Corrections to the Law of Mass Action and Properti

arXiv:cond-mat/0503198v1[cond-mat.stat-mech]9Mar2005CorrectionstotheLawofMassActionandPropertiesoftheAsymptotict=∞StateforReversibleDiffusion-LimitedReactionsR.Voituriez,1,∗M.Moreau,1,†andG.Oshanin1,2,3,‡1LaboratoiredePhysiqueTh´eoriquedesLiquides,Universit´eParis6,4PlaceJussieu,75252Paris,France2Max-Planck-Institutf¨urMetallforschung,Heisenbergstr.3,D-70569Stuttgart,Germany3Institutf¨urTheoretischeundAngewandtePhysik,Universit¨atStuttgart,Pfaffenwaldring57,D-70569Stuttgart,Germany(Dated:February2,2008)Onexampleofdiffusion-limitedreversibleA+A⇋Breactionswere-examinetwofundamentalconceptsofclassicalchemicalkinetics-thenotionof”ChemicalEquilibrium”andthe”LawofMassAction”.Weconsiderageneralmodelwithdistance-dependentreactionrates,suchthatanypairofAparticles,performingstandardrandomwalksonsitesofad-dimensionallatticeandbeingatadistanceμapartofeachotherattimemomentt,mayassociateformingaBparticleattheratek+(μ).Inturn,anyrandomlymovingBparticlemayspontaneouslydis-sociateattheratek−(λ)intoageminatepairofAs”born”atadistanceλapartofeachother.WithinaformallyexactapproachbasedonGardiner’sPoissonrep-resentationmethodweshowthattheasymptotict=∞stateattainedbysuchdiffusion-limitedreactionsisgenerallynotatruethermodynamicequilibrium,butratheranon-equilibriumsteady-state,andthattheLawofMassActionisinvalid.Theclassicalconceptsholdonlyincasewhentheratiok+(μ)/k−(μ)doesnotdependonμforanyμ.PACSnumbers:05.70.Ln;05.40.-a;05.45.-a;82.20.-w2I.INTRODUCTION”ChemicalEquilibrium”(CE)andthe”LawofMassAction”(LMA)aretwocentralconceptsofclassicalchemicalkinetics(see,e.g.,Refs.[1,2]).Invirtuallyeverytext-bookonefinds,regarding,forinstance,thebehaviorofreversibleassociation/dissociationreactionoftheformA+AK+⇋K−B,(1)whereK+andK−aretheforwardandthebackwardrateconstants,respectively,thattheasymptoticstateachievedinclosedsystemsatt=∞isthestateofChemicalEquilibrium-statewithnonetchangeinactivity,orconcentrationwithtimet.Thermodynamically,theconditionofCEistheconditioninwhichthedrivingforcesofthereactioninEq.(1)(oranyotherreversiblereaction)areequalandopposite.Thisconditionimpliesthatnospontaneouschangeisobservedandthat,accordingtothezerothprincipleofthermodynamics,thenetGibbsfreeenergychangeofamixtureofreactantsandproductsvanishes.Kinetically,theconditionofCEistheconditioninwhichtheratesoftheforward,K+a2∞,andthebackward,K−b∞,reactionsareequalandopposite,suchthata∞andb∞-the”equilibrium”concentrationsofAandBspecies,obeytheLMA:−K+a2∞+K−b∞=0ora2∞b∞=K−K+=Keq,(2)withKeqbeingthe”equilibrium”constant,dependentonlyonthethermodynamicpropertiesofthereactivesystem[1,2].ItisimportanttoremarkthatoncetheCEisachieved,theforwardandbackwardreactionscontinuetorun.Itisjustatequilibrium,sincetheratesareequal,thereisnovisibleormeasurablechangeinthesystem.Inthispaperwere-examinethesetwofundamentalconcepts,servingtodefinethecompo-sitionofreactivemixtures(aswellasgeneraltrendsinreactivesystems),incaseofreversiblereactionswhichinvolvediffusiveparticlesandproducts,i.e.theso-calleddiffusion-limitedreactions[3,4,5,6,7,8].Ourfocalquestionsherearewhetherforsuchdiffusion-limitedreactionstakingplaceinclosedsystemstheCEisalwaysatruethermodynamicequilib-riumstate(TES),andwhethertheLMAinEq.(2)alwaysholds.Weemphasizethatit∗Electronicaddress:[e-mail]†Electronicaddress:[e-mail]‡Electronicaddress:[e-mail]3doesnotmeanthattheprinciplesofthermodynamicsarecontested:accordingtothezerothprinciple,foraclosedsystemtheCEalwaysexistsandisbydefinitionthethermodynamicequilibriumstate.Whatismeantherebytruethermodynamicequilibriumstate,ismuchmorerestrictive:withintheconventionalpictureitdesignatesastatefullydescribedbythethermodynamical,asopposedtodynamical,quantities,andwhichsatisfiesthedetailedbalanceequilibrium.ThefactthattheCEisnot,insomecasesdescribedbelow,aTESisthereforefullycompatiblewiththelawsofthermodynamics.NotethatinparticulartheLMAisnotalawofthermodynamics,sinceitreliesonanidealgasapproximationwhichisnotnecessarilyalwaysthecase.Weconcentrateonaparticularreactionscheme-thesimpleassociation/dissociationre-actioninEq.(1),butouranalysiscanbereadilygeneralizedforanyothertypeofreversiblereaction.WeconsiderherearathergenerallatticemodelofreactionsinEq.(1),whichwasfirststudiedanalyticallybyZeldovichandOvchinnikov[18].InthismodeltheAparticlesandproductsBperformstandardrandomwalksonsitesofad-dimensionalhypercubiclat-ticeandtheelementaryreactionratesarelong-rangedanddependentontheinstantaneousdistancebetweenanytwoAparticles;thatis,anypairofAsmayassociate(formingaBparticle)atanymomentoftimetatratek+(μ),whereμistheinstantaneousdistanceseparatingthesetwoparticles.Inturn,anyBparticlemayspontaneouslydissociateatratek−(λ)givingbirthtoageminatepairofAsseparatedbyadistanceλ.Inouranaly-siswesupposethatthebimolecularelementaryreactionratek+(μ)andtheunimolecularelementaryreactionratek−(λ)arearbitrary(integrable)functionsofμandλ.Notealsothatwehavechosenthelatticeformulationjustfortheconvenienceofexposition;ananalo-gouscontinuous-spaceformulationcanbereadilyworkedout.Forthismodel