參考應(yīng)用技術(shù)學(xué)院電子教案

亥姆霍茲函數(shù)及吉熱力學(xué)一 亥姆霍dSQW≥dU－T環(huán)T1＝T2＝T環(huán)＝

Hermannvon(1821年－1894年 也稱為：亥姆霍茲自由能 A的SI單位A吉布斯T1＝T2＝T環(huán)＝p1＝p2＝p環(huán)＝或W’≥d（H－TS） H吉布斯函數(shù)(Gibbsfunction)G

JosiahWillard(1839年－1903年G的SI單位

熵判據(jù)“對 系統(tǒng)或絕熱“

=表示可逆，平衡亥姆霍茲函數(shù) “=”表示可逆，平“<”表示不可吉布斯函數(shù)

“=”表示可“<”表示不可 蒸發(fā)焓為40.69kJmol－1。－1K＝－(1ol)(40.69kJol-

-1)／(373.2K)＝－109.0JK-109.0JK ＝－(1ol)(8.314Jol－1K－1K＝19.JK0J 例3.5.3C6H6的正常為5℃，摩爾熔化焓9916Jmol-1，＝126.8JK1mol＝122.6JK1mol

C6H6(s),101325Pa,C6H6(s),101325Pa,C6H6(l),101325Pa,

Q＝＝{1×[126.8×(278.2-268.2)＋(-9916)＋122.6×(268.2-＝-△U＝Q＋W≈Q＝-△S1＝nCp，m(l)ln(T2／T1＝4.642JK△S2＝35.643JK△S3＝nCp，m(s)ln(T1／T24.488JKJK△A＝△U－T△S＝[-9874-268.2×(-35.50)]J＝-△G＝△H－T△S＝[-9874-268.2×(-35.50)]J＝-熱力學(xué)第三thethirdlawof熱力學(xué)能斯特?zé)岫ɡ?Nernstheat lim(GH)T1906年，Nernst經(jīng)過系統(tǒng)地研了溫lim(G lim(S T

T limST0K

(凝聚系統(tǒng)或△S 熱力學(xué)第三定律的普朗克0K時純物質(zhì)凝聚相的熵為limS(B凝聚相)T0或3純物質(zhì)完美晶體的熵，0K時為零規(guī)定摩爾熵(conventionalmolar，也稱為絕對摩爾熵(absoluteentropyS(B,T)

標準摩爾熵(standardmolareSe(B,T)

Cp,md 0 以Cp/T為縱如圖所S (Cp/T10K以下，用德拜(Debye)TCp，m＝Cm＝aT Cp固 ST

dT C(液+b

TCp(氣 從Cp，m數(shù)據(jù)求HCl(g)

S JKlmmSmSm0＝∑B反應(yīng)的標準摩爾熵[變(standardmolarentroiesomS 8TmS

Z,TaS

A,TbS

B,Tmmm B 8Tmmm B

8B,T .1熱力學(xué) 熱傳導(dǎo)過程的不可逆氣體混合過程的這是度增加的過程，也是熵增加的過程，熱力學(xué)第二定律，凡是自發(fā)的過程都是 熵的統(tǒng)計比。例如：有4個小球分裝在兩個盒子中，總的分

4(4,0)C44444(2,2)C24

C144

(0,4)C04其中，均勻分布的熱力學(xué)概率?(2,2)最大，為6。4 0VN的函數(shù)，兩者之間必定有某SS(Boltzmann認為這個函數(shù)應(yīng)該有如下的對數(shù)形Skln這就是 k＝R／L＝1.381×10－23JK 把熱力學(xué)宏觀量S玻耳茲 上的關(guān)系S=klog(后來普朗克將其改寫為S=kBlog 熵與信只計字數(shù)內(nèi)容，單在Shannon從概

