级物理化学-4、参考已双语eng-yxjwhk

-13thChapterIV ponentnisaconstant,closesystems,2characterdeterminethestateof

ii+2characterdeterminethestateofInthisFirst:Introducetwoconceptsofchemicalpotentialandpartialmolar Second:Describethethermodynamicpropertiesofamixture(perfectgas,idealsolutionanddilutesolution)byusingthechemicalpotentialofasubstance.Third:Givetheconceptsoffugacityandactivity,andextendtheapplicationofthermodynamicfunctiontonon-idealsystems.The'chemicalpotential',the tyonwhichalmostallthemostimportantapplicationsofthermodynamicstochemistryarebased.TheThermodynamicfundamentalequationandchemicalpotentialof ponentsystemsThermodynamicfundamentalToone-phase,onlyvolumework, ponentsystems,U,H,FandG(generalX)areextensiveproperties,Xcanbeexpressedasfunctionsofcharactervariables,wehave:U=U(S,V,n1,n2,…,nrH=H(S,p,n1,n2,…,nrF=F(T,V,n1,n2,…,nrG=G(T,p,n1,n2,…,nr

n1,n2,nrisrespectivelythemolarnumbersofcomponent1,2,,andrFromtotaldifferentialofstatefunction,then(subscriptnj≠meansthatallmolarnumbersexceptniareheldfixed)dniniirdU(U dS(U dVdniniir V

S dp1H

S,V,ndH(

)p,ndS(

)S

ri

dS,p,n

dF(T)V,ndT(V)T,ndV(n idp

G

T,V,ndG(

)p,ndT(

)T

ri

T,p,n FromMaxwell U

HS

UV

FV

V

S

T

FT

GT

Hp

Gp V

S

T

rir(ndniiS,Vrir(ndniiS,V,n(nr(ndniiiS,p,nji r(GT,V,nii T,p,ndUTdSpdVdHTdSVdpdFSdTpdVdGSdTVdp

one-phaseinaequilibriumvolumeworkThethermodynamicfundamentalequationadoptsnotonlysystemwithnvariable,butalsosuitsopenFromH=U+ then:dHdUpdVrSo:dHTdSpdVi

(ni

S,V,nj

dni

pdVrdHTdSVdpi

(ni

S,V,nj

dniUComparisonwith4.1.2:n

H iS,V iS, FromF=U–TS，G=H–TS=U–TS+pVatthesameU H

F

G

n

n

iS,V

iS,

iT,V

iT,

niniT,U H F

iS,V iS, iT,V 由定义可知，i的物理意义是在恒熵恒容以及除物质i外其它物质的量均保持不变(nj≠i)的条件下，体系中组分i物质的量改变Δni后，体系内能改变量ΔU与Δni之比在Δni 0时的极限值。同样可表达成以H，F，G表示的三个i定义式的物质意义。Ingeneral,chemicalpotentialiisthechangerateofthermodynamicsfunctionagainsttheamountofcomponentiwhenthecharactervariablesareconstant.Combineformula4.1.2andformula4.1.4,dUTdSpdVidniidHTdSVdpi

idn

ThedFSdTpdVidnidGSdTVdpidn

fundamentalequationoftheuniformphasesystem, volumeworkonly.iidniiseffectofthechangeofcompositiononthestateiThermodynamicfundamentalequationdependsonthebothcharactervariableandtheamountofcomponenti,whichcanbeusedattheuniformclosesystemwithnivariableoropensystem.Itcanbealsousedinnon-equilibriumadjacentequilibriumregion.Ifthesystemhasotherwork，e.g.electricwork,magnetismetc,addyi idGSdTVdpi yi Tomulti-phasesystemonlyngvolumeworkwithnvariable,thethermodynamicfundamentalequationisthesumofthatofeveryphase,forexample: dGSkdTkVkdpkkdn Whentemperatureandpressureareequalineveryphase,thermodynamicfundamentalequationdGSdTVdpkdn Chemicalpotentialisthestatefunction,intensive withdimensionofJ·mol-1.Thechemicalpotentialofdifferentspeciescannotbecomparedowingtoitsvaluecannotcertain；Chemicalpotentialexpressesthecertainspecies,thereisnottheconceptofchemicalpotentialofsystem；Chemicalpotentialexpressesthecertainspeciesinauniformsystem,itisimpossibletodiscussthechemicalpotentialofthecertainspeciesinmulti-phasesystem;Chemicalpotentialsareintensiveproperties,wecanusemolefractionsxiinsteadofmolar tiesnitoexpressthecompositiondependenceofi.i=i(T，p，n1，n2，…nr)=i(T，p，x1，x2，…xr-1CriterionofChemicaldGSdTVdpidniiw’=0，constantTandp dGiiThencriterionofchemical<0spontaneousii

