L.D.D

AdvancedBuildingPhysics-Thermodynamics

1-PHYSICALQUANTITIESANDMEASUREMENT

Qualityofanobject--->quasi-serialorder(qualitativeorder)--->numberassignedtothe

quasi-serialorder--->quantitativeorder

�Inthedescriptionofaphysicalphenomenon,onlythetermsthatcanbeoperationally

definedmustbeused.Thesearecalledphysicalquantities.

[Physicstextbooks]

�Eachphysicalquantityisdefinedbythesetofoperationsthatareneededtoobtainitsmeasurement.Eachentitythatcannotbeoperationallydefinedisanobjectthatshouldnot

haveaplacewithinphysicaltheories.

[PercyBridgman-manifestoof'ThelogicofmodernPhysics']

�Themajorityofphysicalquantitiesaredefinedbythesetofoperations(practicalof

theoretical)thatareneededtoobtainitsmeasurement.

[Moderateattitude-assumedinthiscourse]

-->Carnap:Atheoreticaltermissignificantifexistsastatementscontainingtsuchthat

fromsandtheremainingcomplexofthetheoryit'spossibletoderiveanobservational

statementthatcouldn'tbederivedwithouts.

(e.g.entropy--->energyflowsfromhighertolowertemperature)

Quantitativedescriptionofreality-->measuringprocess

-Choiceoffundamentalquantities

-Choiceofunitofmeasurement

-Constructionofsamplesofadoptedunits

�Wecall'measureofaquantity'therationbetweenthevalueofthequantityGandthatofaquantityhomogeneoustotheonetobemeasured,UG,thatwechooseasmeasurement

unit.

Tobeadopted,ameasurementunithastomeetthefollowingrequirements:

-Precision

-Accessibility

-Reproducibility

-Invariability(-->sthneedtouseatomicsamplesforthisrequirement)

Note:

-Accuracy:capabilityofaninstrumenttoindicatethetruevalueofmeasuredquantity

-Precision:repetabilityofmeasurementsofthesamequantityunderthesameconditions

-Error,random:statisticalerrorcausedbychanceandnotrecurring

-Error,systematic:persistenterrornotduetochance

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2-THREEVIEWSONTHECOGNITIVEVALUEOFSCIENTIFICTHEORIES:

REALISM,POSITIVISMANDAMENDEDREALISM

�Realisminterpretsscienceasasimpletranscriptionoftheobservations,aliteral

descriptionofnature,a'readingofthegreetbookofnature'.

�Positivisminterpretsscientifictheoriesthatgobeyondimmediateexperience,toolsto

correlateandpredicttheresultsofpossibleexperimentsandnotaneffectivedescriptionof

physicalreality.

�AccordingtotheModifiedrealism,scientifictheoriesareneitherasimplereflectionof

thelawsalreadywritteninnature,norsimplecalculationtools.Theyareman-mademodels

continuouslycomparedwithwhatwecallreality.Itispossibletohavealternative

theoreticaldescriptionsofthesamesetofexperimentalobservations.Thedescriptionthat

possiblyprevailsisnotthedescriptionofreality,butthemodelconsideredmore

appropriatetoexplaintheknownfacts.

[Adoptedmodelinthiscourse-lessextremethantheothers]

(e.g.--->Schroedingerequation:Quantummechanic/Relativity)

�Inductiveinferencesbringfrompremisesaroundparticularcasestoaconclusionhaving

thecharacterofagenerallaworaprinciple.

�Deductiveinferencesstartingfrompremisesgenerallyvalidbringtowhathappensina

particularcase.

�Atheoryiscalledscientificifitisconfutablei.e.ifstartingfromaseriesofhypothesis,it

makes(viadeduction)predictionsthatcanbecomparedwithobservationsandexperiments

andifthiscomparisonallows,inprinciple,tofalsifythetheory.

[CarlPopper-CriterionofDemarcation]

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3-THEAXIOMATIZATIONOFTHERMODYNAMICS

1)Clausius/Kelvin/Caratheodory:thermodynamicsystemtreatedasablackboxthat

exchangesenergywithauxiliarysystemsidealizedasreservesofheatandwork.

-->operationaldefinitionofthermalquantities(U,S,T)startingfrommechanical

macroscopicmeasurableparameters(p,v).

2)Gibbs/Tisza/Callen:theconceptsofinternalenergyandentropyareconsideredasastartingdatumandareusedinordertoprovideadetaileddescriptionofthesystemat

equilibrium.

-->describethethermodynamicequilibriumusingtherelationshipbetweenthevariables

u,s,vratherthantheonebetweenp,v,Tthataredirectlymeasurable.

-->usefulbecausep=p(v,T)isderivablefromu=u(s,v)buttheoppositisnottrue.

�Thermodynamicisthestudyofmacroscopiceffectsofthemyriadsofatomiccoordinates

that,duetostatisticalaverages,donotappearexplicitlyinthemacroscopicdescriptionofa

system.

�Theenergytransferredthroughmodeswhicharenotvisibleatamacroscopiclevelis

calledHeat.(-->becauseifitwasmechanic/thermaltransferiwouldhaveseenitatamacroscopiclevel)

Thermodynamicisaverygeneraltheorythatcanbeapplicabletosystemswithany

mechanical,electricorthermalproperty.Inordertosimplifythestudyofthermodynamics,

weintroducesimplesystems:

a)Macroscopicallyhomogeneous

b)Isotropic(propertiesareindependentofdirection)

c)Electricallyneutral(atamacroscopiclevel)

d)Chemicallyinert

e)Freeofsuperficialeffects

f)Notsubjectedtoelectric,magneticorgravitationalfields(isolated)

Significantparametersforasimplesystem:

-Volume(V)

-Numbersofmoles(NK)(k=1,2...c;c=numberofchemicalcomponentsinthesystem)

�WecanfindtheMolarMassexpressingingramstheatomicnumberorthesumofthe

atomicnumbersoftheindividualcomponents(foramolecule).

