Wei-prater Analysis of Complex Reaction Systems

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The Structure and Analysis of Complex Reaction Systems JAMES WEI AND CHARLES D. PRATER SOCDny Mobil Oil Co., Inc., Research Department, Paulsboro, Neu: Jersey I. Introduction II. Reversible Monomolecular Systems , . . . . . . . . . . . . . . . . . . . . . . .. A. The Rate Equations for Reversible Monomolecular Systems. . . . . . . . . .. B. The Geometry of the System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. C. The Structure of Reversible Monomolecular System
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  TheStructureandAnalysisofComplexReactionSystems JAMESWEI AND CHARLESD.PRATER SOCDny MobilOilCo., Inc., ResearchDepartment,Paulsboro, Neu: JerseyPage I.Introduction204II.ReversibleMonomolecularSystems,........................208A.TheRateEquationsforReversibleMonomolecularSystems...........208B.TheGeometryoftheSystem.......................................213C.TheStructureofReversibleMonomolecularSystems..................243III.TheDeterminationoftheValuesoftheRateConstantsforTypicalReversi-bleMonomolecularSystemsUsingtheCharacteristicDirections........244A.TheTreatmentofExperimentalData...............................244B.ExampleofaThreeComponentSystem:ButeneIsomerizationoverPureAluminaCatalyst247C. An ExampleofaFourComponentSystem257IV.IrreversibleMonomolecularSystems270A.GeometricPropertiesofIrreversibleSystems 270B.ExperimentalProceduresfortheDeterminationofRateConstantsfromCharacteristicDirectionsforIrreversibleSystemsandApplicationstoTypicalExamples................................................285V.MiscellaneousTopicsConcerningMonomolecularSystems................295A.LocationofMaximaandMinimaintheAmountsofVariousSpecies295 B. PerturbationsontheRateConstantMatrix302C.InsensitivityofSingleCurvedReactionPaths to theValuesoftheRateConstants................................................,.309VI.Pseudo-Mass-ActionSystemsinHeterogeneous'Catalysis313A.SomeClassesofHeterogeneousCatalyticReactionSystemswithRateEquationsofthePseudomonomolecularandPseudo-Mass-ActionForm.313 B. SystemswithmorethanaSingleTypeofIndependentCatalyticSite...332C.TheHydrogenation-dehydrogenationofCs-CyclicsoverSupportedPlati-numCatalystasaPseudo-Mass-ActionSystem334QUalitativeFeaturesofGeneralComplexReactionSystems339A.GeneralComments...............................................339 .E. COnstraints.......................................................340.C.TheEquilibriumPointinGeneralComplexReactionSystems343D.LiapounovFunctions344E.IrreversibleThermodynamicsandtheRelationofLiapounovFunctionstotheDirectionoftheReactionPaths349GeneralDiscussionandLiteratureSurvey355 APPENDICES TheOrthogonalCharacteristicSystem.................................364A.TransformationoftheRateConstantMatrix:intoaSymmetricMatrix.364B.TransformationtotheOrthogonalCharacteristicCoordinateSystem....368203  204 .TAMESWEIANDCHARLESD.PRATER C.ProofThattheCharacteristicRootsoftheRateConstantMatrixKNonpositiveRealNumbers.D.TheCalculationoftheInverseMatrix X-L. II.ExplicitSolutionfortheGeneralThreeComponentSystem3i2III.AConvenientMethodforComputingtheCharacteristicVectorsandRootsoftheRateConstantMatrix K. IV. CanonicalForms.V.ListofSymbols.References. I. Introduction Incatalyticandenzymechemistryweoftenencounterhighlysystemsofchemicalreactionsinvolvingseveralchemicalspecies. It is