ClaudeShannon(1916–2001)找人找到的概率是1／50知道住在三層，找到的概率1／10知道房間號碼，則概率加大到1 一，讓乙猜它的花色，規(guī)則是允許乙提問題， S＝-香農(nóng)把這叫做信息熵，它意味著信息量 S

ln

lni 率Pi的平均。如果所有的Pi＝1／N，則上式歸結(jié)為S＝-熵S＝0.469，從而這句話含信息量I＝1—S＝0.531bit加。在一個過程中△I＝—△S，即信息量相當(dāng)于負熵從信息熵 1bit＝kln2＝0.957×10-23JK- 增加一個bit，它的熵減少kln2，這只能對于自然界一切自發(fā)有序(order)的減無序(disorder)信息(information)熵(entropy)吉布斯佯(又稱吉布斯悖論,Gibbs..麥克斯韋(Maxwell's熱力學(xué)(fundamentalequationofU、H、S、A、G、p、V、TH=U+pVA=U-TSG=H-

U＝U(S、H＝H(S、A＝A(T、G＝G(T、(Gibbs－HelmholtzT＝(U／S)V＝(H／S)p－(U／V)S＝－(A／V)T＝(H／p)S＝(G／p)T(G／p)T＝V吉布斯—亥姆霍(Gibbs－HelmholtzGT 1G GSGTS TT T

T T GT p TpAT p Tp麥克斯韋關(guān)系式(Maxwell's

例3.8.1文石和方解石是兩種不同晶型的CaCO3。25℃、100kPa下1mol文石轉(zhuǎn)變?yōu)榉浇馐瘯r，體積增

p1V(文石)dpV(文石)( p2 2p2G3 V(方解石)dpV(方解石22

p1∴△V(p2－p1)＝795J例 (UV)T 因為所以對于理想氣體，有p＝nRT／V (U／V)T＝(TnR／V)－p＝0p＝[nRT／(V－nb(U／V)T＝[TnR／(V－nb)]－p＝n2a／V2 H2O(l),101325Pa,-H2O(l),489.2Pa,-H2O(g),489.2Pa,-

H2O(S),101325Pa,-HH2O(S),475.4Pa,-H2O(g),475.4Pa,-△G2△G4△G1＋△G5△G3＝Vdp≈nRT／pdp＝＝[1×8.314×270.2×ln(475.4＝-4＝-克拉佩龍Gm()≤Gm() ，T，p)＝ T→T＋dT，p→ )＝dGm(

(1799-1864)dGm()＝dGm(dGm()dGm()＝－Sm()dT＋Vm(－Sm()dT＋Vm()dp＝－Sm()dT＋Vm( Sm()

Vm() T

克拉佩龍方程(Clapeyron克勞修△Vm≈Vm(g)≈△vapHm＝dp／dT＝△vapHm／TVm(氣dln{p}vap RTlnp

vapH

1p1

T2

T1ln{p}vapH 克勞修斯－克拉佩(Clausius－Clapeyron3.9.3Trouton規(guī)則和Antoine楚頓規(guī)則(Trouton’srule)。vap

85JK-13.9.4外壓與蒸氣壓的p g

Vm(1)(pp*pg pg**g g

壓。

pep

時，則 p* 0℃時冰的融化焓為6008Jmol－1，冰的19.652c3ol18.018c3ol1℃所需的壓力變dpfusH 6008Jmol-(273.15K)(18.01819.652)10-6m3mol- 1.346107PaK-47.6kPa下苯的沸kJol 47.6kPa31800Jmol- 8.314 K- mol-1 353.2K SecondLawofThesecondlawofthermodynamicsisanexpressionoftheuniversallawofincreasingentropy,statingthattheentropyofanisolatedsystemwhichisnotinequilibr time,approachinga umvalueatequilibrium.ThesecondlawtracesitsorigintoFrenchphysicistSadiCarnot's1824paperReflectionsontheMotivePowerofFire,whichpresentedtheviewthatmotivepower(work)isduetothefallofcaloric(heat)fromahottocoldbody(workingsubstance).Insimpleterms,thesecondlawisanexpressionofthefactthatovertime,ignoringtheeffectsofself-gravity,differencesintemperature,pres ,anddensitytendtoevenoutinaphysicalsystemthatisisolatedfromtheoutsideworld.Entropyisameasureofhowfaralongthisevening-outprocesshasprogressed.Therearemanyversionsofthesecondlaw,buttheyallhavethesameeffect,whichistoexinthephenomenonofirreversibilityinTherearemanywaysofstatingthesecondlawofthermodynamics,butallareequivalentinthesensethateachformofthesecondlawlogicallyimplieseveryotherform.Thus,thetheoremsofthermodynamicscanbeprovedusinganyformofthesecondlawandthirdThe

