AnintroductiontothermodynamicsZhigangSuo,HarvardUniversity,suo@seas.harvard.edu

Iwillkeepupdatingthisgoogledoc,anddrawfiguresinclass.Pleaseaccessthecurrentversiononline,andkeepgoodnotesinclass.

Theplayofthermodynamics

EntropyIsolatedsystemSamplespaceDefinitionofentropyFundamentalpostulateDispersionofinkEquilibriumIrreversibilityFluctuationKineticsConstrainedequilibriumSeparationofphasesInternalvariableBasicalgorithmofthermodynamicsThesecondlawofthermodynamics

Energy,space,matter,chargePotentialenergyDefinitionofenergyFormsofenergyEnergybelongstomanysciencesAnisolatedsystemconservesenergy,space,matter,andchargeAclassificationofsystemsTransferenergybetweenaclosedsystemanditssurroundingsEntropyandenergy

ThermalsystemAfamilyofisolatedsystemsofasingleindependentvariationGeneralfeaturesofthefunctionS(U)PhrasesassociatedwithafamilyofisolatedsystemsDissipationofenergy

Isentropicprocess.ReversibilityThermalcontactDefinitionoftemperatureTwounitsoftemperatureUnitofentropyIdealgaslawVaporpressureSpecificquantitiesGeneralfeaturesofthefunctionT(U)Thermalcapacity

ExperimentalthermodynamicsDivisionoflaborCalorimetryThermometryExperimentaldeterminationofentropy

MeltingEmpiricalfactsPrimitivecurvesRuleofmixtureDerivedcurveEquilibriumofasinglehomogeneousstateEquilibriumoftwohomogeneousstatesTemperature-entropycurveThermalsystemofanonconvexcharacteristicfunctions(u).Metastability

TemperatureasanindependentvariableU(T)andT(U)ThermostatThermalreservoirIsothermalprocessAlgorithmofthermodynamicsforisothermalprocessHelmholtzfunctionMeltinganalyzedusingtheHelmholtzfunction

ClosedsystemAfamilyofisolatedsystemsoftwoindependentvariationsConstant-volumeprocessAdiabaticprocess

ExperimentaldeterminationofthefunctionS(U,V)GeneralfeaturesofthefunctionS(U,V)

IdealgasThemodelofidealgasIdealgaslawEnergyofanidealgasEntropyofanidealgasEnergytransferbyworkandbyheatforafixedamountofanidealgasEntropicelasticityThermoelasticcoupling

OsmosisTheoryofosmosisBalanceofosmosis

PhasesofapuresubstanceEmpiricalfactsPrimitivesurfacesRuleofmixtureDerivedsurfaceEquilibriumofasinglehomogeneousstateEquilibriumoftwohomogeneousstatesEquilibriumofthreehomogeneousstatesCriticalpointMetastabilityEnergy-volumeplaneTemperatureandpressure

AlternativeindependentvariablesEntropyS(U,V)EnergyU(S,V)HelmholtzfunctionF(T,V)EnthalpyH(S,P)GibbsfunctionG(T,P)Constant-pressureandconstant-temperatureprocessAlgorithmofthermodynamicsforconstant-pressureandconstant-temperatureprocessEquilibriumoftwohomogeneousstatesbyequatingtheGibbsfunctionClapeyronequationRegelation

VanderWaalsmodelofliquid-gasphasetransition

IsothermsCriticalpointEnergyEntropyEntropy-energycompetitionMaxwellrule

OpensystemAfamilyofisolatedsystemsofmanyindependentvariationsDefinitionofchemicalpotentialsTwoopensystemsincontactExperimentaldeterminationofthechemicalpotentialofaspeciesofmoleculesinacomplexsystemExtensibilityGibbsfunctionGibbs-Duhemrelation

ChemicalpotentialofaspeciesofmoleculesinapuresubstanceChemicalpotentialofaspeciesofmoleculesinanidealgasHumidityIncompressiblepuresubstanceTheascentofsap

Mixtureofidealgases

ChemicalreactionStoichiometriccoefficientsDegreeofreactionChemicalequilibriumReactionofidealgases

TheplayofthermodynamicsLeadingrole.Thermodynamicsisoftencalledthescienceofenergy.Thisdesignationstealsaccomplishmentsfromothersciences,anddiminishesaccomplishmentsofthermodynamics.Rather,thermodynamicsisthescienceofentropy.Entropyplaystheleadingroleinthermodynamics.Energycrisisisatimelytopic;entropycrisis,timeless.Supportingroles.Inthermodynamics,energyplaysasupportingrole,alongwithspace,matter,andcharge.Indeed,thesesupportingrolesareanalogoustooneanother,andareofequalimportance.Callingthermodynamicsthescienceofenergydistortsthestructureofthesubject,andneglectsobviouslysignificantrolesofspace,matter,andcharge.

Childrenandgrandchildren.Eachofthesesupportingroles,togetherwithentropy,producesachild.Thefourchildren—temperature,pressure,chemicalpotential,andvoltage—arethesecondgenerationofsupportingroles.Theyproducegrandchildren:thermalcapacity,compressibility,coefficientofthermalexpansion,etc.Extras.Therearealsosomeextras:enthalpy,Helmholtzfunction,Gibbsfunction,etc.Theyarecalledthermodynamicpotentials,introducedbyGibbs(1875).Theyareshadowsofentropy.Letnoshadowsobscuretherealthing—entropy.

TheCastofThermodynamicsLeadingrole:EntropySupportingroles:EnergySpaceMatterChargeChildrenofentropyandthesupportingroles:TemperaturePressureChemicalpotentialVoltageGrandchildren:ThermalcapacityCompressibilityCoefficientofthermalexpansion…...Extras:EnthalpyHelmholtzfunctionGibbsfunction…...

Thiscourse.Thiscoursewilldevelopthelogicofentropyfromfirstprinciples,intuitionofentropyfromeverydayexperience,andapplicationofentropyinmanydomains.

Wewillleteverydayexperience,alongwithdiscoveriesandinventions,revealthelongarmofentropy,anditsintriguingplayswiththesupportingroles.Iamwritingthisfileforlecturesthatfocusonthermodynamicsitself,ratherthanitsapplications.Anyoneofthestandardtextbookswillfilltherestofthecoursewithcopiousapplications.Thiscoursedoesnotteachthehistoryofthermodynamics.ItisimpracticaltoteachthermodynamicsbytracingthestepsofCarnot,Clausius,Boltzmann,andGibbs,justasitisimpracticaltoteachcalculusbytracingthestepsofNewtonandLeibniz.Asubjectanditshistoryaredifferentthings.Thussaid,thehistoryofthermodynamicsisinteresting,important,andwell-documented,fullofmomentsoftriumphanddespair.ManyoriginalworksareavailableonlineinEnglish.Iwillplaceafewnamesandyearsinthenotesaslandmarks.Youcanreadthehistoryofthermodynamicsonline,startingwiththeWikipediaentry.

EntropyThedefinitionofentropyrequirestwoideas:isolatedsystemandsamplespace.

IsolatedsystemSystem.Wehavemetthecast.Nowlookatthestage—theworld.Anypartoftheworldiscalledasystem.Therestoftheworldiscalledthesurroundings.Wecanregardanypartoftheworldasasystem.Eventheemptyspacecanbeasystem;thevacuumhostselectromagneticfield.Aprotonandanelectronconstituteasystem,calledahydrogenatom.Ahalfbottleofwaterisasystem.Thesystemiscomposedofwatermoleculesandsomeothermolecules,suchasnitrogen,oxygen,andcarbondioxide.Inthehalfbottleofwater,liquidoccupiessomevolume,andgasfillstherest.Theliquidandthegastogetherconstitutethesystem.Doweincludetheplasticbottleasapartofthesystem?Maybe,ifwedecidetostudythepermeationofwatermoleculesthroughtheplastic.Thedecisionisours.Interactionbetweenasystemanditssurroundings.Asystemanditssurroundingscanhavemanymodesofinteraction.Thehydrogenatomchangestheshapeofitselectroncloudwhentheatomabsorbsoremitsphotons,orwhentheatomissubjecttoanelectricfield.Iholdthehalfbottleofwaterinmyhand.Iseethewaterbecausetheliquid-gasinterfacerefractslight.Ifeelmoistbecausewatermoleculeshitme.Iwarmupthewaterwhenthevibrationofthemoleculesinmyhandcouplesthevibrationofthemoleculesinthewater.When

Idrinkfromthebottle,thebottletransferswatermoleculestomybody.Ishakethebottleandhearthesound.Ipourhoneyintowaterandwatchthemmix.Isolatedsystem.Ourplay—thermodynamics—showsallmodesofinteractionbetweenasystemanditssurroundings.Butournarrativebeginswithsomethingsimpler:anisolatedsystem—asystemthatdoesnotinteractwithitssurroundings.Tomakethehalfbottleofwateranisolatedsystem,Icapthebottletopreventmoleculesfromleakinginandout.Iinsulatethebottleinathermostoblockthevibrationofthemoleculesinmyhandfromcouplingwiththevibrationofmoleculesinthewater.Imakethebottlerigidtofixthevolume.Idonotshakethebottle.Iamalerttoanyothermodesofinteractionbetweenthewaterandthesurroundings.Doesthemagneticfieldoftheearthaffectthewater?Ifitdoes,Iwillfindawaytoshieldthebottleofwaterfromthemagneticfieldalso.Ofcourse,nothingisperfectlyisolated.Likeanyidealization,theisolatedsystemisausefulapproximationofthereality,solongastheinteractionbetweenthesystemandtherestoftheworldnegligiblyaffectsaphenomenonthatIchoosetostudy.Forexample,itmaybetoomuchtroubleformetoisolatethewaterfromgravity.Fewpeoplecaretostudywaterunderthezero-gravitycondition.GravityisimportantifImovethebottlearound,butunimportantifIstudythevaporpressureinthebottle.Exercise.Describeasystemandwhatyouneedtodotomakeitanisolatedsystem.

SamplespaceQuantumstatesofanisolatedsystem.Whenahydrogenatomisisolatedatthesecondenergylevel,theisolatedsystemhaseightquantumstates.Quantummechanicsgovernsallsystems,howevercomplicated.Aquantumstateofthehalfbottleofwaterisacloudofelectronsandpositionsofnuclei.Suchamacroscopicisolatedsystemhasalarge,butfinite,numberofquantumstates.Samplespace.Inthetheoryofprobability,eachtrialofanexperimentisassumedtoresultinoneofmultiplepossibleoutcomes.Eachpossibleoutcomeiscalledasamplepoint.Thesetofallpossibleoutcomesoftheexperimentiscalledthesamplespace.ThenotionofsamplespacecomesfromMises(1919).Anisolatedsystemisan“experiment”inthesenseofthewordusedinthetheoryofprobability.Theisolatedsystemflipsfromonequantumstatetoanother,rapidlyandceaselessly.Eachquantumstateisapossibleoutcome,orasamplepoint,oftheisolatedsystem.Allthequantumstatesoftheisolatedsystemconstitutethesamplespaceoftheisolatedsystem.

Exercise.Whatisthesamplespaceofathrowofacoin?Whatisthesamplespaceofthrowacoinandadiesimultaneously?Whatisthesamplespaceofathrowoftwodiessimultaneously?Howmanypossibleoutcomesdoyougetwhenyouthrow1000dies?Exercise.Whatisthesamplespaceofahydrogenatomisolatedatthesecondenergylevel?Sketchtheelectroncloudsofthequantumstates.

DefinitionofentropyNowenterstheleadingrole—entropy.LetΩbethenumberofquantumstatesofanisolatedsystem.DefinetheentropyoftheisolatedsystembyS=logΩ.Logarithmofanybasewilldo.Forconvenience,wewillusethenaturalbasee.Thenumberesimplifiesthederivativeoflogarithm.Recallafactofcalculus:dlogx/dx=1/x.Foranyotherbaseb,recallthatlogbx=(logbe)(logex),sothatdlogbx=(logbe)/x.Theprefactorclutterstheformulaandservesnopurpose.Entropyisanextensivequantity.WhydowehideΩbehindalog?Considertwoisolatedsystems,AandB.IsolatedsystemAhasonesamplespaceofΩAquantumstates,labeledas{a1,a2,...,aΩA}.IsolatedsystemBhasanothersamplespaceofΩBquantumstates,labeledas{b1,b2,...,bΩB}.Thetwosystemsareseparatelyisolated.Togethertheyconstituteacomposite,whichisalsoanisolatedsystem.Eachquantumstateofthiscompositeisacombinationofaquantumstateofoneisolatedsystem,ai,andaquantumstateoftheotherisolatedsystem,bj.Allsuchcombinationstogetherconstitutethesamplespaceofthecomposite.Thetotalnumberofallsuchcombinationsistheproduct:Ωcomposite=ΩAΩB.Recallapropertyoflogarithm:log(ΩAΩB)=logΩA+logΩB.Thus,theentropyofacompositeoftwoseparatelyisolatedsystemsisthesumoftheentropiesofthetwoindividualisolatedsystems.Wenowseethesignificanceoflogarithm:itturnsaproducttoasum.Theentropyofasystemisthesumoftheentropiesofitsparts,eachpartbeingseparatelyisolated.Suchanadditivequantityiscalledanextensivequantity.

Entropyofapuresubstance.Apuresubstanceisacollectionofalargenumberofasinglespeciesofmolecules(oratoms).Themoleculescanaggregateintovariousforms,calledphases.Forexample,atroomtemperatureandatmosphericpressure,diamondisacrystallinelatticeofcarbonatoms,waterisaliquidofH2Omolecules,andoxygenisagasofO2molecules.Entropyisanextensivequantity.Theentropyofapieceofapuresubstanceisproportionaltothenumberofmoleculesinthepiece.Forapieceofapuresubstance,havingthenumberofmoleculesNandentropyS,theentropyofthesubstancepermoleculeiss=S/N.Laterwewilldescribehowtomeasureentropyexperimentally.Fornow,welookatsomemeasurednumbers.Atroomtemperatureandatmosphericpressure,theentropyofdiamond,leadandwaterare0.3,7.8and22.70,respectively.Astrongsubstance,suchasdiamond,hasasmallvalueofentropy,becauseindividualatomsareheldtogetherbystrongchemicalbonds,whichreducesthenumberofquantumstates.Complexsubstancesgenerallyhavelargerentropiesthansimplesubstances.Forexample,atroomtemperatureandatmosphericpressure,theentropiesforO,O2andO3are19.4,24.7,28.6,respectively.Whenapuresubstancemelts,themoleculestransformfromacrystaltoaliquid.Associatedwiththisphasetransition,theentropytypicallyincreasesbyanumberbetween1to1.5permolecule.Zeroentropy.Theentropyofapuresubstanceisoftentabulated,attheendoftextbooksandonline,asafunctionoftemperatureandpressure.Suchatableoftenassumesanarbitrarystateoftemperatureandpressureasareferencestate.Thetableliststheentropyofthesubstanceatthisreferencestateaszero,andliststheentropyofthesubstanceatanyotherstateoftemperatureandpressurerelativetothereferencestate.Wefollowthispracticewithcaution.Recallthedefinitionofentropy,S=logΩ.Zeroentropyisnotsomethingarbitrary,buthasphysicalsignificance:zeroentropycorrespondstoanisolatedsystemofasinglequantumstate.Exercise.Theentropyofathrowofafairdieislog6.TheentropyofanisolatedsystemislogΩ.Whatistheentropyofathrowofacoin?Whatistheentropyofasimultaneousthrowofacoinandadie?Whatistheentropyofasimultaneousthrowof1000dies?Exercise.12gramsofdiamondhas6.02×1023numberofcarbonatoms.Howmanyquantumstatesarethereinonegramofdiamondatroomtemperatureandatmosphericpressure?

Exercise.Carbonatomscanalsoaggregateinotherforms,suchasgraphene,nanotube,andbuckyball.Learnabouttheseformsonline,andfindtheentropyperatomineachform.

FundamentalpostulateOfourworldweknowthefollowingfacts:

1. Anisolatedsystemhasacertainnumberofquantumstates.DenotethisnumberbyΩ.2. Theisolatedsystemflipsfromonequantumstatetoanother,rapidlyandceaselessly.3. Asystemisolatedforalongtimeflipstoeveryoneofitsquantumstateswithequal

probability,1/Ω.Thus,asystemisolatedforalongtimebehaveslikeafairdie:

1. Thediehassixfaces.2. Thedieisrolledfromonefacetoanother.3. Thedieisrolledtoeveryfacewithequalprobability,1/6.

Fact3oftheworldiscalledthefundamentalpostulate.Thefundamentalpostulatecannotbededucedfrommoreelementaryfacts,butitspredictionshavebeenconfirmedwithoutexceptionbyempiricalobservations.Wewillregardthefundamentalpostulateasanempiricalfact,andusethefacttobuildthermodynamics.Thefundamentalpostulatelinksthermodynamicstoprobability.Ourworldactslikeacompulsivegambler,ceaselesslyandrapidlythrowingfairdies,eachhavinganenormousnumberoffaces.

Probability Thermodynamics

Experiment Rollafairdie Isolateasystemforalongtime

Samplespace 6faces Ωquantumstates

Probabilityofasamplepoint

1/6

1/Ω

Subset Event Subset

Probabilitytorealizeasubset

(numberoffacesinthesubset)/6

(numberofquantumstatesinthesubset)/Ω

Afunctionfromsamplespacetoasetofvalues

Randomvariable

Internalvariable

Exercise.Acheatermakesanunfairdieofsixfaces,labeledasa,b,c,d,e,f.Throughmanythrows,thecheaterfindsthattheprobabilityoffaceais½,theprobabilityoffacefis1/10,andtheotherfourfaceshaveanequalprobability.Whatistheprobabilityofgettingfacebtwiceintwothrows?Ignoranceisbliss.Inthrowingadie,thegamblerdoesnotneedtoknowthematerialthatmakesthedie,orthesymbolsthatmarkthefaces.Allthegamblerneedstoknowaboutthedieisthatithassixfacesofequalprobability.Thesameistrueinthermodynamics.Instudyinganisolatedsystem,wedonotneedtoknowthequantumstatesthemselves(theshapeofthecloudofelectrons,thepositionsofnuclei,orthenumberofprotons).Wejustneedtoknowhowmanyquantumstatesthattheisolatedsystemhas.Thisenormousreductionofinformationiscentraltothesuccessofthermodynamics.Anisolatedsystemisreducedtoapurenumber,thenumberofquantumstates,Ω.Laterwewilllearnhowtocountthenumberofquantumstatesofanisolatedsystemexperimentally.

DispersionofinkEmpiricalfacts.Letuswatchthefundamentalpostulateinaction.Dripadropofinkintoabottleofwater,andtheinkdispersesovertime.Thedispersionofinkisreadilyobservedatmacroscopicscale,butcanalsobeobservedinamicroscope,asdescribedbelow.Theinkcontainspigmentparticlesofsizelessthanamicron.Eachpigmentparticleisbombardedbywatermolecules,rapidlyandceaselessly,fromalldirections.Atanygiventime,thebombardmentsdonotfullycancelout,butresultinanetforcethatmovesthepigmentparticle.Thisrapid,ceaseless,randommotionofaparticleinaliquidwasfirstobservedinamicroscopebyBrown(1827).WikiBrownianmotion.Thermodynamictheory.Individualpigmentparticlesmoveinalldirectionsrandomly.Aftersometime,thepigmentparticlesdisperseinthebottleofwaterhomogeneously.Howcantherandommotionofindividualpigmentparticlescausepigmentparticlescollectivelytodosomethingdirectional—dispersion?

Theanswerissimple.Whentheconcentrationofpigmentparticlesisinhomogeneous,moreparticleswilldiffusefromaregionofhighconcentrationtoaregionoflowconcentration.Thisbiascontinuesuntilthepigmentparticlesaredistributedhomogeneously.Letusrelatethiseverydayexperiencetothefundamentalpostulate.Tomakeadefinitecalculation,weassumethatthepigmentparticlesinwaterarefarapart,sothateachparticleisfreetoexploreeverywhereinthebottleofwater,unaffectedbythepresenceofotherpigmentparticles.Consequently,thenumberofquantumstatesofeachpigmentparticleisproportionaltothevolumeofthewaterinthebottle,V.ThenumberofquantumstatesofNpigmentparticlesisproportionaltoVN.Ontheotherhand,iftheNpigmentparticlesarelocalizedinasmallregion,say,intheinitialdropofinkofvolumeV/70,thenumberofquantumstatesoftheNpigmentparticlesisproportionalto(V/70)N.Aftertheinkisinwaterforalongtime,allquantumstatesareequallyprobable.Thus,theratiooftheprobabilityoffindingtheNpigmentparticlesinvolumeVtotheprobabilityoffindingtheNpigmentparticlesinvolumeV/70is70N.Thisratioisenormousbecauseadropofinkhasalargenumberofpigmentparticles,N.Thisfactexplainswhythepigmentparticlesmuch,muchpreferdispersiontolocalization.Exercise.Thedensityofthepigmentmaterialis1000kg/m3.Assumeeachpigmentparticleisasphere,diameter100nm.Howmanypigmentparticlesaretherein1gofdryink?Aftertheinkisinabottleofwater,volume100ml,foralongtime,whatistheratiooftheprobabilityoffindingallpigmentparticlesinavolumeof10mltotheprobabilityoffindingallpigmentsinthevolumeof100ml?

EquilibriumThedispersionofinkillustratesseveralcharacteristicscommontoallisolatedsystems.Rightafterasmalldropofinkentersthebottleofwater,allthepigmentparticlesarelocalizedinthedrop.Thepigmentparticlesthenstarttodiffuseintothepurewater.Aftersometime,thepigmentparticlesdisperseinthebottleofwaterhomogeneously,andthesystemofthepigmentparticlesinwaterissaidtohavereachedequilibrium.Rightafterisolation,thesystemhasΩquantumstates,flipstosomeofthemmoreoftenthanothers,andissaidtobeoutofequilibrium.Outofequilibrium,theprobabilityfortheisolatedsystemtobeinaquantumstateistime-dependent.Afterbeingisolatedforalongtime,thesystemflipstoeveryoneofitsquantumstateswithequalprobability,1/Ω,andissaidtohavereachedequilibrium.Inequilibrium,theprobabilityfortheisolatedsystemtobeinaquantumstateistime-independent.

Severalphrasesaresynonymous:asystem“isolatedforalongtime”isasystem“flippingtoeveryoneofitsquantumstateswithequalprobability”,andisasystem“inequilibrium”.Wheneverwespeakofequilibrium,weidentifyasystemisolatedforalongtime.Inoroutofequilibrium,anisolatedsystemflipsfromonequantumstatetoanother,ceaselesslyandrapidly.Inoroutofequilibrium,anisolatedsystemhasafixedsamplespaceofΩquantumstates.Theentropyoftheisolatedsystem,S=logΩ,isafixednumberanddoesnotchange,nomatterwhethertheisolatedsystemisinoroutofequilibrium.

IrreversibilitySolongasthebottleisisolated,thehomogeneouslydispersedpigmentparticleswillbeextremelyunlikelytocomebackintoasmallvolume.Onceinequilibrium,theisolatedsystemwillnotgooutofequilibrium.Theisolatedsystemoutofequilibriumissaidtoapproachequilibriumwithirreversibility.Thermodynamicsusesthedirectionoftime,butnotthedurationoftime.Thermodynamicsmakesnouseofanyquantitywithdimensionoftime.Timeentersthermodynamicsmerelytodistinguishbetween“before”and“after”.Irreversibilitygivestimethedirection,thearrowoftime.

FluctuationInequilibrium,thepigmentparticleskeepinceaselessBrownianmotion.Thedistributionofthepigmentparticlesfluctuates.Possiblyallpigmentparticlescanmoveintoasmallvolumeinthebottle.However,theprobabilityoffindinganonuniformdistributionisexceedinglysmall.Thiscoursewillignorefluctuation.

KineticsAsystemisolatedforalongtimeflipstoeveryoneofitsquantumstateswithequalprobability.Howlongislongenough?Thefundamentalpostulateissilentonthisquestion.Thepigmentparticlesdisperseslowerinhoneythaninwater.Thetimeneededtoreachequilibriumscaleswiththeviscosityoftheliquid.Thestudyofhowfastasystemevolvesiscalledkinetics,whichwillnotbestudiedinthiscourse.

