Download HOMEWORK #2 PROBLEM SET DUE: Wednesday, September 21st 1.

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.