中山大学 第2章 均相反应动力学基础

第2章均相反应动力学基础均相反应的基本概念（1）均相反应：参与反应的各物质处于同一相内均相反应动力学：研究各种因素如温度、压力、催化剂、反应物组成等对反应速率、反应产物分布的影响，并确定表达这些影响因素与反应速率之间定量关系的速率方程。

均相反应动力学是解决工业均相反应器选型、操作与设计计算的理论基础。反应速率表示方法

TheRateExpressions均相反应速率定义：单位时间内，单位体积反应混合物中某一组分i的反应量（或生成量）。TheminussignmeansdisappearingSupposeasingle-phasereactionAPThemostusefulmeasureofrateforAisthen

TheRateExpressions例如，有一简单反应A

PCaution1&2(-rA)为一整体符号，恒为正值Supposeasingle-phasereactionaA+bBpP+sSTheratesofallmaterialsarerelatedbyCaution3AccordingtotherateequationCaution4OrFortheidealgaseswhereInaconstant-volumesystemthemeasureofreactionrateofcomponentibecomesCaution5TherateofreactioninitsvariousformsisdefinedasfollowsBasedonunitvolumeofreactionfluidBasedonunitmassofsolidinfluid-solidsystemsBasedonunitinterfacialsurfaceintwo-fluidsystemsorbasedonunitsurfaceofsolidingas-solidsystemsBasedonunitvolumeofsolidingas-solidsystemsBasedonunitvolumeofreactorInhomogenoussystems,thevolumeoffluidinthereactorisoftenidenticaltothevolumeofreactor.InsuchcaseVandVRareidenticalandareusedinterchangeable.Inheterogeneoussystemsallabovedefinitionsofreactionrateencountered,thedefinitionusedinanyparticularsituationoftenbeingamatterofconvenience.Theseintensivedefinitionsofreactionratearerelatedby

W——固体（催化剂）的重量S——相界面面积VP——固相（催化剂）占体积；V——液相占体积；VR­­——反应器的有效体积，VR＝VP＋V。Theconversion

转化率SupposethatNA0isinitialamountofAinreactorattimet=0,andthatNAtheamountpresentattimet.thentheconversionofAintheconstantvolumesystemisgivenby微分上式fortheconstantvolumesystem

式中CA0——A的初始浓度。KineticsofHomogeneousReaction均相反应：反应物系中，所有反应物及生成物（包括催化剂在内）都处于同一相中。影响反应速率的参数：浓度、温度、催化剂等，因此，反应速率与上述这些参数成函数关系。SupposeahomogeneousirreversiblereactionTherateofprogressofthereactioncanbeapproximatedbyanexpressionofthefollowingtypen=α+β（n为总反应级数）Whereα,βarenotnecessarilyrelatedtothestoichiometriccoefficients.Wecallthepowerstowhichtheconcentrationsareraisedtheorderofthereaction.Thus,thereactionis

αthorderwithrespecttoA

βthorderwithrespecttoBnthorderoverall反应的级数：动力学方程中浓度项的幂数，（如，），由实验确定的常数。基元反应级数，等于计量系数值，=a，=b非基元反应，应通过实验来确定级数的大小反映了该物料浓度对反应速率影响的程度。级数越高，表明该物料浓度的变化对反应速率的影响越显著，如是负值，表明抑制反应，使反应速率下降。反应速率常数k

RateConstantkWhentherateexpressionforahomogenouschemicalreactioniswrittenintheforminbook,thedimensionsoftherateconstantkfornth-orderreactionare[time]-1[concentration]1-nwhichforafirst-orderreactionbecomessimply(time)-1反应速率常数值的大小决定了反应速率的高低和反应进行的难易程度不同反应有不同的速率常数，对于同一反应，k随温度、浓度和催化剂的变化而变化。其中，温度是影响反应速率的主要因素之一。Arrhenius’LawFormanyreactions,andparticularlyelementaryreaction,therateexpressioncanbewrittenasaproductofatemperature-dependenttermandacomposition-dependentterm,or(-rA)=f1(temperature)·f2(composition)=k·f2(composition)Forsuchreactionthetemperature-dependentterm,thereactionrateconstant,hasbeenfoundinpracticallyallcasestobewellrepresentedbyArrhenius’Law:k=k0e-E/RTWhere:k0frequencyorpre-exponentialfactorEActivationenergyAttwodifferenttemperatures,Arrhenius’LawindicatesthatActivationenergyandtemperaturedependencyFromArrhenius’Lawaplotoflnkvs1/Tgivesastraightline,withlargeslopeforlargeEandsmallslopeforsmallEReactionswithhighactivationenergiesareverytemperature-sensitive;reactionwithlowenergiesarerelativelytemperature-insensitive.Anygivenreactionismuchtemperature-sensitiveatalowtemperaturethanatahightemperature.FromtheArrhenius’Law,thevalueoffrequencyfactork0doesnotaffectthetemperaturesensitive.等温恒容过程反应动力学方程的建立