ConstrainedequilibriumWhenanisolatedsystemisinequilibrium,anypartoftheisolatedsystemisalsoinequilibrium,solongasthepartismacroscopic.Itmakesnosensetotalkaboutequilibriumatthelevelofafewpigmentparticles,butequilibriumwillprevailinalargenumberofpigmentparticles.Wecandividetheisolatedsystemintomanyparts.Eachpartislargecomparedtoindividualparticles,butsmallcomparedtotheentireisolatedsystem.Weregardeachpartasanisolatedsubsystem.Eachisolatedsubsystemhasitsownsamplespaceofquantumstates.Thus,theentropyofanisolatedsystemisthesumoftheentropiesofallpartsofthesystem.Whenanisolatedsystemisoutofequilibrium,weoftendividetheisolatedsystemintoparts.Forexample,beforetheinkisfullydispersedinthebottleofwater,wemaydividethebottleintomanysmallvolumes.Eachsmallvolumehasalargenumberofpigmentparticles,whichareapproximatelyhomogeneouslydistributed,sothatwecanthinkofeachsmallvolumeasanisolatedsystem,withitsownsamplespaceofquantumstates.Wesaythattheisolatedsystemisinconstrainedequilibrium.

SeparationofphasesEmpiricalfacts.Anisolatedsysteminequilibriumcanbeheterogeneous.Hereisthehalfbottleofwateragain.AsIshakethebottle,watermovesandbubblespop.AfterIstopshaking,thehalfbottleofwaterbecomesapproximatelyanisolatedsystem.Rightaftertheisolation,thehalfbottleofwaterisstilloutofequilibrium.Afterbeingisolatedforsometime,thehalfbottleofwatercalmsdownatmacroscopicscale.Inequilibrium,somewatermoleculesformtheliquid,andothersformthevapor.Watermoleculesinthebottlearesaidtoseparateintotwophases,liquidandvapor.Thermodynamictheory.Wearenotreadytodevelopafulltheoryofphaseseparation,butbeginwithafewideashere,andpickthemuplater.Theisolatedsystemflipsamongasetofquantumstates,whichconstitutethesamplespaceoftheisolatedsystem.DenotethenumberofquantumstatesoftheisolatedsystembyΩ.ThehalfbottleofwaterhasatotalofMwatermolecules.Thenumberofwatermoleculesinthevapor,N,cantakeoneofasetofvalues:{0,1,...,M}. WhenthenumberofwatermoleculesinthevaporisfixedatN,theisolatedsystemflipsamongquantumstatesinasubsetofthesamplespace.DenotethenumberofquantumstatesinthissubsetbyΩ(N).Thus,

Ω(0)+Ω(1)+…+Ω(M)=Ω.Probability.Afterthebottleisisolatedforalongtime,everyquantumstateinthesamplespaceisequallyprobable,sothattheprobabilitytoobserveNwatermoleculesinthevaporisΩ(N)/Ω.Inequilibrium,themostprobableamountofmoleculesinthevapor,N,maximizesthefunctionΩ(N).Fromprobabilityto(almost)certainty.Thehalfbottleofwaterisamacroscopicisolatedsystem,andhasasamplespaceofalargenumberofquantumstates.Inequilibrium,thefluctuationinthenumberofwatermoleculesinthevaporisexceedinglysmall,andtheobservedamountofwatermoleculesinthevaporiswelldescribedbytheamountNthatmaximizesthefunctionΩ(N).Thefluctuationinthenumberofmoleculesinthevaporisnegligiblecomparedtothetotalnumberofmoleculesinthevapor.ThisobservationindicatesthatthefunctionΩ(N)hasasharppeak.

InternalvariableConstraintinternaltoanisolatedsystem.Fixingthenumberofwatermoleculesinthevaporinthehalfbottleofwaterisanexampleofaconstraintinternaltoanisolatedsystem.Theconstraintcanbemaderealbyplacingasealbetweentheliquidandvapor.Withtheseal,thetwopartsoftheisolatedsystemcanseparatelyreachequilibrium,butarenotinequilibriumwitheachother.Theisolatedsystemisinconstrainedequilibrium.Whentheconstraintisremoved,thenumberofwatermoleculesinthevaporcanchange,andiscalledaninternalvariable.Letusabstractthisexampleingeneralterms.Samplespace.Anisolatedsystemhasasetofquantumstates,whichconstitutethesamplespace.DenotethetotalnumberofquantumstatesoftheisolatedsystembyΩ.Subsetofsamplespace.LetXbeasetofvalues:X={x1,...,xn}.AninternalvariableisafunctionthatmapsthesamplespacetothesetX.Inthetheoryofprobability,suchafunctioniscalledarandomvariable.WhenaconstraintinternaltotheisolatedsystemfixestheinternalvariableatavaluexinthesetX,theisolatedsystemflipsamongquantumstatesinasubsetofthesamplespace.DenotethenumberofquantumstatesinthissubsetbyΩ(x).

Theinternalvariabledissectsthesamplespaceintoafamilyofsubsets.Anytwosubsetsinthefamilyaredisjoint.Theunionofallthesubsetsinthefamilyisthesamplespace.Thus,Ω(x1)+…+Ω(xn)=Ω.Probability.Aftertheconstraintisremovedforalongtime,theisolatedsystemflipstoeveryoneofitsΩquantumstateswithequalprobability,andtheinternalvariablecantakeanyvalueinX.Inequilibrium,theprobabilityfortheinternalvariabletotakeaparticularvaluexinXisΩ(x)/Ω.Equilibrium.Amacroscopicisolatedsystemhasasamplespaceofalargenumberofquantumstates.ThefunctionΩ(x)hasasharppeak.Aftertheconstraintisremovedforalongtime,theobservedvalueoftheinternalvariableiswelldescribedbythevaluexthatmaximizesthefunctionΩ(x).Thermodynamicsstudiesanisolatedsysteminconstrainedequilibrium,anddeterminesthemostprobablevalueoftheinternalvariable.Thermodynamicsneglectsthefluctuationinaninternalvariable,anddoesnotstudythekineticsofhowtheisolatedsystemapproachesequilibrium. Irreversibility.Rightafterbeingisolated,thesystemisoutofequilibrium.Astimemovesforward,theisolatedsystemevolvestowardequilibrium,andtheinternalvariablechangesinasequenceofvaluesthatincreasethefunctionΩ(x).Solongasthesystemisisolated,thechangeintheinternalvariableisirreversible.

BasicalgorithmofthermodynamicsFunctionΩ(x).AninternalvariableisafunctionthatmapsthesamplespaceofanisolatedsystemtoasetX.WhentheinternalvariabletakesavaluexinthesetX,theisolatedsystemflipsamongthequantumstatesinasubsetofthesamplespace.DenotethenumberofquantumstatesinthesubsetbyΩ(x).FunctionS(x).DefineS(x)=logΩ(x).Becauselogarithmisanincreasingfunction,maximizingΩ(x)isequivalenttomaximizingS(x),andincreasingΩ(x)isequivalenttoincreasingS(x).ThefunctionS(x)standsfor“thelogarithmofthenumberofquantumstatesinthesubsetofthesamplespaceofanisolatedsystemwhenaninternalvariableisfixedatavaluex”.Thisfunctioniscentraltotheapplicationofthermodynamics,butisunnamed.Forbrevity,IwillcallthefunctionS(x)thesubsetentropy.Basicalgorithmofthermodynamics.Hereishowweuseentropyinthermodynamics.

1. Constructanisolatedsystemwithaninternalvariablex.

2. IdentifythefunctionS(x).3. Equilibrium.FindthevalueoftheinternalvariablexthatmaximizesthefunctionS(x).4. Irreversibility.Changethevalueoftheinternalvariablexinasequencethatincreases

thefunctionS(x).

ThesecondlawofthermodynamicsThebasicalgorithmisoneofmanyalternativestatementsofthelawoftheincreaseofentropy,orthesecondlawofthermodynamics.Forentertainment,wewilllaterlistsomehistoricalstatementsofthesecondlawofthermodynamics.Theymaysoundlikeancientphilosophicalpronouncements.Theysoundmysteriousnotbecausetheyaremoreprofoundthanthebasicalgorithm,butbecausetheymissbasicfactsoftheworld(e.g.,therapidandceaselessflipsamongquantumstates,andthefundamentalpostulate).HereisonesuchstatementmadebyClausius(1865),inthesamepaperinwhichhemadeupthewordentropy:Theentropyoftheuniversetendstoamaximum.Wewilltaketheworduniversetomeananisolatedsystem.Thisstatementisadoptedbynumeroustextbooks.Thestatementiselegantbutconfusing.TheentropyofanisolatedsystemisS=logΩ,whereΩisthetotalnumberofquantumstatesintheentiresamplespaceoftheisolatedsystem.Theentropyofanisolatedsystemisafixednumber,notafunctionthatcanchangevalues.Totalkaboutamacroscopicchangeofanisolatedsystem,weneedtoidentifyaninternalvariable.Whenaconstraintinternaltotheisolatedsystemfixestheinternalvariableatavaluex,theisolatedsystemflipsamongquantumstatesinasubsetofthesamplespace.ThenumberofquantumstatesinthissubsetisΩ(x),andtheentropyofthissubsetisS(x)=logΩ(x).Whentheconstraintinternaltotheisolatedsystemisremoved,theinternalvariablexcanchangevalues.ItisthesubsetentropyS(x)thattendstoamaximum.Wheneleganceandclarityconflict,wegoforclarity.Thiscoursedoesnotstudythehistoryofthermodynamics.WedonotattempttoreadthemindofClausiusanddecipheroldpronouncements.Rather,wewillusethebasicalgorithmtodirectcalculationandmeasurement.Inparticular,thebasicalgorithmwillletuscountthenumberofquantumstatesexperimentally.

Energy,space,matter,chargeThebasicalgorithmcallsforinternalvariables.Nowenterstheparticularshowysupportingactor—energy.Energyservesasaninternalvariableinthermodynamics,alongwithspace,matter,andcharge.

PotentialenergyAnappleweighsabout1Newton.WhenIpickuptheapplefromtheground,theapplereachesabout1meterhighandaddsabout1Jouleofenergy.Thisformofenergyiscalledthepotentialenergy(PE).Frommechanicsyouhavelearnedthefact:PE=(weight)(height).Potentialenergyhastheunitofforcetimeslength,(Newton)(meter).ThisunitofenergyiscalledtheJoule.Theheightisrelativetosomefixedpoint,suchastheground.Thus,potentialenergyisarelativequantity.Foragivenheight,thepotentialenergyisproportionaltotheamountofmaterial.Thus,potentialenergyisalsoanextensivequantity.Indeed,allformsofenergyarerelativeandextensive.Frommechanicsyouhavelearnedanotherfact: weight=(mass)(accelerationofgravity).Anapplehasamassabout0.1kg.Theaccelerationofgravityisabout10m/s2.Thus,theweightoftheappleis(0.1kg)(10m/s2)=1Newton.

DefinitionofenergyEnergyiswhateverthatcanliftaweighttosomeheight.Bythisdefinition,energyisconserved,relative,andextensive.Thedefinitionofenergycallsforaction.Itisuptopeopletodiscoverenergyinitsvariousforms,andinventwaystoconvertenergyfromoneformtoanother.Howdoweknowthatsomethingoffersaformofenergy?Justtestifthissomethingcanbeconvertedtoliftaweighttosomeheight.

FormsofenergyKineticenergy.Myhandnowreleasestheapple.Justaftertherelease,theappleis1meterhighandhaszerovelocity.Thefallingapplethenlosesheight,butgainsvelocity.Theenergyassociatedwiththevelocityofamassiscalledthekineticenergy(KE).Thefallingappleconvertpotentialenergytokineticenergy.MechanicstellsusthatKE=(½)(mass)(velocity)2.Theconservationofmechanicalenergy.Thefrictionbetweentheappleandtheairisnegligible.Mechanicstellsyouthat,astheapplefalls,thesumofthepotentialenergyandthekineticenergyisconstant—thatis,PE+KE=constant.Thepotentialenergyandthekineticenergyaretwoformsofmechanicalenergy.Theaboveequationsaysthatmechanicalenergyisconservedwhenfrictionisnegligible.Exercise.Whatisthevelocityoftheapplejustbeforehittingtheground?Exercise.Atigerjumps1mhigh.Whatisthevelocityofthetigerjustbeforeithitstheground?Exercise.DerivetheconservationofmechanicalenergyfromNewton’ssecondlaw.Thermalenergy(internalenergy).Watchtheapplefallagain.Theapplefallsfromaheight,andgainsavelocityjustbeforehittingtheground.Afterhittingtheground,theapplebumps,rolls,andthenstops.Whathappenstoallthatpotentialenergyandkineticenergy?Justbeforetheapplehitstheground,allpotentialenergyhasconvertedtokineticenergy.Aftertheapplehitsthegroundandcomestorest,allthekineticenergyoftheappledisperses,ordissipates,intothemotionofthemoleculesinsidetheappleandtheground.Wesaythattheappleandthegroundgainsthermalenergy(TE),alsocalledinternalenergy.WegeneralizetheprincipleoftheconservationofenergytoTE+PE+KE=constant.Thewordthermalisanadjectivethatdescribesphenomenarelatedtomicroscopicinteractionandmotion.Thermalenergyisjustthepotentialenergyandkineticenergyatmicroscopicscale.WedesignatePEandKEasthepotentialenergyandkineticenergyatmacroscopicscale.

InHeatconsideredasaModeofMotion,publishedin1863,JohnTyndalldescribednumerousexperimentsthattestedthehypothesisofheatasaformofenergy.WikiJuliusRobertvonMayer.WikiJamesPrescottJoule.Thefollowingpassageistakenfromthebook.Abullet,inpassingthroughtheair,iswarmedbythefriction,andthemostprobabletheoryofshootingstarsisthattheyaresmallplanetarybodies,revolvingroundthesun,whicharecausedtoswervefromtheirorbitsbytheattractionoftheearth,andareraisedtoincandescencebyfrictionagainstouratmosphere.Electricalenergy.Electricalenergy(EE)takesmanyforms.Onewaytouseelectricalenergyistheresistiveheating.Avoltageofanelectricoutletmoveselectronsinametalwire,andtheresistanceofthemetalconvertstheelectricalenergyintothermalenergy.Recallvoltage=(resistance)(current).TheelectricalenergyisEE=(resistance)(current)2(time)=(voltage)(current)(time).Chemicalenergy.Chemicalenergy(CE)hasalwaysbeenfamiliartohumansintheformoffireandfood.Wewilllearnhowtomeasurechemicalenergylaterinthecourse.Exercise.Describehowonecantestifelectrostaticsoffersaformofenergy.Exercise.Findthenutritionenergyofabanana.Ifthisnutritionenergyisfullyconvertedtothepotentialenergyofthebanana,whatwillbetheheightofthebanana?

EnergybelongstomanysciencesEnergyplayspartsinmanysciences.Allhavemuchtoclaimaboutenergy:formsofenergy,storesofenergy,carriersofenergy,conversionofenergyfromoneformtoanother,andflowofenergyfromoneplacetoanother.Associatedwiththesewords—forms,stores,carriers,conversion,andflow—areagreatvarietyofinventionsanddiscoveries.Examplesincludefire,food,blood,wind,rivers,springs,capacitors,waterwheels,windmills,steam,engines,refrigerators,turbines,generators,batteries,lightbulbs,andsolarcells.Theseyouhavelearned,andwilllearnmore,frommanycourses(includingthisone),aswellasfromdailylife.Thesefactsdonotoriginatefromthermodynamics,butwewillusethemjustasweusefactsincalculus.Wedonotsteallinesfromothersciences;weborrow.Ithasbeencommontoletthesupportingactor—energy—todominatetheplayofthermodynamics.Wewillavoidthispitfall.Wewillletenergyplayitssupportingrole,alongwithspace,matter,andcharge.

tothermal tomechanical toelectrical tochemical

thermal heatexchanger engine thermocouple reaction

mechanical friction turbine generator fracture

electrical resistor compressor capacitor chargingbattery

chemical fire,food muscle battery reaction

Exercise.Whatisthefunctionofaturbine?Howdoesitwork?Linkyouranswertoavideoonline.Exercise.Whatisthefunctionofagenerator?Howdoesitwork?Linkyouranswertoavideoonline.

Anisolatedsystemconservesenergy,space,matter,andchargeBesideenergy,wenowaddafewothersupportingroles:space,matter,andcharge.Imakeahalfbottleofwaterintoanisolatedsystem.Iclosethebottlesothatwatermoleculescanneitherenternorleavethebottle.Ithermallyinsulatethebottletostopanyenergytransferbyheat.Idonotsqueezethebottle,sothatthevolumeofthebottleisfixed.Thereisalsonotransferofchargebetweenthesystemanditssurroundings.Westatetheprinciplesofconservation:anisolatedsystemconservesenergy,space,matter,andcharge.Thermodynamicswillusetheseconservedquantitiesasinternalvariables.Ofcourse,internalvariablesneednotberestrictedtoconservedquantities.Allconservedquantitiesobeysimilarmathematics,andareconvenienttostudyinparallel.Exercise.Whatarespace,matter,andcharge?Doyouknowwhyeachisconserved?

AclassificationofsystemsDependingonthemodesofinteractionbetweenthesystemsandtheirsurroundings,weclassifysystemsintoseveraltypes.

transfermatter

transferspace

transferenergy

isolatedsystem no no no

thermalsystem no no yes

closedsystem no yes yes

opensystem yes yes yes

Thermalsystem.Athermalsysteminteractswithitssurroundingsinonemode:transferenergy.Forexample,abottleofwaterisathermalsystem.Wecapthebottletopreventmoleculestoleakinorout.Wemakethebottlerigidtofixitsvolume.Wecanstillchangetheenergyofthewaterbyplacingthebottleoveraflame,orbyshakingthebottle.Closedsystem.Aclosedsystemanditssurroundingsdonottransfermatter,buttransferspaceandenergy.Consideracylinder-pistonsetupthatencloseswatermolecules.Somewatermoleculesformaliquid,andothersformavapor.Wecanmakethewatermoleculesinsidethecylinderclosedsystembysealingthepiston,sothatnomoleculeswillleakinorout.Thecylinder-pistonsetupinteractswiththerestoftheworldintwoways.First,whenweightsareaddedontopofthepiston,thepistonmovesdownandreducesthevolumeinsidethecylinder.Second,whenthecylinderisbroughtoveraflame,theflameheatsupthewater.Opensystem.Anopensystemanditssurroundingstransfermatter,space,andenergy.Abottleofwater,oncethecapisremoved,isanopensystem.Exercise.Describeamethodtokeepwaterhotforalongtime.Whatcanyoudotoprolongthetime?Whatmakeswatereventuallycooldown?Exercise.Foreachtypeofsystemlistedabove,giveanexample.Ineachexample,describeallmodesofinteractionbetweenthesystemanditssurroundings.

TransferenergybetweenaclosedsystemanditssurroundingsAhalfbottleofwater.Hereisthehalfbottleofwateragain.Icapthebottletopreventmoleculesfromleakinginorout.Ishakethebottle,orjusttouchit.Inbothcases,myhandtransfersenergytothebottle.Inshakingthebottle,myhandtransfersenergytothebottlebywork,throughforcetimesdisplacement.Intouchingthebottle,myhandtransfersenergytothebottlebyheat,throughmolecularvibration.Thehalfbottleofwaterisaclosedsystem.Workandheataretwowaystotransferenergybetweentheclosedsystemanditssurroundings.

Transferenergybywork.Wehavelearnedmanywaysofdoingworkinmechanicsandelectrodynamics.Wenowrecalltwoexamples.Theexpansionofagas.Aweightisplacedontopofapiston,whichsealsacylinderofgas.Forbrevity,byaweightwemeantheforceactingonthepistonduetobothablockofmassandthepressureofthesurroundingair:weight=mg+P0A.Heremismass,gistheaccelerationofgravity,P0isthepressureofsurroundingair,andAistheareaofthepiston.Assumethatthepistonmoveswithnofriction.Thebalanceoftheforcesactingonthepistonrelatestheforceoftheweight,F,tothepressureofthegasinthecylinder,P:weight=PA.Whenthepistonraisesitsheightbydz,thegasexpandsitsvolumebydV=Adzanddoesworktotheweight:(weight)dz=PdV.Theexpansionofthegasraisestheweight.Resistiveheating.Asasecondexample,aresistorisplacedinacontainerofwater.WhenavoltageVisappliedtothetwoendsoftheresistor,anelectriccurrentIgoesthroughtheresistor,andtheworkdonebythevoltageperunittimeisIV.Theelectricalworkheatsthewater.Transferenergybyheat.Energytransfersbyheatinseveralways.Conduction.Energycangothroughamaterial.Atamacroscopicscale,thematerialremainsstationary.Atamicroscopicscale,energyiscarriedbytheflowofelectronsandvibrationofatoms.Convection.Energycangofromonesystemtoanotherwiththeflowofafluid.Thiswayofenergytransferinvolvesthetransferofmatterbetweensystemsandispresentforanopensystem.Radiation.Energycanbecarriedbyelectromagneticwaves.Becauseelectromagneticwavescanpropagateinvacuum,twosystemscantransferenergywithoutbeinginproximity.Signconvention.Givenaclosedsystem,weadoptthefollowingsignconvention.

● Q>0,energytransferbyheattotheclosedsystemfromthesurroundings.● W>0,energytransferbyworkfromtheclosedsystemtothesurroundings.● ΔU>0,increaseoftheinternalenergyoftheclosedsystem.

Someauthorsadoptothersignconventions.Youcanadoptanysignconvention,solongasyoumakeyoursignconventionexplicitinthebeginningofathought,anddonotchangethesignconventioninthemiddleofthethought.Thefirstlawofthermodynamics.Foraclosedsystem,thefirstlawofthermodynamicsstatesthatQ=W+ΔU.Internalenergyisapropertyoftheclosedsystem.Neithertransferenergybyworknortransferenergybyheatisapropertyoftheclosedsystem;theyaremethodsofthetransferofenergybetweentheclosedsystemanditssurroundings.Wealreadyknowhowtomeasuretheinternalenergyandtheenergytransferbywork.Thefirstlawofthermodynamicsdefinestheenergytransferbyheat.Wedoourbesttoidentifyvariousprocessesthattransferenergybywork.Energytransferbyheatisthenthetransferofenergythatwedonotbothertocallwork.Thus,thefirstlawofthermodynamicsisnotreallyalaw;itjustdefinestheenergytransferbyheat.Inoldliterature,heatwassometimesusedasasynonymforthermalenergy.Wewillsticktothemodernusage.Heatisamethodofenergytransfer,whereasthermalenergyisaformofenergy.Thetwoconceptsaredistinct.Onecanincreasethethermalenergyofasystembyworkorheat.

EntropyandenergyNowthatyouhaveseenbothentropyandenergy,youarereadytocritiquethefollowingextractfromapaperbyafounderofthermodynamics(Clausius1865).WemightcallSthetransformationalcontentofthebody,justaswetermedthemagnitudeUitsthermalandergonalcontent.ButasIholdittobebettertoborrowtermsforimportantmagnitudesfromtheancientlanguages,sothattheymaybeadoptedunchangedinallmodernlanguages,IproposetocallthemagnitudeStheentropyofthebody,fromtheGreekwordτροπη),transformation.Ihaveintentionallyformedthewordentropysoastobeassimilaraspossibletothewordenergy;forthetwomagnitudestobedenotedbythesewordsaresonearlyalliedintheirphysicalmeanings,thatacertainsimilarityindesignationappearstobedesirable.

1. Theenergyoftheuniverseisconstant.2. Theentropyoftheuniversetendstoamaximum.

Exercise.Critiquethisextract.Taketheword“universe”tomeananisolatedsystem.Doyouagreethatentropyandenergyarenearlyalliedintheirphysicalmeanings?Howcantheentropyofanisolatedsystemincrease?

ThermalsystemNowwehaveanall-starcastofactors.Letuswatchthemplay,actbyact.Letactonestart:theunionofentropyandenergyproducesachild—temperature.Thechildissoprodigiousthatitismuchbetterknownthanitsparents,entropyandenergy.

