Page - 8 - in Advanced Chemical Kinetics

Incaseof reversible complexchemical reactions, XN i¼1 αSiAi⇔Θ ∓ i ⇔ XN i¼1 βSiAi (8) The concentration of activated complex can thus be obtained by applying the equilibrium conditions, i.e., Θ∓i � �¼ k∓i XN i¼1 αSiAi (9) where∓ refers to theactivatedcomplex. 2.Reactionrate A stoichiometric vector γρ of the reactionmechanism (1) is an n� dimensional vector with coordinatesγρi=βρi�αρi, that is,“gainminus loss” in theρth elementary reaction. Inmatrix form, it takesa form S¼ γρ1; γρ2;…;γρi h i (10) The chemical composition of the substances is given by the molecular matrixM, with the elementmij as a number of atoms of the jth element in the ith component.M is a (Nc�Ne) matrix,whileNc is the number of reacting components lying in themixture consisting ofNe; thechemical elementsandthe lawofconservationofatomssay Mnc¼Vc constant vectorð Þ: (11) The totalnumberofanymolesof ciatomscanbemeasuredbyusing the relation MTnc¼ne (12) Here,MT is the transposedmolecularmatrix andnc andne are the component amount (mol) andtheamountof thechemical elements (mol) in thecolumnvector form. The dynamics of the involved concentration species can bemeasuredwhenwemeasure the rateof formationof theproductsordeformationanddisappearanceof the reactants. Finally, the rateof reactionwill takea form X cð Þ¼ _c¼dc dt ¼SρWρ cð Þ (13) HereWρ(c):Wρ cð Þ¼ kρ Tð Þ Q i c αρi i is thereactionrate functionof theρthstep(i.e., thedifference betweenthe rateof forwardWþρ cð ÞandbackwardW�ρ cð Þ reactions). Advanced Chemical Kinetics8