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An Introduction to Thermal Physics

,withattribution, ,includingre-postingonanypublicWebsite, ,exceptforfiguresthatareattributedtoothe rsources, 2000, ,mechanically,anddi usivelywithitsenvironment exchangingenergy,volume, , 2000, ,whichmeasuretheexpansionofmercury(forhi ghertemperatures)andalcohol(forlowertemp eratures).Thedialthermometertotherightme asurestheturningofacoilofmetal, ajunctionoftwometals scones, 2000, ( C)Pressure (atmospheres) 200 ( ).Thethreedatasetsareforthreedi erentamountsofgas(air) ,thepressureisalinearfunctionoftemperatu rethatextrapolatestozeroatapproximately 280 C.(Moreprecisemeasurementsshowthatthezer o-pointdoesdependslightlyontheamountofga s,buthasawell-definedlimitof Casthedensityofthegasgoestozero.)Copyrig htc 2000, =LPiston area =AVolume =V=LA! , 2000, , ,downthelengthofthemolecule, 2000, bed-spring , , 2000, 2000,Addison-Wesley.

An Introduction to Thermal Physics Daniel V. Schroeder Weber State University This collectionof figures and tables is provided for the personal and classroom use of students and instructors. Anyone is welcome to download this document and save a personal copy for reference. Instructors are welcome to incorporate these figures

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1 ,withattribution, ,includingre-postingonanypublicWebsite, ,exceptforfiguresthatareattributedtoothe rsources, 2000, ,mechanically,anddi usivelywithitsenvironment exchangingenergy,volume, , 2000, ,whichmeasuretheexpansionofmercury(forhi ghertemperatures)andalcohol(forlowertemp eratures).Thedialthermometertotherightme asurestheturningofacoilofmetal, ajunctionoftwometals scones, 2000, ( C)Pressure (atmospheres) 200 ( ).Thethreedatasetsareforthreedi erentamountsofgas(air) ,thepressureisalinearfunctionoftemperatu rethatextrapolatestozeroatapproximately 280 C.(Moreprecisemeasurementsshowthatthezer o-pointdoesdependslightlyontheamountofga s,buthasawell-definedlimitof Casthedensityofthegasgoestozero.)Copyrig htc 2000, =LPiston area =AVolume =V=LA! , 2000, , ,downthelengthofthemolecule, 2000, bed-spring , , 2000, 2000,Addison-Wesley.

2 U=Q+ ,thevol-umeofthegaschangesby V(anegativeamount)andtheworkdoneonthegas (assumingquasistaticcom-pression)is P 2000, area =AForce =F x V= A xVolumePressurePViVfArea=P (Vf Vi)Area= , 2000,Addison-Wesley.(a)(b) 2000, ,thePVgraphisaconcave-uphyperbola, , 2000, (calledanadiabat) 2000, (K) (H2) , (1948).Copyrightc 2000, (K)200300400 LeadAluminumDiamond3 RHeat capacity (J/K) (datapoints)foronemoleeachofthreedi , cientlyhightemperatures, , ,ed.,ThermophysicalPropertiesofMatter(Pl enum,NewYork,1970).Copyrightc 2000, ,themagicianmustsummonupnotonlytheenergy Uoftherabbit,butalsosomeadditionalenergy ,equaltoPV, ,H=U+ 2000, xArea = 2000, ! ,itsweepsoutacylinderwhosevolumeis4 r2 .Copyrightc 2000,Addison-Wesley.!Box 1 Box 2xArea = , 2000, , , (1994).Copyrightc 2000, T( K)kt(W/m K) zFluidArea = :twoparallelsurfacesslidingpasteachother , ,thefluidflowislaminar:Atthemacroscopics calethefluidmovesonlyhorizontally, 2000, ,therewillbedi usion,anetflowofmolecules, 2000, !

3 Mi-crostates ofasetofthreecoins(whereHisforheadsandTi sfortails).Copyrightc 2000,Addison-Wesley.! , 2000, ,anysystemwithaquadraticpotentialenergyf unctionhasevenlyspacedenergylevelssepara tedinenergybyhf, , 2000, :#1#2#3 Energy:000100010001200020002110101011 Oscillator:#1#2#3 ,containingatotalofzero,one,two, 2000, ,qASolidBNB, , 2000, AqB B total= A B0 16 28281 35 21632 64 15903 103 101004 1526905 2113636 280128462 =6+6 160qA ,eachcontainingthreeoscillators, 2000, AqB B total01100 1081130099 1083245150 98 10853 4545100 97 10896 106841 1011460 106940 1011461 106939 101151100qA total( 10114) ,with300and200oscillatorsrespectively, 2000, ,q few hundredN,q few ,con-tainingafewhundredoscillatorsandene rgyunits(left)andafewthousand(right).