Acomposedsystemismadebytheunion(U)ofdisjointsimplesystems(i.e.systemshaving

intersectionequaltozero).Thesimplesystemsarecalledsub-systems.

�Theparametersthatinacomposedsystemhavevalueequaltothesumofthevalues

assumedinthesinglesubsystems,arecalledExtensiveParameters.(-->theothersareintensive)

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�TheMolarFractionofthechemicalspecieskisdefinedastheratiobetweenthemolesof

aconstituentandthesumofmolesofalltheconstituents.

Thedefinitionofathermodynamicsystemrequiresthatwemakeitdistinguishablefromthe

restoftheuniverse,thatwetraceaboundarythatrealizesthisseparation.

�Anywallorsurfaceorphysicaleffectthatpreventtheexchangeofanextensivequantity,iscalledRestrictive(withrespecttothisquantity).

Leibnitz:'Principleofconservationofenergy'.

-->Relatedtothesumofkineticandpotentialenergyforamaterialpointsubjectedtothe

earth'sgravitationalfield(frictionsareconsiderednegligible).

[1/2mv2+mgz=costant]

Bohr:'Restrictedvalidity'oftheprincipleofconservationofenergyatsubatomiclevel

Pauli:Hypotesisofconservationofenergyatsubatomiclevelandexistenceofanewparticle

Fermi:NewparticlediscoveredbyPauliiscalled'Neutrino'.Noelectriccharge,massequal

tozero.[1956].

Consideringmacroscopicsystemsasaggregatesofelectronsandnuclei,subjectedto

interactionsforwhichisvalidtheconservationofenergy,wecanformulatesome

hypothesis:

�POSTULATEA:amacroscopicsystemhasineachstateawell-definedenergy,subjectedto

theprincipleofconservation

--->energyisastatefunction

--->energyisconserved

Whenwearetreatingsimplesystems,weareconsideringonlytheenergyboundtothe

hiddencoordinates,thatwecallinternalenergy(U).

�Foranysimplethermodynamicsystemitisnecessarytodefineareferencestatetowhich

isarbitrarilyassignedavalueofinternalenergyequaltozero(U0=0).

--->Foranyotherstatetheinternalenergywillbedeterminedasthedifferencebetween

theinternalenergyofthestateweareconsideringandtheinternalenergyofthereference

state.

�POSTULATEB:weassumethatalsotheinternalenergyisanextensivevariable,suchasV

andNK.

--->Theinternalenergyofacompositesystemisthesumoftheinternalenergiesofthe

subsystemscomponents

Itisimportantatthispointthatinternalenergycanbemeasurable.Inordertomeasurethe

energyofasystemweneedtobesurethatthisoneisdefined(i.e.itdoesn'tchangeduring

themeasurement).

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Forinternalenergy,weneedtohaveawallthatpreventstheexchangeofenergythrough

thehiddenway.

--->1)Considerasystemwithiceandliquidwater;shake(mechanicalenergyprovided)-->

fromstateAtostateB.

--->2)Sameexperimentbutwithheating(nomechanicalwork).

PostulateA:energyin1and2hasincreasedofthesamequantity.

--->3)Sameexperimentof2butwiththickerglass:A-->A.Ifwemeasuretheinternal

energyfasterthanthevelocityoftransmissionofheat,wecanassumetohaveobtaineda

measureofaquantitywhichisnotchangedduringthemeasurement.

�WecallAdiabatic(oradiathermic)thewallsthatareimpermeabletoheat.

�WecallDiathermic(orheatconductor)thewallsthatarepermeabletoheat.

Therearerestrictivewallsinrelationtoeachoftheextensiveparameters:

-Energy:wallrestrictivetothetransferofworkandheat

-Volume:fixedandrigidwalls

-Numberofmoles:impermeablewalls(orsemipermeable).

�Opensystem:canexchangewiththerestoftheuniversemassandenergy

�Closedsystem:cannotexchangemass;canexchangeenergy

�Isolatedsystem:cannotexchangemassnorenergy

Tosimplifyourstudies,wewillconsiderforeachsystemonlystatesthatareparticularly

easytodescribe,saidequilibriumstates.

�POSTULATEI:simplesystemscanbeinsomeparticularstatesthat,atamacroscopic

level,arecompletelydeterminedbythevaluesofinternalenergyU,volumeVandnumber

ofmolesNk.Wecallequilibriumstatesthosparticularstates.

--->Thosevariables(U,V,Nk)areindependentfromthepasthistoryofthesystem

-Operativedefinitionofinternalenergyandheat

Considerametalcontainercontainingwaterinliquidandsolidform.

Transformation1:

Adiabaticwall.Amixerproducesmechanicalenergy.SystemfromstateMtostateN.We

haveprovidedmechanicalworkbutworknolongerappearsinmacroscopicform:ithas

beentransferredbyhiddencoordinatesandtransformedintointernalenergy.

ThesysteminthefinalstateNhasgotanhigherinternalenergy.

�Heatisthermalenergytransferredfromasystemtoanother.Whenwerefertothe

'content'ofenergyofabodyorsystemwearereferringtotheinternalenergy.

Wecanrepeattheexperimentseveraltimes.Theworkdonebyexternalforcestobringthe

systemfromstateMtostateNisalwaysthesame.

