Consider a wellinsulated pistoncylinder assembly On the 005 m piston rests two 5000kg blocks The initial temperature is 500 K The ambient pressure is 5 bar One mole of an ideal gas is contained in the cylinder The gas is compressed in a process in w ID: 50272
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3.Asalespersonclaimsthatamysteriousblackbox,withnomovingparts,cantakeaninletstreamofidealgasat2kg/sand4barand50‰andcool0.5kg/sofitto�10‰and1barwhiletherestofthegasisraisedto70‰at1bar.Isthispossible?Explain.(i)Massisconserved(ii)Energyisconserved:0:5kg=sCP(�10�50)K=1:5kg=sCP(70�50)K(iii)Doesthesecondlawhold?Foranadiabaticprocess,dS dtuniv=0:5kg=s(S2�S1)+1:5kg=s(S3�S1)S2�S1=CplnT2=T1�RlnP2=P1S3�S1=CplnT3=T1�RlnP3=P1dS dtuniv=0:5kg=sCP(ln263+3ln343�4ln323)+2Rln4SecondlawholdsaslongasCP218R,whichistrueforallimaginablegases.4.Calculatethemaximumworkandthemaximumnon-expansionworkthatcanbeobtainedfromthefreezingofsupercooledwaterat�5‰and1.0atm.Thedensitiesofwaterandiceare0.999and0.917gcm�3,respectivelyat�5‰.ThesimplestprocedureistocalculatethechangeinGandAbyusingtherelationshipA=U�TSamdG=H�TS.FromthermodynamictablesweknowthatH=�6008J/molforfreezingat0‰,CPforliquidwateris75J/(molK),andCPforiceis36.4J/(molK).ThecomputationofHandSwasdiscussedinclass.Asazerothapproximation,wecouldtakethatU=Hbecausecondensedphasesareinvolved.WecoulddobetterbyusingthedensitiestondthePdVcontributiontoH.H(268K)=H(273K)+(CP(solid)�CP(liquid))(�5K)U(268K)=H(268K)�1atm(Vm(solid)�Vm(liquid))S(268K)5.OnemoleofHeisheatedfrom200‰to400‰ataconstantpressureof1atm.GiventhattheabsoluteentropyofHeat200‰is810JK�1mol�1,andassumingHeisaperfectgas,commentonthespontaneityoftheprocess.S=S0+CPln673=473Thespontaneityoftheprocesswillbedeterminedbyhowtheheatingisdone.Iftheheatingisachievedbyputtingthegasintoaheatbathat400‰,thenStot=Ssys+Ssurr�0:Ssurr=�Hsys=673;Hsys=CP(200K)6.Derivetherelations:(i)Cp�Cv=T�@P @TV�@V @TP;@H @TP�@U @TVUsethedenitionH=U+PV@H @TP�@U @TV=@U @TP+P@V @TP�@U @TVToobtainanexpressionfor�@U @TPweproceedasfollowsdU=@U @TVdT+@U @VTdVDividingbydTandapplyingconstantpressuregives@U @TP=@U @TV+@U @VT@V @TP:ThuswehaveCP�CV=@U @VT@V @TP+P@V @TPUsingtheresultthatwasprovedinclass@U @VT=T@P @TV�P 9.Thevaporpressureofzincvarieswithtemperatureaslogp(mmHg)=�6850 T�0:755logT+11:24andthatofliquidzincaslogp(mmHg)=�6620 T�1:255logT+12:34:Calculate(a)theboilingpointofzincAttheboilingpoint,thevaporpressureoftheliquidbecomestheatmosphericpressure,whichis760mmofHg.log760=�6620 T�1:255logT+12:34Wesolvethisbythemethodofsuccessiveapproximation.IdropthelogTtermtobeginwithandsolveforT.ThisgivesmeaninitialguessfortheroottobeT=699:85.IusethisguessrootinthelogTandsolvetheequationlog760=�6620 T�1:255log699:85+12:34forT,whichyieldsT=1124:1.IrepeatthisprocessafewtimesandndthatT=1181.Thereareothermethodstosolvethis-graphicalandusingacomputeralgebrasystemaretwopossibilities.(b)thetriplepointAtthetriplepoint,thesolid-vaporandliquid-vaporcoexistencecurvesintersect.Thismeans�6620 T�1:255logT+12:34=�6850 T�0:755logT+11:24230 T�0:5logT+1:10=0Solvingthisequationwendthatthetemperatureatthetriplepointiscloseto700.OneusesthisTinoneoftheequationstondtheP.(c)theheatofevaporationattheboilingpoint2:303logp(mmHg)=lnP=2:303�6620 T�1:255logT+12:34:dlnP=2:303RdT6620 RT2�1:255 RT=2:303R6620�1:255T RT2dTComparingthiswiththeClausius-Clapeyronequation,dlnP=H=RT2dT,showsthatH=2:303R(6620�1:255T).SubstitutetheboilingpointT=1181inH=2:303R(6620�1:255T)andobtainH.(d)theheatoffusionTheheatoffusion,whichistheenthalpyofmelting,maybeobtainedfromsubtractingtheheatofevaporationfromtheheatofsublimation.Theheatofsublimationisobtainedbyfollowingthesameprocedureasinpart(c)forthesublimationcurve.(e)thedi erenceintheCpsofsolidandliquidzinc.IfHhastheformA+BT,thenitimpliesthatB=CP.FromthetemperaturedependentpartofHforevaporationandsublimation,wecanobtainthedi erenceintheCP's.10.ThePlanckfunctiondenedasY=�H T+SisathermodynamicpotentialsimilartotheGibbsenergy.Obtaina\TdS"typeequationfordYandfromitaMaxwellrelation.Also,whatisthethermodynamiccriterionforspontaneityintermsofY.dY=H T2dT�dH T+dS=H T2dT�1 T(dH�TdS)WeknowthatdH=TdS�VdPsothatdY=H T2dT�V TdPComparingthiswithdY=@Y @TPdT+@Y @PTdP