AfamilyofisolatedsystemsofasingleindependentvariationCharacteristicfunctionS(U).Athermalsystemanditssurroundingsinteractinonemodeonly:transferenergy.WhenthethermalenergyofthethermalsystemisfixedatavalueU,thethermalsystembecomesanisolatedsystem.DenotethenumberofquantumstatesofthisisolatedsystembyΩ(U).AstheenergyUofthethermalsystemvaries,thefunctionΩ(U),oritsequivalent,S(U)=logΩ(U),characterizesthethermalsystemasafamilyofisolatedsystems,capableofoneindependentvariation.WecallS(U)thecharacteristicfunctionofthethermalsystem.Laterwewilldeterminethisfunctionbyexperiment—thatis,wewillcountthenumberofquantumstatesofeachmemberisolatedsystemexperimentally.Hydrogenatom.Ahydrogenatomchangesitsenergybyabsorbingphotons.Whenisolatedataparticularvalueofenergy,thehydrogenatomhasafixedsetofquantumstates.Eachquantumstateinthesetischaracterizedbyadistinctelectroncloudandspin.Thehydrogenatomisathermalsystem.Itscharacteristicfunctionhasbeencomputedinquantummechanics:Ω(−13.6eV)=2,Ω(−3.39eV)=8,Ω(−1.51eV)=18,...…ThedomainofthefunctionΩ(U)isasetofdiscretevaluesofenergy:−13.6eV,-3.39eV,-1.51eV,…TherangeofthefunctionΩ(U)isasetofintegers:2,8,18,….Forthehydrogenatom,thevaluesofenergyhavelargegaps.

Ahalfbottleofwater.Ahalfbottleofwaterisathermalsystem.Wecantransferenergytothewaterinmanyways,bytouch,fire,shake,andelectriccurrent,etc.Wesealthebottletopreventanytransferofmatter.Wemakethebottlerigidtofixthevolume.Foracomplexsystemlikeahalfbottleofwater,thevaluesofenergyaresocloselyspacedthatweregardtheenergyofthesystemasacontinuousrealvariable.ThecharacteristicfunctionS(U)isacontinuousfunction.Exercise.Describeafewmoreexamplesofthermalsystems.

GeneralfeaturesofthefunctionS(U)Energy-entropyplane.Inaplane,wedrawenergyandentropyastwoperpendicularaxes,withenergyasthehorizontalaxisandentropyastheverticalaxis.Drawingtwoaxesperpendicularlyisaconvention,andhasnoexperimentalsignificance.Whatdoesitevenmeantosaythatenergyisperpendiculartoentropy?Theworldworkswellifwedrawthetwoaxeswithanarbitraryangle,ornotdrawthematall.Butwewillfollowtheconventionanddrawthetwoaxesperpendicularly.Inlinearalgebra,wecallenergyUandentropyStwoscalars.Apairofthevaluesofenergyandentropy(U,S)iscalledavector.Theenergy-entropyplaneiscalledatwo-dimensionalvectorspace.FeaturesoftheS(U)curve.WecharacterizeathermalsystemwithafunctionS(U),whichisacurveintheenergy-entropyplane.Ofcourse,differentthermalsystemshavedifferentcharacteristicfunctions.Severalfeaturesarecommontoallthermalsystems.WewilllistthesecommonfeaturesinmathematicaltermsofthecurveS(U)here,andwillrelatethemtoexperimentalobservationsasweprogress.

1. Becauseenergyisrelative,thecurveS(U)cantranslatehorizontallywithoutaffectingthebehaviorofthethermalsystem.

2. Becauseentropyisabsolute,thecurvestartsatS=0,andcannotbetranslatedupanddown.

3. Thebehaviorofathermalsystemisoftenindependentofthesizeofthesystem.Forexample,1kgofwaterbehavesthesameas2kgofwater.Bothenergyandentropyislinearinthesizeofthethermalsystem.Aswechangethesizeofthethermalsystem,thecurveS(U)changessize,butkeepstheshape.

4. Asentropyapproacheszero,thecurveS(U)approachesthehorizontalaxisvertically.Thatis,asS→0,dS(U)/dU→∞.

5. Themoreenergy,themorequantumstates.Thus,Ω(U)isanincreasingfunction.Becauselogarithmisanincreasingfunction,S(U),isalsoanincreasingfunction.Thatis,theslopeofthecurveS(U)ispositive,dS(U)/dU>0.

6. ThecurveS(U)isconvexupward.Thatis,theslopedS(U)/dUdecreasesasUincreases,orequivalently,d2S(U)/dU2<0.

PhrasesassociatedwithafamilyofisolatedsystemsAsingleisolatedsystem.Forasingleisolatedsystem,asnotedbefore,severalphrasesaresynonymous:asystem“isolatedforalongtime”isasystem“flippingtoeveryoneofitsquantumstateswithequalprobability”,andisasystem“inequilibrium”.Wheneverwespeakofequilibrium,weshouldidentifyasystemisolatedforalongtime.Associatedwitheachisolatedsystemisasetofquantumstates—thesamplespace.Theisolatedsystemflipstoitsquantumstatesceaselesslyandrapidly.Outofequilibrium,theisolatedsystemflipstosomeofitsquantumstatesmoreoftenthanothers.Inequilibrium,theisolatedsystemflipstoeveryoneofitsquantumstateswithequalprobability.Afamilyofisolatedsystems.Athermalsystemisafamilyofisolatedsystemscapableofoneindependentvariation—energy.Eachmemberinthisfamilyisadistinctisolatedsystem,hasafixedvalueofenergy,andflipsamongthequantumstatesinitsownsamplespace.Wedescribeafamilyofisolatedsystemsusingseveraladditionalphrases.Thesephrasesareapplicabletoanyfamilyofisolatedsystems,buthereweintroducethesephrasesusingafamilyofisolatedsystemsthatconstituteathermalsystem.Thermodynamicstate.Inafamilyofisolatedsystems,amemberisolatedsysteminequilibriumiscalledathermodynamicstate,orastateofequilibrium,ofthefamily.Wenowusetheword“state”intwoways.Anisolatedsystemhasmanyquantumstates,butasinglethermodynamicstate.Athermodynamicstateissynonymoustoanisolatedsysteminequilibrium.Inathermodynamicstate,theisolatedsystemflipstoeveryoneofitsquantumstateswithequalprobability.

Athermalsystemisafamilyofisolatedsystems.Eachmemberisolatedsystemcorrespondstoonethermodynamicstateofthethermalsystem,specifiedbyavalueofenergy.Asenergyvaries,thethermalsystemcanbeinmanythermodynamicstates.EachthermodynamicstatecorrespondstoapointonthecurveS(U).Thermodynamicprocess.Asequenceofthermodynamicstatesiscalledareversiblethermodynamicprocess,orquasi-equilibriumprocess.Eachthermodynamicstateinthisprocesscorrespondstoadistinctisolatedsysteminequilibrium.Athermodynamicprocessissynonymoustoafamilyofisolatedsystems,eachbeinginequilibrium.Athermalsystemiscapableofonetypeofthermodynamicprocess:changingenergy.Aftereachchangeofenergy,weisolatethethermalsystemlongenoughtoreachequilibrium.Functionofstate.Wespecifyathermodynamicstateofthethermalsystembyavalueofenergy.Energyiscalledafunctionofstate.Thewordstateheremeansthermodynamicstate,notquantumstate.Afunctionofstateisalsocalledathermodynamicproperty.Entropyisalsoafunctionofstate,soisthenumberofquantumstates.Forathermalsystem,wewillsoonintroducefourotherfunctionsofstate:temperature,thermalcapacity,Massieufunction,andHelmholtzfunction.Equationofstate.OncetheenergyUisfixed,athermalsystembecomesanisolatedsystemofafixedsamplespace,sothattheentropySisalsofixed.Consequently,givenathermalsystem,theenergyandentropyarerelated.TherelationS(U)iscalledanequationofstate.Again,thewordstateheremeansthermodynamicstate,notquantumstate.Ingeneral,anequationofstateisanequationthatrelatesthermodynamicpropertiesofafamilyofisolatedsystems.Forathermalsystem,wewillsoonintroduceseveralequationsofstate,inadditiontoS(U).

DissipationofenergyEmpiricalfacts.WhenIshakeahalfbottleofwater,energytransfersfrommymuscletothewater,andincreasesthetemperatureofthewater.Icanalsotransferenergytothewaterusinganelectriccurrent.Thistransferofenergygoesthroughseveralsteps.Anelectricoutlettransfersenergybyworktoametalwire,wherethevoltageoftheelectricoutletmoveselectronsinthemetalwire.Theresistanceofthemetalwireheatsupthemetalwire.Letussaythatthemetalwireisimmersedinthewater,andtransfersenergybyheattothewater.Icanalsodropaweightintothebottleofwaterfromsomeheight.Theweightcomestorestinthewaterandheatsupthewater.

Inthethreeexamples,theenergystartsintheformof,respectively,chemicalenergyinthemuscle,electricalenergyintheoutlet,andthepotentialenergyintheweightatsomeheight.Aftertransferringintothewater,chemical,electrical,andpotentialenergyconvertstothermalenergyofthewater.Thechemical,electrical,andpotentialenergyaresaidtodissipateintothermalenergy.Thermodynamicanalysis.Tospeakofthedissipationofenergy,weidentifythermalenergyandotherformsofenergyinasingleisolatedsystem.Theisolatedsystemconservesenergy,butdissipatestheotherformsofenergyintothermalenergy.Solongasthesystemisisolated,thedirectionofdissipationisirreversible,fromotherformsofenergytothermalenergy,nottheotherwayaround.Thermalenergyiscalledlow-gradeenergy.Wenextanalyzethefallingweightintowaterusingthebasicalgorithm. Constructanisolatedsystemwithaninternalvariable.Thebottleofwaterandtheweighttogetherconstituteanisolatedsystem.Beforetheweightdrops,thethermalenergyofwaterisU0,andthepotentialenergyoftheweightisPE.Accordingtotheprincipleoftheconservationofenergy,theenergyoftheisolatedsystemisfixed.Aftertheweightcomestorestinwater,thepotentialenergyoftheweightvanishes,andthethermalenergyofthewaterisU0+PE.Thus,thethermalenergyofthewater,U,isaninternalvariableoftheisolatedsystem,increasingfromU0toU0+PE.Findthesubsetentropyasafunctionoftheinternalvariable.Thebottleofwaterisathermalsystem,characterizedbyafunctionS(U),whichisanincreasingfunction.LettheentropyoftheweightbeSweight,whichistakentobeunchangedafterfallingintothewater.TheentropyoftheisolatedsystemisthesumS(U)+Sweight.Maximizethesubsetentropytoreachequilibrium.Beforetheweightdrops,theentropyoftheisolatedsystemisS(U0)+Sweight.Aftertheweightcomestorestinthewater,S(U0+PE)+Sweight.ThelawoftheincreaseofentropyrequiresthatS(U0+PE)+Sweight>S(U0)+Sweight.BecauseS(U)isanincreasingfunction,theaboveinequalityholdsifthepotentialenergyoftheweightchangestothermalenergyinthewater.Thepotentialenergyoftheweightissaidtodissipateintothethermalenergyinthewater.Theisolatedsystemmaximizestheentropywhenthepotentialenergyoftheweightfullychangestothethermalenergyofthewater,theweightcomestorest,andtheisolatedsystemreachesequilibrium.Increasethesubsetentropytoseeirreversibility.Thereversechangewouldviolatethelawoftheincreaseofentropy,andthereforeviolatethefundamentalpostulate.Theweight,afterrest

inwater,willnotdrawthermalenergyfromthewaterandjumpup.Dissipation—theconversionofpotentialenergytothermalsystem—isirreversible.Whatmakethermalenergylow-gradeenergyisitshighentropy.Theirreversibilityisunderstoodfromthemolecularpicture.Thermalenergycorrespondstomolecularmotion,whereasthejumpingupoftheweightcorrespondstoallmoleculesaddingvelocityinonedirection.Theformercorrespondstomorequantumstatesthanthelatter.Theisolatedsystemchangesinthedirectionthatincreasesthenumberofquantumstates.Consequently,thepotentialenergydissipatesintothermalenergy,nottheotherwayaround.TheKelvin-Planckstatementofthesecondlawofthermodynamics.Thisanalysisconfirmsageneralempiricalobservation.Itisimpossibletoproducenoeffectotherthantheraisingofaweightbydrawingthermalenergyfromasinglethermalsystem.ThisobservationiscalledtheKelvin-Planckstatementofthesecondlawofthermodynamics.Exercise.Usethebasicalgorithmtoanalyzeheatingwaterbyanelectriccurrent.Exercise.Usethebasicalgorithmtoanalyzeheatingbyfriction.

Isentropicprocess.ReversibilityAfallingapplelosesheight,butgainsvelocity.Solongasthefrictionofairisnegligible,thesumofthepotentialenergyandkineticenergyoftheappleisconstant,andthethermalenergyoftheappleisalsoconstant.Wemodeltheapple,togetherwithapartofspacearoundtheapple,asanisolatedsystem.Theheightoftheappleistheinternalvariableoftheisolatedsystem.Whentheappleisataparticularheight,z,theisolatedsystemflipsamongasetofquantumstates,andhasacertainentropy,S(z).TheprocessoffallingkeepsthesubsetentropyS(z)fixed,independentoftheinternalvariablez.Suchaprocessiscalledanisentropicprocess.Anisentropicprocessofanisolatedsystemisreversible.Wecanarrangeasetuptoreturntheappletoitsoriginalheightwithoutcausinganychangetotherestoftheworld.Forexample,wecanlettheapplefallalongacircularslide.Solongasfrictionisnegligible,theapplewillreturntothesameheight.Ofcourse,frictionisinevitableinreality;wehaveneverseenanapplegoupanddownaslideforalongtime.Theappleinevitablystopsaftersometime.Butafrictionlessprocesscanbeausefulidealization.Forexample,theplanetEarthhasbeenmovingaroundtheSunforaverylongtime.Foranisolatedsystem,thetwowords,isentropicandreversible,areequivalent.Bothadjectivesdescribeanisolatedsystemthatkeepsthenumberofquantumstatesunchangedwhenaninternalvariablechanges.Anisentropic(orreversible)processisalsocalledaquasi-equilibrium

process.Toavoidincreasingentropy,theprocessmustbeslowenoughfortheisolatedsystemtohavealongenoughtimetoreachequilibriumateverypointalongtheprocess.Exercise.Describeanotherisentropicprocess.

ThermalcontactWehavejustanalyzedthedissipationofenergyfromahighgradetoalowgrade.Wenowlookthetransferenergyfromonethermalsystemtoanotherthermalsystem.Twothermalsystemsaresaidtobeinthermalcontactifthefollowingconditionshold.

1. Thetwothermalsystemsinteractinonemodeonly:transferenergy.2. Thetwothermalsystemstogetherformanisolatedsystem.

Empiricalfacts.Theprincipleoftheconservationofenergyrequiresthatanisolatedsystemshouldhaveafixedamountofenergy.Itallowsarbitrarypartitionofenergybetweenthetwothermalsystems,solongasthesumoftheenergiesofthetwothermalsystemsremainsconstant.However,oureverydayexperienceindicatestwofacts.

1. Whentwothermalsystemsareinthermalcontact,energytransfersfromonesystemtotheothersystem,one-wayandirreversible.

2. Aftersometime,theenergytransferstops,andthetwothermalsystemsaresaidtoreachthermalequilibrium.

Thermodynamicanalysis.Wenowtracetheseempiricalfactstothefundamentalpostulate.Weusethebasicalgorithmtoanalyzethermalcontact.Constructanisolatedsystemwithaninternalvariable.Thetwothermalsystems,AandB,togetherconstituteanisolatedsystem.Wecallthisisolatedsystemthecomposite.LettheenergiesofthetwothermalsystemsbeUAandUB.Thecompositeofthetwothermalsystemsisanisolatedsystem,andhasafixedamountofenergy,denotedbyUcomposite.Energyisanextensivequantity,sothatUcomposite=UA+UB.Consequently,theisolatedsystemhasasingleindependentinternalvariable,say,theenergyofoneofthethermalsystems,UA.Findthesubsetentropyasafunctionoftheinternalvariable.Thetwothermalsystemsarecharacterizedbytwofunctions,SA(UA)andSB(UB).OncetheinternalvariableUAisfixed,thecompositeflipsamongasubsetofitsquantumstates.DenotethesubsetentropybyScomposite(UA).Entropyisanextensivequantity,sothatScomposite(UA)=SA(UA)+SB(UB).

Equilibrium.WhenonethermalsystemgainsenergydUA,theotherthermalsystemlosesenergybythesameamount,sothatdScomposite(UA)=(dSA(UA)/dUA-dSB(UB)/dUB)dUA.Afterbeingisolatedforalongtime,thecompositereachesequilibrium,andthesubsetentropymaximizes,dScomposite(UA)=0,sothatdSA(UA)/dUA=dSB(UB)/dUB.Thisequationistheconditionofthermalequilibrium,anddeterminestheequilibriumpartitionofenergybetweenthetwothermalsystemsAandB.Irreversibility.Priortoreachingequilibrium,thesubsetentropyofthecompositeincreasesintime,dS>0,sothatIfdSA(UA)/dUA>dSB(UB)/dUB,thendUA>0,andthermalsystemAgainsenergyfromthermalsystemB;IfdSA(UA)/dUA<dSB(UB)/dUB,thendUA<0,andthermalsystemAlosesenergytothermalsystemB.Theseinequalitiesdictatethedirectionofenergytransfer.Inthermalcontact,thedirectionofenergytransferisonewayandirreversible.Thisanalysisconfirmsthetwoempiricalfactsofthermalcontact.

1. Withoutlossofgenerality,assumethat,rightafterthermalcontact,dSA(UA)/dUA>dSB(UB)/dUB.OuranalysisshowsthatsystemAgrainsenergyfromsystemB.Thedirectionoftheenergytransferisone-wayandirreversible.

2. WefurtherassumethatSA(UA)andSB(UB)areconvexfunctions.Consequently,assystemAgainsenergy,theslopedSA(UA)/dUAdecreases.AssystemBlosesenergy,theslopedSB(UB)/dUBincreases.Ouranalysisshowsthatthetransferofenergycontinuesuntilthetwoslopesareequal,whenthetwothermalsystemsreachthermalequilibrium.

DefinitionoftemperatureDefinethetemperatureTby1/T=dS(U)/dU.BothSandUareextensivethermodynamicproperties.ThedefinitionensuresthatTisanintensivethermodynamicproperty.

Wecanusethewordtemperaturetoparaphrasetheaboveanalysisofthermalcontact.1. Whentwothermalsystemsarebroughtintothermalcontact,energytransfersonlyin

onedirection,fromthesystemofhightemperaturetothesystemoflowtemperature.ThisobservationiscalledtheClausiusstatementofthesecondlawofthermodynamics.

2. Aftersometimeinthermalcontact,energytransferstops,andthetwosystemshavethesametemperature.Thisobservationiscalledthezerothlawofthermodynamics.

Butwaitaminute!AnymonotonicallydecreasingfunctionofdS/dUwillalsoserveasadefinitionoftemperature.Whatissospecialaboutthechoicemadeabove?Nothing.Itisjustachoice.Indeed,allthatmattersistheslopedS/dU.Whatwecalltheslopemakesnodifference.Rangeoftemperature.Forathermalsystem,thefunctionS(U)isamonotonicallyincreasingfunction.Themoreenergy,themorequantumstates,andthemoreentropy.Thus,thedefinition1/T=dS(U)/dUmakestemperaturepositive.Usuallyweonlymeasuretemperaturewithinsomeinterval.Extremelylowtemperaturesarestudiedinthescienceofcryogenics.Extremelyhightemperaturesarerealizedinstars,andotherspecialconditions.Whatcanyoudofortemperature?Anessentialsteptograspthermodynamicsistogettoknowthetemperature.Wedefinetemperaturebyanalyzinganeverydayexperience—thermalcontact.Howdoestemperaturerisesupasanabstractionfromeverydayexperienceofthermalcontact?Howdoestemperaturecomesdownfromtheunionofenergyandentropy?Letmeparaphraseabetter-knownBostonian.Andso,myfellowthermodynamicist:asknotwhattemperaturecandoforyou—askwhatyoucandofortemperature.Exercise.Writeanessaywiththetitle,Whatistemperature?Howdoestemperaturerisesupasanabstractionfromeverydayexperienceofthermalcontact?Howdoestemperaturecomesdownfromtheunionofenergyandentropy?Exercise.Hotnessandhappinessaretwocommonfeelings.Wemeasurehotnessbyanexperimentalquantity,temperature.Canwedosoforhappiness?Why?

TwounitsoftemperatureRecallthedefinitionoftemperature,1/T=dS/dU.Becauseentropyisadimensionlessnumber,temperaturehastheunitofenergy,Joule.Butadifferentunitfortemperature,Kelvin,iscommonlyused.Theconversionfactorbetweenthetwounitsoftemperature,JouleandKelvin,iscalledtheBoltzmannconstantk,definedby1.380649×10−23Joule=1Kelvin.

Wiki2019redefinitionoftheSIbaseunits.TheoldandnewdefinitionsofKelvinstartat28:40ofthisvideo.ModernCelsiusscale.Temperatureisdefinedbytheequation1/T=dS(U)/dU.Anymonotonicfunctionofthistemperaturedefinesanarbitraryscaleoftemperature.Herewewilljustmentionacommonlyusedscale,theCelsiusscaleC,definedbyC=T−273.15K(Tintheunitofkelvin).ThismoderndefinitionoftheCelsiusscalediffersfromthehistoricaldefinition.Specifically,themeltingpointandtheboilingpointofwaterarenolongerusedtodefinethemodernCelsiusscale.Rather,thesetwotemperaturesaredeterminedbyexperimentalmeasurements.Theexperimentalvaluesareasfollows:watermeltsat0°Candboilsat99.975°C.ThemeritofusingCelsiusineverydaylifeisevident.Itfeelsmorepleasanttohearthattoday’stemperatureis20degreeCelsiusthan293.15Kelvin,or404.34×10−23Joule.Exercise.HowwasCelsiusscaleoriginallydefined?Whatmadetheinternationalcommitteetoredefineit?Exercise.Howwouldyoudefendthechoiceoftheconversionfactor,1.380649×10−23Joule=1Kelvin?

UnitofentropyDimensionlessentropy.Theentropyofanisolatedsystem,S,isdefinedbythenumberofquantumstatesoftheisolatedsystem,Ω,asS=logΩ.Theentropyisadimensionlessnumber.Aunitofentropy.Byahistoricalaccident,however,theentropyisgivenaunit.Topreservetheequation1/T=dS/dU,whenweusetemperatureintheunitofKelvin,wemultiplythedimensionlessentropybyk.TheentropyisthenreportedintheunitofJK-1.WriteS=klogΩ.TheBoltzmannconstant,k=1.380649×10−23Joule/Kelvin,doesnohonortothethreegreatscientists.Itisunfortunatethatweassociateanunsightlynumberwiththethreegreatnames.Thenumberisjusttheconversionfactorbetweenthetwounitsoftemperature,JouleandKelvin.

Thisunitofentropymaygiveanimpressionthattheconceptofentropydependsontheconceptsofenergyandtemperature.Thisimpressioniswrong.Aswehaveseen,forasamplespaceofafinitenumberΩofsamplepoints,entropyislogΩ.Entropyandenergyaretwoindependentconcepts.Whereasentropyisameasureofthesizeofasamplespaceofanisolatedsystem,energyisjustoneofnumerousquantitiesthatcanserveasinternalvariables.Temperatureisthechildoftheunionofentropyandenergy,1/T=dS/dU.

IdealgaslawkTistemperatureintheunitofenergy.TheBoltzmannconstantkhasnofundamentalsignificance;itmerelydefinesaunitoftemperaturemoremanageableineverydayuse.Foranyfundamentalresult,ifTisinunitsofKelvin,theproductkTmustappeartogether.OftenpeoplecallkTthermalenergy.Thisdesignationisingeneralincorrect.Thereisnoneedtogiveanyotherinterpretation:kTistemperatureintheunitofenergy.Idealgaslaw.Thechangeofunitisillustratedwiththeidealgaslaw.TheidealgaslawtakestheformPV=NT.wherePisthepressure,Vthevolume,Nthenumberofmolecules,andTthetemperatureintheunitofenergy.Historically,thelawofidealgaseswasdiscoveredempiricallybeforethediscoveryofentropy.Laterwewillshowthatthetwodefinitionsoftemperature,PV=NTand1/T=dS(U)/dU,arethesame.WhenTisintheunitofKelvin,theidealgaslawbecomesthatPV=NkT.AvogadroconstantNA.Accordingtothe2019redefinitionoftheSIbaseunits,theAvogadroconstantNAisdefinedby6.02214076×1023items=1moleofitems.Ifweusemoleataunitfortheamountofgas,anduseKelvinasaunitfortemperature,theidealgaslawbecomesthatPV=nRT,wherenistheamountofthegasintheunitofmoleand R=kNA=8.314JK

-1mole-1.