--->UN-UM=WINAdiabatic(operativedefinitionofinternalenergy)

--->UN-UM=-Wout

Adiabatic

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Transformation2:

ThesamestatetransitionM--->Ncanbeachievedbyputtingthesystemincontactwitha

flameorawarmerbody.WhenstateNisreached,Uisincreasedofthesameamountofthe

firsttransformation,becausethesystemstillundergoesthesametransformationfromMto

NandforthepostulateA.

Nowwehaveawaytomeasuretheheattransfer:

QIN=UN-UM(operativedefinitionofheat)

(where:UN-UM=WINAdiabatic)

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4-FIRSTLAWOFTHERMODYNAMICS:STATEMENTFORACLOSEDSYSTEM

Inthegeneralcaseinwhichthesystemexchangesenergycontemporaneouslyasheatand

workalongatransformationA-->B,makingtheassumptionthattheprincipleof

conservationofenergyisstillvalidandrememberingthattheheatisdefinedastheamount

ofenergytransferrednotaswork,wecandefineheat:

QIN=ΔU-WIN=ΔU+WOUT(operativedefinitionofheatinthegeneralcase)

Wecannowdefine:

�Firstlawofthermodynamicsforaclosedsystem:

ΔU=QIN+WIN

ΔU=QIN-WOUT

IfwemovethesystemfromanequilibriumstateAtoandequilibriumstateBinanadiabatic

way,theworkisthesameforalltheadiabaticpaths(AandBset).

--->Theadiabaticworkisastatefunction(itisonlyfunctionoftheinitialstateAandthe

finalstateB).

WAdiabatic=f(A,B)

RememberingthatΔU=UB-UA=-WOUT

Adiabatic=WINAdiabatic

--->InternalenergyisdefinedasafunctionsuchasΔUdependsonlyontheinitialandfinal

stateofthetransformation.

--->Uisastatefunction(itdoesnotdependonthepath)

Viceversa,QisNOTastatefunction.

AlsoWisNOTastatefunction.

Soheatandworkareformsofenergiesintransit,nolongerdistinguishableoncetheprocessisover,whenthey'reconvertedininternalenergyofasystem(i.e.inenergy

associatedwiththehiddencoordinates).

�Firstlawofthermodynamicsforaclosedsystem(whentherearemultipleexchangesof

heatandwork):

ΔU=∑QIN+∑WIN

ΔU=∑QIN-∑Wout

� Wedefinequasistaticatransformationcompletelyformedofanorderedsequenceof

(infinite)equilibriumstates.

Inpracticeweareabletoapproximateaquasistatictransformationwithasequenceof

equilibriumstatesmakingthetransformation'slowly'applyingactionsontheconstraints.

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With'slowly'weintendslowinrelationtotherelaxationtimeofthesystem(timeneededto

thesystemtoturnbackinequilibriumafterthatthechangeoftheconstraintshasmovedit

fromitsinternalequilibrium).

Thetransformationcanbeconsideredquasistaticift>>τ(timeinwhichthe

transformationoccursismuchgreaterthantherelaxationtimeofthesystem).

(e.g.compressionofagas-->applyresistanceonpiston).

-Formulationofworkforclosedsystemsandquasistatictransformation

Considerathermodynamicsystemmadeofthegasinsidecontainercylinder+piston.

Assumethat:-Thesystemisclosed

-Duringthetransformationthere'savariationinvolume

-Thetransformationisquasi-static.(-->pressuremustbeuniforminsidethe

system)

Theinfinitesimalworkdonebythesystemcanbeexpressedas:

Sinceinthiscasetheforceandthedisplacementoccuralongthesamedirection,wecan

simplywrite:

W=Fdx=(F/A)Adx=pdV

weseethat:δW>0whendV>0

Thismeansthattheworkispositivewhenthesystem(gas)expands.

� Infinitesimalworkincaseofquasi-statictransformationscanbeexpressedas:

δW=pdV=δWout

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5-FUNDAMENTALPROBLEMOFTHERMODYNAMICSANDSECONDLAW

Consideranisolatedcomposedsystemmadeoftwosimplesystemscontainedinacylinder

andseparatedbyapiston.

-Systemisisolated-->can'texchangeenergynormasswiththeexternalenvironment

-Thetwosubsystemscanexchangeextensivevariables

1.Cylinderandpiston:rigid,fixed,adiabatic,waterproof.

2.Removeaconstraint-->transformation-->newequilibriumstate-->newvaluesofU,V,Nk

�Fundamentalproblemofthermodynamics:determinetheequilibriumstatetowards

whichanisolatedcomposedsystemevolves,whensomeofitsinternalconstraintsare

removed.

Tryingtodefinethestateoftheisolatedcomposedsystemunderconsideration,wetryto

understandhowmanyparametersarenecessarytodescribeitandhowmanyequationscan

bewrittenundertheexistingconstraints.Inthiswaywecandeterminethenumberoffree

variables:

�Thedifferencebetweenthetotalnumberofvariablesandthenumberofconstraints

equationsdeterminesthenumberoffreevariables,calledYi

cdenotesthenumberofchemicalcomponentsinourIsolatedComposedSystemandthe

constraintindicatingitsbeingisolatedrequires:

NSCIj=Nj1+Nj2=cost=Xc(j=1,2,..c)

Assumingthatinsidethesystemtherewon’tbechemicalreactions,itwillalsobevalid:

USCI=U1+U2=cost=Xc+1

VSCI=V1+V2=cost=Xc+2

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Wehaveintotal2c+4variablesdescribingthestateofthetwosubsystemsandc+2constraintequations.Thenumberoffreevariablesisequaltothedifferencebetweenthe

totalnumberofvariableslessthenumberofconstraintequations.

f=(2c+4)-(c+2)=c+2�

--->fromthisappearsthatfvariablescanhavearbitraryvalues.TheothervariableswillhavevaluesdeterminedbytheequationsandbytheconstantsXk.��

� Byvaryingthefreevariableswithintheirrangeofvariability,weobtainthevirtualstatesofthesystem,(i.e.thestatescompatiblewiththeconstraintsandwiththevaluesofthe

fixedvariablesXk).

