納米結(jié)構(gòu)體系物理化學性質(zhì)的理論研究方法與實例課件

StatisticalThermodynamicsandChemicalKineticsStateKeyLaboratoryforPhysicalChemistryofSolidSurfaces廈門大學固體表面物理化學國家重點實驗室Lecture9May12,2003StatisticalThermodynamicsandChapter9ComplexReactions9.1Exactanalyticsolutionsforcomplexreactions9.1.1IntroductionMostchemicalprocessesarecomplex,i.e.theyconsistofanumberofcoupledelementaryreactions.Thesecomplexreactionscanbedividedintoseveralclasses:(1)opposingorreversiblereactions,(2)consecutivereactions,(3)parallelreactions,and(4)mixedreactions.Inthischapter,weexaminemethodsfordeterminingexactanalyticsolutionsforthetimedependenceofconcentrationsofspeciesinvolvedincomplexreactions.Chapter9ComplexReactionsReversiblereactionsInreversiblereactionsoropposingreactions,theproductsoftheinitialreactioncanproceedtore-formtheoriginalsubstances.Achemicalexampleofthisisthecis-transisomerizationof1,2-dichloroethylene:(9-1)Assuch,thesimplestreversiblereactionisoftheform

(9-2)andisfirstorderineachdirection.Thedifferentialequationforthismechanismis(9-3)(9-4)9.1.2ReversiblereactionsIfitisassumedthatbothA1andA2arepresentinthesystemattimet=0,thatis,[A1]=[A1]0and[A2]=[A2]0,thenatanytimeafterwordsthetotalamountofreactantremainingandnewproductformedmustequaltheinitialamountofthereactantsbeforereactions.Hence,

[A1]0+[A2]0=[A1]+[A2](9-5)Solvingfor[A2],weobtain

[A2]

=[A1]0+[A2]0-[A1](9-6)andsubstitutingthisintoequation(9-3)yields(9-7)IfitisassumedthatbothA1Tofindthesolutionofequation(9-7)weintroduceavariablem,definedas(9-8)Thisallowsustorewriteequation(9-7)as(9-9)whichwemaythenintegrate(9-10)Thesolutionis(9-11)TofindthesolutionofequatIfonly[A1]ispresentinthesysteminitially,att=0,thenthesolutionreducesto(9-12)whichisjust(9-13)Usingthemassconservationconstraint[A1]0=[A1]+[A2],wecanobtainthesolutionfor[A2]:

(9-14)Whenequilibriumisreached,theindividualreactionsmustbebalanced;inotherwords,thereactionABmustoccurjustasfrequentlyasthereversereaction.Theforwardandreversereactionsoccuratthesamerate.Ifonly[A1]ispresentinthConsequently,forreaction(9-2)atequilibrium,(9-15)and(9-16)andwehavethefollowingdefinitionoftheequilibriumconstantKeqexpressedintermsofrateconstants:

(9-17)Thesameargumentcanbeextendedtoareversiblereactionthatoccursinmultiplestages.

Consequently,forreaction(First-orderreversiblereactionsinvolvingtwostepsReversiblereactionsmaybedistinguishedbynumberofstagesandthenumberofinitialreactantsinvolvedinthereaction.Here,weconsiderthecompletederivationforfirstorderreversiblereactionsinvolvingonlytwostages,thatis,(9-18)Thekineticequationsforthesystemare

(9-19)(9-20)(9-21)First-orderreversibleWeassumethatatt=0,[A1]=[A1]0,and[A2]0=[A3]0=0andthattheamountsofA1,A2andA3whichhavereactedatalatertimesatisfytheequation:

[A1]0=[A1]+[A2]+[A3](9-22)Bytheprincipleofdetailedbalance,wehaveand(9-23,24)Usingequations(9-22)-(9-24)gives(9-25)So(9-26)Weassumethatatt=0,[A1]=[ASubstitutingequation(9-26)intoequation(9-19)weobtainthenewfirst-orderdifferentialequaion:

(9-27)Solvingthiseuationusingstandardmethods,wehave(9-28)Substitutingequation(9-22),(9-26)and(9-28)intothisexpression,wehave,uponsimplification,(9-29)

Substitutingequation(9-26)iAst,

(9-30)(9-31)(9-32)andthesystemistheninastateofequilibrium.Themisuseoftheprincipleariseswhentheinter-mediateA2isdifficulttodetect,sothatanexperimentalistmightthinkthatA1A3isanelementaryreaction.Thustheequilibriumconstantfor[A3]e/[A1]emightbetakentobek1/k-2.However,sinceA1A3isnotanelementaryreaction,thisconclusionisincorrect.