ThisquantityRiscalledtheuniversalgasconstant,andistheproductoftwohistoricalaccidents:theKelvinunitfortemperature,andtheAvogadrounitforamountofitems.Thename“theuniversalgasconstant”soundspretentioustothemodernear.

VaporpressurePartialpressure.Theairaroundusisamixtureofmanyspeciesofmolecules,includingnitrogen,oxygen,carbondioxide,andwater.WikiatmosphereoftheEarth.LetVbethevolumeofamixtureofgas,andNwaterbethenumberofwatermoleculesinthevolume.Thepartialpressureofwaterinthemixture,Pwater,isdefinedbyPwaterV=PwaterkT.Vaporpressure.Thevaporpressureofwateristhepartialpressureofwateratwhichthevaporequilibrateswithacondensedphaseofwater,liquidwaterorice.Thevaporpressuredependsontemperature.Atroomtemperature,thevaporpressureisapproximately3kPa.At100degreesCelsius,thevaporpressureisapproximately100kPa.Wikivaporpressure.Wikivaporpressureofwater.Whenthepartialpressureofwaterisbelowthevaporpressure,theliquidwateroriceevaporates.Whenthepartialpressurethevaporpressure,thewatermoleculesinthegaswillcondense.Relativehumidity.Atagiventemperature,whenamoistairisinequilibriumwiththeliquidwater,wesaythattheairissaturatedwithwater.Ifaircontainsfewerwatermoleculesthanthesaturatedairdoes,thenumberofwatermoleculesintheairdividedbythenumberofwatermoleculesinthesaturatedairiscalledtherelativehumidity.WriteRH=N/Nsatu.Whenthevaporismodeledasanidealgas,therelativehumidityisalsogivenbyRH=P/Psatu,Exercise.Estimatethenumberofwatermoleculesinthevaporinthehalfbottleofwater.Whatisthevolumeofthevapordividedbythenumberofmolecules?Comparethisvaluetothevolumeofanindividualwatermolecule.

SpecificquantitiesBothmassMandvolumeVareextensivequantities.Theirratiosdefinetwointensivequantities:theratioofmasstovolumeM/Viscalledthemassdensity,andtheratioofvolumetomassV/Miscalledthespecificvolume.WealsoreportvolumepermoleculeV/N,orvolumepermoleofmolecules,V/n.SometextbooksgiveV/M,V/N,andV/ndistinctsymbols.Wewilldenotethemallbyv,andletthecontexttellthedifference.WewillsimilarlydenotespecificentropyS/M,entropypermoleculeS/N,andentropypermoleofmoleculesS/nbys.WedenotespecificenergyU/M,energypermoleculeU/N,andenergypermoleofmoleculesU/nbyu.Themassofamoleofwateris18g.Atroomtemperatureandatmosphericpressure,themassdensityofwateris1000kg/m3.Exercise.Howwouldyoudefendthechoiceoftheconversionfactor,6.02214076×1023items=1moleofitems?Exercise.Calculatethemassdensity,specificvolume,volumepermolecule,andvolumepermoleofwatermoleculesatatemperatureof200°Candapressureof100kPa.Exercise.ThegravityoftheEarthpullsmoleculesofgastowardtheEarth,buttheentropyofthegasdispersesthemoleculesintothespace.Assumethatthetemperatureisconstant,howdoesthedensityofaspeciesofmoleculesinthegaschangewiththeelevation?

GeneralfeaturesofthefunctionT(U)Recallthedefinitionoftemperature,1/T=dS(U)/dU.WeplotthefunctionT(U)ontheenergy-temperatureplane.Weuseenergyasthehorizontalaxis,andtemperatureastheverticalaxis.Eachpointonthecurverepresentsathermalsystemisolatedataparticularvalueofenergy.Thatis,eachpointonthecurverepresentsathermodynamicstateofthethermalsystem.NoteseveralgeneralfeaturesofthecurveT(U).

1. Theenergyisdefineduptoanadditiveconstant,sothatthecurveT(U)canbetranslatedalongtheaxisofenergybyanarbitrarilyamount.

2. ThetemperatureTstartsatabsolutezero,ispositive,andhasnoupperbound.3. Forthetimebeing,weassumethatT(U)isanincreasingfunction.

ThesefeaturesofT(U)shouldbecomparedwiththoseofS(U).

Exercise.AthermalsystemcanalsobecharacterizedbyfunctionS(T).DiscussthegeneralfeaturesofthecurveS(T)ontheentropy-temperatureplane.

ThermalcapacityWhentheenergyofathermalsystemchangesbydU,thetemperaturechangesbydT.Definethethermalcapacityby1/C=dT(U)/dU.Becauseenergyisanextensivequantityandtemperatureisanintensivequantity,thermalcapacityisanextensivequantity.Thermalcapacityisafunctionofstate.Whenwereporttemperatureintheunitofenergy,thermalcapacityisdimensionless.WhenwereporttemperatureintheunitKelvin,thethermalcapacityhastheunitJK-1.Thethermalcapacityisafunctionofenergy,C(U).Weassumethatthethermalcapacityispositive.ThisassumptionisequivalenttothatthefunctionT(U)haspositiveslope,andthatthefunctionS(U)isconvex.Recallthedefinitionoftemperature,1/T=dS(U)/dU.Thermalcapacityisalsocalledheatcapacity.Thermalcapacityisafunctionofstate,andisindependentofthemethodofenergytransfer.Forexample,wecanaddenergytoahalfbottleofwaterbywork,suchasshakingthebottle,orelectricheating.Oncewecommittothemodernusageofthewordheatasamethodofenergytransfer,itisinappropriatetonameafunctionofstateusingthewordheat.Specificthermalcapacity.Theenergyneededtoraisethetemperatureofaunitmassofasubstancebyadegreeiscalledthespecificthermalcapacityofthesubstance.Liquidwaterhasapproximatelyaconstantspecificthermalcapacityof4.18J/g/K.Icehasapproximatelyaconstantspecificthermalcapacityof2.06kJ/kg/K.Exercise.(a)Weimmersea100Wlightbulbin1kgofwaterfor10minutes.Assumethatallelectricenergyappliedtothebulbconvertstotheinternalenergyofwater.Howmuchdoesthetemperatureofthewaterincrease?(b)Howmuchdoesthetemperatureofthewaterincreasewhenanapplefallsfromaheightof1mintothewaterandconvertsallitspotentialenergytotheinternalenergyofthewater?Similarexperimentswereconductedin1840stoestablishthatheatisaformofenergy.Exercise.Whatarethechangesinenergyandentropyof1kgofwaterwhenraisedfromthefreezingtotheboilingtemperature?Exercise.When1kgofwateratthefreezingpointismixedwith2kgofwaterattheboilingpoint,whatisthetemperatureinequilibrium?Whatisthechangeofentropyassociatedwiththismixing?

Exercise.Forasolidoraliquid,thechangeofvolumeissmallwhentemperatureincreases.Wemodelthesolidorliquidasathermalsystem,andmeasurethetemperature-energycurveexperimentally.Forasmallrangeoftemperature,thethermalenergyisapproximatelylinearintemperature,U(T)=CT,wherethethermalcapacityCistakentobeaconstant.DerivethecharacteristicfunctionS(U).

Experimentalthermodynamics

DivisionoflaborSomethingsareeasytocalculatetheoretically,othersareeasytomeasureexperimentally.Adivisionoflaborimprovestheeconomicsofgettingthingsdone.Muchofthermodynamicsisonlysensibleintermsofthedivisionoflaborbetweentheoryandexperiment,andbetweenpeopleandmachines.Formostisolatedsystems,countingquantumstatesexperimentallyisfarmoreeconomicthancomputingthemtheoretically. Experimentandtheory(Bryan,Thermodynamics,1907).Itismaintainedbymanypeople(rightlyorwrongly)thatinstudyinganybranchofmathematicalphysics,theoreticalandexperimentalmethodsshouldbestudiedsimultaneously.Itishoweververyimportantthatthetwodifferentmodesoftreatmentshouldbekeptcarefullyapartandifpossiblestudiedfromdifferentbooks,andthisisparticularlyimportantinasubjectlikethermodynamics.

CalorimetryTheartofmeasuringthermalenergyiscalledcalorimetry.Adevicethatmeasuresthermalenergyiscalledacalorimeter.Calorimetryhasbecomeafineartofhighsophistication,anditistoomuchofatangenttotalkaboutcurrentpracticeofcalorimetryinabeginningcourseinthermodynamics.Allweneedtoknowisthatthermalenergyismeasuredroutinely.Ifyouwishtohaveaspecificmethodinmind,justthinkofanelectricheater.Theelectricenergyis(current)(voltage)(time),assumedtobefullyconvertedintothermalenergy.

ThermometryTheartofmeasuringtemperatureiscalledthermometry.Adevicethatmeasurestemperatureiscalledathermometer.Temperatureaffectsallpropertiesofallmaterials.Inprinciple,anypropertyofanymaterialcanserveasathermometer.Thechoiceisamatterofconvenience,accuracy,andcost.Herearetwocommonlyusedthermometers.

Mercury-in-glassthermometer.Amercury-in-glassthermometerreliesonapropertyofmercury:thevolumeexpandsastemperatureincreases.Thus,avolumeindicatesatemperature. Gasthermometer.Anidealgasobeystheequationofstate:PV=NkT,ThisequationrelatestemperatureTtomeasurablequantitiesP,V,andN.Thus,anidealgascanserveasathermometer,calledthegasthermometer.Exercise.Beforeyouacceptanidealgasasathermometer,describeamethodtomeasurethenumberofmoleculesinabottleofgas.Adivisionoflabor.Howdoesadoctordeterminethetemperatureofapatient?Certainlyshedoesnotcounttheenormousnumberofquantumstatesofherpatient.Instead,sheusesathermometer.Letussaythatshebringsamercury-in-glassthermometerintothermalcontactwiththepatient.Uponreachingthermalequilibriumwiththepatient,themercuryexpandsacertainamount,givingareadingofthetemperatureofthepatient.Themanufacturerofthethermometermustassignavolumeofthemercurytoatemperature.Thishecandobybringingthethermometerintothermalcontactwithaflaskofanidealgas.Hedeterminesthetemperatureofthegasbymeasuringitsvolume,pressure,andnumberofmolecules.Also,byheatingorcoolingthegas,hevariesthetemperatureandgivesthethermometeraseriesofmarkings.Anyexperimentaldeterminationofthethermodynamictemperaturefollowsthesebasicsteps:

1. Forasimplesystem,formulateatheorythatrelatestemperaturetoameasurablequantity.

2. Usethesimplesystemtocalibrateathermometerbythermalcontact.3. Usethethermometertomeasuretemperaturesofanyothersystembythermalcontact.

Steps2and3aresufficienttosetupanarbitraryscaleoftemperature.ItisStep1thatmapsthearbitraryscaleoftemperaturetothethermodynamictemperature.Ourunderstandingoftemperaturenowdividesthelaborofmeasuringtemperatureamongadoctor(Step3),amanufacturer(Step2),andatheorist(Step1).Onlythetheoristneedstocountthenumberofquantumstates,andonlyforaveryfewidealizedsystems.

ExperimentaldeterminationofentropyThisisamagicofthermodynamics.Wecancountexperimentallythenumberofquantumstatesofanisolatedsystemofanycomplexity,knowingnothingaboutthequantumstatesthemselves.Weillustratethemethodusingathermalsystem. Hereisastatementofthetask.Givenathermalsystem,measureitscharacteristicfunctionS(U).Recallthedefinitionoftemperature,dS(U)=T-1dU.CountingthenumberofquantumstatesrequiresacombinationofthermometrytomeasureTandcalorimetrytomeasureU.ExperimentalmeasurementofthefunctionT(U).Weaddenergytothethermalsystemby,say,anelectricheater.WemeasurethechangeinenergyUby(time)(resistance)(current)2.Ateachincrementofenergy,weisolatethesystem,waituntilthesystemreachesequilibrium,andmeasuretemperatureT.TheseincrementalmeasurementsdeterminethefunctionT(U). Determinationofentropy.Recallthedefinitionoftemperature:dS(U)=T-1dU.ThisequationrelatesthefunctionS(U)toexperimentallymeasurablequantity,UandT.OncethefunctionT(U)ismeasuredexperimentally,anintegrationdeterminesthefunctionS(U).Inthisintegration,setS=0asT→0.Thatis,atthegroundstate,thenumberofquantumstatesislow,andissettobeone.Thisisastatementofthethirdlawofthermodynamics.Ontheenergy-entropyplane,thecurveS(U)approachesthehorizontalaxisvertically.Often,themeasurementonlyextendstoatemperaturemuchaboveabsolutezero.Assumethatthemeasurementgivestheenergy-temperaturecurveintheintervalbetweenT0andT.Uponintegrating,weobtainS(T)-S(T0).SuchanexperimentleavesS(T0)undetermined.RecallthatthefunctionS(U)characterizethethermalsystemasafamilyofisolatedsystems.Thus,wecancountthenumberofquantumstatesofeachmemberisolatedsysteminthefamily.Debyemodel(1912).ToillustratethedeterminationofthecharacteristicfunctionS(U),considertheDebyemodel.Nearabsolutezero,theinternalenergyofasolidtakestheformU=aT4,whereaisaconstant.Debyeobtainedthisexpressionfromamicroscopicmodel.HereweregardtheU(T)asacurveobtainedfromexperimentalmeasurement.

Inverttheaboveequation,andwehaveT=(U/a)1/4.IntegratingdS=T-1dU,weobtainthatS(U)=(4/3)a1/4U3/4.WehaveusedtheconditionS=0atT=0.Exercise.CalculatethefunctionC(U)fortheDebyemodel.

Melting

EmpiricalfactsIceandliquidarecalledtwophasesofwater.Themeltingoficeiscalledaphasetransition.Afixednumberofwatermoleculesisathermalsystem.Itsthermodynamicstatesarecapableofoneindependentvariation.Werepresentthethermodynamicstatesbyacurve,T(U),onthetemperature-energyplane,withenergyasthehorizontalaxis,andtemperatureastheverticalaxis.

Solid.Inablockofsolidwater,ice,watermoleculesformacrystal—aperiodiclatticeofwatermolecules.Individualwatermoleculesvibratearoundtheirlatticesites,butchangeneighborsrarely.Theblockoficeusuallycontainsmanygrainsofthesamecrystal.Betweentwograins,alayerofwatermoleculesdonotbelongtothelatticeofeithergrain,andiscalledagrainboundary.Thegrainboundaryisthin,andthethicknessisnomorethanafewmolecules.Consequently,thewatermoleculesinthegrainboundarycontributenegligiblytotheextensivequantities(i.e.,entropy,energy,andvolume)oftheblock.Asablockoficereceivesenergy,itstemperatureincreases.Icehasanapproximatelyconstantvalueofspecificthermalcapacity:cs=2.06kJ/kg/K.

Thesolidphasecorrespondstoalineofthisslopeintheenergy-temperatureplane.Eachpointonthelinecorrespondstoonethermodynamicstateofthesolidphase.Inthisthermodynamicstate,oncethegrainboundariesareneglected,allpartsoftheblockaresimilar.Theblockoficeissaidtobeinahomogeneousthermodynamicstate.Liquid.Inabottleofliquidwater,watermoleculesdonotformalattice.Thewatermoleculestouchoneanotherandchangeneighborsreadily.Astheliquidreceivesmoreenergy,thetemperatureincreasesagain.Waterhasanapproximatelyconstantvalueofspecificthermalcapacity:cf=4.18kJ/kg/K.Thesubscriptfstandsforflüssigkeit,theGermanwordforliquid.Theliquidphasecorrespondstoalineofthisslopeintheenergy-temperatureplane.Eachpointonthelinecorrespondstoonethermodynamicstateoftheliquidphase.Inthisthermodynamicstate,allpartsoftheliquidinthebottlearesimilar.Thebottleofwaterissaidtobeinahomogeneousthermodynamicstate.Mixtureofsolidandliquid.Whenthetemperatureincreasestoacertainlevel,astheblockoficereceivesmoreenergy,itstemperatureremainsfixed,butsomewatermoleculesbecomeliquidwater.Thefixedtemperatureiscalledthemeltingtemperature,Tm.Forwater,themeltingtemperatureisTm=273.15K.Amixtureoficeandwater,whenisolated,conservestheamountofenergyandthenumberofwatermolecules.Afterbeingisolatedforalongtime,thetwophasescoexistinequilibrium:somewatermoleculesformthesolid,andtheremainingwatermoleculesformtheliquid.Thesolidandtheliquidaretwothermalsystems,inthermalcontactandinthermalequilibrium.Thecoexistentsolidandtheliquidhavethesametemperature,Tm.Amixtureofsolidandliquidinequilibriumiscalledaheterogeneousthermodynamicstate.Whenamixtureofsolidandliquidequilibrate,denotetheenergypermoleculeinthesolidbyus,andtheenergypermoleculeintheliquidbyuf.OfatotalnumberofNmoleculesinthemixture,Nsmoleculesformthesolid,andNfmoleculesformtheliquid.ThetotalenergyofthemixtureisU=Nsus+Nfuf.Thisequationiscalledtheruleofmixture.Wecanaddenergytothemixture,whilekeepingthetotalnumberofwatermoleculesinthemixturefixed.Astheenergyofthemixtureincreases,somewatermoleculesgofromthesolidtotheliquid.

Thus,meltingcorrespondstoahorizontallineintheenergy-temperatureplane.Thelevelofthelineisfixedatthemeltingtemperature,Tm.Thelineendsontheleftwhenallmoleculesformthesolid,andendsontherightwhenallmoleculesformtheliquid.Latentenergy.Definethelatentenergypermoleculebythejumpinenergy,uf−us.Often,wereportlatentenergyperunitmass,orspecificlatentenergy.Wewillusethesamesymboluforboththeenergypermoleculeandenergyperunitmass.Thetwoquantitiesareconvertedbyrecallingthat6×1023watermolecules=1moleofwatermolecules=18gramsofwatermolecules.Thespecificlatentenergyofwaterisuf−us=334kJ/kg.Thelatentenergyiscommonlyknownbyitshistoricalname,latentheat(Black,1750).Thishistoricalnameisnolongerappropriate.Latentenergyisajumpinenergy(afunctionofstate),andshouldnotbenamedusingthewordheat,amethodofenergytransfer.Wecanmeltablockoficebywork,suchasmovingtheblockagainstfriction.Thatis,wecanformtheconceptoflatentenergyandmeasureitexperimentallywithouteverinvokingheat.FunctionT(U).Insummary,apuresubstancenearmeltingischaracterizedbyacurveonthetemperature-energyplaneusingfourquantities:themeltingtemperatureTm,thelatentenergyuf−us,thespecificthermalcapacityofthesolidcs,andthespecificthermalcapacityoftheliquidcf.Belowthemeltingtemperatureandabovethetemperature,thefunctionT(U)iscurved,andbothcsandcfchangewithtemperature.Inmanyestimateinvolvingwater,wewillassumeapproximateconstantvalues,cs=2.06kJ/kg/Kandcf=4.18kJ/kg/K.TheapproximatefunctionT(U)summarizesathermodynamicmodelofwater.Exercise.Calculatetheenergyneededtobring1kgoficeat-50degreeCelsiustoliquidwaterat50degreeCelsius.Exercise.1kgoficeatthefreezingtemperatureismixedwith1kgofwaterattheboilingtemperature.Themixtureisinsulated.Whatwillbethetemperatureinequilibrium?

PrimitivecurvesWenowtracetheexperimentalobservationofmeltingbacktothefundamentalpostulate.Wemodelfixedamountofapuresubstanceasathermalsystem.Thismodelassumesthatthepuresubstancecanchangeenergy,butignoresthatthepuresubstancecanalsochangevolume.Themodelisagoodapproximationforsolidandliquid,butnotforgas. InphaseA,theenergypermoleculeisuA,theentropypermoleculeissA,andthecharacteristicfunctionissA(uA).Similarly,wemodelphaseBasathermalsystemofcharacteristicfunctionsB(uB).ThefunctionssA(uA)andsB(uB)aretwocurvesintheenergy-entropyplane.Thetwo

curvesarecalledtheprimitivecurvesofthepuresubstance.Eachpointonaprimitivecurvecorrespondstoahomogeneousthermodynamicstateofthepuresubstance.

RuleofmixtureAmixtureoftwohomogeneousstates.Wenowconsideramixtureoftwohomogeneousstates:state(uA,sA)isapointononeprimarycurve,andstate(uB,sB)isapointontheotherprimarycurve.ThemixturehasatotalofNmolecules,ofwhichNAmoleculesareinstateA,andNBmoleculesareinstateB.DenotethenumberfractionsofthemoleculesinthetwohomogeneousstatesbyyA=NA/N,yB=NB/N.BothyAandyBarenonnegativenumbers.Thenumberofmoleculesinthemixtureisconserved:N=NA+NB.DividingtheaboveequationbyN,weobtainthatyA+yB=1.Themixtureisalsoathermalsystem.Letuandsbetheenergyandentropyofthemixturedividedbythetotalnumberofmolecules.Energyisanextensivevariable,sothattheenergyofthemixtureisthesumoftheenergiesofthetwohomogeneousstates:Nu=NAuA+NBuB.DividingtheaboveequationbyN,weobtainthatu=yAuA+yBuB.Thesameistrueforentropy:s=yAsA+yBsB.Graphtherulesofmixture.Theserulesofmixturecanbegraphedontheenergy-entropyplane.HomogeneousstateAisapoint(uA,sA)ontheprimitivecurvesA(uA).HomogeneousstateBisapoint(uB,sB)ontheprimitivecurvesB(uB).Themixtureisapoint(u,s)onthelinesegmentjoiningthetwopoints(uA,sA)and(uB,sB),locatedatthecenterofgravity,dependingonthefractionofmoleculesyAandyBallocatedtothetwophases.Ingeneral,themixture(u,s)isapointofftheprimitivecurves,andmaynotbeastateofequilibrium.

Amixtureofanynumberofhomogeneousstates.Nowconsideramixtureofanynumberofhomogenousstates.Thehomogeneousstatescanbeononeprimitivecurve,oronbothprimitivecurves.Forexample,consideramixtureofthreehomogeneousstates,A,B,andC.ApuresubstancehasatotalofNmolecules,ofwhichNA,NB,andNCmoleculesareinthethreehomogeneousstates.DenotethenumberfractionsbyyA=NA/N,yB=NB/N,andyC=NC/N.HereyA,yBandyCarenon-negativenumbers.Themixtureconservesthenumberofmolecules:NA+NB+NC=N.DividethisequationbyN,andweobtainthatyA+yB+yC=1.Thethreehomogeneousstates,(uA,sA),(uB,sB),and(uC,sC),arethreepointsononeortwoprimitivecurves.Thethreepointsformatriangleintheenergy-entropyplane.Theenergyandenergypermoleculeofthemixture(u,s)aregivenbyu=yAuA+yBuB+yCuC,s=yAsA+yBsB+yCsC.Themixture,(u,s),isapointintheenergy-entropyplane,locatedatthecenterofgravityinthetriangle,dependingonthefractionofmoleculesyA,yBandyCallocatedtothethreehomogeneousstatesattheverticesofthetriangle.Ingeneral,themixture(u,s)isapointofftheprimitivecurves.Neglectthespatialarrangementofpiecesofhomogeneousstates.Therulesofmixturedependonthenumberofmoleculesineachhomogenousstate,butnotonhowthepiecesofhomogenousstatesarearrangedinspace.Therulesofmixturealsoneglectmoleculesattheinterfacesbetweenpiecesofhomogenousstates.Themoleculesattheinterfaceshavetheirownthermodynamicproperties,differentfromthoseofthehomogeneousstates.Theinterfacescontributetoenergyandentropynegligibly,solongasthepiecesofthehomogenousstatesaremuchlargerthanthesizeofindividualmolecules.Convexhull.Eachpointontheprimitivecurvescorrespondstoahomogenousstate.Amixturecorrespondstoapointatthecenterofgravityofsomenumberofhomogeneousstates.Allpossiblemixturesofarbitrarynumbersofhomogeneousstatesconstitutearegionintheenergy-entropyplane.Eachmixtureintheregionmaynotbeastateofequilibrium.Incidentally,inthelanguageofconvexanalysis,theenergy-entropyplaneiscalledavectorspace,andeachpointintheplaneiscalledavector.Allthehomogeneousstatesontheprimary

curvesdefineasetofvectors.Amixtureiscalledaconvexcombinationofthehomogenousstates,andthesetofallmixturesiscalledtheconvexhullofthehomogenousstates.

