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)
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