importantpurposeofchemicalkineticsto,exploreandtodescribethetionsbetweentheamountsofthevariou~speciesduringthecourseofreaction,andtorelatetheconcentrationchangestoaminimalnumberconcentrationindependentparametersthatcharacterizethereactiontem.Reactionkineticsprovideanimportantpartoftheofhighlycoupledsystemsand,inaddition,providethemethodforingtheirbehavior.Asiswellknownfrompreviousattempts,thebehaviorevenlinearsystemscontainingasfewasthreereactingspeciesiscomplicatedtomaketheirbasicdynamicbehaviordifficulttoChemicalkineticsalsoplaysabasicroleinthestudyofthecatalyticactivity.Studiesofthecataly~tandreactantsintheaofappreciableover-allreaction,suchasstudiesoftheelectronic TH(U''',p ofcatalyticsolidsoropticalstudiesofadsorbedmolecularspeciescanvidevaluableinformationaboutthesematerials.Inmostcases,kineticdataareultimatelyneededtoestablishtherelationandofanyinformationderivedfromsuchstudiestothecatalyticreactionForexample,aparticularadsorbedspeciesmaybeobservedandbyaspectraltechnique;yetitneednstplayanyessentialroleincatalyticreactionsinceadsorptionisamoregeneralphenomenoncatalyticactivity.Ontheotherhand,kineticsstudiescanprovide'tionaboutthevariation,asafunctionofexperimentalconditions,relativenumberofadsorbedspeciesthatplayabasicroleintheConsequently)suchinformationmaymakeitpossibletoidentify if any,oftheadsorbedspeciesstudiedbytheuseofadirecttechniquearerelevanttothereaction.Asanotherexample,whenaremadeofthesolidstatepropertiesofagivencatalyticsolid,thetionastowhich,ifany,ofthesepropertiesarerelatedtocatalyticmustultimatelybeansweredintermsofconsistencywiththebehaviorofthereactionsystem. ANALYSISOFCOMPLEXREACTIONSYSTEMS 205 Theinformationneededaboutthechemicalkineticsofareactionsystemisbestdeterminedintermsofthestructureofgeneralclassesofsuchsys-tems.Bystructurewemeanqualitativeandquantitativefeaturesthatare common tolargewell-definedclassesofsystems.Fortheclassesofcom-plexreactionsystems to bediscussed in detailinthisarticle,thestructuralapproachleadstotworelatedbutindependentresults.First,descriptivemodelsandanalysesaredevelopedthatcreateasoundbasisforunder-standingthemacroscopicbehaviorofcomplexaswellassimpledynamicsystems.Second,thesedescriptivemodelsandtheproceduresobtainedfromthemleadtoanewandpowerfulmethodfordeterminingtherateparametersfromexperimentaldata.Thestructuralanalysisisbestap-proachedbyageometricalinterpretationofthebehaviorofthereactionsystem.Suchadescriptioncanbereadilyvisualized.Thestructuralapproachwillalsocontributetotheanalysisofthether-modynamiesofnonequilibriumsystems.Itistheaimandpurposeofthermodynamicstodescribestructuralfeaturesofsystems in termsofmacroscopicvariables.Unfortunately,classicalthermodynamics is con-cernedalmostentirelywiththeequilibriumstate;itmakesonlyweakstatementsaboutnonequilibriumsystems.Thenonequilibriumthermo-dynamicsofOnsager (1), Prigogine (2), andothersintroducesadditionalaxiomsintoclassicalthermodynamicsinanattempttoobtainstrongerandmoreusefulstatementsaboutnonequilibriumsystems.Theseaxiomslead,however,toanexpressionforthedrivingforceofchemicalreactionsthatdoesnotagreewithexperienceandthatisonlyapplicable,asanapproximation,tosmalldeparturesfromequilibrium.AwayinwhichthissituationmaybeimprovedisoutlinedinSectionVII.Themajorpartofthisarticlewillbedevotedtoaparticularclassofreac-tionsysterne+narnely,monomolecularsystems.Areactionsystemof (n) molecularspeciesiscalledmonomolecular if thecouplingbetweeneachpairofspecies is byfirstorderreactionsonly.Theselinearsystemsaresatisfactoryrepresentationsformanyrateprocessesovertheentirerangeofreactionandarelinearapproximationsformostsystemsinasufficientlysmallrange.Theyplayaroleinthechemicalkineticsofcomplexsystemssomewhatanalogoustotheroleplayedbytheequationofstateofaperfectgasinclassicalthermodynamics.Consequently)anunderstandingoftheirbehaviorisaprerequisiteforthestudyofmoregeneralsystems.Twosubclassesofmonomolecularsystemswillbediscussed:reversibleirreversiblemonomolecularsystems.Areactionsystemwillbecalledreversiblemonomolecularifthecouplingbetweenspeciesisbyreversibleorderreactionsonly.Atypicalexampleofareversiblemonomolecular