DerivedcurveTheregionofallpossiblemixtures(i.e.,theconvexhull)isboundedfromabovebyasinglecurve,calledthederivedcurve.Thederivedcurveisformedbyrollingtangentlinesontheprimitivecurves.Atangentlinecantouchtheprimitivecurvesatonepointortwopoints,butnotthreeormorepoints.Thederivedcurveisconvex.Sofarenergyandentropyplaysimilarroles.Allwehaveinvokedisthattheyareextensivequantities.Wenextapplythebasicalgorithm.Afixedamountofmixtureofafixedamountofenergyisanisolatedsystem.Theisolatedsystemhasanenormousnumberofinternalvariables:thenumberofhomogeneousstates,thelocationofeachhomogeneousstateononeoftheprimitivecurves,andthenumberfractionofmoleculesallocatedtoeachhomogeneousstate.Theisolatedsystemhasafixedamountofenergyu,representedbyaverticallineontheenergy-entropyplane.Theverticallineintersectsthederivedcurveatonepoint,whichmaximizethesubsetentropy.Astheenergyofapuresubstancevaries,thederivedcurverepresentsallthermodynamicstatesofthepuresubstance.

EquilibriumofasinglehomogeneousstateIfalinetangenttoonepointonaprimitivecurvedoesnotcutanyprimitivecurves,thispointbelongstothederivedcurve.Thesetofallsuchpointsiscalledthecurveofabsolutestability.Thetangentlinecanrollontheprimitivecurvetochangetheslopeofthetangentline.Thus,thecurveofabsolutestabilityhasonedegreeoffreedom.Recall1/T=ds(u)du.Theslopeofthetangentlinecorrespondstotemperature.

EquilibriumoftwohomogeneousstatesIfalinetangenttotwopointsontheprimitivecurvesdoesnotcutanyprimitivecurves,thestraight-linesegmentconnectingthetwotangentpointsbelongstothederivedcurve.Thestraight-linesegmentiscalledatieline.Thecommontangentlinecannotrollontheprimitivecurves,isfixedintheenergy-entropyplane,andhasnodegreeoffreedom.Thetwotangentpointsattheendsofthetielinearecalledthelimitsofabsolutestability.LetthetangentlinetouchoneprimitivecurvesA(uA)atpoint(uA,sA),andtouchtheotherprimitivecurvesB(uB)atpoint(uB,sB).Thetwotangentpointscorrespondtothetwohomogeneousstatesinequilibrium.Theslopeofthetangentlinedefinesthemeltingtemperature.Thus,1/Tm=(sB-sA)/(uB-uA)=dsB(uB)/duB=dsA(uA)/duA.Giventhetwoprimitivecurves,sA(uA)andsB(uB),theaboveequationssolveforthemeltingtemperatureTm,aswellasthetwohomogeneousstatesinequilibrium,(uA,sA)and(uB,sB).

Temperature-entropycurveWenowsketchthecurveT(S)nearthemeltingtemperature.RecalldS=T-1dU=C(T)T-1dT.Icemeltsattemperature273.15K.Atthemeltingtemperature,thespecificthermalcapacityforiceiscs=2.06kJ/kg/K.Thus,theslopeofthetemperature-entropycurveforicenearthemeltingtemperatureiscs/Tm=(2.06kJ/kg/K)/(273.15K)=0.0075kJ/kg/K

2. Atthemeltingtemperature,thespecificthermalcapacityforliquidwateriscf=4.18kJ/kg/K.Thus,theslopeofthetemperature-entropycurveforliquidwaternearthemeltingtemperatureiscf/Tm=(4.18kJ/kg/K)/(273.15K)=0.015kJ/kg/K

2.

Recallthatsf-ss=(uf-us)/Tm.Thespecificlatentenergyis334kJ/kg.Thechangeinspecificentropyissf-ss=(334kJ/kg)/(273.15K)=1.2kJ/kg/K.

Exercise.Canthemodelofthermalsystemdescribesthreehomogenousstatesinequilibrium?Exercise.Forwater,wehavesketchedcurvesT(u)andT(s)nearthemeltingtemperature.Nowcalculateandsketchthecurves(u)nearthemeltingtemperature.Exercise.Forwater,themeltingtemperatureis273.15K,thespecificlatentenergyis334kJ/kg.Calculatetheratioofthenumberofquantumstatesintheliquidtothatinthesolid.

Thermalsystemofanonconvexcharacteristicfunctions(u).Inhindsight,weshouldnothaveacceptedsoreadilythatthecharacteristicfunctionofathermalsystem,s(u),isaconvexfunction,orthatu(T)isanincreasingfunction,orthatthermalcapacityispositive.Infact,athermalsystemmayhaveanonconvexprimarycurves(u).Wecanformatangentlinetouchingtwopointsonthecurves(u).Thetwopointscorrespondtoequilibriumoftwohomogeneousstates.Itturnsoutthatasolid-liquidtransitionismodeledwithtwoconvexprimitivecurves,butaliquid-gastransitionismodeledwithasinglenonconvexprimitivecurve.Wewillseethiseffectclearlylaterinamodelthatallowsthepuresubstancetovarybothenergyandvolume.

MetastabilityAprimitivecurvemaycontainaconvexpartandanon-convexpart.Thepointseparatingthetwopartsiscalledtheinflectionpointincalculus,andiscalledthelimitofmetastabilityinthermodynamics.Ifaconvexpartoftheprimitivecurveliesbelowthederivedcurve,thepartoftheprimitivecurveisbeyondthelimitofabsolutestability.Eachpointofthispartoftheprimitivecurveiscalledametastablestate.Ametastablestateisstableinregardtocontinuouschanges

ofstate,butisunstableinregardtodiscontinuouschangesofstate.Watchavideoonsupercooledwater.

TemperatureasanindependentvariableAthermalsystemhasasingleindependentvariation.Sofarwehavespecifiedthethermodynamicstatesofthethermalsystemusingenergyasanindependentvariable.Anyoneofthefunctionsofstatecanserveasanindependentvariable.Apopularchoiceistemperature.

U(T)andT(U)Asdescribedabove,wecanmeasuretheenergy-temperaturecurveofathermalsystem,T(U).Forthetimebeing,weassumethatthefunctionT(U)isanincreasingfunction.Thatis,onreceivingenergy,athermalsystemincreasestemperature.Fromcalculuswehavelearnedthatanymonotonicfunctionisinvertible.WritetheinversefunctionasU(T).Thetwofunctions,U(T)andT(U),correspondtothesamecurveontheenergy-temperatureplane,andcontainthesameinformation.

ThermostatAthermostatisadevicethatmeasurestemperatureandswitchesheatingorcoolingequipmenton,sothatthetemperatureiskeptataprescribedlevel.Thermostatsarewidelyusedinrefrigeratorsandhome-heatingand-coolingunits.Sous-vide(/suːˈviːd/;Frenchfor"undervacuum")isamethodofcooking.Food(e.g.,apieceofmeat)issealedinanairtightplasticbag,andplacedinawaterbathforalongertimeandatalowertemperaturethanthoseusedfornormalcooking.Thetemperatureisfixedbyafeedbacksystem.Becauseofthelongtimeandlowtemperature,sous-videcookingheatsthefoodevenly;theinsideisproperlycookedwithoutovercookingtheoutside.Theairtightbagretainsmoistureinthefood.

ThermalreservoirAthermalreservoirisathermalsystemofafixedtemperature.Weusethethermalreservoirtofixthetemperatureofanotherthermalsystembythermalcontact.Theotherthermalsystemhasamuchsmallerthermalcapacitythanthethermalreservoir,sothatthetemperatureofthereservoir,TR,isfixedasthethermalreservoirandtheotherthermalsystemexchangeenergy.Inthermalequilibrium,theotherthermalsystemhasthesametemperatureasthethermalreservoir.Wecanrealizeathermalreservoirbyusingalargetankofwater.Whenwaterlosesorgainsasmallamountofthermalenergy,thetemperatureofwaterisnearlyunchanged.

Thesituationisanalogoustoawaterreservoir.Awaterreservoirhasalotofwater.Itswaterlevelremainsnearlyunchangedwhenwetakeasmallcupofwaterfromthereservoir.Thethermalreservoirisathermalsystem,andinteractswiththerestoftheworldinonemode:transferenergy.WhentheenergyofthereservoirisfixedatavalueUR,thereservoirbecomesanisolatedsystem,andhasacertainnumberofquantumstates,ΩR(UR).AsURvaries,thefunctionΩR(UR),oritsequivalent,SR(UR)=logΩR(UR),characterizesthereservoirasafamilyofisolatedsystems.Whenthereservoirisinthermalcontactwithasmallthermalsystem,thecompositeofthereservoirandthesystemisanisolatedsystemandhasaconstantenergy,Ucomposite.LetUbetheenergyofthesmallthermalsystem.Theenergyofthecompositeisasumofparts:Ucomposite=UR+U.Recallthedefinitionoftemperature,1/TR=dSR(UR)/dUR.BecauseTRisconstant,integrating,wefindthatSR(UR)=SR(Ucomposite)-(Ucomposite-UR)/TR.Thus,athermalreservoirisathermalsystemcharacterizedbyalinearfunctionSR(UR).

IsothermalprocessAprocessthatoccursataconstanttemperatureiscalledanisothermalprocess.Forexample,thesous-videcookinginanisothermalprocess.Thetemperatureisfixedwhilethefoodcooks.Wecanfixthetemperaturebyathermostat,orbyathermalreservoir.Sofarasthefoodisconcerned,themethodoffixingtemperaturemakesnodifference. Wemodelanisothermalprocessbyasystemoftwovariables:energyUandaninternalvariablex.WhenbothUandxarefixed,thesystemisanisolatedsystemhavingacertainnumberofthequantumstates,Ω(U,x).LetS(U,x)=logΩ(U,x).Wefixthetemperatureofthesystembythermalcontactwithareservoir.Thecompositeofthesystemandthereservoirisanisolatedsystem.Theentropyofthecompositeisthesumoftheentropiesofthereservoirandthesystem:Scomposite=SR(Ucomposite)-U/T+S(U,x).Theisolatedsystemhastwointernalvariables:theenergyUinthesystem,andtheinternalvariablex.ThetotalenergyofthecompositeUcompositeisfixed.

Theisolatedsystemisinthermalequilibrium,sothat∂Scomposite/∂U=0.Thisconditionrecoversafamiliarcondition:1/TR=∂S(U,x)/∂U.Thatis,inthermalequilibrium,thereservoirandthethermalsystemhavethesametemperature.WewilldropthesubscriptRandwritethetemperatureasTinthefollowing.ThefunctionS(U,x)isknown,theaboveconditionofthermalequilibriumdefinesthefunctionU(T,x).WecanalsowritetheentropyasafunctionS(T,x).Massieufunction.TheenergyofthecompositeUcompositeisfixed,sothatSR(Ucomposite)isaconstant.Thecompositeisanisolatedsystemofasingleinternalvariable,x.TheprocessproceedstochangextoincreaseScomposite,orequivalently,increasethefunctionJ=S(T,x)-U(T,x)/T.ThisfunctioniswrittenasJ(T,x).Thisfunctioncontainsquantitiesofthethermalsystemalone,andwasintroducedbyMassieu(1869).Intheisothermalprocess,thetemperatureisnotavariable,butisfixedbythethermalreservoir.Asidefromanadditiveconstant,theMassieufunctionisthesubsetentropyofanisolatedsystem:thecompositeofathermalsystemandathermalreservoir.

AlgorithmofthermodynamicsforisothermalprocessWenowparaphrasethebasicalgorithmofthermodynamicsforanisothermalprocess.

1. Constructathermalsystemwithaninternalvariablex.2. IdentifythefunctionJ(T,x).3. Equilibrium.FindthevalueoftheinternalvariablexthatmaximizesthefunctionJ(T,x).4. Irreversibility.Changethevalueoftheinternalvariablexinasequencethatincreases

thefunctionJ(T,x).

HelmholtzfunctionBecausethetemperatureisconstantandpositive,maximizingtheJ(T,x)isthesameasminimizingthefollowingfunction:F=U−TS.ThisfunctioniswrittenasF(T,x).Thisfunctioncontainsquantitiesofthethermalsystemalone,andiscalledtheHelmholtzfunction,orfreeenergy.ThefunctionwasintroducedbyGibbs(1865).NotethatUhasanarbitraryadditiveconstant,whichalsoappearsinF.

ObservethatF=-J/T.Whenthesystemisheldatafixedtemperature(i.e.,inthermalequilibriumwiththereservoir),ofallvaluesoftheinternalvariablex,themostprobablevalueminimizestheHelmholtzfunctionF(T,x).Inthisminimization,temperatureisnotavariable,butisfixedbythethermalreservoir.Shadowoftherealthing.TheHelmholtzfunctioncomesfromthesubsetentropyofanisolatedsystem:thecombinationofasystemandathermalreservoir.TheHelmholtzfunctioncontainsnonewfundamentalprinciple,andisashadowofentropy.Inthepracticeofthermodynamics,theHelmholtzfunctionissocommonlyusedthatmanypeopleareenamoredbytheshadow,andforgettherealthing—entropy.DerivativeofHelmholtzfunction.Atfixedx,dF=dU-TdS-SdT.Recallthedefinitionoftemperature,dU=TdS.WehavethatdF=-SdT. Thus,-S=∂F(T,x)/∂T.BecauseSispositive,FdecreasesasTincreases.BecauseSincreasesasTincreases,FisaconvexfunctionofT.WecanparaphrasethealgorithmofthermodynamicsforisothermalprocessinusingtheHelmholtzfunctionF(T,x).Letususethealgorithmtoanalyzemelting.

MeltinganalyzedusingtheHelmholtzfunctionLettheHelmholtzfunctionpermoleculeinthesolidbefA(T),andthatintheliquidbefB(T).TheHelmholtzfunctionpermoleculeinamixtureofsolidandliquidisf=yAfA(T)+yBfB(T).Thechangeofphaseismodeledasanisothermalprocess.RecallthatyA+yB=1.Thefractionofmoleculesinthesolid,yA,istheindependentinternalvariable,whichisvariedtominimizetheHelmholtzfunctionofthemixture.ThetwocurvesfA(T)andfB(T)aredecreasingandconvexfunctions.TheequationfA(T)=fB(T).

determinesthemeltingtemperatureTm.WhenT<Tm,theHelmholtzfunctionofthemixtureminimizesifallmoleculesfreeze.WhenT>Tm,theHelmholtzfunctionofthemixtureminimizesifallmoleculesmelt.RecallthedefinitionoftheHelmholtzfunction,f=u−Ts.TheconditionofequilibriumofthesolidandliquidgivesthatuA-TmsA=uB-TmsB.Thisexpressionrecoverswhatwehaveobtainedbefore.TheHelmholtzfunctiongivesusaslickderivationoftheequilibriumcondition,butpushesusastepawayfromtheleadingrole,entropy.

ClosedsystemNowentersasecondsupportingrole—volume.Inthermodynamics,energyandvolumeplayanalogoussupportingroles.

AfamilyofisolatedsystemsoftwoindependentvariationsAclosedsystemanditssurroundingsdonottransfermatter,buttransferenergyandvolume.Considerahalfcylinderofwatersealedwithapiston.Abovethepistonisaweight,andbeneaththecylinderisafire.Insidethecylinder,liquidoccupiessomevolume,andvaporoccupiestherest.Thewatermoleculesinsidethecylinderconstituteaclosedsystem.CharacteristicfunctionS(U,V).LetUbetheenergyandVbethevolumeofaclosedsystem.WhentheenergyUandvolumeVarefixed,theclosedsystembecomesanisolatedsystem.Forthehalfcylinderofwater,wefixUandVbythermallyinsulatethecylinderandlockthepositionofthepiston.DenotethenumberofquantumstatesofthisisolatedsystembyΩ(U,V).AsUandVvary,thefunctionΩ(U,V),oritsequivalent,S(U,V)=logΩ(U,V),characterizestheclosedsystemasafamilyofisolatedsystems,capableoftwoindependentvariations,UandV.Thermodynamicstate.Eachmemberisolatedsystem,inequilibrium,definesathermodynamicstateoftheclosedsystem,specifiedbyfixedvaluesofthetwothermodynamicproperties,UandV.Intheenergy-volumeplane,apointrepresentsathermodynamicstate.Athermodynamicstateissynonymoustoamemberisolatedsysteminequilibrium.Thermodynamicprocess.Acurveintheenergy-volumeplanerepresentsathermodynamicprocess.Athermodynamicprocessissynonymoustoasequenceofmemberisolatedstates,eachbeinginequilibrium.Aclosedsystemisafamilyofisolatedsystems.Athermodynamicprocessoftheclosedsystemisasubfamilyofisolatedsystems.Becausetheclosedsystemiscapableoftwoindependentvariations,thereareinfinitelymanytypesofthermodynamicprocesses.

Energy-volume-entropyspace.Thecharacteristicfunctionofaclosedsystem,S(U,V),isasurfaceintheenergy-volume-entropyspace,withenergyandvolumeasthehorizontalaxesandentropyastheverticalaxis.CharacterizingaclosedsystembyasurfaceS(U,V)intheenergy-volume-entropyspacestartswithGibbs(1873).Forafunctionoftwovariables,S(U,V),recallafactofcalculus:dS(U,V)=(∂S(U,V)/∂U)dU+(∂S(U,V)/∂V)dV.Furtherrecallthemeaningsofthetwopartialderivativeincalculus:∂S(U,V)/∂U=(S(U+dU,V)-S(U,V))/dU,∂S(U,V)/∂V=(S(U,V+dV)-S(U,V))/dV.DrawaplanetangenttothesurfaceatapointonthesurfaceS(U,V).Thetangentplanehastheslope∂S(U,V)/∂UwithrespecttotheUaxis,andtheslope∂S(U,V)/∂VwithrespecttotheVaxis.Wenextinterpretthetwopartialderivativesintermsofexperimentalmeasurements.

Constant-volumeprocessWhenthevolumeVisfixedandtheenergyUisvaried,theclosedsystembecomesathermalsystem.Forthehalfcylinderofwater,welockthepositionofthepiston,sothatthevolumeinthecylinderisfixed.Weallowthewatertoreceiveenergyfromthefire,sothattheenergyinthecylinderisvaried.Intheenergy-volumeplane,theconstant-volumeprocesscorrespondstoalineparalleltotheenergyaxis.Forathermalsystem,wehavedefinedtemperatureTby1/T=∂S(U,V)/∂U.Thisequationrelatesonepartialderivativetoanexperimentallymeasurablequantity—temperatureT.

AdiabaticprocessWenextlookattheotherpartialderivative,∂S(U,V)/∂V.Wemakeboththecylinderandthepistonusingmaterialsthatblockthetransferofmatterandblockthetransferofenergybyheat.Butwecanmovethepistonandtransferenergybywork.

Whenaclosedsystemtransfersenergywithitssurroundingsbyworkbutnotbyheat,theclosedsystemissaidtoundergoanadiabaticprocess.Whenweaddweights,thegaslosesvolumebutgainsenergy.Addingweightscompressesthegasandcausesadiabaticheating.Whenweremoveweights,thegasgainsvolumebutlosesenergy.Removingweightsexpandsthegasandcausesadiabaticcooling.Thehalfcylinderofwaterandtheweighttogetherconstituteanisolatedsystem.Theisolatedsystemhastwointernalvariables,thevolumeofthecylinder,V,andtheenergyinthewatermolecules,U.Thetwointernalvariablesarerelated.Thepistonisassumedtomovewithnofriction.LetPbethepressureinsidethecylinder,andAbetheareaofthepiston.Theweightappliesaforcetothepiston,PA.Whenthepistonmovesupbyadistancedz,thevolumeinsidethecylinderincreasesbydV=Adz,thepotentialenergyoftheweightincreasesbyPAdz=PdV,andtheenergyofthewaterincreasesbydU.Theenergyoftheisolatedsystemisthesumofthethermalenergyofthewaterandthepotentialenergyoftheweight.Theisolatedsystemconservesenergy,dU+PdV=0.ThisequationrelatesthetwointernalvariablesUandV.Whenthepistonmoves,theheightofweightchanges,buttheentropyoftheweight,Sweight,doesnotchange.AtfixedUandV,theentropyoftheisolatedsystemisthesumoftheentropyofthewatermoleculesandtheentropyoftheweight:S(U,V)+Sweight.WhenUandVvary,thesubsetentropymaximizesinequilibrium,sothatT-1dU+(∂S(U,V)/∂V)dV=0.Theisolatedsystemundergoesanisentropicprocess.Thisisentropiccondition,togetherwiththeconservationofenergy,dU+PdV=0,yieldsthatP/T=∂S(U,V)/∂V.Thisequationrelatesthepartialderivativetoexperimentallymeasurablequantities—pressurePandtemperatureT.TheratioP/Tisthechildoftheunionofentropyandvolume,justasthetemperatureisthechildoftheunionofentropyandenergy.Pressureisanintensivethermodynamicproperty.Exercise.WhenIstretcharubberbandrapidly,energyhasnotimetodiffuseout,andtherubberbandapproximatelyundergoesadiabaticheating.Mylipscanfeelanincreaseofthetemperatureoftherubberband.Aftersometime,someenergydoesdiffuseout,andthetemperatureoftherubberbandbecomesthesameasthatofmylips.WhenIrapidlyreleasethestretch,therubberbandapproximatelyundergoesadiabaticcooling,andmylipsfeeladecreaseintemperature.Trytheseexperimentsyourself.

ExperimentaldeterminationofthefunctionS(U,V)Theaboveequationsgive

dS=(1/T)dU+(P/T)dV.ThisequationrelatesthecharacteristicfunctionS(U,V)toexperimentallymeasurablequantities,U,V,P,T.Byincrementallychangingtheenergyandvolume,wecanmeasurethefunctionS(U,V).Forexample,wecanprescribeavalueofthevolume,V1,regardtheclosedsystemasathermalsystem,andmeasurethecurveS(U,V1)usingtheprocedurethesameasthatforanythermalsystem.Thatis,wemeasurethefunctionT(U,V1)byincrementallyaddingenergybutfixingvolumeV1.Wethenintegrate1/T=∂S(U,V1)/∂U,startingwithS(U,V1)=0asT(U,V1)→0.Wethenprescribeanothervalueofthevolume,V2,etc.Thesemeasurementsdetermineafamilyofcurvesontheenergy-entropyplane,S(U,V1),S(U,V2),...Eachcurvecharacterizesathermalsystemofafixedvolume.Becausetheenergyisarelativequantity,eachcurvecantranslatehorizontallybyanarbitraryamountwithoutaffectingthebehaviorofthethermalsystem.Buttwocurvesofdifferentvolumes,S(U,V1)andS(U,V2),shouldnottranslaterelativetoeachotherbyanarbitraryamount.Rather,thefamilyofcurvesconstituteasinglesurface,S(U,V),intheenergy-volume-entropyspace.Theentiresurfacecantranslatealongtheaxisofenergybyonearbitraryamountwithoutaffectingthebehavioroftheclosedsystem.WecaneliminatethearbitrarytranslationbetweenthetwocurvesS(U,V1)andS(U,V2)byintegratingP/T=∂S(U,V)/∂V.Thisprocedurerequiresmeasurementsofbothpressureandtemperature.ThefunctionS(U,V)characterizetheclosedsystemasafamilyofisolatedsystems.Thus,wecanexperimentallycountthenumberofquantumstatesforeachmemberisolatedsysteminthefamily.

GeneralfeaturesofthefunctionS(U,V)NotethefollowingfeaturesofthesurfaceS(U,V)commontoallclosedsystems.

1. Becauseenergyisrelativetoanarbitraryreference,thesurfaceS(U,V)cantranslateinthedirectionofenergybyanarbitraryamountwithoutaffectingthebehavioroftheclosedsystem.

2. Becauseentropyisabsolute,thesurfacestartsatS=0,andcannotbetranslatedupanddown.

3. Volumeisalsoabsoluteandpositive.4. WhenaplaneistangenttothesurfaceS(U,V)atapoint,thetwoslopesofthetangent

planerepresent1/TandP/T.5. ForeachfixedV,asSapproacheszero,S(U,V)isacurvethatapproachestheUaxis

vertically.Thatis,∂S(U,V)/∂UapproachesinfinityasSapproacheszero.