=0reversible>0processcannot

chemicalpotentialusedastheAnswer:Theapplicationconditionoftwocriterionisdifferent,e.g.,criterionofchemicalpotentialmaybeusedinothermacroscopicalconfinedconditions,butGibbsfunctiononlyinconstantTandp.w’≠dGSdTVdpiconstantTandconstantp, dn

R

ToreversibleprocesswithconstantTandp,havingexpansionworkdn i T,p Formula4.1.11indicatesthatthedecreaseofchemicalpotentialofsystemturnsintothe umofnon-expansionworkofcoursewhentheequilibriumisbuilt.Deduction:asystemwith ponentandmulti-phase,inconstantTandp:dGdGkkikiR

Example:SubstanceAundergoesainfinitesimalamountsdnmolestransportfromphasetophaseinconstantTandp,thenthechangeofGibbsfunction:dGdndn FromFig.we dn

dn Then( Therefore Theresultillustrates:SubstanceAflowsspontaneouslyfromaphasewithhigherAtoaphasewithlowerA;ThisflowAAcontinueuntilthe ofsubstanceAinphasehasbeenequalizedtotheofsubstanceAinphaseAAAdifferenceinchemical

isthedrivingforcetheflowofchemicalspeciesifromonephasetoThestatefunctionTdetermineswhetherthereisthermalequilibriumbetweenphases;thestatefunctionpdetermineswhetherthereismechanicalequilibriumbetweenphases;thestatefunctionidetermineswhetherthereismaterialequilibriumbetweenphases.Partial Thermodynamicfunction ponentThestatefunctionsofsystemsV，U，H，S，F，G(generallyisextensiveproperties.Iftheamountofsubstancei ponentsystemisni,and

is tiesofsubstances,thentheexpressionof tiesofthermodynamicfunctionisV V

S

；U

U；F

F

H

H；G nnnnnnnni

i

i

i

i

inn designation,themolar ty,theyareallintensiveproperties.nnrniTomixturecomposedbythesame Lni

i

Example:Therelationbetweenvolumeandconcentrationinmixsolutionofwaterandalcohol(20℃,pΘ)ofalcohol/(%inweight)//Volumesumofwaterandalcohol/mlVolumeofmixing/ml/Definitionofpartial Tosimplephase,onlyvolumework, ponentsystems,V,U,H,S,FandG(generalL)areextensiveproperties,LdependsonnotonlyTandp,butalson,thenthetotaldifferentialofLL(T、p、n1，n2…,nr):dLL dTL

L T p

n i T i1 iT,i

LLLiniT,jThesubscriptniinthefirsttwopartialdifferentialindicatesthatallmolenumbersareheldconstant;thesubscriptnj≠iindicatesthatallmolenumbersexceptniareheldconstant.one-phaseLiispartial tiesofsubstanceintheconstantT、p、LLLiniT,j物理意义：在恒温恒压下，除物质i以外的其它物质的量都保持不变的条件下，加Δni的物质而引起体系广度量L的增量ΔL与Δni的比值在Δni 0时的极限值，或者可以理解为恒温恒压下，除物质i外的其它物质的量保持不变时，在充分大的体系加入1mol物质i所引起体系广度量L的改变数Physicalmeanings:partialmolar tiesaretherateofchangeoftheextensive tiesofsystemwithrespecttoniatconstantT,pandnj≠i.Itlshowthethe tiesofsystemrespondstotheconstantT,pandnj≠iadditionofitotheExample:partialmolarvolumei Vi iT,

whereVisthesolution’svolumeandthepartialdifferentialistakenwithT,p,andallmolenumbersexceptniheldconstant.Exercise:Acertain ponentgasmixtureobeystheequationofstatep(V-n1b1-n2b2)=(n1+n2)RT,whereb1andb2areconstants.FindV1andV2forthismixture. Fromp(V-n1b1-n2b2)=(n1+n2)RT,thenV=[(n1+n2)RT Therefore:VV

pb VV p 2 1T,2

1 2T,1 substanceistheslopeofthea

b

tiesvarywiththecomposition,asshownbythedifferentslopesatAmountofA,

compositionsaandTheslopeatbisnegativenotesthattheoverallvolumeofthesampledecreaseasAisadded.Somecommonpartial n

