Shapiroetal.callit“k”butthiscanbeconfusedwithBoltzmann’sconstantwhichcomesupeverywhereinstatisticalmechanicssoIprefertousetheGreeklettergammahere).Whyshouldthespeedofsoundberelatedtothecharacteristicatomic/molecularspeedsinagas?Wellsoundwavesadvancebetweenalternatecompressionsandrarefactionsbycollisionsbetweengasmolecules,soitmakessensethatthespeedwithwhichsoundcouldpropagatewouldbeofthesameorderofmagnitudeasthecharacteristicatomicormolecularspeeds.Atomic/MolecularEnergyDistributionSofarwehaveusedtheatomic/molecularMaxwell‐Boltzmannspeeddistribution

tocalculatesomecharacteristicspeedsforanidealgasofatomsormoleculesofmassmattemperatureT.Howeverwearealsooftenveryinterestedintheenergydistributionfortheseparticles.Thiswecaneasilyderive.Thekineticenergyisgivenby

ε = 12mv

2 .Whatwearetryingtocalculateisessentiallyachange‐of‐variablesfromatomic/molecularspeedvtoenergyE.Sincethereisaone‐to‐onemappingfromspeedtoenergy,thisisastraightforwardprocess.Byourdefinitionofprobabilitydistributionswemusthave

g(v)dv = g(ε)dε .Re‐arrangingweget

g(v) = g(E)dE /dv .Nowwecaneasilycalculate

dE /dv = mv ,sowegetfinally

gE (E) = g(v) /mv Fromthisweget,finally

gE (E) = 2π

1kBT( )

3 / 2Ee−E / kBT

Youcanverifythefactthatthisisaproperlynormalizeddistributionandthatit

givesameanenergywhichobeystheequipartitionfunction,ifyoulike(Iwillleavethisasanexerciseforyou,eagerstudent).