s dlawthatrefersto Inasstema rocessthatoccurswilltendtoincreasethetotalentropyoftheuniverse.Thus,whileasystemcanundergosomephysicalprocessthatdecreasesitsownentropy,theentropyoftheuniverse(whichincludesthesystemanditssurroundings)mustincreaseoverall.(Anexceptiontothisruleisareversibleor"isentropic"process,suchasfrictionlessadiabaticcompression.)Processesthatdecreasetotalentropyoftheuniverseareimpossible.Ifasystemisatequilibrium,bydefinit processesoccur,andthereforethesystemisat umentropy.AlsoduetoRudolfClausiusisthesimlestformulationofthesecondlaw,theheatformulationorClausiusstatement:HeatgenerallycannotspontaneouslyflowfrommaterialatlowertemperaturetoamaterialathigherInformally,"Heatdoesn'tflowfromcoldtohot(withoutinput)",whichisobviouslytruefromeverydayexperience.Forexampleinarefrigerator,heatflowsfromcoldtohot,butonlywhenaidedbyanexternalagent(i.e.thecompressor).Notethatfromthemathe ade heatflowsfromcoldtohothasdecreasingentropy.Thiscanhappeninanon-isolatedsystemifentropyiscreatedelsewhere,suchthatthetotalentropyisconstantorincreasing,asrequiredbythesecondlaw.Forexample,theelectricalenergygoingintoarefrigeratorisconvertedtoheatandgoesouttheback,representinganetincreaseinentropy.Theexceptiontothisisinstatisticallyunlikelyeventswhereparticles