Ast,TodetermineK=[A3]e/[A1]ecorrectly,theequilibriumofeachelementaryreactionmustbeconsidered,thatis,equations(9-23)and(9-24)mustbeused.ThecorrectexpressionofKcanbefound:

(9-33)

Anotherexampleofafirst-orderreversiblereactioninvolvingonlytwostagesisthecyclicreaction

TodetermineK=[A3]e/[A1]ecFirstandsecondorderreversiblereactions.Areversiblereactionmaybeofmixedorder,suchas,

AnexampleisN2O4

2NO2.Therateexpressionforthistypeofreactionis

(9-34)Tofindthesolution,weintroduceaprogressvariable,x=[A1]0-[A1](9-35)Equation(9-34)thusbecomes

FirstandsecondorderOr(9-36)Welet(9-37)andSothat(9-38)Then(9-39)Thesolutionofequation(9-39)is(9-40)WhereOr9.1.3ConsecutiveReactionsIrreversiblereactionscanbedefinedasthosewhichstartwithaninitialreactantandproduceproductsorintermediatesgenerallyinonlyonedirection.Consecutivereactionsaresequentialirreversiblereactions.Therearetwoclassesofconsecutivereactions:Thosewhicharefirstorderandthosewhicharemixedfirstorderandsecondorder.

9.1.3ConsecutiveReactionsFirst-orderconsecutivereactions.

.1First-orderwithtwosteps.

Considertheconsecutivefirst-orderreactioninvolvingtwostages:(9-42)Thismechanismcanbedescribedbythefollowingsetofrateexpressions:(9-43)(9-44)(9-45)First-orderconsecutivTheconcentrationofA1isobtainedafterintegrationas

(9-46)TheconcentrationofA2iscomputedfromequation(9-44)(9-47)Solvingthisequation,weobtainthetimedependenceof[A2]:(9-48)If[A2]0=0att=0,then(9-49)TheconcentrationofA1isobtTheconcentrationofA3canbedeterminedfromconservationofmass:[A1]0=[A1]+[A2]+[A3](9-50)Hencewehave(9-51)Simplifyingthisexpressiongives(9-52)

TheconcentrationofA3canbe

.2First-orderwiththreesteps.

Thesystemofdifferentialequationsforafirst-orderconsecutivereactioninvolvingthreesteps,viz.:(9-53)canbeintegratedinawaysimilartothetwo-stepcase.ThedifferentialequationsforA1andA2andtheirsolutionsdonotdifferfromthoseobtainedinthetwo-stepcase,whilethedifferentialequationforA3is(9-54)Substitutingequation(9-49)for[A2]intoequation(9-54)gives(9-55).2First-orderwithtIntegratingthislinearequationsubjecttotheinitialconditionthat[A3]=0att=0,weobtain(9-56)Then(9-57)

Integratingthislinearequati

Higherorderconsecutivereactions.

Somehigherorderconsecutivereactionsmaybefirstorderinonestepandsecondordertheinsecond,suchas(9-58)Orbothstepsmaybesecondorderconsecutivereactions,asin(9-59)Thedifferentialequationsformostofthesesystemsofreactionsarenonlinearandgenerallytheyhavenoexactsolutions.Nevertheless,analyticsolutionsforsuchsystemscanbeobtainediftimeiseliminatedasavariable.Toillustratethismethod,considerthereactionsequencedescribedbythesystemofequations(2-58):Higherorderconsec(9-60)(9-61)(9-62)Tosolvethissystemofequationswedivideequation(9-61)by(9-60)(9-63)IfweletK=k1/k2,then(9-64)