4 Asthesizeofthesystemincreases, q 1020, 2000, !NNe N 2 NErrorlnN!NlnN 1 101579 sapproximation( )toexactvaluesforN=1,10, 2000,Addison-Wesley. maxq/2 Width =q/ NFull scale (high-temperaturelimit). 2000, , 2000, = 2mU x pxLLpPosition spaceMomentum independent ,theybecomewiderinmomentumspace, 2000, , 2000, ,VA,NUB,VB, ,eachconfinedtoafixedvolume, 2000, ,asafunctionoftheenergyandvolumeofgasA(w iththetotalenergyandtotalvolumeheldfixed ).Ifthenumberofmoleculesineachgasislarge , 2000, 2000, , , 2000, erentgases, ,eachgasexpandstofillthewholecontainer, 2000, ASA/kqB BSB/k ,multiplicities,andentropiesofasystemoft woEinsteinsolids,onewith300oscillatorsan dtheotherwith200, 2000, (units ofk) (qA=60),thetotalentropyisamaximumsoitsgr aphhasahorizontaltangent; (forinstance,atqA=12),thesolidwhosegraph hasthesteepertangentlinetendstogainenerg yspontaneously; 2000, Normal Miserly Enlightened ( )fora normal systemthatbecomeshotter(moregenerous)asi tgainsenergy;a miserly systemthatbecomescolder(lessgenerous)asi tgainsenergy.

5 Andan enlightened systemthatdoesn 2000, ,initialUB, 2000, K300 K1500 J SA= 3 J/K SB= +5 ,itsentropydecreasesby3 ,itsentropyincreasesby5 2000, (Q)thatleavesahotobjectisrequiredtocarry someentropy(Q/T) , 2000,Addison-Wesley.! ,consistingofNmicroscopicmagneticdipoles ,eachofwhichiseither up or down ;theydonotinteractwiththeirneighbors(exc epttoexchangeenergy).Copyrightc 2000, B(forthe up state)and+ B(forthe down state).Copyrightc 2000,Addison-Wesley. Up Down + B B0 EnergyN U/ BM/N S/kkT/ BC/Nk100 99 492 . 484 .. ,N .Themultiplicity ,whiletheentropySiskln .Thelasttwocolumnsshowthetemperatureandt heheatcapacity, 2000,Addison-Wesley. 100 0 50 100S/kU/ B 2000, (Thisgraphwasplot-tedfromtheanalyticform ulasde-rivedlaterinthetext; )Copyrightc 2000, BkT/ B 11 20 101020kT/ B 11 10M/N 5105kT/ (com-putedfromtheanalyticformulasderived laterinthetext).

6 Copyrightc 2000,Addison-Wesley. 11xtanhx 1 2 ,theargumentxofthehyperbolictangentis 2000, 1/T(K 1) ( B/kT)Curie s DPPH (ina1:1complexwithbenzene),takenatB= (with = B),whilethedashedlineisthepredictionofCu rie slawforthehigh-temperaturelimit.(Because thee ectivenumberofelementarydipolesinthisexp erimentwasuncertainbyafewpercent,thevert icalscaleofthetheoreticalgraphshasbeenad justedtoobtainthebestfit.) , , ,JournalofChemicalPhysics68,5225(1978).C opyrightc 2000, ,VA,SAUB,VB, 2000, 2000, ,considertheprocessintwosteps:chang-ingU whileholdingVfixed, 2000,Addison-Wesley. U VStep 1 Step :veryfastcompressionthatcreatesinternald isequilibrium, 2000,Addison-Wesley.! 2000, ,NA,SAUB,NB, 2000,Addison-Wesley. A , 2000, 3,q= 3, = 10N= 4,q= 2, = (representedbyabox)tothisverysmallEinste insolidwhileholdingtheentropy(ormultipli city)fixed,wemustremoveoneunitofenergy(r epresentedbyadot).