6. Themoreenergyandvolume,themorequantumstates.Thus,Ω(U,V)isanincreasingfunctionwithrespecttobothUandV.Becauselogarithmisanincreasingfunction,S(U,V),isalsoanincreasingfunction.Thatis,theslopes∂S(U,V)/∂Uand∂S(U,V)/∂Varepositive.

7. Forthetimebeing,weassumethatS(U,V)isaconvexfunction.

Idealgas

ThemodelofidealgasAbottleofvolumeVcontainsNmoleculesandenergyU.Themoleculesarecalledanidealgasiftheyarefarapartonaverage.Themoleculesfly,collide,andseparate.WhenU,V,Narefixed,thegasisanisolatedsystem.DenotethenumberofquantumstatesofthisisolatedsystembyΩ(U,V,N).Becausethemoleculesarefarapart,eachmoleculecanflytoanywhereinthebottleasifallothermoleculesarenotthere.Consequently,thenumberofquantumstatesofeachmoleculeisproportionaltothevolumeV,andthenumberofquantumstatesofNmoleculesisproportionaltoVN.TheproportionalfactordependsonUandN,butnotonV.WriteΩ(U,V,N)=VNf(U,N),wheref(U,N)isafunctionofUandN.Bydefinition,theentropyisS=klogΩ,sothatS(U,V,N)=NklogV+klogf(U,N).

IdealgaslawInsertingtheaboveexpressionforentropyintothegeneralequationP/T=∂S(U,V,N)/∂V,weobtainthatPV=NkT.Thisequationofstateiscalledtheidealgaslaw.Historically,theidealgaslawwasdiscoveredexperimentally,beforethediscoveryofentropy.Afterthediscoveryofentropy,theidealgaslawisderivedfromthissimplemodel.

EnergyofanidealgasInsertingtheexpressionforentropyintothedefinitionoftemperature,1/T=∂S(U,V,N)/∂U,weobtainthat1/T=∂(logf(U,N))/∂U.ThisequationshowsthattheenergyUisafunctionoftemperatureTandnumberN,andisindependentofthevolumeV.Theenergyisanextensiveproperty,andislinearinthenumberofmoleculesN.Write U=Nu(T). Hereu(T)isthethermalenergypermolecule.Thisfunctionhasbeendeterminedformanyspeciesofmolecules.DefinethethermalcapacitypermoleculeundertheconditionofconstantvolumebycV=du(T)/dT.Ingeneral,thethermalcapacityisafunctionoftemperature,cV(T).Thus,dU=NcV(T)dT.Asanapproximation,theenergyistakentobelinearintemperature,andcVisaconstantindependentoftemperature.

EntropyofanidealgasRecallthegeneralequationforaclosedsystemdS=(1/T)dU+(P/T)dV.Insertingtheequationsspecifictoanidealgas,dU=NcV(T)dTandPV=NkT,weobtainthatdS=(NcV(T)/T)dT+(Nk/V)dV.AssumingthatcVisindependentoftemperatureandintegrating,weobtainthatS(T,V)=NcVlog(T/T0)+Nklog(V/V0)+S(T0,V0).Entropyisafunctionofstate.Athermodynamicstateoftheclosedsystemisspecifiedbytwoindependentvariables,sayTandV.Intheaboveequationforidealgases,thefirsttermisduetothechangeintemperature,andthesecondisduetothechangeinvolume.

EnergytransferbyworkandbyheatforafixedamountofanidealgasAfixedamountofanidealgasisaclosedsystem,characterizedbytwoequations:PV=NkT,dU=NcV(T)dT.TheenergytransferbyworkfromthegastothesurroundingsisdW=PdV.ThefirstlawofthermodynamicsdefinestheenergytransferbyheatfromthesurroundingstothegasasdQ=dU+PdV.Thetemperature,pressure,volume,energy,andentropyarefunctionsofstate.Neithertheenergytransferbyworknortheenergytransferbyheatisafunctionofstate;theydependonprocess.Wenextconsiderseveralprocesses. Constant-volumeprocess.Subjecttoaconstantvolume,thegasdoesnowork.Theenergytransferbyheatequalsthechangeininternalenergy:dQ=NcV(T)dT.WhenthetemperaturechangesfromT1toT2,assumingthethermalcapacityisindependentoftemperature,theenergytransferbyheatisQ=NcV(T2-T1).Constant-pressureprocess.WhenthevolumechangesbydV,thegasdoesworkdW=PdV.WhenthevolumechangesfromV1toV2underaconstantpressureP,thegasgoesworkW=P(V2-V1).Ingeneral,theenergytransferbyheatbothchangestheinternalenergyandvolume:dQ=dU+PdV.Subjecttoaconstantpressure,thisexpressionbecomesthatdQ=d(U+PV).

Foranidealgasunderaconstantpressure,theenergytransferbyheatisdQ=N(cV+k)dT.ThequantitycV+kisthethermalcapacitypermoleculeunderconstantpressure,andisdesignatedascP.Isothermalprocess.Aconstant-temperatureprocessisalsocalledanisothermalprocess.Becauseoftheidealgaslaw,theisothermalprocessischaracterizedbyacurvePV=constant.Foranidealgas,theisothermalprocessdoesnotchangetheinternalenergy,sothattheenergytransferbyheattotheclosedsystemisthesameastheworkdonebytheclosedsystem:dW=dQ=(NkT/V)dV.WhenthevolumechangesfromV1toV2,theenergytransferbyheatisW=Q=NkTlog(V2/V1).Adiabaticprocess.Aclosedsystemissaidtoundergoanadiabaticprocessifthesystemisthermallyinsulated,sothatnoenergytransfersbyheatbetweentheclosedsystemandthesurroundings:dU+PdV=0.Foranidealgas,thisequationbecomescV(T)dT+(kT/V)dV=0.Wefurtherassumethatthethermalcapacityisindependentoftemperature.IntegrationyieldscVlogT+klogV=constant,orTVb=constant,whereb=k/cV.Theconstantisdeterminedbyonestateintheprocess,saytheinitialstate(Ti,Vi).Thus,constant=TiVi

b.TheresultTVb=constantexplainsadiabaticcoolingwhenagasexpandsinanadiabaticprocess.RecalltheidealgaslawPV=NkT.TheadiabaticprocessalsoobeysthatPVb+1=constant.

WhenthevolumechangesbydV,theadiabaticprocesstransfersenergybywork:dW=PdV=(constant)V-b-1dV.IntegrationyieldsW=(constant/b)(V1

-b-V2-b)

Exercise.AssumethatanidealgashasaconstantthermalcapacitypermoleculecV.Onemoleoftheidealgaschangesfromaninitialstate(Pi,Vi)toafinalstate(Pf,Vf)alongastraightlineonthe(P,V)plane.Calculatetheenergytransferbyworkandenergytransferbyheat.Calculatethechangesininternalenergyandentropy.Exercise.Anidealgasundergoesanadiabaticprocessfromaninitialstate(Pi,Vi)toafinalstateofvolumeVf.Calculatethevolumeofthefinalstate,Vf.Calculatetheenergytransferbyworkintheadiabaticprocess.Exercise.Inaheliumgas,eachmoleculeconsistsofasingleheliumatom.Forsuchagasofsingle-atommolecules,cV=1.5k.Calculatehowtemperaturechangeswhenpressuredoublesinanadiabaticprocess.

EntropicelasticityWhenaspringmadeofsteelispulledbyaforce,thespringelongates.Whentheforceisremoved,thespringrecoversitsinitiallength.Thiselasticityisduetoadistortionoftheelectroncloudofatoms.Suchelasticityiscalledenergeticelasticity. Abagofairactslikeaspring.Thevolumedecreaseswhenthepressureincreases,andrecoverswhenthepressuredrops.Thiselasticityclearlydoesnotresultfromdistortionofatomicbondsinthemolecules,butfromthechangeofthenumberofquantumstateswithvolume.Suchelasticityiscalledentropicelasticity.

ThermoelasticcouplingThebagofairalsoillustratesthermoelasticcoupling.DefineelasticmodulusbyB=-V∂P/∂V.Thisdefinitionisincomplete;weneedtospecifywhatistakentobeconstantwhenwetakethepartialderivative.Letusconsidertwoexamples.

Isothermalelasticmodulus.Undertheisothermalcondition,temperatureisconstant.Recalltheidealgaslaw,P=NkT/V.Takingthepartialderivative,weobtaintheisothermalelasticmodulusBT=P.Adiabaticelasticmodulus.Undertheadiabaticcondition,PVb+1=constant.Takingthepartialderivative,weobtaintheadiabaticelasticmodulusBad=(b+1)P.Agasisstifferundertheadiabaticconditionthanundertheisothermalcondition.

OsmosisConsiderNparticlesdispersedinabagofwaterofvolumeV.Theparticlesaredifferentfromwatermolecules,andcanbeofanysize.Theparticlescanbeofmanyspecies.Whentheparticlesaremolecules,wecallthemsolutes.Whentheparticlesaresomewhatlarger,sayfrom10nmto10µm,wecallthemcolloids.Thebagisimmersedinatankofpurewater.Thebagismadeofasemipermeablemembrane:watercanpermeatethroughthemembranebuttheparticlescannot.

TheoryofosmosisThephysicsofthissituationisanalogoustotheidealgas,providedthattheconcentrationoftheparticlesisdilute.Everyparticleisfreetoexploretheentirevolumeinthebag.Thenumberofquantumstatesofthewater-particlesystemscalesasΩ∝VN.Aswaterpermeatesthroughthemembrane,thevolumeofthebagVchanges.RecallS=klogΩandtheP/T=∂S(U,V)/∂V,weobtainthatP=kTN/V.Thispressureiscalledtheosmoticpressure.Thelawofosmosisisidenticaltothelawofidealgas.

BalanceofosmosisTheosmoticpressurecanbebalancedinseveralways.Forexample,thetensioninthemembranecanbalancetheosmosispressure.Onecanalsodisperseparticlesinthetankofwateroutsidethebag.Thedifferenceintheconcentrationofparticlesinthebagandthatofparticlesinthetankcausesadifferenceinthepressuresinthebagandinthetank.Thedifferenceinpressurecanbebalancedbythetensioninthemembrane.Asyetanotherexample,weplacearigid,semi-permeablewallintheliquid,withtheparticlesononeside,butnottheother.Waterisonbothsidesofthewall,butalcoholisonlyononeside.Themoleculesoftheliquidcandiffuseacrossthewall,buttheparticlescannot.Fortheparticlestoexploremorevolume,theliquidmoleculeshavetodiffuseintothesidewhereparticlesare.Ifthisexperimentwerecarriedoutinthezero-gravityenvironment,theinfusionwouldcontinueuntilthepureliquidisdepleted.Inthegravitationalfield,however,theinfusionstopswhenthepressureinthesolutionbalancesthetendencyoftheinfusion.

Phasesofapuresubstance

EmpiricalfactsPhases.Apuresubstanceconsistsofasinglespeciesofmolecules,andcanformthreephases:solid,liquid,andgas.(Apuresubstancemayhavemorethanthreephases.Forexample,waterhasmultiplesolidphases.Forthetimebeing,ignorethisfact.)Ineachphase,thepuresubstancecanchangethermodynamicstatebychangingtwoindependentthermodynamicproperties,suchastemperatureandpressure.Inthesolidphase,moleculesformaperiodiclattice,calledacrystal.Manysmallgrainsofthecrystalformabulksolid.Individualmoleculesvibrateneartheirsitesinthelattice,andrarelyjumpoutofthesites.Intheliquidphase,moleculesstilltouchoneanother,butdonotformaperiodiclattice.Moleculeschangeneighborsreadily.Inthegasphase,moleculesonaveragearefarapart.Theyfly,collide,andseparate.Thermodynamicstatesoftwoindependentvariations.Afixedamountofapuresubstanceisaclosedsystem.Itsthermodynamicstatesarecapableoftwoindependentvariations.Forexample,afixedamountofapuresubstancecanbeanisolatedsystemofafixedvolumeVandafixedenergyU.Theisolatedsystemhasacertainnumberofquantumstates.Denotethe

numberofquantumstatesbyΩ(U,V),andwriteS(U,V)=logΩ(U,V).RecalltheidentitydS(U,V)=(1/T)dU+(P/T)dV.Steamtables.Thedataforwatermoleculesarepresentedbytables,calledthesteamtables.Inconstructingthesteamtables,theenergyandentropyoftheliquidatthetriplepointaretypicallysettobezero.Theextensivequantities,S,U,V,arelistedasvaluesperunitamountofthesubstance.

Temperature-pressureplane.Temperatureandpressurearebothintensivequantities.Eachpointonthetemperature-pressureplanerepresentsapairoftemperatureandpressure.Inthetemperature-pressureplane,wemarktriplepointandcriticalpoint,alongwiththreephaseboundaries.Suchadiagramiscalledaphasediagramofapuresubstance.Seephasediagramofwater.Theexperimentalsignificanceofthephasediagramisexplainedasfollows.Wewillignoremanyphasesoficeathighpressure.Equilibriumofasinglehomogeneousstate.Whenthesubstanceequilibratesinahomogenousstate,thethermodynamicstatecorrespondstoapointinthetemperature-pressureplane.Inthevicinityofasinglehomogeneousstateinequilibrium,thethermodynamicstatecanchangecontinuouslybyindependentchangeoftemperatureandpressure.Thesteamtablesthenlistvolume,energy,andentropyasfunctionsoftwoindependentvariables,temperatureandpressure.Forexample,atT=400KandP=100kPa,watermoleculesformagas,withspecificvolume,specificenergy,andspecificentropy:u=2967.85kJ/kgv=3.10263m3/kgs=8.5434kJ/kg/KEquilibriumoftwohomogeneousstates.Whenthesubstanceequilibratesasamixtureoftwohomogeneousstates,theyhavethesametemperatureandthesamepressure.Inthetemperature-pressureplane,allthermodynamicstatesoftwohomogeneousstatesfallona

curve,calledaphaseboundary.Apointonthephaseboundaryrepresentsmanythermodynamicstates;eachthermodynamicstateisamixtureofthetwohomogeneousstatesofsomeproportion.Therearethreephaseboundaries:gas-liquid,liquid-solid,andsolid-gas.Forsuchaphaseboundary,givenapressure,thesteamtableslistthetemperature,aswellasthespecificvolumes,energy,andentropyofthetwohomogeneousstates.Forexample,atP=100kPa,liquidwaterandgaseouswaterequilibrateattemperature99.62degreeCelsius.Thespecificvolumes,energies,andentropiesofthetwophasesareuf=417.33kJ/kg,ug=2506.06kJ/kgvf=0.001043m

3/kg,vg=1.69400m3/kg

sf=1.3025kJ/kg/K,sg=7.3593kJ/kg/KEquilibriumofthreehomogenousstates.Whenthesubstanceequilibratesasamixtureofthreehomogeneousstates,theyhavethesametemperatureandthesamepressure.Inthetemperature-pressureplane,allthermodynamicstatesofthreehomogeneousstatescollapsetoasinglepoint,calledthetriplepoint.Thetriplepointrepresentsmanythermodynamicstates;eachthermodynamicstateisamixtureofthethreehomogeneousstatesofsomeproportions.Forwatermolecules,thetriplepointoccursatT=273.16KP=611.657PaThespecificvolumes,energies,andentropiesofthethreehomogeneousstatesinequilibriumarelistedinthefollowingtable.

volumem3/kg

energykJ/kg

entropykJ/kg/K

solid 0.001091 -334 -1.2

liquid 0.001000 0 0

gas 206.132 2375.33 9.1562

Criticalpoint.Weaddafewmorefactsofthephaseboundariesinthetemperature-pressureplane.Thethreephaseboundariesmeetatthetriplepoint.Theliquid-solidphaseboundaryextendsindefinitelyasthepressureincreases.Thegas-solidphaseboundaryextendsastemperatureandpressurereduces,butmustterminatewhentemperatureapproachesabsolutezero.Thegas-liquidphaseboundaryterminatesatapoint,calledthecriticalpoint.Thus,aliquidstatecancontinuouslychangetoagaseousstate,withoutcrossingtheliquid-gasphaseboundary.Thecriticalpointisathermodynamicstate.

Forwatermoleculesatthecriticalpoint,thefunctionsofstatetakethefollowingvalues:T=374.1degreeCelsiusP=22.089MPau=2029.58kJ/kgv=0.003155m3/kgs=4.4297kJ/kg/KNoequilibriumoffourormorehomogeneousstates.Apuresubstancecannotequilibrateasamixtureoffourormorehomogeneousstates.Pressure-volumeplane.Pressureisanintensivevariable,butvolumeisanextensivevariable.Weusethespecificvolumevasthehorizontalaxis,andthepressurePastheverticalaxis.EachpointintheP-vplanecorrespondstoathermodynamicstateofaunitmassofapuresubstance,saywater.AllthermodynamicstatesofafixedtemperaturecorrespondtoacurveintheP-vplane,calledanisotherm.ConsideranisothermofT=300K.Atahighpressure,waterisintheliquidphase.Thevolumeoftheliquidincreasesasthepressuredrops.

Temperature-entropyplane.

Exercise.Foraunitmassofwatermolecules,athermodynamicstateisspecifiedbytwoofthermodynamicpropertiesamongmany,suchastemperatureT,pressureP,specificenergyu,specificvolumev,specificentropys,specificenthalpyh,andqualityx.Foreachofthethermodynamicstatesspecifiedbelow,usethesteamtablestodeterminetheotherthermodynamicproperties.(a)T=100degreesCelsius,x=0.9.(b)T=100degreesCelsius,P=10kPa.(c)T=100degreesCelsius,P=500kPa.(d)T=50degreesCelsius,v=0.00050m2/kg.

PrimitivesurfacesWenowdescribethethermodynamictheoryofphasesofapuresubstanceduetoGibbs(1873).BeforeGibbsdevelopedthistheory,theempiricalfactsofpuresubstancesdescribedabovewereknown.Theyweresonumerousandcalledforatheory.Energy,volume,andentropyareextensiveproperties.Letu,v,andsbetheenergy,volume,andentropyofthesubstancepermolecule.Thevariablesu,v,sformtheaxesofathree-dimensionalspace.Becausethethermodynamicstateofafixedamountofapuresubstancehastwoindependentvariations,oncethesubstanceisspecifiedinastatebythevaluesofenergyuandvolumev,thevalueofentropysisfixed.Forthetimebeing,letusrepresenteachphasebyitsownenergy-volume-entropyrelation,s(u,v),correspondingtoasurfaceintheenergy-volume-entropyspace.Thethreephasescorrespondtothreesurfaces.Gibbscalledthemtheprimitivesurfaces.Apointononeofthethreeprimitivesurfacescorrespondstoahomogeneousstateofthesubstance.

RuleofmixtureConsidertwohomogeneousstates,AandB,whichcanbetwopointseitherononeprimitivesurface,orontwoprimitivesurfaces.AmixtureofthetwohomogeneousstateshasatotalofNmolecules,ofwhichNAmoleculesareinonehomogeneousstate,andNBmoleculesareintheotherhomogeneousstate.DenotethenumberfractionsofthemoleculesbyyA=NA/NandyB=NB/N.Thetotalnumberofmoleculesinthemixtureisconserved:N=NA+NB.DividethisequationbyN,andweobtainthatyA+yB=1.BothyAandyBarenon-negativenumbers. Denotetheenergies,volumes,andentropiesofthetwohomogeneousstatesby(uA,vA,sA)and(uB,vB,sB).Letu,v,andsbetheenergy,volume,andenergyofthemixturedividedbythetotalnumberofmolecules.Energyisanextensivevariable,sothattheenergyofthemixtureisthesumoftheenergiesofthetwohomogeneousstates:Nu=NAuA+NBuB.DividethisequationbyN,andweobtainthatu=yAuA+yBuB.Thesameruleappliestovolumeandentropy:v=yAvA+yBvB,s=yAsA+yBsB.Therulesofmixturehaveagraphicinterpretationintheenergy-volume-entropyspace.Thetwohomogeneousstates,(uA,vA,sA)and(uB,vB,sB),aretwopointsononeortwoprimitivesurfaces.Themixture,(u,v,s),isapointonthelinejoiningthetwopoints(uA,vA,sA)and(uB,vB,sB),locatedatthecenterofgravity,dependingonthefractionofmoleculesyAandyBallocatedtothetwohomogeneousstates.Ingeneral,themixture(u,v,s)isapointofftheprimitivesurfaces.Wecangeneralizetherulesofmixturetoamixtureofthreehomogeneousstates.ApuresubstancehasatotalofNmolecules,ofwhichNA,NB,andNCmoleculesareinthethreehomogeneousstates.DenotethenumberfractionsbyyA=NA/N,yB=NB/N,andyC=NC/N.HereyA,yBandyCarenon-negativenumbers.Themixtureconservesthenumberofmolecules:

NA+NB+NC=N.DividethisequationbyN,andweobtainthatyA+yB+yC=1.Thethreehomogeneousstates,(uA,vA,sA),(uB,vB,sB),and(uC,vC,sC),arethreepointsononeortwoprimitivecurves.Thethreepointsformatriangleintheenergy-volume-entropyspace.Theenergy,volume,andentropypermoleculeofthemixture(u,v,s)aregivenbyu=yAuA+yBuB+yCuC,v=yAvA+yBvB+yCvC,s=yAsA+yBsB+yCsC.Themixture,(u,v,s),isapointintheenergy-volume-entropyspace,locatedatthecenterofgravityinthetriangle,dependingonthefractionofmoleculesyA,yBandyCallocatedtothethreehomogeneousstatesattheverticesofthetriangle.Ingeneral,themixture(u,v,s)isapointofftheprimitivesurfaces.Wecanfurthergeneralizetherulesofmixturetoamixtureofanynumberofhomogenousstates.Nowconsiderallpossiblemixturesofarbitrarynumbersofhomogeneousstates.Giventheprimitivesurfaces,therulesofmixturecreateasetofpoints,whichconstituteasolidfigureintheenergy-volume-entropyspace.Ingeneral,eachpointinthesolidfigurerepresentsamixture.Inthelanguageofconvexanalysis,theenergy-volume-entropyspaceiscalledavectorspace,andeachpointinthevectorspaceiscalledavector.Allthehomogeneousstatesontheprimitivesurfacesformasetofvectors.Amixtureiscalledaconvexcombinationofthehomogenousstates,andthesetofallmixturesiscalledtheconvexhullofhomogenousstates.

DerivedsurfaceGibbscalledtheupper-boundsurfaceofallmixturesthederivedsurface.Thissurfaceisderivedbyrollingplanestangenttotheprimitivesurfaces.Intheenergy-volume-entropyspace,atangentplanecantouchtheprimitivesurfacesatone,two,orthreepoints,butnotfourormorepoints.Whenatangentplanetouchesthethreeprimitivesurfacesatthreepoints,thethreetangentpointsarethevertexofatriangle.Fromeachedgeofthetrianglewerolloutthetangentplanetotouchtwoprimitivesurfacesattwopoints.Thetwotangentpointsaretheendsofastraight-linesegment,calledthetieline.

Fromeachvertexofthetriangle,weretainaconvexpartofaprimitivesurface.Thederivedsurfacehasasinglesheet,andisaconvexsurface.Sofar,thethreequantities—energy,volume,andentropy—playsimilarroles.Allwehaveinvokedisthattheyareextensivequantities.Wenextisolateafixedamountofapuresubstancebyfixingenergyandvolume.Thisisolatedsystemhasanenormousnumberofinternalvariables:thenumberofhomogeneousstates,thelocationofeachhomogeneousstateonaprimitivesurface,andthenumberfractionofmoleculesallocatedtoeachhomogeneousstate.Whenafixedamountofasubstanceisisolatedwithafixedenergyandafixedvolume,thesubstanceingeneralisamixture,correspondingtoapointonaverticalline.Suchamixtureingeneralisnotinathermodynamicstate.Theisolatedsystemreachesequilibrium—thatis,reachathermodynamicstate—atthepointwheretheverticallineintersectsthederivedsurface.

EquilibriumofasinglehomogeneousstateIfaplanetangenttoonepointonaprimitivesurfacedoesnotcutanyprimitivesurfaces,thispointoftheprimitivesurfacebelongstothederivedsurface.Thepointcorrespondstoasinglehomogenousstateinequilibrium.Gibbscalledthesetofallsuchpointsthesurfaceofabsolutestability.Thetangentplanecanrollontheprimitivesurfacetochangethetwoslopesofthetangentplaneindependently.Theslopesofthetangentplanedeterminesthetemperatureandpressureofthestate.Thus,thesurfaceofabsolutestabilityhastwodegreesoffreedom.