partialmolarofsubstance iT,p

j

partialmolarinternalofsubstance T,p

j H

partialmolarofsubstance T,p

j Si n

partialmolarofsubstance iT,p

jSomecommonpartial F iT,p

j

partialmolarHelmholtzofsubstanceG iT,p

j

partialmolarGibbsenergyofsubstanceiCVCV

T,p

j

partialmolarisochoricthermalofsubstance Cpp p T,p

partialmolarisobaricthermalj ofsubstance1Onlytheextensive tyintheuniformsystemhaspartial ties,andintensive tyhasnotpartial MacroscopicconfinedconditionsT,p,nj≠iareheldPartial tiesofpuresubstanceisthetiesofpureTherearen'tpartial tiesofsystem,only tiesofsubstanceiin

tiesarealwayspositive,butpartialtiesneednotbe.Somepartial tiesbedetermined,e.g.Vi,CV,i,Cp,i;andothercan’tbe,e.g.Ui,Hi,Fi,Gi.齐次函数及其Euler定理若函数yf(x1x2xr的自变量x1x2

都扩大同的倍时，函数yf(x1x2,xr的值将随着扩大m倍，则称此函数是关于这些自变量x1x2,xr的m次齐函数。Eurler定理：函数yf(x1x2xr)是关于x1x2rr的m次齐函数的充分必要条

xiii

定理若函

yf(x1x2xr)x1x2

是关

x1,x2,,

的m-1次下列函数是否是齐次函数，若是齐次函数 是几次齐函数f x

是，1/2次齐（2）f

x22y23z

是，-2次齐函）fxy（4）fx3xy

是，1次齐函不是次齐Theaddingnatureofpartial rIf isthe tyofpuresubstancei,ingeneral,rnon-ideal

L

n m,nAnyaextensive tyLofsystemisaoddfunctionaboutn1，n2，…,nrinwhichpoweris1,fromtheoremof Euler,then：rL n

L

n

i

niT,p

j

rirTheformulas4.2.2illustratesthatanyaextensive tyLofuniformsystemequalstothesumofproductofeverysubstancesniandthecorrespondingtopartialmolar tyLiApplicationaddingnatureofpartialmolar tiestotheotherthermodynamicfunctions,Forexample:rVr

GniGi rrCVniCV

HniHiGTSnii rCpniCp,ir

FniFiGpVnii rSr

UniUiHpVniiTS Therelationsofthermodynamicfunctionin ponentsystems: GiUiTSipViHi (Ui

(Hi

V(iV

S(i

(i

T relationsin ponentsystem, onlymolar tiesisreplacedbythecorrespondingtopartialmolar

G(G

[(HTS)]ii

T,p,n

T,p,n(H

T,p,n

T,p,n

Hi

(Ui

[(U

T,p,n [(U) (CV

T,p,n

T,p,n

VTherelationnatureofpartial Li=Li(T,p,n1,n2,…,nr)isaoddfunctionaboutn1,n2,…,nrwhichpoweris0,accordingtotheoremof Linj

j

jT,

k Ljnjn

iT,

kAboutformulaindicatesthatintheuniformsystem,therearerofpartialmolar tiesLj(j=1,2,…r),andpartialdifferentialofpartialmolar tyagainsttheamountofsubstanceidependonothervalueintheuniformsystem.Vi/cm3Vi/cm3·mol-xV1

xV2 1 2 x2T, x2T, Thevaluesofslopeoftwocurvesarealwaysbeinginoppositionorzeroatthesametime(SeenfromThemeasurementofpartialmolardiagrammatizingMeasure (V ofsolutioncomposedbyAand

T,First,wepreBaresolutionsatthedesiredTandp,allwhichcontainafixednumberofmolesofcomponentAbutvaryingvaluesofnB.MeasurethevolumeofsolutionwiththeaddingofPlotthemeasuredsolutionvolumesVversusBATheslopeoftheV–versus-nBcurveatBAcompositionisthenBforthatVAcanbecalculated A VA 令Vm为平均摩尔体积，根据偏摩尔量的集合 xVxVV(VVnm1nm

则(Vm VV

T, 2VVx(Vm 2 T,V

(1

)(Vm

T,

截距法测定偏摩尔p点的切线在x20轴上的截距即为组分1的偏摩尔体V1，而在x2=1轴上的截距即为组分2的偏V2,。用此法可求出各种浓度下的V1V2 yticVabncn2 2Fromdefinition,wehave: (V2

b2n

T,Fromaddingnatureofpartial ties,we V1

(abnn2...)nTheGibbs-DuhemrFromaddingnatureofpartial ties,wer niLii1dL (nidLiLidniBecauseL=L(T，p，n1，n2，…,nr),dLL

n iii

i1 iT,

BecauseLisastatefunction,t