7 Copyrightc 2000, S U V,NmechanicalvolumepressurePT= S V U,Ndi usiveparticleschemicalpotential T= S N U, , 2000, , 2000, reservoir,ThCold reservoir,TcEngineQhQcWQhQc(a)(b)(c)(d) :(a)isothermalexpansionatThwhileabsorbin gheat;(b)adiabaticexpansiontoTc;(c)isoth ermalcompressionatTcwhileexpellingheat;a nd(d) (a)andwiththecoldreservoirduringstep(c). Copyrightc 2000, 2000, , 2000, reservoir,ThCold reservoir, , , 2000, 2000, ,shownduringthepowerstrokewhenthehotpist onismovingoutwardandthecoldpistonisatres t.(Forsimplicity,thelinkagesbetweenthetw opistonsarenotshown.)Copyrightc 2000, reservoirCold reservoirQhQcW1234 PumpBoilerTurbineCondenser(Water)(Steam) (Water + steam) (nottoscale), ,whereitissteam, 2000, ( C)(bar)(kJ)(kJ)(kJ/K)(kJ/K) ,where1bar=105Pa ,andaremeasuredrelativetoliquidwateratth etriplepoint( ).ExcerptedfromKeenanetal.

8 (1978).Copyrightc 2000, ( C)P(bar) (kJ)28753074327834883705S(kJ/K) (kJ)28663069327534863703S(kJ/K) (kJ)28283051326434793698S(kJ/K) (kJ)2994323134573682S(kJ/K) (kJ)309733743625S(kJ/K) (kJ)215130813444S(kJ/K) , (1978).Copyrightc 2000, reservoirCold reservoirWPressureVolumeQhQc1234 Condenser(Liquid)(Gas)(Liquid + gas) (nottoscale) ,gas, 2000, , 2000, (bar)( C)(kJ)(kJ)(kJ/K)(kJ/K) (atitsboilingpointforeachpressure).Allva luesarefor1kgoffluid,andaremeasuredrelat ivetoanarbitrarilychosenreferencestate,t hesaturatedliquidat 40 (1995).Copyrightc 2000, ( C)P(bar) (kJ)274284295S(kJ/K) (kJ)269280291S(kJ/K) (kJ)276287S(kJ/K) (gaseous) , (1995).Copyrightc 2000, (toroomtemperatureissu cientifitisnitrogenoroxygen) , 2000, (approximatelyhori-zontal,atintervalsof4 00J/mol)andinversioncurve(dashed) ,socoolingoccursonlytotheleftoftheinvers ioncurve, (1975)andWoolleyetal.

9 (1948).Copyrightc 2000, (K)P(bar)501001502001001502000 Temperature(K)77(liq.)77(gas)10020030040 05006001bar 3407216128565800871711,63514,57317,55410 0bar 19464442817411,39214,49217, (injoules) (1994).Copyrightc 2000, (lightgray), (darkgray) 2000, ExchangerCompressorMixingchamberStill4He bath3 HeConstriction(few mK)( K)(1 K)(300 K) ,attwodi erentvaluesofthemagneticfieldstrength.( )Themagneticcoolingprocessconsistsofanis othermalincreaseinthefieldstrength(step1 ),followedbyanadiabaticdecrease(step2).C opyrightc 2000, , 2000, ,themagicianneednotsummonuptheentireenth alpy,H=U+ ,equaltoTS,canflowinspontaneouslyasheat; themagicianmustprovideonlythedi erence,G=H TS, 2000, ,addPV;togetFfromUorGfromH, 2000, +PV , (graphite).Bubblesofhydrogen(toosmalltos ee)formatthenegativeelectrode(left)while bubblesofoxygenformatthepositiveelectrod e(right).

10 Copyrightc 2000,Addison-Wesley. G= 237 kJ(electrical work)P V= 4 kJ (pushingatmosphere away)T S= 49 kJ(heat) U= 282 ,49kJofenergyenterasheat(T S),sotheelectricalworkrequiredisonly237k J: G= H T erencebetween Hand UisP V=4kJ, 2000,Addison-Wesley.+ ,hydrogenandoxygengaspassthroughporousel ectrodesandreacttoformwater, 2000,Addison-Wesley.(electrical work)(heat)394 kJ78 kJ U= 316 ,thesystem senergydecreasesby316kJanditsentropyincr easesby260 ,thesystemcanabsorb78kJofheatfromtheenvi ronment;themaximumworkperformedistherefo re394kJ.(Becausenogasesareinvolvedinthis reaction,volumechangesarenegligibleso U Hand F G.)Copyrightc 2000, ,surroundingamagneticspecimen,connectedt oapowersupplythatcanchangethecurrent, 2000, , 2000, (reservoir)V,U,S,P,T2V,2U,2S,P, ,energy,andentropyasonerabbit, 2000, ,holdingthetemperatureandpressurefixed,t hesystem sGibbsfreeenergyincreasesby.

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