IS  206 JAMESWEIANDCHARLESD.PRATER wheretheithmolecularspeciesisdesignated Ai. Areactionsystemwillbecalledirreversiblymonomolecular if someofthespeciesareconnectedtootherspeciesbyfirstorderreactionsthatareirreversible.Thepresenceofãã..completelyirreversiblestepsimpliesaninfinitechangeinfreeenergyandisconsequentlyanidealization.Nevertheless,manyreactionscontain...stepswithasufficientlylargechange :in freeenergysothatirreversibility····isanexcellentapproximationforthemexcept :in theneighborhoodofequilibriumpoint.Thetypeofapproachtobeusedanditsadvantageovertheconventaonaj,approachisillustrated in SectionII,Aby~briefdiscussionoftheproofdeterminingthevalueoftherateconstantsfromexperimentaldatareversiblemonomolecularsystems.Ourdiscussionofmonomolecularsystemswillalsoprovideinformationaboutanimportantclassof nonlinear reactionsystems,weshallcallpseudomonomolecularsystems.Pseudomonomoleculararereactionsystemsinwhichtheratesofchangeofthevariousspeciesgivenbyfirstordermassactionterms,eachmultipliedbythe same tionofcompositionandtime.Forexample,therateequationsforathreecomponentreversiblepseudomonomo,cularsystemare d~l = ¢{_ (021 + 031)al + 8 12a2 + 818 a a} da2. at = ¢{ 02lal- (8 12 + 8 3Z)a2 + 023a aj da a{ I -]-= ¢ Oolal + 8 3Z az- (8 13 + 8 2a)aa M <@ InEq.(2), a, istheamountofthespecies Ai,Bij isthenseuoo-rate-consraforthereactionfromthejthtotheithspeciesandisindependentamountsofthevariousspecies,and ¢ issomeunspecifiedfunctionamountsofthevariousspeciesandtime.Thisconceptmaybegeneralizedtogivepseudo-mass-actionsystems.Thesearedefinedterns in whichallratesofchangeofthevariousspeciesaregivenbyactiontermsof various integralordereachmultipliedbytheofcompositionandtime.Pseudomonomolecularsystemsandpseudo-mass-actionsystemsarisewhenthereactionsystemcontainsquantitiesofintermediatethatarenotdirectlymeasuredandthatconsequently,donot ANALYSISOFCOMPLEXREACTlONSYSTEMS 207explicitlyintherateexpressions.Theseunmeasuredspeciesmayincludeadsorbedspeciesontheactivesitesofasolidcatalyst j hence,heterogeneouscatalyticsystemswilloftenfollowratelawsofthepseudo-mass-actionform.Thischaracteristicofmanyheterogeneouscatalyticsystemsmakesitpossibletosimplifytheirtreatmentbyseparatingtheproblemintotwoparts,eachofwhichcanbeindependentlystudied.ThemassactionpartcanbestudiedasifthesystemwereahomogeneousreactionbetweenthemeasuredspeciesaswillbeshowninSectionII,B,2,iandSectionVI.Hence,contrarytofirstimpressions,theunderstandingandformulationofmassactionkineticsforhighlycoupledsystemsplayanimportantroleintheunderstandingofheterogeneouscatalyticreactions.Someconditionsthatleadtopseudo-mass-actionkineticsinheterogeneouscatalysiswillbediscussedinSectionVI.Thisarticle is designedtoserveamultiplicityofpurposesandunfor-tunatelydoesnotescapetheweaknessesinherentinsuchmultiplicity.Somecommentsonthehandlingandapplicationofthismaterialmayproveusefultothereader.Manyofthosewhomightfindusefulapplicationfortheresultsandmethodspresentedhereinmayhaveonlyalimitedacquaintancewiththelinearalgebraused in thedetailedapplications.Consequently,mostofthislinearalgebraispresentedintermsofthegeometricalconceptsarisingfromthekineticproblem. Thereader,therefore,neednothavespecializedpreparationinlinearalgebraandtheneedtoconsultworksonabstractalgebraisminimized.Detailedexamplesaregiventoprovidepracticeintheuseoftheprocedures.Matrixnotationisusedforthemanip-ulationofthegeometricalinterpretation;thecomputationproceduresforthematrixoperationsarepresentedinfootnoteswheretheyfirstoccurinthetext.ThedevelopmentofthemainideasarepresentedinSections