-PostulateII:Secondlawofthermodynamics

TheIIPostulateexpressestheprincipleofmaximumentropyforanisolatedcomposed

systemandcorrespondstothesecondlawofthermodynamics.

�POSTULATEII:Isolatedcomposedsystemstendtoaquiescentstate,calledstateof

thermodynamicequilibrium,inwhichthefreevariablesassumeconstantvaluesspecifiedas

solutionsofaproblemofextreme.

Anentropyfunctionisassignedtoeachsimplesystem:

Sa=Sa(Xia)=Sa(Ua,Va,Nja)�

TheentropyfunctionoftheIsolatedcomposedsystemisthesumoftheentropiesofthe

Simplesystemscomposingthesystem:

(SS.C.I.)(U1,U2,...Un,V1,V2,...Vn,Nj1,Nj2,...Njn)=

=S1(U1,V1,Nj1)+S2(U2,V2,Nj2)....+Sn(Un,Vn,Njn)

ThestableequilibriumstateistheoneforwhichtheEntropyoftheIsolatedcomposed

system(SS.C.I.)assumesmaximumvalue.

Notethattheentropy(SS.C.I.)variesinrelationtothevariationofthedistributionofthe

extensivevariablesbetweenthesubsystems.

--->wecanreformulatethesecondlawasitfollows:

�POSTULATEII:inanIsolatedcomposedsystem,atequilibrium,theextensiveparameters

Ua,Va,Nja(thosewhicharefreetochange)ofthesubsystemsassume,amongallthe

possiblevaluescompatiblywiththeconstraints,thosevaluesthatmaximisetheentropy

(SS.C.I.)oftheisolatedcomposedsystem.

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Considerathermodynamicoperationinwhichsomeconstraints,withintheisolated

composedsystem,areremoved.Thedomainofvirtualstatesisincreased.

So,theentropycanincreaseoratleastcanremainconstant.

--->consequenceofthePostulateII:

ThepostulateIIanditsconsequenceareourstatementoftheSecondlawof

thermodynamics.ThereareotherstatementsreferredtoasPhenomenologicalstatements

ofthesecondlaw:

� Firstphenomenologicalstatementofthesecondlaw:

Noprocessispossiblewhosesoleresultisthetransferofheatfromabodyoflower

temperaturetoabodyofhighertemperature[Clausiusstatement]

� Secondphenomenologicalstatementofthesecondlaw:

Noprocessispossibleinwhichthesoleresultistheconversionofthermalenergyfroma

body,intomechanicalwork[Kelvin/Planckstatement]

� Thirdphenomenologicalstatementofthesecondlaw:

Theenergyoftheuniverseisconserved,theentropyincreases[Clausiusstatement]

� POSTULATEIII:Theentropyofasimplesystemisafunctionwhichiscontinuous,

differentiableandmonotonicallyincreasingwithinternalenergyU

--->ThispropertyisNOTvalidforacomposedsystem!

AsaconsequenceofthePostulateIII,forasimplesystemwecanextractfromS=S(U,V,Nk)

functiontheinverseU=U(S,V,Nk)function;thisisalsocalledfundamentalrelation:

� S = S(U,V,Nk)[Fundamentalrelationinformofentropy]

� U=U(S,V,Nk)[Fundamentalrelationinformofenergy]

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� POSTULATEIV:Theentropyofasimplesystemisequaltozerowhenthefollowing

conditionisverified:

(i.e.atabsolutezerotemperature)

Note:entropyisproportionaltothenaturallogarithmofthenumberofpossible

microstates.S=kln(microstatesnumber)

--->atabsolutezeroasystemisinamacroscopicstate(atabsolutezeroallthemolecules

areattheminimumenergylevel,andthenumberofmicrostatesisequalto1).

-HomogeneityofSfunction

ConsiderasimplesystemS=S(U,V,Nj).Supposetodivideitin2subsistems,only

conceptually,withamathematicalsurfacethatseparatesthem.Consideritasacomposed

system.

-Addittivity:S(U,V,Nj)=S1(U1,V1,Nj1)+S2(U2,V2,Nj2)

Choosethe2subsystemssothattheyhavethesamedimensions:

U1=U2V1=V2Nj1=Nj2

Sinceweareconsidering2portionsofthesamesimplesystem,forwhichthefunctional

formofentropyisthesame,itresults:S1=S2

(a)

Besides:

(b)

Comparing(a)and(b),weget:

or,whichisthesame:

Ifwerepeattheprocessbydividingthroughmathematicalsurfacesthesysteminnparts

(withnintegerorreal),weobtain:

Ingeneral,wecanthereforesaythattheEntropyisahomogeneousfunctionoforder1,i.e.:

[perogniλεR]

Soforeveryλthatbelongstothedomainofrealnumbers.

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6-EXTENSIVE,INTENSIVEVARIABLESANDTHEIRRELATIONSHIP

Forasimplesystem,Uisanhomogeneousfunctionoforder1.

