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HOMEWORK#2PROBLEMSET DUE:Wednesday,September21st 1.(a.)Whydoesatri-atomicgashaveagreaterenergycontentpermolethanadi-atomic oneatthesametemperature?(b.)Whathappenswhenthetemperatureisraised?(c.)Ifthe temperaturegoestoohigh,whathappenseventually?(Hint:LookatH2’sCVfromlecture.) 2.(a.)UsingtheformulafortheMaxwell-Boltzmannspeeddistribution(pleaseconsultthe lecturenotes)derivethefactthattheaveragespeedormeanvelocityis (b.)Isthisgreaterorsmallerthanthemostprobablespeed?Explainwhy. 3.(a.)Whydoesahelium-filledballoonfloatawayuponitsrelease?(b.)Isthereanylimitto howhighitwillgo,andifsowhatitisandwhy?(c.)Whenitisheated,weassumeT(the temperature)changesinside,butisPchanging,Vchanging,orboth?(Assumegasisideal.) 4.Recalltheformulaforvolumeexpansion/contractionwithtemperatureoffluidswe learnedwhenwealsodiscussedsolids.Derive“beta,”thethermalcoefficient,foragas,with theuseoftheidealgaslaw.(Youmayassumeisobaricprocessforsimplicity;V,Tchange. Youwillalsoneedtoknowtheinitialtemperature.Assumethatitis0°C,butconverttoK!) 5.Thesunhasapoweroutputofabout3.846×1026W.Ifthesunwereaballof(ideal)gas atafixedpressurePof2.5×1016Pa,then(a.)howmuchwoulditsvolumeneedtohave changedoveritslifetime,toaccountforthatmuchpower?Assumethesunisaspherefor yourcalculationofthe“initial”volume,witharadiusof6.957×108m.(b.)Duringsuchan expansionwoulditengulfEarth?(c.)Canastar“dowork”onitssurroundingsexpanding? 6.A2.00-molsampleofadi-atomicidealgasexpandsslowlybutadiabaticallyfroma pressureof5.00atmandavolumeof12.0Ltoafinalvolumeof30.0L.(a.)Whatisthefinal pressureofthegas?(b.)Whatistheinitialandfinaltemperature?(c.)FindQ,W,andΔEint. SOLUTIONS 1.(a.)Atri-atomicgashasmoredegreesoffreedom,usuallyvibrationalontopofrotational (b.)Whenthetemperatureisraisedmoredegreesoffreedomopenupbeyondtranslational (c.)Moleculeisrippedapart(otheracceptableanswers:monoatomic,dissociation,plasma) 2.(a.) Inthisnotationheref(v)meansN_v(v) ThisisinAppendix6ofRexandThornton (b.)Thisisbiggerthanmostprobablespeed,becauseofdistribution’sshape(high-vtail). 3.(a.)Heliumislessdensethanair(partialcreditfor“lighter”).(b.)Thereisalimit.Itcan’t gomuchhigherthanpointintheatmospherewherethedensityoftheairisevenlessthan helium.Youcanalsorefertotemperature(colderhigherupmeanstheheliumwillget denser)orpressureargumentsforfullcredit.Balloonmaypopnotjustsink:occurswhen theoutsidepressurebecomestoolowcomparedtoinside,andtherubbercan’ttakeit anymore.(c.)Becauserubbercanstretch,everythingischanging.Thevolumeisnotfixed. Thevolumeincreasesfirst,butthenthepressurerisesoncetherubbercan’tstretchmore. 4.deltaV=beta*V_i*deltaT V_f–V_i=beta*V_i*(T_f–T_i) nRT_f/P_f–nRT_i/P_i=beta*nRT_i/P_i*(T_f–T_i),usingtheidealgaslaw T_f/P_f–T_i/P_i=beta*T_i/P_i*(T_f–T_i),cancelingnR(constant) T_f/P–T_i/P=beta*T_i/P*(T_f–T_i),becauseP_i=P_f=P(isobaricprocess) T_f–T_i=beta*T_i*(T_f–T_i) 1=beta*T_i=>beta=1/T_i=1/(273K)=0.003663Kelvin^-1 5.(a.)V_i=(4/3)*pi*(6.957×108m)^3=1.410×1027m^3 Power=Work/Time=P*deltaV/Time=2.5×1016Pa*(V_f–V_i)/1.419×1017s (whereIhaveconverted~4.5billionyearsintoseconds) 3.846×1026W=2.5×1016Pa*(V_f–1.410×1027m^3)/1.419×1017s (V_f–1.410×1027m^3)=3.846×1026W*1.419×1017s/(2.5×1016Pa) (V_f–1.410×1027m^3)=2.183×1027J/Pa[=J/(N/m^2)=(N-m)/(N/m^2)=m^3 =>V_f=3.593×1027m^3.Volumehaschangedbyfactorofover2.5! (b.)V_f=(4/3)*pi*r^3=>r^3=(3/4)*V_f/pi=>r=[(3/4)*V_f/pi]^(1/3)=9.501×108m, whichisonly1.366timesbigger.TheearthisaLOTfurtheraway(150×109m!) (c.)No,thisproblemwasunrealistic,andjustanexercisewiththeequations,because pushingonvacuumisnotdoingwork,eventhoughwejusttriedtocalculateawork.Itis likeafreeexpansion.There’snothingtopushagainstinspace,analogoustoapiston. 6.