Equation(9-64)canberearrangedto(9-65)(9-66)Integratingequation(9-66)overtherespectivelimits,i.e.,(9-67)gives(9-68)Equation(9-64)canberearran9.1.4ParallelreactionsParallelreactionsaredefinedastwoormoreprocessesinwhichthesamespeciesparticipateineachreactionstep.Themostcommoncasesofparallelreactionsare:(1)thoseinwhichtheinitialreactantdecomposesintoseveraldifferentproducts;(2)thoseinwhichtheinitialreactantsaredifferent,butyieldthesameproducts;and(3)thoseinwhichasubstancereactswithtwoormoreinitialreactants.9.1.4ParallelreactionsParalFirstorderdecaytodifferentproductsConsiderthemechanism(9-69)Att=0theinitialconcentrationsofthefourcomponentsare[A1]=[A1]0;[A2]=[A3]

=[A4]

=0(9-70,71)Thedifferentialequationforthesystemcanbewritten,asexemplifiedfor[A1],(9-72)Integratingequation(9-72)yields(9-73)FirstorderdecaytodTosolvefor[A2]wesubstituteequation(9-73)intoequation(9-74),(9-74)Anduponintegratingweobtain(9-75)Similarlyweobtain(9-76)(9-77)Tosolvefor[A2]wesubstituItfollowsthattherelativerateconstantscanbedeterminedbymeasuringtherelativeproductyields:(9-78)Thisratiodefinesthebranchingratioforthereaction;notethatthisbranchingrationisindependentoftime.Theaboveexamplecanbeeasilygeneralizedtoreactionsofasinglereactantintondifferentproducts.Wethushave(9-79)(9-80)Itfollowsthattherelative

Firstorderdecaytothesameproducts.Considerthereactionsequence

(9-81)TherateexpressionsforthedisappearanceofA1andA3andtheappearanceofA2are(9-82)(9-83)

(9-84)Att=0,[A1]=[A1]0,[A3]=[A3]0,and[A2]=.2FirstorderdecaytoTheequationsdescribingthetimedependenceforeachcomponentare(9-85)(9-86)Since(9-87)Uponintegrationweobtain

(9-88)or(9-89)TheequationsdescribingthetParallelsecond-orderreactionsInthecaseofparallelsecond-orderreactions,(9-90)Twodifferentialequationscanbewritten,(9-91)(9-92)Conservationofmassdemands[A4]=[A2]0-[A2](9-93)[A5]=[A3]0-[A3](9-94)[A1]0-[A1]=[A4]

+[A5]=[A2]0-[A2]+[A3]0-[A3](9-95)Parallelsecond-orderEliminatingtimeasavariable,yields(9-96)whichcanbeintegratedundertheinitialconditionthat[A3]=[A3]0and[A2]=[A2]0attimet=0.Thisgives(9-97)Fromthemassconservationrelationwecandetermine[A5]bysubstitutingequation(9-94)intoequation(9-97).Rearrangingthengives(9-98)EliminatingtimeasavariablTheexpressionfor[A1]isobtainedfromequation(9-95)[A1]=[A1]0-[A2]0-[A3]0+[A2]+[A3](9-99)Substitutingequation(9-96)intoequation(9-99)gives(9-100)Usingthisexpressionfor[A1],weobtainadifferentialequationfor[A2]:(9-101)(9-102)Theexpressionfor[A1]isobIntegrating(9-102),weobtain(9-103)Thereisnoexplicitsolutiontothisintegral,butwecanconsidersomelimitingcases.

CaseI.k1>>k2

Underthisconditiontherationk2/k1isapproximatelyzero,thusequation(9-103)reducestothesimpleform(9-104)(9-105)Integrating(9-102),weobtaiRearrangingandsolvingfor[A2]tyields(9-106)CaseII.k1=k2

Underthisconditiontherationk2/k1

=1,thusequation(9-103)canbeintegrable,viz.(9-107)(9-108)Rearrangingandsolvingfor[Assignments:

Considerthereaction

att=0,[A1]=[A1]0,[A2]=[A3]=…=[An-1]=[An]=0.Deriveanexpressionthatwilldescribetheconcentrationoftheintermediatesexistingatanytimetduringthereaction.[E.Abel,Z.Phys.Chem.A56,558(1906).]Considerthefirst-ordercyclicreaction