SincewehaveshownthatUisastatefunction,weknowthatthedifferentialofUexists:

dU=(δU/δS)V,NjdS+(δU/δV)S,NjdV+Σ(δU/δNj)S,V,Nk≠jdNj

Forasimplesystem,wecandefinethefollowingquantities:

T=δU/δS)V,Nj

p=-δU/δV)S,Nj

μj=δU/δNj)S,V,Nk

sowecannowintroduce:

�Differentialformofthefundamentalrelationintermsofenergy:

dU=TdS-pdV+ΣμjdNj

Temperature,pressureandchemicalpotentialaretheintensivethermodynamicvariables

thatcanbedefinedforasimplesystem.

Considernowasystemwithconstantnumberofmoles,i.e.nomassexchangewithoutside

andnochemicalreactionsinside(-->closedsymplesystem).

Nj=constant∀j--->dNj=0∀j

so,fromthedifferentialformofthefundamentalrelationweget:

dU=TdS-pdV

whileaccordingtothefirstprincipleforclosedsystems,foraninfinitesimaltransf.weget:

dU=δQIN-δWOUT

forquasistatictransformationsofaclosed,simplesystem:

δWOUT=+pdV

--->thus:dU=δQIN-pdV

�Consideringquasistatictransformationsandsimple,closedsysteminwhichnochemical

reactionstakeplace,wheredSisthevariationofentropyofthesystemthatexchangesthe

heatδQINandTisitstemperature:

[δQIN=TdS]or[dS=δQIN/T]

SinceTisalwaysgreaterthanzero,thisrelationtellsusthatifthere'sheatphysically

enteringthesystem,thenanincreaseofentopywilltakeplace,andviceversa.

[δQIN>0-->dS>0][δQIN<0-->dS<0]

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Notethatasimplesystemcanreduceitsentropy(itisn'tincontraddictionwiththesecond

principle,asthesystemisnotisolated)ifanothersimplesystemincreasesitsentropyina

quantitythatensuresΔSS.C.I≥0.

Foropenedsystems,inthepresenceofmassexchange,thefollowingrelationisvalid:

dS=δQIN/T+dSmin(wheredSm

inistheentropyassociatedtothemassentering)

SinceT=δU/δSandUishomogeneousofdegree1,Tishomogeneousofdegree0.

--->Following,demonstration:

T(λS,λV,λNj)=λ0T(S,V,Nj)=T(S,V,Nj)

infact,being:

df(λx)=d[λnf(x)]=λndf(x)andd(λx)=λd(x)

f'(λx)=df(λx)/d(λx)=λndf(x)/λdx=λ(n-1)!"($)!$

=λ(n-1)f'(x)

--->thederivativeofanhomogeneousfunctionofdegree'n'isanhomogeneousfunctionof

degree'n-1'.

� Wecallintensivequantitiesthethermodynamicquantitiesthatarehomogeneousof

degreezero,asT,pandμ.

Notethatwhenasimplesystemdoublesitsdimensions,anextensivequantitydoublesits

value,whileanintensivequantityremainsunchanged.

-Equationsofstateoftheenergyformulation

� Therelationsthatexpresstheintensivevariablesasfunctionsoftheextensiveonesarecalledstateequations.

1)T=δU/δS)V,Nj=T(S,V,Nj)

2)p=δU/δV)S,Nj=p(S,V,Nj)

3)μj=δU/δN)V,S,Nk≠j=μj(S,V,Nj)

Whenallthestateequationsareknown,wehaveacompleteknowledgeofthesystemfrom

thethermodynamicpointofview.

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-Molarquantitiesandtheirrelation

� Themolarquantitiesarerelatedtoamoleofthesubstanceinexam.

Considerasimplemono-componentsystem(i.e.consistingofasinglechemicalspecies).

TakingintoaccountthehomogeneitypropertyofthefunctionSandusing1/Nas

multiplicativefactor,weget:

&'S(U,V,N)=S(

(' ,

*' ,

'')=S(u,v,1)=S(u,v)

fromwhich:S(U,V,N)=Ns(u,v)

andsimilarly:U(S,V,N)=Nu(s,v)

where:

s=+'isthemolarentropy

v=*'isthemolarvolume

u=('isthemolarinternalenergy

Differentiatingu=u(s,v),weobtain:

du=δu/δs)vds+δu/δv)sdv

where:

δu/δs)v=N/Nδu/δs)v=δU/δS)V,N=Tδu/δv)s=N/Nδu/δv)s=δU/δV)S,N=-p

fromtheselasttworelations,wecanwrite:

du=Tds-pdv

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7-DERIVATESOFIMPLICITFUNCTIONS

....

IfweconsiderthefunctionsS(U,V,Nj)andU(S,V,Nj)relatedtoasimplesystemandwetake

intoaccountquasi-statictransformationsforwhichisvalid:

dU=TdS-pdV+ΣμjdNj

itfollows:

1)δS/δU)V,Nj=1/(δU/δS)V,Nj=1/T

This,togetherwiththeassumptionthatS,forsimplesystem,ismonotonicallyincreasing

withU,leadsustoconcludethatT(Kelvin)isarealnumbergreaterthanzero.

Continuingwiththederivations,wegetthefollowingtworelations:

2)δS/δV)U,Nj=-δS/δU)V,NjδU/δV)S,Nj=-1/T(-p)=p/T3)δS/δNk)V,S,Nj≠k=-δS/δU)V,NjδU/δNk)S,V,Nj≠k=-1/Tμk=-μk/T

Andsince:

dS=δS/δU)V,NjdU+δS/δV)S,NjdV+ΣδS/δNk)S,V,Nj≠kdNk

thenwecanobtainthe:

� Differentialformofthefundamentalrelationinentropicform

dS=1/TdU+p/TdV-Σ(μk/T)dNk

Notethatthisrelationisvalidforquasistatictransformayionsandforsimplesystems

(becauseScanonlybeinvertedifthesystemissimple,asaconsequenceofentropy

postulates).