II, IV,A,VI,A,andVII.Thedetailedexamplesarecontained in SectionsIII,lV,E,andVI,Eandarenotnecessaryforthemaindevelopment.Theseexamplesarebuiltaroundthedeterminationofrateconstantsfromexperi-mentaldata.Thisshouldnotbeconsideredtomeanthatthisistheonly,oreventhemostimportant,usethatcanbemadeofthisapproachtoreac-t.ionrateproblems.Thereaderunfamiliarwithlinearalgebrashould,onthefirstreadingofthemaindevelopment,ignorethealgebraicformalismasmuchaspos-sibleandthinkintermsofthegeometricinterpretations.InthisrespectVIisthemosttedioussinceitinvolvesconsiderablealgebraic * Thegeometricalapproach, in termsofthekineticproblem, to linearalgebrashouldthisusefulbranchofmathematicsmoreappealingtotheexperimentalist.Infact,easewithwhichtheresultsandmethodsmaybevisualised in geometricaltermsitanaturalmathematicsfortheexperimentalist.  n + k m2 fl2...- (I' kim) a m.'.. + kmna.. j= (5) 208 JAMESWEIANDCHARLESD.PRATERANALYSISOFCOMPLEXREACTIONSYSTEMS 209 manipulation.Thissectionis,however,ofimportancetotheinvestigator in heterogeneouscatalysis.ThereaderfamiliarwithlinearalgebramayobtainthemainpointsofthedevelopmentfromSectionII,A,Section II,B,2,c,d,e,g,j, SectionH,C,AppendixI,SectionIV,A,SectionVI,A,andSectionVII..Therelationoftheresultsofpreviousinvestigationstotheresultspre-sentedinthisarticleisbestunderstoodagainstabackgroundofthecom-pletepicture. It isforthisreasonthatfewreferencestopreviouswork will begivenbeforeSectionVIII,whichcontainsahistoricalsurveyandadiscussionoftherelationsofpreviousresultstotheresultspresentedinthisarticle.ThestructureofEqs.(4)leadstothegeneralizationforn-componentsystems, n -(I' k j ) al + k 12 a2... + k1ma m j= II.ReversibleMonomolecularSystems dan at n ...-(I' k j n) an j =1 A. THERATEEQUATIONSFORREVERSIBLEMONOMOLECULAR wheretheabsenceofrateconstantsoftheform k ii fromeachsummationterm is signifiedbythenotation r i.e., lJlj=l k ii isthesumoftheratecon-stants le ii forall j from1to n except j = i.Thegeneralsolution(3-6)toasetoflinearfirstorderdifferentialequa-tionssuchasEqs.(5)iswellknown;itis 1. TheGeneralSolution Lettheithspeciesofamonomolecularreactionsystembedesignated Ai andtheamountby ai. Lettherateconstantforthereactionofithspeciestothejthspeciesbe k ji, i.e., Ai~A j; therewillbenoratestantsoftheform k ii. Usingthissystemofnotation,themostthree-componentmonomolecularreactiondystemis k 21· al = C10 + Cne- Altããã + cl(m_l)e-.... -ltããã + cl(n_l)e-.... ~lt a2 = C20 + c21e- Alt... + c2(m_l)e-Am- !... + ~(n_l)e-An_t! A~,~, Aa Therateofchangeoftheamountofeac~peciesinscheme(3)iswhere Cj; and Ai areconstantparametersrelatedtotherateconstants.Proceduresforcalculatingthevaluesoftheconstants (c, A) fromknownvaluesoftherateconstantscanbefoundinmanystandardworksonchemi-calkineticsorordinarydifferentialequations(3-6).Usingthevaluesoftheconstants (c, A) determinedbytheseproceduresthetimecourseofthereaction-thatis,theamount ct; asafunctionoftime-----iseasilycomputed.Buttheinverseprocessofdeterminingtherateconstants le ji fromtheex-perimentallyobservedtimecourseofthereactionhaspresenteddifficulties.TherightsideofthesetofEqs.(4)iswrittensothatthevariousareinnumericalorder+-c-, a2, andthen as. ThenegativetermontheoftheithequationofEqs.(4) is thesumofthereactionratestheithspeciesandtheremainingtermsarethereactionratesofspeciesbacktotheithspecies. 2.DifficultiesinDeterminingtheValuesoftheRateConstantsfromExperimentalData Therateconstants k i; maybedetermineddirectlyfromtherateEqs,bymeasuringthe initial ratesofformationofthevariousspecies Ai Ai. Thedifficultiesencounteredinobtainingtheaccuracy :'~'J L U. inthechemicalanalysesforpointssufficientlyclosetozerotime
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