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Notethat,foranidealgas,β=1/Tandκ=1/P,sothatequationreducestoRNotethat,inequation,κistheisothermalcompressibilityBydifferentiating:dA= dU TdS SdTBut∴dU=TdS pdV∴dA= SdT pdVIfaninfinitesimalchangetakesplaceinasystemofconstantTandV,thus:Forirreversible process,Arease(dA)t,V≤0Forreversibleprocess,AisconstantTherefore,theyaredH=dU+pdVdG=dH–TdS=dU+pdVTdSAlsodU=dqrev+ dwrevandTdS=dqrevConsidertheentropySasafunctionoftemperatureandvolume:S=S(T,V):=+∂EquatingEqs(1)and(2)throughthedTterm givesdSImportant:theseequationsrelatetheentropychangeofasystemtothechangesinotherproperties:dh,du,dp,dvItismorepracticaltousethe criterion(dA)t,V≤DerivationofTdsequations:The1stLaw:Theworkisgivenby:δQ δW=dUδW=PdVForareversibleprocess:TdS=δQSubstituting gives:TdS=dU+PdVOronaperunitmassbasis:Tds=du+Pdv•Tds=du+pdv,alsoTds=dhvdp≡TdsForastaticsystemthisismorefamiliarlywritten asde=Tds pdρ(Acc5)AnotherofMaxwell’sequationsforaWewillusethefundamentalthermodynamicidentity.(Eventheirderivationisbasedona reversibleprocess.)ThisManuscriptinvolvesanotherwayofderivingtheThirdsTdSequationapplyingthesecondlawofthermodynamicstogetherwithequations alreadyderivedandintroducedfromthederivationsofT&Vwhichisalsoanapplicationofthesecondlawofthermodynamics∂SdTTVTdS=CdT+T∂p TheTdSequationsenablesustocalculatethechangeofentropyduringvariousreversibleprocessesintermsofeitherdVanddT,ordPanddT,ordVanddP, andevenintermsofdirectlyWehavealreadyshownthattheexpansioncoefficientofanidealgasis1/T,andtheisothermalcompressibilityofanidealgasis1/P cpdTvdP=(2) dPcpdvDerivationofTdsequations:The1stLaw:Theworkisgivenby:δQ δW=dUδW=PdVForareversibleprocess:TdS=δQ Substitutinggives:TdS=dU+PdVOronaperunitThislectureisaboutthemostimportantTdSequationsThisisoneofasetofrelationscalledMaxwell’s equations.⇒ .C.dS=VdT+∂p.Tds=du+pdv,alsoTds=dhvdp.Therefore,theyareindependentoftheprocesses.∂T.V.SdV∂V.T.Weapplythe definitionoftheheatcapacitytothefirsttermandaMaxwellrelationtothesecond,andobtainTheserelationscanbeusedforreversibleaswellasirreversible processes∂TdVordU=TdS pdV+μdNasanaidtomemorizingtheoftemperature,pressure,andchemicalpotentialfromtheconsiderationThese equationscanbeused,forexample,tocalculate,byintegration,thechangeofentropybetweenonestateandanother,providedthattheequationofstateisknown inorderthatwecanevaluatethepartialderivativesTheTdSEquationsSubstitutingfordUandTdSdG=dqrev+dwrev+pdVTdS=dwrev+pdVLetThis ManuscriptinvolvesanotherwayofderivingtheThirdsTdSequationapplyingthesecondlawofthermodynamicstogetherwithequationsalreadyderivedand introducedTakingthederivativeyieldsdh=Tds+vdPTds==dhvdPImportant:theseequationsrelatetheentropychangeofasystemtothechangesinother properties:dh,du,dp,dv.dh=du+Pdv+vdP.