theenergyofcoldparticlesenoughthatthesidegetscolderandthehotsidegetshotter,foraninstant.Sucheventshavebeenobservedatasmallenoughscalewherethelikelihoodofsuchathinghappeningislargeenough.ThemathematicsinvolvedinsuchaneventaredescribedbyfluctuationAthirdformulationofthesecondlaw,byLordKelvin,istheheatengineformulation,orKelvinstatement:Itisimpossibletoconvertheatcomple yintoworkinacyclicprocess.Thatis,itisimpossibletoextractenergybyheatfromahigh-temperatureenergysourceandthenconvertalloftheenergyintowork.Atleastsomeoftheenergymustbepassedontoheatalow- rgys .T ,a atengine h100%e isthermodynamicallyimpossible.Thermodynamicsisatheoryofmacroscopicsystemsandthereforethesecondlawappliesonlytomacroscopicsystemswithwell-definedtemperatures.Forexample,inasystemoftwomolecules,thereisanon-trivialprobabilitythattheslower-moving("cold")moleculetransfersenergytothefaster-moving("hot")molecule.Suchtinysystemsareoutsidethe ofclassicalthermodynamic, heycanbeinvestigatedinquantumthermodynamicsbyusingstatisticalmechanics.Foranyisolatedsstemwithamassofmorethana ico robabilitiesobservingadecreaseinentropyapproachEnergyThesecondlawofthermodynamicsisanaxiomof eat,entropy,andthedirectionnwhichthermodynamicprocessescanoccur.Forexample,thesecondlawimpliesthatheatdoesnotspontaneouslyflowfromacoldmaterialtoahotmaterial,butitallowsheattoflowfromahotmaterialtoacoldmaterial.Roughlyspeaking,thesecondlawsaysthatinanisolatedsystem,concentratedenergydispersesovertime,andconsequentlylessconcentratedenergyisavailabletodousefulwork.Energydispersalalsomeansthatdifferencesintemperature,pressure,anddensityevenout.Againroughlyspeaking,thermodynamicentropyisameasureofenergydispersal,andsothesecondlawiscloselyconnectedwiththeconceptofentropy.HistoryofThefirsttheoryontheconversionofheatintomechanicalworkisduetoNicolasLéonardSadiCarnotin1824.Hewasthefirsttorealizecorrectlythattheefficiencyofthisconversiondependsonthedifferenceoftemperaturebetweenanengineanditsenvironment.RecognizingthesignificanceofJamesPrescottJoule'sworkontheconservationofenergy,RudolfClausiuswasthefirsttoformulatethesecondlawin1850,inthisform:heatdoesnotspontaneouslyflowfromcoldtohotbodies.Whilecommonknowledgenow,thiswas tothecalorictheor ofheat oularatthetime,whichconsideredheatasaliquid.FromtherehewasabletoinferthelawofSadiCarnotandthedefinitionofentropy(1865).Establishedinthe19thcentury,theKelvin- nckstatementoftheSecondLawsays,"Itisimpossibleforanydevicethatoperatesonacycletoreceiveheatfromasinglereservoirandprod ofwork."ThiswasshowntobeequivalenttothestatementofTheErgodichypothesisisalsoimportant approach.Itsaysthat,overlongperiodsoftime,thetimespentinsomeregionofthephasespaceofmicrostateswiththesameenergyisproportionaltothevolumeofthisregion,i.e.thatallaccessiblemicrostatesareequallyprobableoverlongperiodoftime.Equivalently,itsaysthattimeaverageandaverageoverstatisticalensemblearetheUsingquantummechanicsithasbeenshownthatthelocalannentropyisatits umvaluewithanextremelyhighprobability,thusprovingthesecondlaw.[5]Theresultisvalidforalargeclassofisolatedquantumsystems(e.g.agasinacontainer).Whilethefullsystemispureandhasthereforenoentropy,theentanglementbetweengasandcontainergivesrisetoanincreaseofthelocalentropyofthegas.Thisresultisoneofthemostimportantachievementsofquantumthermodynamics.InformalThesecondlawcanbestatedinvarioussuccinctways,Itisimpossibletoproduceworkinthesurroundingsusingacyclicprocessconnectedtoasingleheatreservoir(Kelvin,1851).Itisimpossibletocarryoutacyclicprocessusinganengineconnectedtotwoheatreservoirsthatwillhaveasitsonlyeffectthetransferofa tyofheatfromthelow-temperaturereservoirtothehigh-temperaturereservoir(Clausius,1854).IfthermodynamicworkistobedoneatafiniteenergymustbeThirdLawofThethirdlawofthermodynamicsisastatisticallawofnatureregardingentropyandtheimpossib ofre hinga oftemperature.Themostcommonenunciationofthirdlawofthermodynamicsis:“Asasystemapproachesabsolutezero,allprocessesceaseandtheentropyofthesystemapproachesaminimumvalue.”Notethattheminimumvalueisnotnecessarilyzero,althoughitisalmostalwayszeroinaperfect,purecrystal;seethearticleResidualentropyformoreinformation.Theessenceofthepostulateisthatthesystemnearabsolutezerodependsonlyonthetemperature(i.e.tendstoaconstantindeendentl oftheother arameters.'ThethirdlawwasdevelopedbyWaltherNernst,duringtheyears1906-1912,andisthussometimesreferredtoasNernst'stheoremorNernstspostulate.Thethirdlawofthermodynamicsstatesthattheentropyofasystematzeroisawell-definedconstant.Thisisbecauseasstematzerotemeratureexistsinits roundstatesothatitsentropyisdeterminedonlybythedegeneracyofthegroundstate;or,itstatesthat"itisimpossiblebyanyprocedure,nomatterhowidealised,toreduceanysystemtotheabsolutezerooftemperatureinafinitenumberofoperations".'AnalternativeversionofthethirdlawofthermodynamicsasstatedbyGilbertN.LewisandMerleRandallin1923:“Iftheentropyofeachelementinsome(perfect)crystallinestatebetakenaszeroattheabsolutezerooftemperature,everysubstancehasafinitepositiveentropy;butattheabsolutezerooftemperaturetheentropymay ezero,anddoesso einthecaseo erectcrstallinesubstances.”ThisversionstatesnotonlyΔSwillreachzeroat0Kelvin,butSitselfwillalsoreachzero,atleastforperfectcrystallinesubstances.(Thisstatementisnowknowntohavesomerareexceptions.)Insimpleterms,theThirdLawstatesthattheentropyofmostpuresubstancesapproacheszeroastheabsolutetemperatureapproacheszero.Thislawprovidesan thedeterminationofentropy.Theentropydeterminedrelativetothispointistheabsoluteentropy.Aspecialcaseofthisissystemswithauniquegroundstate,suchasmostcrystallattices.TheentropyofaperfectcrystallatticeasdefinedbyNernst'stheoremiszero(ifit