EquilibriumoftwohomogeneousstatesIfaplanetangenttotwopointsontheprimitivesurfacesdoesnotcutanyprimitivesurfaces,thestraight-linesegmentconnectingthetwopointsbelongstothederivedsurface.Thestraight-linesegmentiscalledatieline.Thecommontangentplanecanrollonthetwoprimitivesurfacestochangeitsslopebyasingledegreeoffreedom.Asthecommontangentplanerolls,thetielinesformadevelopablesurface,andthetwotangentpointstraceouttwocurvesontheprimitivesurfaces.Gibbscalledthetwocurvesthelimitsofabsolutestability.

EquilibriumofthreehomogeneousstatesIfaplanetangenttothreepointsontheprimitivesurfacesanddoesnotcutanyprimitivesurfaces,thetriangleconnectingthesethreepointsbelongstothederivedsurface.Thetangentplanehasnodegreeoffreedomtoroll,andisfixedintheenergy-volume-entropyspace.

CriticalpointGibbs(1873)introducedthetheoryofcriticalpoint.HecitedapaperbyAndrews(1869),whichreportedtheexperimentalobservationofasubstancechangingcontinuouslyfromaliquidtoagas.Gibbsthenwrote,“...thederivedsurfacewhichrepresentsacompoundofliquidandvaporisterminatedasfollows:asthetangentplanerollsupontheprimitivesurface,thetwopointsofcontactapproachoneanotherandfinallyfalltogether.Therollingofthedoubletangentplanenecessarilycometoanend.thepointwherethetwopointsofcontactfalltogetheristhecriticalpoint.”Insummary,Gibbsmodeledapuresubstancewithtwoprimitivesurfaces:aconvexsurfaceforthesolidphase,andanonconvexsurfacefortheliquidandgasphases.YoucanwatchavideoontheGibbssurface.

MetastabilityAprimitivesurfacemaycontainaconvexpartandanon-convexpart.Thecurveseparatingthetwopartsiscalledthelimitofmetastability.Ifaconvexpartoftheprimitivesurfaceliesbelowatangentplaneofthederivedsurface,thepartoftheprimitivesurfaceisbeyondthelimitofabsolutestability.Eachpointofthispartoftheprimitivesurfaceiscalledametastablestate.Gibbsnotedthatsuchastateisstableinregardtocontinuouschangesofstate,butisunstableinregardtodiscontinuouschangesofstate.

Energy-volumeplaneGibbsprojectedthederivedsurfaceontothevolume-entropyplane.Hedrewthetriangleforthestatesofcoexistencethreephases,limitsofabsolutestability,limitsofmetastability,andcriticalpoint.Hedidnot,however,drawtheprimitivesurfacesandthederivedsurfaceintheenergy-volume-entropyspace.Maxwelldrewthesurfaceintheenergy-volume-entropyspaceinalatereditionofhistextbook,TheoryofHeat.Planckprojectedthederivedsurfaceontheenergy-volumeplane(Figure4)inhistextbook,TreatiseonThermodynamics.Planckmadeamistakeofaddingacriticalpointforthesolid-liquidtransition.Suchacriticalpointdoesnotexist.HereIsketchtheprojectionofthederivedsurfacetotheenergy-volumeplane.Bothenergyandvolumeareextensivequantities,sothateachthermodynamicstatecorrespondstoadistinctpointintheenergy-volumeplane.

Exercise.Findthedatafortodrawthewater-steamdomeontheenergy-volumeplane.Includetheequilibriumice-water-steamtriangle,thecriticalpoint,andseveraltielines.

TemperatureandpressureIntheabove,wehavedevelopedtheentiretheoryusingonlythreefunctionsofstate:energy,volume,andentropy.Eachisanextensivequantity,andobeystheruleofmixture.Wenextdiscusstherolesoftheothertwofunctionsofstate:temperatureandpressure.Theyareintensivequantities,andobeydifferentmathematicalrules.Writethederivedsurfaceasafunctions(u,v).Recallthemeaningsfortheslopesofthesurfaces(u,v):1/T=∂s(u,v)/∂u,P/T=∂s(u,v)/∂v.Whenthesubstanceequilibratesinamixtureoftwohomogeneousstates,atangentplanecontactstheprimitivesurfacesattwopoints,(uA,vA,sA)and(uB,vB,sB).Thetangentplanehasthesameslopesatthetwopoints,sothatthetwohomogeneousstateshavethesametemperatureandpressure:1/T=∂sA(uA,vA)/∂uA=∂sB(uB,vB)/∂uB,P/T=∂sA(uA,vA)/∂vA=∂sB(uB,vB)/∂vB.Thetangentplanecutstheverticalaxisofentropyatsomepoint.Theinterceptcanbecalculatedusingthequantitiesateitherthestate(uA,vA,sA)orthestate(uB,vB,sB),sothatsA-(1/T)uA-(P/T)vA=sB-(1/T)uB-(P/T)vB.Theaboveequationstranscribethegeometricalexpressionsoftheconditionofequilibriumintoanalyticalexpressions.

Exercise.Wehavedescribedtheconditionsoftheequilibriumofthreehomogeneousstatesingeometricterms.Transcribetheseconditionsinanalyticexpressions.

Alternativeindependentvariables

EntropyS(U,V)WehavesofarusedenergyUandvolumeVasindependentvariablestospecifythermodynamicstatesofaclosedsystem.Withthischoice,thethermodynamicsofaclosedsystemischaracterizedbythefunctionS(U,V).Theothertwofunctionsofstate—temperatureT(U,V)andpressureP(U,V)—aredeterminedfromtheslopes:1/T=∂S(U,V)/∂U,P/T=∂S(U,V)/∂V.Thus,oncethefunctionS(U,V)isdeterminedbyexperimentalmeasurement,thefunctionT(U,V)isdeterminedbycalculation.Theotherwayaroundisuntrue.OncewemeasurethefunctionT(U,V),thefunctionT(U,V)doesnotletuscalculateS(U,V).ThefunctionS(U,V)letsuscalculateallfunctionsdefinedbyit,includingtemperature,pressure,thermalcapacity,latentenergy,enthalpy,compressibility,andmanymore.Thisisanenormousreductionofexperimentalwork.Thisfactrevealsthelongarmofentropy.Wenextconsiderotherchoicesofindependentvariables.Themathematicsofchangeofvariablesmaybringconvenience,butaddsnonewphysics.

EnergyU(S,V)RecallthatS(U,V)isanincreasingfunctionwithrespecttoU.WecanthusinvertthisfunctiontoobtainthefunctionU(S,V).Thisinversionispurelymathematical,andaddsorlosesnoinformation.BothfunctionsS(U,V)andU(S,V)characterizethesameclosedsystem,andcorrespondtothesamesurfaceintheenergy-volume-entropyspace.RearrangetheequationdS=(1/T)dU+(P/T)dVasdU=TdS-PdV.ThisequationinterpretsthepartialderivativesofthefunctionU(S,V):T=∂U(S,V)/∂S,-P=∂U(S,V)/∂V.

Onepartialderivativereproducesthedefinitionoftemperature,andtheotherpartialderivativecomesfromthemechanicsofadiabaticprocess.Thischoiceofindependentvariables,SandV,placesthetemperatureandpressureinthesymmetricrolesofthetwoslopesofthesurfaceU(S,V).ThechoicecomesfromGibbs(1873)andhasbeenadoptedinmanytextbooks.Thischoiceofindependentvariablesisconvenienttodiscussanisentropicprocess,butentropyisrarelyusedasanindependentvariableinexperiments.Wewillnotconsiderthischoiceanyfurther.

HelmholtzfunctionF(T,V)Manyexperimentschoosetemperatureandvolumeasindependentvariables.Asnotedbefore,wecansetthetemperatureofasystemasanindependentvariablebybringingtheclosedsysteminthermalcontactwithathermalreservoir.DefinetheHelmholtzfunctionbyF=U-TS.NotethatdF=dU-TdS-SdT.Thisequation,togetherwiththeequationdS=(1/T)dU+(P/T)dV,givesdF=-SdT-PdV.ThisequationsuggeststhattheHelmholtzfunctionoftheclosedsystemshouldbeafunctionoftemperatureandvolume,F(T,V),withpartialderivatives-S=∂F(T,V)/∂T,-P=∂F(T,V)/∂V.Thus,oncethefunctionF(T,V)isknown,theaboveequationsdeterminethetwofunctionofstatesS(T,V)andP(T,V).Maxwellrelation.Forafunctionoftwoindependentvariables,F(T,V),recallanidentityincalculus:∂(∂F(T,V)/∂T)/∂V=∂(∂F(T,V)/∂V)/∂T.Weobtainthat∂S(T,V)/∂V=∂P(T,V)/∂T.

ThisequationiscalledaMaxwellrelation.ThermalcapacityCV(T,V).Whenthevolumeofaclosedsystemisfixed,theclosedsystembecomesathermalsystem.Recallthedefinitionofthethermalcapacity:CV=∂U(T,V)/∂T.ThesubscriptVindicatesthatthevolumeisfixed.FunctionS(T,V).Wehavejustregardtheentropyasafunctionoftemperatureandvolume,S(T,V).Recallafactincalculus:dS=(∂S(T,V)/∂T)dT+(∂S(T,V)/∂V)dVRecallthedefinitionoftemperature:dS=T-1dUatconstantvolume.Thus,∂S(T,V)/∂T=T-1∂U(T,V)/∂T=CV(T,V)/T.Thisrelation,alongwiththeMaxwellrelation∂S(T,V)/∂V=∂P(T,V)/∂T,givesthatdS=(CV(T,V)/T)dT+(∂P(T,V)/∂T)dV.ThisrelationindicatesthatwecandeterminethefunctionS(T,V)bymeasuringthetwofunctionsU(T,V)andP(T,V).FunctionU(T,V).Theaboveequation,alongwiththeequationdS=(1/T)dU+(P/T)dV,givesthatdU=CV(T,V)dT+(T∂P(T,V)/∂T-P)dV.Thisequationsuggestsanidentity:∂U(T,V)/∂V=T∂P(T,V)/∂T-P.

EnthalpyH(S,P)Onceagainconsiderahalfcylinderofwater.Aweightisplacedontopofthepiston,andthewaterisinthermalcontactwithafire.Constant-pressureprocess.LetPbethepressureinsidethecylinder,andAbetheareaofthepiston.Thepistonmoveswithoutfriction.Thepressureinthecylinderpushesthepistonup.ThebalanceofforcesrequiresthattheweightabovethepistonshouldbePA.Whentheweightisconstant,thepressureinsidethecylinderisalsoconstant.

Whentheweightisataheightz,thevolumeofthecylinderisV=Az,andthepotentialenergyoftheweightisPAz=PV.Thewaterandtheweighttogetherconstituteathermalsystem.Theenergyofthecompositeisthesumoftheinternalenergyofthewatermoleculesinthecylinder,U,andthepotentialenergyoftheweight,PV:H=U+PV.ThequantityHiscalledtheenthalpyofthewatermoleculesinthecylinder.Thus,thesameexperimentalsetupcanbeviewedasaclosedsystemorathermalsystem.Enthalpyisusedtomeasureenergytransferbyheattoaclosedsysteminaconstant-pressureprocess.FunctionH(S,P).NowletthepressurePbeanindependentvariable.Recalltheproductruleincalculus,andweobtainthatdH=dU+PdV+VdP.CombiningwithdS=(1/T)dU+(P/T)dV,weobtainthatdH=TdS+VdP.Weregardtheenthalpyasafunctionofentropyandpressure,H(S,P).Theaboveequationinterpretsthepartialderivatives:T=∂H(S,P)/∂S,V=∂H(S,P)/∂P.Entropyisrarelyusedasanindependentvariableinpractice,sothefunctionH(S,P)isseldomuseful.

GibbsfunctionG(T,P)Gibbsfunctionofaclosedsystem.ThehalfcylinderofwaterandtheweightabovethepistontogetherconstituteathermalsystemofenergyU+PV.TheHelmholtzfunctionofthisthermalsystemisG=U+PV-TS.ThisquantityiscalledtheGibbsfunction.TheenergyUhasanarbitraryadditiveconstant,whichalsoappearsinG.TheGibbsfunctionisanextensivequantity,andisafunctionofstate.PartialderivativesoftheGibbsfunction.Notethat

dG=dU+PdV+VdP-TdS-SdT.CombiningwithdS=(1/T)dU+(P/T)dV,weobtainthatdG=-SdT+VdP.WeregardtheGibbsfunctionasafunctionofpressureandtemperature,G(T,P).Theaboveequationinterpretsthepartialderivatives:-S=∂G(T,P)/∂T,V=∂G(T,P)/∂P.TheseequationssuggestaMaxwellrelation:-∂S(T,P)/∂P=∂V(T,P)/∂T.ThermalcapacityCP(T,P).Whenthepressureofaclosedsystemisfixedbyaweight,thecompositeoftheclosedsystemandtheweightbecomesathermalsystem.DefinetheenthalpyH=U+PV.Recallthedefinitionofthethermalcapacity:CP=∂H(T,P)/∂T.ThesubscriptPindicatesthatthepressureisfixed.FunctionS(T,P).Wehavejustregardtheentropyasafunctionoftemperatureandpressure,S(T,P).Recallafactincalculus:dS=(∂S(T,P)/∂T)dT+(∂S(T,P)/∂P)dPRecallthedefinitionoftemperature:dS=T-1dHatconstantpressure.Thus,∂S(T,P)/∂T=T-1∂H(T,P)/∂T=CP(T,P)/T.Thisrelation,alongwiththeMaxwellrelation∂S(T,P)/∂P=-∂V(T,P)/∂T,givesthatdS=(CP(T,P)/T)dT-(∂V(T,P)/∂T)dP.ThisrelationindicatesthatwecandeterminethefunctionS(T,P)bymeasuringthetwofunctionsH(T,P)andV(T,P).FunctionH(T,P).Theaboveequation,alongwiththeequationsH=U+PVanddS=(1/T)dU+(P/T)dV,givesthatdH=CP(T,P)dT+(V-T∂V(T,P)/∂T)dP.Thisequationsuggestsanidentity:

∂H(T,P)/∂P=V-T∂V(T,P)/∂T.

Constant-pressureandconstant-temperatureprocessConsideraclosedsystemwithaninternalvariablex.Forexample,theclosedsystemcanbeahalfcylinderofwatersealedbyafrictionlesspiston,andtheinternalvariablecanbethenumberofwatermoleculesinthevaporinsidethecylinder.Thewatermoleculesareinmechanicalequilibriumwiththeweightabovethepiston.Theweightisconstant,andisrelatedtothepressureinsidethecylinderasPA.ThewatermoleculesareinthermalequilibriumwithathermalreservoirofconstanttemperatureT.Weidentifythecompositeofthewatermolecules,theweight,andthethermalreservoirasanisolatedsystemwiththreeinternalvariables:theinternalenergyofthewatermoleculesU,thevolumeenclosedbythecylinderV,andx.TheenergyoftheisolatedsystemisUcomposite=U+PV+UR.HereUistheinternalenergyofthewatermolecules,PVisthepotentialenergyoftheweight,andURistheinternalenergyofthethermalreservoir.Theisolatedsystemconservesenergy,sothatUcomposite=constant.Thewatermoleculesconstituteaclosedsystem,characterizedbyafunctionS(U,V,x).TheentropyoftheweightSweightisconstantThethermalreservoirreservoirisathermalsystemofconstanttemperatureT,characterizedbyafunctionSR(UR),sothatSR(UR)=SR(Ucomposite)+(UR-Ucomposite)/T.TheentropyofthecompositeisScomposite=S(U,V,x)+Sweight+SR(Ucomposite)-(U+PV)/T.Theisolatedsystemisinthermalandmechanicalequilibrium,sothat∂Scomposite/∂U=0and∂Scomposite/∂V=0.WhenpressurePandtemperatureTareconstant,thetwoconditionsrecoverthefamiliarconditions:1/T=∂S(U,V,x)/∂UandP/T=∂S(U,V,x)/∂V.GiventhefunctionS(U,V,x),thesetwoequationssolvethefunctionU(T,P,x)andV(T,P,x).WecanalsowritetheentropyasafunctionS(T,P,x).

Thus,xistheonlyremaininginternalvariable.ThebasicalgorithmrequiresxtochangetoincreaseScomposite,orequivalently,toincreasethefunctionY=S-(U+PV)/T.InmaximizingthisfunctionY(T,P,x),TandParefixedbythethermalreservoirandtheweight,andonlyxisvariable.Asidefromadditiveconstants,thisthefunctionY(T,P,x)isthesubsetentropyofanisolatedsystem:thecompositeofaclosedsystem,aweightthatfixesthepressureP,andathermalreservoirthatfixesthetemperatureT.BoththefunctionJ=S-U/TandthefunctionY=S-(U+PV)/TwereintroducedbyMassieu(1869),andthefunctionYwasextensivelyusedlaterbyPlanck.WewillcallJtheMassieufunction,andYthePlanckfunction.

Algorithmofthermodynamicsforconstant-pressureandconstant-temperatureprocessWenowparaphrasethebasicalgorithmofthermodynamicsforaconstant-pressureandconstant-temperatureprocess.

1. Constructaclosedsystemwithaninternalvariablex.2. IdentifythefunctionY(T,P,x).3. Equilibrium.FindthevalueoftheinternalvariablexthatmaximizesthefunctionY(T,P,x).4. Irreversibility.Changethevalueoftheinternalvariablexinasequencethatincreases

thefunctionY(T,P,x).MaximizingtheMassieufunctionisequivalenttominimizingU+PV-ST.ThisistheGibbsfunctionG(T,P,x).Inthisminimization,TandPareconstant,andxisvariable.TheabovealgorithmcanbeparaphrasedintermsofminimizingG(T,P,x).

EquilibriumoftwohomogeneousstatesbyequatingtheGibbsfunctionTwohomogeneousstatesinequilibriumhaveequalvaluesoftemperatureTandpressureP,buthavedifferentvaluesofenergy,volume,andentropypermolecule.Thephaseboundarybetweentwophasesofapuresubstancehasasingledegreeoffreedom.WecanregardTastheindependentvariable.Alongthephaseboundary,P,sA,sB,uA,uB,vA,andvBarefunctionsofT.

LettheGibbsfunctionpermoleculeinhomogeneousstateAbegA(T,P)=uA+PvA-TsA.LettheGibbsfunctionpermoleculeinhomogeneousstateBbegB(T,P)=uB+PvB-TsB.TheGibbsfunctionpermoleculeofamixtureofthetwohomogeneousstatesisg=yAgA(T,P)+yBgB(T,P).Thechangeofphaseismodeledasaprocessofconstanttemperatureandconstantpressure.RecallthatyA+yB=1.Thefractionofmoleculesinonehomogenousstate,yA,istheindependentinternalvariable,whichisvariedtominimizetheGibbsfunctionofthemixture.TheconditionofequilibriumisgA(T,P)=gB(T,P)ThisequationisthesameasuA+PvA-TsA=uB+PvB-TsB.Thisconditionreproduceswhatwehaveobtainedbymaximizingentropy.

ClapeyronequationRecalltheidentities:dgA=-sAdT+vAdP,dgB=-sBdT+vBdP.Alongthephaseboundary,thetwophaseshavetheequalvalueoftheGibbsfunction,sothatdgA=dgB.Thus,-sAdT+vAdP=-sBdT+vBdP.Rearranging,weobtainthatdP/dT=(sB-sA)/(vB-vA).ThisresultiscalledtheClapeyronequation.

Liquid-solidphaseboundary.Therightsideoftheequationisapproximatelyindependentoftemperature.Thus,theliquid-solidphaseboundaryisapproximatelyastraightline,withtheslopegivenbytherightsideoftheClapyronequation.Forwatermolecules,thespecificvolumeissmallerinliquidwaterthaninice,vl-vs=-0.09m

3/kg.Thespecificentropyislargerinliquidwaterthaninice:sl-ss=0.09kJ/K/kg.InsertingthesevaluesintotheClapyronequation,weobtaintheslopefortheice-waterphaseboundary:dP/dT=13MPa/K.Liquid-gasandsolid-gasphaseboundaries.

Regelation

VanderWaalsmodelofliquid-gasphasetransitionTheidealgasmodelrepresentsrealgaseswellathightemperaturesandlowpressures,whenindividualmoleculesarefarapartonaverage.However,atlowtemperaturesandhighpressures,whenthemoleculesarenearcondensation,theidealgasmodelgreatlydeviatesfromthebehaviorofrealgases.WenowdescribeanequationofstateduetovanderWaals(1873):(P+aN2/V2)(V-Nb)=NkT,whereaandbareconstantsforagivensubstance.Whena=0andb=0,thevanderWaalsequationreducestotheequationofidealgases.Wenextexaminethephysicalsignificanceofthetwomodifications.

(FiguretakenfromtheWikipediapageonvanderWaalsequation)

IsothermsWritethevanderWaalsequationasP=NkT/(V-Nb)-aN2/V2.ForafixednumberofmoleculesNandaconstanttemperatureT,thisequationcorrespondstoacurveonthepressure-volumeplane.Thecurveiscalledanisotherm.Anisothermatahightemperatureisamonotoniccure.Anisothermalatalowtemperaturehasaminimumandamaximum.

CriticalpointThecriticalpointtakesplaceontheisothermwheretheminimumandthemaximumcollide,sothat∂P(T,V)/∂V=0,∂2P(T,V)/∂V2=0.Thus,thecriticalpointsatisfythreeequations:P=NkT/(V-Nb)-aN2/V2,-NkT/(V-Nb)2+2aN2/V3=0,2NkT/(V-Nb)3-6aN2/V4=0.Solvingthesethreeequations,weobtainthat

Vc=3Nb,kTc=8a/(27b),Pc=a/(27b

2).Theseequationsexpressthecriticalvolume,temperature,andpressureintermsoftheconstantsaandb.

EnergyRecallanidentityforaclosedsystem:∂U(T,V)/∂V=T∂P(T,V)/∂T-P.Weobtainthat∂U(T,V)/∂V=aN2/V2.Integrating,weobtainthatU(T,V)=-aN2/V+Nu(T).Hereu(T)isthesameasthethermalenergypermoleculeforanidealgas.ThetermaN2/Vrepresentstheeffectofcohesionbetweenthemolecules.ThevanderWaalsmodelassumesthatthecohesionreducesthethermalenergybyanamountproportionaltothenumberdensityofthemolecules.Thisseemstobeareasonablefirst-orderapproximation.InthevanderWaalsmodel,thethermalcapacityisindependentofvolume:CV=∂U(T,V)/∂T=Ndu(T)dT=NcV(T),wherecV(T)isthesameasthethermalcapacitypermoleculeinanidealgas.

EntropyRecallanotheridentityforaclosedsystem:dS=(CV(T,V)/T)dT+(∂P(T,V)/∂T)dV.ForthevanderWaalsmodel,thisequationreducestodS=(NcV(T)/T)dT+(kN/(V-Nb))dV.ThetermNbrepresentstheeffectoffinitevolumeofthemolecules.Atafixedtemperature,thenumberofquantumstatesisproportionalto(V-Nb)N.Intheabove,wehavestartedfromthevanderWaalsequation,andexamineditsconsequencesforenergyandentropy.SincethephysicalinterpretationofthetwotermsaN2/VandNbarequitereasonable,wemayaswellusethemasastartingpointtoderivethevanderWaalsequation.

Entropy-energycompetitionAfixednumberofmoleculesformsaclosedsystem,whichwecharacterizeusingtheHelmholtzfunctionF(T,V).RecallthatdF=-SdT-PdV.Atafixedtemperature,theHelmholtzfunctionisafunctionofvolume,F(V).PlotthefunctionF(V)asacurveintheplanewithVasthehorizontalaxisandFastheverticalaxis.WhenthevolumechangebydV,theHelmholtzfunctionchangesbydF=-PdV.Thus,-PistheslopeofthecurveF(V).BecausePispositive,FdecreasesasVincreases. Whenthepressuredecreasesasthevolumeincreases,thecurveF(V)isconvexdownward.Whenthepressureincreasesasthevolumeincreases,thecurveF(V)isconvexupward.Thatis,anonmonotonicP(V)curvecorrespondstoanonconvexF(V)curveHowdoesthevanderWaalsmodelproduceanonconvexHelmholtzfunction?RecallthedefinitionoftheHelmholtzfunction,F=U-TS.ForthevanderWaalsmodel,theHelmholtzfunctiontakestheformF(V)=-aN2/V-NkTlog(V-Nb).HerewehavedroppedadditivetermswhicharepurelyfunctionsofT.EntropyandenergycompetetobendthecurveF(V).Thesecondtermcomesfromtheentropyofthemolecules,andisconvexdownward,whichstabilizesahomogeneousstate.Thefirsttermcomesfromthecohesionofthemolecules,andisconvexupward,whichdestabilizesahomogeneousstate.Theentropytendstodispersemolecules,buttheenergytendstoattractmoleculestogether.Atahightemperature,theentropyprevailsforallvaluesofvolume,sothattheentirecurveF(V)isconvexdownward,andtheentirecurveP(V)ismonotonic.Atalowtemperature,theentropyprevailsforsmallandlargevaluesofvolume,sothatonlythesepartsofthecurveF(V)isconvexdownward,andonlythesepartsofthecurveP(V)ismonotonic.