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8-CONDITIONSOFEQUILIBRIUMFORISOLATEDCOMPOSEDSYSTEMS

Thesecondlawofthermodynamicsassertsthatthestateofstableequilibriumistheonefor

whichtheentropyoftheIsolatedcomposedsystem(SS.C.I.)hasmaximumvalue.Theaimof

thischapteristotransformthisconditionontheentropyintoconditionsontheintensive

variables.

1-ThermalequilibriumforIsolatedcomposedsystem

Considertwosubsystems,separatedbyarigid,fixed,impermeableandconductivewall.

Theycanexchangeheatbetweenthembutnotmassnorheatwiththeoutside.

Theyareinequilibrium(U1,U2,V1,V2,Nj1,Nj2)

Constraints:-V1,V2=const.

-Nj1,Nj2=const.

-USCI=U1+U2=const.--->dU1+dU2=0

Undertheseboundaryconditions,theentropyfunctionofthecomposedsystemisgivenby:

SSCI=S1(U1,V1,Nj1)+S2(U2,V2,Nj2)=S1(U1)+S2(U2)=SSCI(U1,U2)

--->SSCIisafunctionofU1andU2separatelyandnotofUSCI

CASEA:StableEquilibrium

TheIsolatedcomposedsystemisinstableequilibrium,withrigid,impermeableand

conductivewalls.SSCIhasmaximumvalue.

WechooseU1asafreevariable.ThevirtualdisplacementfromtheequilibriumstateisdU1

anditcantakeallthevaluescompatiblewiththeconstraints(positive,zeroornegative).

WhateveristhevalueofdU1,dSSCImustbezero.

dSSCI=0∀dU1

dsSCI=dS1+dS2=(δS1/δU1)dU1+(δS2/δU2)dU2=

=(1/T1)dU1+(1/T2)dU2=dU1(1/T1-1/T2)=0∀dU1

InordertohavedSSCIequaltozeroforanyvalueofdU1,itmustbe:1/T1=1/T2-->T1=T2

� Conditionofthermalequilibrium

T1=T2istheconditionofequilibriumforanIsolatedsystemformedbytwosubsystems

separatedbyarigid,impermeableandconductivewall.

L.D.D

CASEB:Transformation

Considerthesameprevioussystembutwithadiabaticwallbetweenthetwosubsystems,so

thattheyhaveinitiallydifferenttemperaturesT1andT2.SupposethatT1>T2.

--->Makethewallconductive-->transformationtillreachthermalequilibrium

Thesystemisnowchoosinganewsetofvirtualstatesinordertoreachmaximumentropy.

Betweentheinitalandfinalstate,asaconsequenceofthePostulateII,itmustbe:

SSCIf-SSCI

i=ΔSSCI≥0.

Assumingthetwostatesiandfveryclose-->ΔSSCI≈dSSCI=(1/T1-1/T2)dU1≥0AssumedthatT1>T2-->(1/T1-1/T2)<0-->dU1<0

-->Thesystemathighertemperaturelosesenergy

TheinternalenergyflowsasheatspontaneouslyfromthebodiesathigherTtotheonesat

lowerTandnotviceversa.

--->Itisnotpossibletorealiseatransformationwhichonlyresultistotransferenergyas

heatfromalowertemperaturesystemtoanhigherone(asseeninthePhenomenologicalstatementofthesecondprinciple,[Clausius]).

2-Mechanicalandthermalequilibrium

CASEA:Stableequilibrium

Thecomposedsystemisalwaysisolated.Iftheseptumdividingtheisolatedsystemis

conductive,movableandimpermeable,wehave:

-V1+V2=const.--->dV1+dV2=0

-U1+U2=const.--->dU1+dU2=0

-Nj1,Nj2=const.--->dNj1=0,dNj2=0

Sincethetwosubsystemscanexchangebothheatandwork,U1andV1areindependentof

eachother.Theentropyoftheisolatedcomposedsystemis:

SSCI=S1(U1,V1,Nj1)+S2(U2,V2,Nj2)

-->dSSCI=(δS1/δU1)dU1+(δS1/δV1)dV1+(δS2/δU2)dU2+(δS2/δV2)dV2=0

-->dSSCI=(1/T1)dU1+(p1/T1)dV1+(1/T2)dU2+(p2/T2)dV2=0

-->dSSCI=(1/T1-1/T2)dU1+(p1/T1-p2/T2)dV1=0

L.D.D

ForarbitraryvaluesofdU1anddV1,itmustbe-->dSSCI=0,thatimplies:

1)1/T1=1/T2--->T1=T22)p1/T1=p2/T2-->p1=p2

CASEB:Transformation

Considerthesameprevioussystembutwithfixedwallbetweenthetwosubsystems,sothat

theyhaveinitiallydifferentpressuresp1andp2.Supposethatp1>p2.

--->Makethewallmovable-->transformationtillreachequilibrium

Thesystemisnowchoosinganewsetofvirtualstatesinordertoreachmaximumentropy.

Betweentheinitalandfinalstate,asaconsequenceofthePostulateII,itmustbe:dSSCI≥0AddingtheassumptionthatsincethebeginningT1=T2andthattemperautresremain

constant:

dSSCI=(p1/T1-p2/T2)dV1≥0

andsincep1/T1-p2/T2≥0---->dV1≥0

-->Thesubsystemathigherpressureexpands

3-Thermalequilibriumandwithrespecttothepassageofmatter

CASEA:Stableequilibrium

Thecomposedsystemisalwaysisolated.Theinternalwallisrigid,fixed,conductiveand

permeabletoasinglechemicalspecies(1).