Assumethatinitiallyatt=0,[A]=[A]0,[B]=[C]=0,andtheamountsofA,BandCalwaysfollow[A]0=[A]+[B]+[C]Usingthedetailedbalancedmethod,solvefor[A],[B]and[C].Assignments:ConsiderthereacStatisticalThermodynamicsandChemicalKineticsStateKeyLaboratoryforPhysicalChemistryofSolidSurfaces廈門大學固體表面物理化學國家重點實驗室Lecture9May12,2003StatisticalThermodynamicsandChapter9ComplexReactions9.1Exactanalyticsolutionsforcomplexreactions9.1.1IntroductionMostchemicalprocessesarecomplex,i.e.theyconsistofanumberofcoupledelementaryreactions.Thesecomplexreactionscanbedividedintoseveralclasses:(1)opposingorreversiblereactions,(2)consecutivereactions,(3)parallelreactions,and(4)mixedreactions.Inthischapter,weexaminemethodsfordeterminingexactanalyticsolutionsforthetimedependenceofconcentrationsofspeciesinvolvedincomplexreactions.Chapter9ComplexReactionsReversiblereactionsInreversiblereactionsoropposingreactions,theproductsoftheinitialreactioncanproceedtore-formtheoriginalsubstances.Achemicalexampleofthisisthecis-transisomerizationof1,2-dichloroethylene:(9-1)Assuch,thesimplestreversiblereactionisoftheform

(9-2)andisfirstorderineachdirection.Thedifferentialequationforthismechanismis(9-3)(9-4)9.1.2ReversiblereactionsIfitisassumedthatbothA1andA2arepresentinthesystemattimet=0,thatis,[A1]=[A1]0and[A2]=[A2]0,thenatanytimeafterwordsthetotalamountofreactantremainingandnewproductformedmustequaltheinitialamountofthereactantsbeforereactions.Hence,

[A1]0+[A2]0=[A1]+[A2](9-5)Solvingfor[A2],weobtain

[A2]

=[A1]0+[A2]0-[A1](9-6)andsubstitutingthisintoequation(9-3)yields(9-7)IfitisassumedthatbothA1Tofindthesolutionofequation(9-7)weintroduceavariablem,definedas(9-8)Thisallowsustorewriteequation(9-7)as(9-9)whichwemaythenintegrate(9-10)Thesolutionis(9-11)TofindthesolutionofequatIfonly[A1]ispresentinthesysteminitially,att=0,thenthesolutionreducesto(9-12)whichisjust(9-13)Usingthemassconservationconstraint[A1]0=[A1]+[A2],wecanobtainthesolutionfor[A2]:

(9-14)Whenequilibriumisreached,theindividualreactionsmustbebalanced;inotherwords,thereactionABmustoccurjustasfrequentlyasthereversereaction.Theforwardandreversereactionsoccuratthesamerate.Ifonly[A1]ispresentinthConsequently,forreaction(9-2)atequilibrium,(9-15)and(9-16)andwehavethefollowingdefinitionoftheequilibriumconstantKeqexpressedintermsofrateconstants:

(9-17)Thesameargumentcanbeextendedtoareversiblereactionthatoccursinmultiplestages.

Consequently,forreaction(First-orderreversiblereactionsinvolvingtwostepsReversiblereactionsmaybedistinguishedbynumberofstagesandthenumberofinitialreactantsinvolvedinthereaction.Here,weconsiderthecompletederivationforfirstorderreversiblereactionsinvolvingonlytwostages,thatis,(9-18)Thekineticequationsforthesystemare

(9-19)(9-20)(9-21)First-orderreversibleWeassumethatatt=0,[A1]=[A1]0,and[A2]0=[A3]0=0andthattheamountsofA1,A2andA3whichhavereactedatalatertimesatisfytheequation:

[A1]0=[A1]+[A2]+[A3](9-22)Bytheprincipleofdetailedbalance,wehaveand(9-23,24)Usingequations(9-22)-(9-24)gives(9-25)So(9-26)Weassumethatatt=0,[A1]=[ASubstitutingequation(9-26)intoequation(9-19)weobtainthenewfirst-orderdifferentialequaion:

(9-27)Solvingthiseuationusingstandardmethods,wehave(9-28)Substitutingequation(9-22),(9-26)and(9-28)intothisexpression,wehave,uponsimplification,(9-29)