实验得到不同反应时间的各物料浓度数据或者测定压力、密度、折射率等换算成各物料的浓度数据处理，有积分法和微分法单一反应TheSingleReactions

一级反应(不可逆单分子一级反应）

IrreversibleUnimolecular-TypeFirst-OrderReactionConsiderthereactionA

Patconstant-volumeandconstant-temperatureprocesses,thefirst-orderrateequationisForthisreaction.SeparatingandintegratingweobtainxA=(CA0-CA)/CA0Aplotofln(1/1-XA)orln(CA0/CA)vs.t,asshownontheright,givesastraightlinethroughtheoriginforthisformofrateofequation.二级反应

IrreversibleBimolecular-TypeSecond-OrderReactionConsiderthereactionA+APwithcorrespondingrateequationWhichonintegrationyieldsConsiderthereactionA+BP.Thedefinitionsecond-orderdifferentialequationbecomesNotingthattheamountsofAandBthathavereactedatanytimetareequalandgivenbyCA0xA,wemaywriteaboveequationintermsofxAasLettingM=CB0/CA0betheinitialmolarratioofreaction,weobtain

WhichonseparationandformalintegrationbecomesAfterbreakdownintopartialfractions,integrationandrearrangement,thefinalresultinanumberofdifferentformisThefiguresshowtwoequivalentwaysofobtainingalinearplotbetweentheconcentrationfunctionandtimeforthissecond-orderratelaw.n级反应

Irreversiblenth-orderReaction

ConsiderthereactionnAP.Whenthemechanismofreactionisnotknown,weoftenattempttofitthedatawithnth-orderrateequationoftheformWhichonseparationandintegrationyieldsTheorderncannotbefoundexplicitlyfromtheequation,soatrial-anderrorsolutionmustbemade.Thisisnottoodifficult,however.Justselectavaluefornandcalculatek.Thevalueofnwhichminimizesthevariationinkisthedesiredvalueofn.气相反应A

3P为一级反应，速度常数k=0.5min-1，反应在恒容间歇式反应器中进行，求1min后体系的总压，进料状况如下：a)纯A，0.1Mpa；b)纯A，1Mpa；c)10%的A和90%的I（惰性气体）混合物，1MPa解：对比（1）、（2），得

当t=0，

=

0积分（3），得

等温恒容不可逆反应的速率方程积分式可逆反应

ReversibleReaction可逆反应是指正方向、逆方向同时以显著速度进行的反应，也叫对峙反应。First-orderReversiblereactionLetusconsidertheopposedunimolecular-typedreactionKC=K=k1/k2equilibriumconstantStartingwithaconcentrationratioM=CR0/CA0therateequationisNowatequilibriumdCA/dt=0.HencewefindtheequilibriumconstantofAatequilibriumconditionstobeCombiningtheaboveequationsweobtain,intermsoftheequilibriumconversionThismaybelookedonaspseudofirst-orderirreversiblereactionwhichonintegrationgivesAplotof–ln(1-xA/xAe)vs.t,asshownintheFig.,givesastraightline二级可逆反应

Second-orderReversibleReaction

Forthebimolecular-typesecond-orderreactionWiththerestrictionthatCA0=CB0andCR0=CS0,theintegratedrateequationforAandBisasfollowsAplotcanthenbeusedtotesttheadequacyofthiskinetics等温恒容可逆反应速率方程积分式