MaxwellruleAconvex-upwardpartoftheF(V)curvecorrespondstoaphasetransition.DrawalinetangenttotheF(V)curveattwopointsAandB.Thetwopointscorrespondtotwohomogeneousstatesinequilibriumatthesamepressure,Psatu.Thispressurecorrespondstotheslopeofthetangentline:FA-FB=Psatu(VB-VA).Thisequationcanbeinterpretedonthepressure-volumeplane:FA-FBistheareaunderthecurveP(V)betweenstatesAandB,andPsatu(VB-VA)istheareaofarectangle.TheequalityofthetwoareasrequiresthatPsatubeplacedatthelevelthatequatethetwoshadedareas.ThisconstructioniscalledtheMaxwellrule.OfthetotalofNmolecules,NAmoleculesareinhomogeneousstateA,andNBmoleculesareinhomogeneousstateB.Themixtureofthetwohomogeneousstatescorrespondtoapointonthetangentline,locatedatthecenterofgravityaccordingtoNAandNB.

OpensystemNowentersanothersupportingrole—thenumberofaspeciesofmolecules.Inthermodynamics,thenumberofaspeciesofmolecules,energy,andvolumeplayanalogoussupportingroles.

AfamilyofisolatedsystemsofmanyindependentvariationsAnopensystemanditssurroundingstransferenergy,volume,andmolecules.Themoleculescanbeofmanyspecies.Toillustratebasicideas,considerthatonlytwospeciesofmolecules,1and2,transferbetweentheopensystemanditssurroundings.Allotherspeciesofmoleculesareblocked.LettheenergybeU,volumebeV,thenumberofmolecularspecies1beN1,andthenumberofmolecularspecies2beN2.WhenU,V,N1,N2arefixed,theopensystembecomesanisolatedsystem.DenotethenumberofquantumstatesofthisisolatedsystembyΩ(U,V,N1,N2).AsU,V,N1,N2vary,thefunctionΩ(U,V,N1,N2),oritsequivalent,S(U,V,N1,N2)=logΩ(U,V,N1,N2),characterizestheopensystemasafamilyofisolatedsystems.Thefamilyofisolatedsystemshasfourindependentvariations,U,V,N1,N2.

DefinitionofchemicalpotentialsForthefunctionoffourvariables,S(U,V,N1,N2),recallafactofcalculus:dS=(∂S(U,V,N1,N2)/∂U)dU+(∂S(U,V,N1,N2)/∂V)dV+(∂S(U,V,N1,N2)/∂N1)dN1+(∂S(U,V,N1,N2)/∂N2)dN2.Whenweblockthetransferofthemoleculesbetweentheopensystemanditssurroundings,butallowthetransferofenergyandvolume,theopensystembecomesaclosedsystem.Wehavealreadyrelatedtwopartialderivativestoexperimentallymeasurablequantities:1/T=∂S(U,V,N1,N2)/∂U,P/T=∂S(U,V,N1,N2)/∂V.Theothertwopartialderivativesareusedtodefinethechemicalpotentials:-µ1/T=∂S(U,V,N1,N2)/∂N1,-µ2/T=∂S(U,V,N1,N2)/∂N2.Theratioµ1/Tisthechildoftheunionoftheentropyandthenumberofmolecularspecies1,andtheratioµ2/Tisthechildoftheunionoftheentropyandthenumberofmolecularspecies2,justasthetemperatureisthechildoftheunionofentropyandenergy,andastheratioP/Tis

thechildoftheunionofentropyandvolume.Thechemicalpotentialsareintensivefunctionsofstate.WritedS=(1/T)dU+(P/T)dV-(µ1/T)dN1-(µ2/T)dN2.Flexibilityindefiningchemicalpotentials.Wehavealreadymentionedtheflexibilityindefiningtemperature:anymonotonicallydecreasingfunctionofthederivative∂S(U,V,N1,N2)/∂Ucanbeusedtodefinetemperature.Thisenormousflexibilitycomesaboutbecausetemperaturehasnodefinitionoutsidethermodynamics.Thechoiceadoptedhere,1/T=∂S(U,V,N1,N2)/∂U,isahistoricalaccident.Wedonothaveanyflexibilityindefiningpressure;weinsistthatthedefinitionofpressureshouldrecoverthatinmechanics,force/area.WehaveshownthatP/T=∂S(U,V,N1,N2)/∂V.Indefiningchemicalpotentials,onceagainwehaveenormousflexibility,becausechemicalpotentialshavenodefinitionoutsidethermodynamics.Whatreallymattersisthatthederivatives∂S(U,V,N1,N2)/∂U,∂S(U,V,N1,N2)/∂V,∂S(U,V,N1,N2)/∂N1,and∂S(U,V,N1,N2)/∂N2playanalogousroles.Allthesepartialderivativesareequallysignificantbecausethermodynamicsisaplayofmaximization.Evenifwechoosenottocallthesederivativesbyanyname,wewillstillbedoingthesameexperimentandthesamecalculation.TheparticulardefinitionofchemicalpotentialsadoptedherecomesfromGibbs(1875),andisjustanamegiventoapartialderivative.Wedonotneedanyreasontogiveaparticularnametoachild.Forthisdefinitionofchemicalpotential,wewillfindareasonforthepresenceofthenegativesign,butwecannotfindanyreasonforthepresenceoftemperature.Notethat∂S(U,V,N1,N2)/∂N1isapurenumber.Thenumbermeanstheincreaseofthenumberofquantumstatesassociatedwithaddingonemoleculeofspecies1,whilekeepingtheenergy,volume,andnumberofmoleculesofspecies2fixed.Thequantityhasclearsignificance.Gibbstwistedthisnumberintoaquantitytohavetheunitofenergy/amount.Hewasperhapstooenamoredwiththesupportingactor,energy.Hisreasonwastwisted,buthisdefinitionhasstuck.Usageofwords.Whenwespeakofachemicalpotential,weshouldnameboththemoleculespeciesandtheopensystem.Forexample,wespeakofthechemicalpotentialofwatermoleculesinapieceofcheese,orthechemicalpotentialofwatermoleculesinaglassofwine.WealsospeakofthechemicalpotentialofcarbondioxideinabottleofCocaCola.Wedenotethechemicalpotentialofmolecularspecies1inopensystemAbyµ1,A.

Whenwespeakoftemperature,weonlyneedtonametheplace.Forexample,wespeakofthetemperatureofapieceofcheese,orthetemperatureofaglassofwine.Thisdifferenceinusagecomesfromsomethingbasic:theworldhasmanyspeciesofmolecules,butonlyonespeciesofenergy.

TwoopensystemsincontactTwoopensystems,AandB,exchangeenergy,volume,andtwomolecularspecies,1and2.TheopensystemAischaracterizedbyafunctionSA(UA,VA,N1,A,N2,A),andthesystemBischaracterizedbyanotherfunctionSB(UB,VB,N1,B,N2,B).NotethatN1,Adenotesthenumberofmoleculesofspecies1insystemA.Wemakethecompositeofthetwoopensystemsintoanisolatedsystem.TheprinciplesofconservationrequirethatUA+UB=constant,VA+VB=constant,N1,A+N1,B=constant,N2,A+N2,B=constant.Hereweassumethatthetwospeciesofmoleculesdonotundergoachemicalreaction,sothatthenumberofmoleculesineachspeciesisconserved.Thecompositeisanisolatedsystemoffourindependentinternalvariables,UA,VA,N1,A,N2,A.Whentheinternalvariablesarefixedatparticularvalues,theisolatedsystemcanonlyflipinasubsetofthesamplespace.DenotethesubsetentropybyScomposite(UA,VA,N1,A,N2,A).Entropyisanextensivequantity,sothatScomposite(UA,VA,N1,A,N2,A)=SA(UA,VA,N1,A,N2,A)+SB(UB,VB,N1,B,N2,B).WhentheinternalvariableschangebydUA,dVA,dN1,A,dN1,B,thesubsetentropychangesbydScomposite=(1/TA-1/TB)dUA+(PA/TA-PB/TB)dVA+(-µ1,A/TA+µ1,B/TB)dN1,A+(-µ2,A/TA+µ2,B/TB)dN2,A.Equilibrium.ThefourinternalvariablesUA,VA,N1,A,N2,Acanchangeindependently.Inequilibrium,thesubsetentropymaximizes,dScomposite=0,sothatTA=TB,PA=PB,µ1,A=µ1,B,µ2,A=µ2,B.

Irreversibility.Outofequilibrium,thesubsetentropyincreasesintime,dScomposite(UA,VA,NA)>0.Considerasituationwherethetwoopensystemsareinpartialequilibrium,TA=TB,PA=PB,µ1,A=µ1,B,butnotinequilibriumwithrespecttothetransferofmolecularspecies2.TheinequalitydScomposite(UA,VA,NA)>0reducesto(-µ2,A+µ2,B)dN2,A>0.Thus,molecularspecies2transferfromthesystemofhighchemicalpotentialtothesystemoflowchemicalpotential.Thepresenceofthenegativesigninthedefinitionofchemicalpotentialleadstothisverbalconvenience.

ExperimentaldeterminationofthechemicalpotentialofaspeciesofmoleculesinacomplexsystemHowdoweexperimentallymeasurethechemicalpotentialofaspeciesofmoleculesinacomplexsystem?Whentwosystemscanexchangeenergyandaspeciesofmolecules,thefundamentalpostulatedictatesthatthetwosystemsreachequilibriumwhentheyhavethesametemperatureandthesamechemicalpotentialofthespeciesofmolecules.Consequently,oncethechemicalpotentialofaspeciesofmoleculesinonesystemisdetermined,thesystemcanbeusedtodeterminethechemicalpotentialofthesamespeciesofmoleculesinothersystems.Forexample,wecandeterminethechemicalpotentialofwatermoleculesinaflaskcontainingapurewatervaporasafunctionoftemperatureandpressure,µ(T,P).Wenowwishtomeasurethechemicalpotentialofwatermoleculesinaglassofwine.Wecanbringthewineintocontactwithaflaskofwatervapor.Thecontactismadewithasemipermeablemembranethatallowswatermoleculestogothrough,butblocksallotherspeciesofmolecules.Wethenallowthewinetoequilibratewiththewatervaporintheflask,sothatbothenergyandwatermoleculesstopexchangingbetweenthewineandtheflask.Thetwosystemshavethesametemperatureandthesamechemicalpotential.Areadingofthechemicalpotentialofwaterintheflaskgivesthechemicalpotentialofwaterinthewine.Molecularreservoir.Wecanfixthechemicalpotentialofaspeciesofmoleculesinasystembylettingittransfersthespeciesofmoleculestoamolecularreservoir.Thesituationisanalogoustofixingtemperature.Themolecularreservoirhasafixedchemicalpotentialofthespeciesofmolecules.Forexample,alargetankofanaqueoussolutionofsaltisamolecularreservoirofwater.Asmallamountofwatermoleculescangoinandoutofthetank,thoughthevapor.Thesaltevaporatesnegligibly.Thechemicalpotentialofwaterinthesolutionisfixed.

ExtensibilityInmathematics,afunctionZ(X,Y)iscalledahomogeneousfunctionifforanynumberathefollowingrelationholds:aZ(X,Y)=Z(aX,aY).Takederivativewithrespectivetoa,andweobtainthatZ(X,Y)=X∂Z(X,Y)/∂X+Y∂Z(X,Y)/∂Y.Thismathematicalidentityholdsforanyhomogeneousfunction.Wenextapplythismathematicalidentitytoanopensystemoftwospeciesofmolecules,characterizedbythefunctionS(U,V,N1,N2).NotethatS,U,V,N1,N2areextensiveproperties.Ifweamplifyeveryextensivepropertybyafactorofa,thecharacteristicfunctionobeystherelation:aS(U,V,N1,N2)=S(aU,aV,aN1,aN2).Thus,thecharacteristicfunctionS(U,V,N1,N2)isahomogeneousfunctionoffourindependentvariables.Takederivativewithrespectivetoa,andweobtainthatS=U/T+PV/T-N1µ1/T-N2µ2/T.ThefourpartialderivativesofthefunctionS(U,V,N1,N2)definefourintensiveproperties.Whenweamplifyeveryextensivepropertybyafactorofa,theopensystemincreasessizeproportionally,butalltheintensivepropertiesremainunchanged.

GibbsfunctionRecallthedefinitionoftheGibbsfunction:G=U-TS+PV.Ofthefivequantitiesontheright-handsideoftheaboveequation,onlyenergyUhasanarbitraryadditiveconstant.ThesameadditiveconstantappearsintheGibbsfunction.ThedefinitionoftheGibbsfunction,alongwiththeequationdS=(1/T)dU+(P/T)dV-(µ1/T)dN1-(µ2/T)dN2,givesthat

dG=-SdT+VdP+µ1dN1+µ2dN2.ThisequationsuggeststhattheGibbsfunctionberegardedasafunctionG(T,P,N1,N2),withthepartialderivatives-S=∂G(T,P,N1,N2)/∂T,V=∂G(T,P,N1,N2)/∂P,µ1=∂G(T,P,N1,N2)/∂N1,µ2=∂G(T,P,N1,N2)/∂N2.

Gibbs-DuhemrelationForanopensystemoffourextensiblevariables,U,V,N1,N2,oncethesizeofthesystem(i,e.,thetotalnumberofmolecules)isfixed,thesystemhasonlythreeindependentvariations.Thus,thefourintensivequantities,T,P,µ1,µ2,cannotbeindependentvariables.Wenextderivearelationamongthefourintensivequantities.ThedefinitionoftheGibbsfunction,G=U-TS+PV,alongwiththeequationS=U/T+PV/T-N1µ1/T-N2µ2/T,givesthatG=µ1N1+µ2N2.Takingderivative,weobtainthatdG=µ1dN1+µ2dN2+N1dµ1+N2dµ2.Thisequation,alongwiththeequationdG=-SdT+VdP+µ1dN1+µ2dN2,givesthat-SdT+VdP-N1dµ1-N2dµ2=0.Thisequation,calledtheGibbs-Duhemrelation,relatesthechangesinthefourintensivequantities,T,P,µ1,µ2.

ChemicalpotentialofaspeciesofmoleculesinapuresubstanceChemicalpotentialisanintensiveproperty.Wehavespecifiedathermodynamicstateofapuresubstancebytwointensiveproperties,temperatureandpressure.Thus,forapuresubstance,thethreeintensiveproperties,chemicalpotential,temperature,andpressure,arenotindependentproperties.Wenowderivearelationbetweenthem.

WemodelapieceofapuresubstanceasanopensystemofacharacteristicfunctionS(U,V,N).Thus,dS=(1/T)dU+(P/T)dV-(µ/T)dNThisequationdefinesthetemperatureT,thepressureP,andthechemicalpotentialµ.Wecanincreasethenumberofmoleculesthepiecewithoutchangingthefunctionsofstatepermolecule,u,v,s,andwithoutchangingTandP.WhenweadddNnumberofmoleculestothepiece,theextensivefunctionsofstatechangebydS=sdN,dU=udN,anddV=vdN.TheequationdS=(1/T)dU+(P/T)dV-(µ/T)dNbecomess=(1/T)u+(P/T)v-(µ/T).Rearranging,wefindthatµ=u+Pv−Ts.ThisequationshowsthatthechemicalpotentialofaspeciesofmoleculesinapuresubstanceequalstheGibbsfunctionpermoleculeofthepuresubstance.Wehavealreadylearnedhowtomeasurethefunctions(u,v)forapuresubstanceexperimentally.Onces(u,v)isdetermined,soisthechemicalpotential.Ofthefivequantitiesontheright-handsideoftheaboveequation,onlyenergyuhasanarbitraryadditiveconstant.Thesameadditiveconstantappearsinthechemicalpotentialofthemolecularspeciesinthepuresubstance.Recallthatds=(1/T)du+(P/T)dv.Thisequation,alongwithµ=u+Pv−Ts,givesthatdµ=−sdT+vdP.Thisequationsuggeststhatweregardthechemicalpotentialasafunctionoftemperatureandpressure,µ(T,P),withthepartialderivatives∂µ(T,P)/∂T=−s,∂µ(T,P)/∂P=v.TheseresultsrecoverthesimilarequationswhenweregardthechemicalpotentialofaspeciesofmoleculesinapuresubstanceastheGibbsfunctionpermoleculeofthesubstance.

ChemicalpotentialofaspeciesofmoleculesinanidealgasAnidealgas,ofNnumberofasinglespeciesofmoleculesinaflaskofvolumeV,issubjecttopressurePandtemperatureT.Recalltheequationsoftheidealgasmodel:PV=NkT.dU=NcV(T)dTdS=(NcV(T)/T)dT+(Nk/V)dV.Thechemicalpotentialµ=u+Pv−Tscanbewrittenasµ(T,P)=µ(T,P0)+kTlog(P/P0).HereP0isanarbitrarypressure.Atafixedtemperature,thisexpressiondeterminesthechemicalpotentialofaspeciesofmoleculesinanidealgasuptoanadditiveconstant.

HumidityAtagiventemperature,whenamoistairisinequilibriumwiththeliquidwater,wesaythattheairissaturatedwithwater.Ifaircontainsfewerwatermoleculesthanthesaturatedairdoes,thenumberofwatermoleculesintheairdividedbythenumberofwatermoleculesinthesaturatedairiscalledtherelativehumidity.WriteRH=N/Nsatu.Whenthevaporismodeledasanidealgas,therelativehumidityisalsogivenbyRH=P/Psatu,wherePisthepartialpressureofwaterintheunsaturatedgas,P<Psatu.Writethechemicalpotentialofwaterintheairasµ=kTlog(P/Psatu)=kTlog(RH),withtheunderstandingthatthechemicalpotentialisrelativetothatofthewatermoleculesinasaturatedwateratthesametemperature.Thelungisalwayssaturatedwithwatervaporatthebodytemperature(37C),buttheatmosphericairmaynotbe.Inwinter,thecoldairoutsidehaslowwatercontentevenat100%relativehumidity.Whenthecoldairentersawarmroom,therelativehumidityintheroomwill

reducebelow100%atroomtemperature.Wewillfeeluncomfortable.Also,waterinsidethewarmroomwillcondenseoncoldwindowpanes.

IncompressiblepuresubstanceInmanyapplicationsofliquids,thepressureissmall,sothatthevolumepermoleculeintheliquid,v,istakentobeindependentofthepressure,andtheliquidiscalledincompressible.Recallthechemicalpotentialforapuresubstance,µ=u+Pv−Ts.Foranincompressiblesubstance,u,v,andsarefunctionsofT,andareindependentofP.Weobtainthatµ(T,P)=u(T)-Ts(T)+Pv(T).Thechemicalpotentialofaspeciesofmoleculesinanincompressibleliquidislinearinpressure.Often,weassumethatthespecificvolumeisalsoindependentoftemperature.

TheascentofsapHowdoesatreetransportliquidwaterfromthegroundtothetop?Unlikeananimal,thetreedoesnothaveahearttopumpliquid.Thetreedoeshaveasystemoftubes,calledxylums,toconductliquidwater.Thetensilestress.Letusviewacolumnofwaterasafreebody.LetAbethecross-sectionalareaofthecolumn.Thepressureinthewateratthetopofthetree,Ptop,exertsaforceAPtop.Thepressureinthewateratthebottomofthetree,Pbottom,exertsaforceAPbottom.ThecolumnofwaterweighsρghA,whereρisthemassdensity,gistheaccelerationofgravity,andhistheheightofthetree.ThebalanceofthethreeforcesrequiresthatPtop=Pbottom-ρgh.Thebottomofthetreeistakentoequilibratewiththewaterinthesoilthroughtherootsofthetree.Assumethatthesoilissaturatedwithwater,sothatPbottom=Psatu.Weestimatetheothertermfora100mtalltree.ρgh=(1000kg/m3)(10m/s2)(100m)=1MPa.Thepressureatthetopofthetreeisenormousandnegative:Ptop=-0.9MPa.

Anegativepressuremeansatensilestress.Thus,thewaterrisesbyatensileforceappliedatthetopofthetree.Whatappliesthetensilestress?Thelowhumidity!Atthetopofthetree,thechemicalpotentialofwaterintheliquidinsidethetreeisµ=(Ptop-Psatu)v.Thechemicalpotentialofwatermoleculesintheairisµ=kTlog(RH).Whentheliquidwaterequilibrateswiththegaseouswaterintheair,thetwochemicalpotentialsareequal,sothatρgh=-(kT/v)log(RH).Thevolumeperwatermoleculeisv=.AttemperatureT=300K,kT=1.38.thus,kT/v=…

Mixtureofidealgases

Chemicalreaction

StoichiometriccoefficientsAtatmosphericpressure,above100degreeCelsius,watermoleculesformagas.Atthispressureandtemperature,oxygenmoleculesalsoformagas,sodohydrogenmolecules.Whenthethreespeciesofmoleculesareenclosedinthesamecontainer,moleculesflyandcollide.Aftercollision,twomoleculesseparateonsomeoccasions,butformnewmoleculesonotheroccasions.LetusfocusonthechemicalreactionO2+2H2=2H2O.Eachoxygenmoleculeconsistsoftwooxygenatoms,eachhydrogenmoleculeconsistsoftwohydrogenatoms,andeachwatermoleculeconsistsoftwohydrogenatomsandoneoxygenatom.Inthisreaction,theoxygenmoleculesandhydrogenmoleculesarecalledthereactants,andthewatermoleculesarecalledthereactionproducts.

Achemicalreactionrecombineatomsfromonesetofmolecules,thereactants,toanothersetofmolecules,thereactionproducts.Thereactionconservesthenumberofeachspeciesofatoms,butchangesthenumberofeachspeciesofmolecules.Oneoxygenmoleculeandtwohydrogenmoleculesreacttoformtwowatermolecules.Thecoefficientsinfrontofthemoleculesensurethateveryspeciesofatomsisconserved.Thesecoefficientsarecalledthestoichiometriccoefficients.Onceatomiccompositionofeverymoleculeisknown,thestoichiometriccoefficientsaredeterminedbyconservingeveryspeciesofatoms.Exercise.AreactionfundamentaltolifeonEarthisphotosynthesis.Plantsabsorbcarbondioxideandwatertoproduceglucose.Determinethestoichiometriccoefficientsofthereaction.

DegreeofreactionConsiderareactionoffourspeciesofmolecules,A,B,C,andD:2A+3B=5C+7D.Theequationindicatesthestoichiometriccoefficientsofthereaction.Denotethedegreeofreactionbydx,suchthatthenumbersofthefourspeciesofmoleculesA,B,C,andDchangebydNA=-2dx,dNB=-3dx,dNC=5dx,dND=7dx.

ChemicalequilibriumWeassumethatthereactiontakesplaceinacylindersealedwithapiston,sothatboththetemperatureTandthepressureParefixedasthereactionproceeds.LettheGibbsfunctionbeG(T,P,NA,NB,NC,ND).Whenthereactionadvancesbydxundertheconditionofconstanttemperatureandconstantpressure,theGibbsfunctionchangesbydG=(5µC+7µD-2µA-3µB)dx.ReactiongoesinthedirectionthatreducestheGibbsfunction.Thus,thereactionmovestoward,dx>0,when2µA+3µB>5µC+7µD.

Thereactionmovesbackward,dx<0,when2µA+3µB<5µC+7µD.Thereactionreachesequilibriumwhen2µA+3µB=5µC+7µD.

ReactionofidealgasesWhenallspeciesofmoleculesformidealgases,eachspeciesobeystheidealgaslaw:PA=cAkT,wherecA=NA/VistheconcentrationofspeciesA.Thechemicalpotentialofspeciesinthecylinderisµ(T,PA)=µ(T,P0)+kTlog(PA/P0).(cC

5cD7)/(cA

2cB3)=K.