V1,V2=const.--->dV1,dV2=0

U1+U2=const.--->dU1+dU2=0

Nj1+Nj2=const.--->dNj1+dNj2=0

dSSCI=1/T1dU1-µj1/T1dNj1+1/T2dU2-µj2/T2dNj2=

=dU1(1/T1-1/T2)+dNj1(µj2/T2-µj1/T1)=0

Theequilibriumconditionwillbe:

1)1/T1=1/T2--->T1=T22)µj1/T1=µj2/T2--->µj1=µj2

L.D.D

CASEB:Transformation

Considerthesameprevioussystembutwithimpermeablewallbetweenthetwo

subsystems,sothattheyhaveinitiallydifferentchemicalpotentialsµj1andµj2.

Supposethatµj1>µj2.

--->Makethewallpermeable(tothechemicalspecies1)

--->transformationtillreachequilibrium

Thesystemisnowchoosinganewsetofvirtualstatesinordertoreachmaximumentropy.

dSSCI=(µj2-µj1)/TdNj1

andbeingdSSCI≥0,ifµj1>µj2thendNj1<0

Therefore,thechemicalspecies1goesfromthesystemwithhigherchemicalpotentialto

thesystemwithlowerchemicalpotential(relativetothecomponent1).Thedifference

betweenchemicalpotentialsinthetwosubsystemsgeneratesaforceactingonthetransfer

ofmatter.

Note:osmosisphenomenon

Experimentallywecanseethatthesolventmovestowardhigherconcentrationsofthe

soluteor(whichisthesame)towardslowerconcentrationsofsolvent.

--->themoleculesofanysubstancedissolvedinwaterexertpressurethattendstoincrease

thespaceattheirdisposal;thispressureiscalledosmoticpressureanddenotedbyπ.

� Ifyoudissolveinanequalvolumeofwaterthesamenumberofmoleculesofdifferent

substances,thesolutionsobtainedinthiswayhaveallthesameosmoticpressure,aslong

asmaintainedatthesametemperature.

Note:Vant'Hoff'slaw

πV=nRT(V=solventvolume,n=numberofmolesofsolute,R=universalgasconstant)

--->lawexperimentallyverifiedwithPfeffercell

L.D.D

9-THEIDEALGASMODEL

Assumptionsoftheidealgasmodel:

1-Eachmoleculeisconsideredasapoint,withzerovolume

2-Moleculesinteractbycontact.Inparticular,thecontactinteractionsareintheformof

elasticcollisions

3-Thegasisnotsubjecttothepresenceofexternalfields,suchasgravitationalor

electromagneticfields

Also,weassumetoconsideramonocomponentsystem(asinglechemicalspecies).

Wecan,withallthesesimplifications,write:

� FundamentalrelationforIdealgases

S=S(U,V,N)=Nf(U/N)+NRln[(V/V0)(N0/N)]+(N/N0)S0(R=8.314[kj/kmolK])

whereU0,N0,V0,S0arethevaluesoftheextensivevariablesinthereferencestate.

Sisastatefunction-->weareinterestedinthevariationbetween2states

--->chooseS0asreferencestate(fundamentalrelationisvalidalsointhisstate)

S0=S(N0,V0,U0)=N0f(U0/N0)+NRln[(V0/V0)(N0/N0)]+(N0/N0)S0

therefore--->f(U0/N0)=0

Notethattheequationisnotvalidatlowtemperatures(itdoesnotsatisfythePostulateIVwhichrequiresS=0atT=0).

-Derivationofthefundamentalequationwithrespecttotheinternalenergy

Rememberingthat:

1/T=δS/δU)V,N

-->derivethefundamentalequationwithrespecttothethreevariablesU,N,Vstarting

fromtheinternalenergy

δS/δU)V,N=δ/δU[Nf(U/N)]=N(δ/δU)f(U/N)=N(df/du)(du/dU)=N(df/du)(1/N)=df/du=df/du(u)--------->1/T=df/du(u)

Sowecanconcludethatforanidealgasthetemperatureisafunctiononlyofthemolar

internalenergyu(andviceversa).

�Firstequationofstateofidealgases:u=u(T)

Note:U=Nu=Nu(T)

--->thetotalinternalenergydependsonTandN

L.D.D

-Derivationofthefundamentalequationwithrespecttothevolume

Rememberingthat:

p/T=δS/δV)U,N

--->derivethefundamentalequationwithrespecttothevolumeV

S=S(U,V,N)=Nf(U/N)+NRln[(V/V0)(N0/N)]+(N/N0)S0

δS/δV)U,N=NR1/[(V/V0)(N0/N)][N0/(V0N)]=(NR)/V

--->p/T=(NR)/V

�Secondequationofstateofidealgases:pV=NRT

Note

1:ifwedividebythenumberofmoles-->pv=RT(v=V/N)

2:ifwedividebythemass

---->pvsp=NRT/M=RT/(M/N)=RT/Mm=R0T--->pvsp=R0T

(R0=R/Mm=characteristicconstantoftheparticulargas)

-Coefficientofexpansion,compressibilityandspecificheat

-Molarheat:amountofenergythatmustbeprovidedintheformofheat,inorderto

increaseof1Kthetemperatureofonemoleofthesubstancecomposingthebodyunder

consideration

-Thermalcapacity:amountofenergythatmustbeprovidedasheat,inordertoincreaseof

1Kthetemperatureofthewholebody

Notethatmolarheat,specificheatandthermalcapacitydependbothon:

-Thetypeoftransformationx

-Thepoint(state)alongthetransformation

L.D.D

Examples:

1-Molarheatalongaquasistatictransformationforaclosedsystem

cx=1/N[(TdS)/dT]x=T(ds/dT)x[J/(molK)]

2-Molarheatalongatransformationatconstantpressure(isobaric)cp=1/N(dQ

in/dT)p[J/(molK)]

3-Molarheatalongatransformationatconstantvolume(isochoric)

cv=1/N(dQin/dT)v[J/(molK)]

-Mayer'srelation

Connectsthemolarheatatconstantpressurecpandthemolarheatatconstantvolumecv

totheidealgasconstantR.