（产物起始浓度为0）动力学的实验和数据处理

KineticsExperimentanditsData

AnalysisArateequationcharacterizestheratereaction,anditsformmayeitherbesuggestedbytheoreticalconsiderationorsimplytheresultofanempiricalcurve-fittingprocedure.Inanycase,thevalueoftheconstantsoftheequationcanonlybefoundbyexperiment;predictivemethodsareinadequateatpresent.Thedeterminationoftherateequationisusuallyatwo-stepprocedure;firsttheconcentrationdependencyisfoundatfixedtemperatureandthenthetemperaturedependenceoftherateconstantsisfound,yieldingthecompleterateequation.Equipmentbywhichempiricalinformationisobtainedcanbedividedintotwotypes,thebatchandflowreactors.Thebatchreactor（釜式反应器或间歇式反应器）issimplyacontainertoholdthecontentswhiletheyreact.Theexperimentalbatchreactorisusuallyoperatedisothermallyandatconstantvolumebecauseitiseasytointerprettheresultsofsuchruns.Thisreactorisarelativelysimpledeviceadaptabletosmall-scalelaboratoryset-ups（装置）,anditneedsbutlittleauxiliary（辅助设施）equipmentorinstrumentation.Thus,itisusedwheneverpossibleforobtaininghomogenouskineticdata.Theflowreactor(连续式反应器)isusedprimarilyinthestudyofthekineticsofheterogeneous(非均相)reactions.Planningofexperimentsandinterpretationofdataobtainedinflowreactorsareconsideredinlaterchapters.Therearetwoproceduresforanalyzingkineticdata,theintegralandthedifferentialmethods(积分法和微分法).Intheintegralmethodofanalysisweguessaparticularformofrateequationand,afterappropriateintegrationandmathematicalmanipulation,predictthattheplotofacertainconcentrationfunctionversustimeshouldyieldastraightline.Thedataareplotted,andifareasonablygoodstraightlineisobtained,thentherateequationissaidtosatisfactorilyfitthedata用积分法分析实验数据

Analyzingtheexperimentdatabyintegralmethod

Half-timet1/2method半衰期法Forasinglenth-orderreactionIntegratingforn1givesDefiningthehalf-timeofreaction,t1/2,asthetimeneededfortheconcentrationofreactantstodroptoone-halftheoriginalvalue,weobtainThisexpressionshowsthataplotoflogt1/2vs.logCA0givesastraightlineofslope1-n,asshowninFig2.7.1Thehalf-timemethodrequiresaseriesofruns,eachatadifferentinitialconcentration,andshowsthatthefractionalconversioninagiventimeriseswithincreasedconcentrationforordersgreaterthanone,dropswithincreasedconcentrationfororderslessthanone,andisindependentofinitialconcentrationforreactionsoffirstorder.用微分法分析实验数据

Analyzingtheexperimentdatabydifferentialmethod

Thedifferentialmethodofanalysisdealsdirectlywiththedifferentialrateequationtobetested,evaluatingalltermsintheequationincludingthederivativedCi/dt,andtestingthegoodnessoffitoftheequationwithexperiment.用微分法分析实验数据

微分法求动力学方程是直接利用某一类动力学方程的微分式，以反应速度对浓度的函数作图，然后与实测的数据相拟合的一种方法。一般也是设法把图形线性化，把实验数据代人。若得出一直线，便认为所假设的动力学方程是正确的。否则，重新选定另一个动力学方程进行猜算，直到得出一条直线为止。

Theprocedureisasfollows.1)PlottheCAvs.tdata,andthenbyeyecarefullydrawasmoothcurvetorepresentthedata.Thiscurvemostlylikelywillnotpassthroughalltheexperimentalpoints/2)Determinetheslopeofthiscurveatsuitablyselectedconcentrationvalues.Theseslopes-dCA/dt=(-rA)aretheratesofreactionatthesecompositions3)Nowsearchforarateexpressiontorepresentthis(-rA)vs.CAdata,eitherbya)pickingandtestingaparticularrateform,(-rA)=kf(CA)b)testingannth-orderform(-rA)=kCAnbytakinglogarithmsoftherateequation在处理实验数据时，最小二乘法特别适用于如下形式的方程式：

k,α,β是待测定的，为此，可对上式取对数：写成：根据最小二乘法法则：

当a0、a1、a2取得最佳值时，有：也就是用分别对a0,a1,a2求偏导数并令其为零，可以得到方程组|：这是一个三元一次方程组，可以用行列式求解复合反应

MultipleReactionSinglereactionrequiresonlyonerateexpressiontodescribeitskineticsbehaviorwhereasmultiplereactionsrequiremorethanonerateexpression.Multiplereactionscanbeconsideredtobecombinationsoftwoprimarytypes:parallelreactionsandseriesreactions.收率：Yield,fractionyield得率：Operationyield选择性：Selectivity一级平行反应

First-orderParallelReactionConsiderthedecompositionofAbyeitheroneoftwopaths:WithcorrespondingrateequationsIntegratingtheequation(1)att=0,CA=CA0，CP=CP0，CS=CS0

Integratingtheequation(2)Integratingtheequation(3)一级连串反应

IrreversibleFirst-orderReactioninseriesWeconsiderconsecutiveunimolecular-typefirst-orderreactionsuchasWhoserateequationsforthethreecomponentsareLetusstartwithaconcentrationCA0ofA,noPandSpresent,andseehowtheconcentrationsofthecomponentschangewithtime.ByintegrationofEq.(1)wefindtheconcentrationofAtobeTofindthechangeconcentrationofP,substitutetheconcentrationofAfromEq.(4)intothedifferentialequationgoverningtherateofchangeR,Eq.(2);thusWhichisafirst-orderlineardifferentialequation(一阶线形微分方程）oftheformBymultiplyingthroughwiththeintegratingfactor(积分因子)R=thesolutionisApplyingthisgeneralproceduretotheintegrationofEq.(5),wefindthattheintegratingfactoris.Theconstantofintegrationisfoundtobe–k1CA0/(k2-k1)fromtheinitialconditionsCR0=0att=0,andthefinalexpressionforthechangingconcentrationofP(6)Notingthatthereisnochangeintotalnumberofmoles,thestoichiometryrelatestheconcentrationsofreactingcomponentsbywhichwithEqs.(4)and(6)givesThus,wehavefoundhowtheconcentrationsofcomponentsofA,P,andSvarywithtime.Ifk2ismuchlargerthank1,Eq.(7)reducestoInotherwords,therateisdeterminedbyk1orthefirststepofthetwo-stepreaction.Ifk1ismuchlargerthank2,thenWhichisafirst-orderreactiongovernedbyk2,theslowerstepinthetwo–stepreaction.Thus,ingeneral,foranynumberofreactionsinseriesitisthesloweststepthathasthegreatestinfluenceontheoverallreactionrate(7)toptAsmaybeexpected,thevalueofk1andk2alsogovernthelocationandmaximumconcentrationofP.ThismaybefoundbydifferentiatingEq.(6)andsettingdCP/dt=0.ThetimeatwhichthemaximumconcentrationofPoccursisthusThemaximumconcentrationofPisfoundbycombiningEqs.(6)andtheabovetogiveIfk1=k2,theconcentrationofPandSshouldbere-foundPage16figure2.4.3自催化反应

AutocatalyticReactionAreactioninwhichoneoftheproductsofreactionactsascatalystiscalledanautocatalyticreaction.ThesimplestsuchreactionisA+PP+PforwhichtherateequationisBecausethetotalnumberofmolesofAandPremainunchangedasAisconsumed,wemaywritethatatanytimeC0=CA+CP=CA0+CP0=constantThus,therateequationbecomesRearrangingandbreakingintopartialfraction,weobtainForanautocatalyticreactioninbatchreactorsomeproductPmustbepresentifthereactionistoproceedatall.StartingwithaverysmallconcentrationofP,weseequalitativelythattheratewillriseasPisformed.Atotherextreme,whenAisjustaboutuseduptheratemustdroptozero.Thisresultisgiveninfig.below,whichshowsthattheratefollowsaparabola,whichamaximumwheretheconcentrationsofAandPareequal.反应前后分子数变化的气相反应

IsothermalGas-phaseReactionwithvarying-molecular变容反应系统：反应前后分子数发生变化，如果过程恒压，则为变容反应系统;恒容变压过程：分于数发生变化的气相反应，如果反应器的容积恒定，其结果使反应系统的总压变化，称之为恒容变压过程。膨胀因子δ和膨胀率εA

膨胀因子δA的意义是反应物A每消耗lmol时，引起整个物系总物质的量的增加或减少值。

aA+bBpP+sSδA的大小只取决于化学计量式本身，与是否存在惰性气体