-->Nmoles(const.)ofidealgas+infinitesimalquantityofheat(atconstantpressure)

δQ=NcpdTFirstlawofthermodynamics:δQ=dU+δW

Considerthatforanidealgas,molarinternalenergydependsonlyontemperature

--->dU=cvdT-->dU=NcvdT

so-->NcpdT=NcvdT+pdV(butfortheequationofstate:pdV=NRdT)

--->NcpdT=NcvdT+NRdT

�Mayer'srelationbetweenmolarheats(foridealgas):cp=cv+R[J/(molK)]

NotethatMayer'srelationcanalsobeexpressedfortheheatcapacities

Cp=Cv+NR[J/K]

-Coefficientofexpansion,compressibilityandspecificheatforidealgases

Foridealgases,somecoefficientsassumeparticularvalues.

Coefficientofthermalexpansion

α=(1/V)(δV/δT)p,N=(1/V)[δ(NRT/p)/δT]p,N=(1/V)(NR/p)=NR/NRT=1/T

Coefficientofisothermalcompressibility

KT=-(1/V)(δV/δp)T,N=-(1/V)(-NRT/p2)=1/p

NotethatαandKT,andalsothedifferencecp-cv=R,foridealgases,donotdependontheparticulargasweareconsidering.(cpandcvdoessingularlydepend)

--->weintroducetheadiabaticexponent

γ=cp/cv>1

L.D.D

-Theteoremofequipartitionofenergy

Atordinarytemperatures,thevibrationdoesnotcomeintoplayindeterminingthespecific

heat.Itfollowsthereforethatweconsideronlytranslationalenergy,evaluatingthe

contributioninthedeterminationofthemolarheatsatconstantvolume:

cv=(1/N)(δQ/dT)v

�Thetheoremstatesthateachdegreeoffreedommovementofamoleculeleadstoa

contributionequaltoR/2tothevalueofcv

1)MonoatomicMoleculeThreetranslationaldegreeoffreedom.Locationofthemoleculeisuniquelydeterminedby

3spatialcoordinates.

cv=(3/2)R

cp=cv+R=(3/2)R+R=(5/2)R

γ=cp/cv=5/3

2)DiatomicMoleculeFivedegreesoffreedom.Assumefixeddistancebetweenatoms.6coordinatesforagiven

positionfortwoatoms,butthedistancebetweenthemisfixed,so:

cv=(5/2)R

cp=cv+R=(5/2)R+R=(7/2)R

γ=cp/cv=7/3

2)Non-linearpolyatomicMoleculeAssumethreeatoms.9Cartesiancoordinatesforagivenposition,butlinkedwith3stiffness

conditions-->6.Theadditionofafourthatomlinkedtotheothersrisetothreenew

coordinatesbutalsotothreenewstiffnessconditions.Sothenumberofdegreesof

freedomisstill6.Thisistrueforanynumberofaddedatoms,so:

cv=(6/2)R=3R

cp=cv+R=3R+R=4R

γ=cp/cv=4/3

Notethatinrealexperience,weshouldknowthatcpandcvalsodependontemperature.

Athightemperatures,thevaluesobtainedbytheformulacanbenotcorrect.

-Formonoatomicgaseswehaveagoodapproximationofthebehaviourwiththemodel

-Fordiatomicgasesweshoulddistinguishbetweencasesatordinarytemperaturesand

casesathightemperatures

-Forpolyatomicgasesthevaluesobtaineddiffersignificantlyfromtherealvalues

(molecularcomplexity-->increaseofpossiblevibrationalmodes)

L.D.D

-Fundamentalequationinparametricform

Consideramonocomponent,closedsystemconsistingofanidealgasandsubjectedto

quasi-statictransformations.

u=u(T)

du=δQin-δWout=δqin-pdv

bydefinition:cv=(1/N)(dQin/dT)V

atconstantvolume:pdv=0--->du=δqin

-->cv=(1/N)(δQin/dT)=(1/N)(δU/δT)V=δu/δT)V=du/dT=cv(T)

Fromabove,sinceuisafunctiononlyofT,itfollowsthatalsocv,beingthederivativeofa

functionofTalone,isafunctionofTalone.

-->du=cv(T)dT

Theresultisvalidalonganutransformation,withoutbeinglimitedonlytotransformations

atconstantV(uisastatefunctions-->doesnotdependonthetransformation).

Integratingbothsides,weobtain:

but recalling that df / du = 1 / T,

from which:

�Thesetofequations1and2istheFundamentalequationinparametricform,with

parameterT.

L.D.D

EliminatingTbetweenthetwoequations,weobtainarelationbetweenonlytheextensive

variablesS=S(U,V,N),whichisafundamentalequation.

Then,ifcvisconstant,theparametricequationbecomes:

DividingbyNweobtainthemolarexpressions:

soweobtain:

Expressionofmolarentropysvalidforanideal

gaswithconstantcv(andsoalsoconstantcp)