Page - 9 - in Advanced Chemical Kinetics

3.Linearalgebraandgraphtheory Inchemical engineering, themathematicalmethodsofgraphtheoryhavefoundwideapplica- tions in complex chemical reactions and in a sequence of uni (ormulti) or parallel reacting events.Agraph is a combinationof nodes (points) and edges (lines) [2],while a cyclic graph involves finite sequencesofedgeswith thesinglenode (fromwhere itbeginsandends). Similarly, related to any combination of reaction, a tree can be defined as a sequence of noncyclicgraphedges. Ina spanning tree, certain intermediatemay formfromother interme- diatesafterasequenceof transformationsbutdoesnotagreetocounteranytworeactionswith thesamestep (e.g., +1and�1)nor tworeactions startedwith the same intermediates (e.g.,�1 and+2,or+1and�3), Spanning trees can be described in terms of “forward” (generated by a sequence of forwarding reactions), “backward” (generated by a sequence of reverse reactions), and “combined” spanning trees (generated by a sequence of both forward and backward reac- tions).Asingle-route,n-stepsNs (edges) reactionmechanismhasNint (intermediates)nodes, suchasNint=Ns=N. The total numbersof spanning trees areN 2 in any reaction,while the forwardNfandbackwardNb spanning treesareNand thenumbersof combinedspanning treesNcare Nc¼NN�2ð Þ¼N2�2N (14) In a chemical reaction, the overall reaction can be found bymultiplying the reactions with certain coefficients, the so-calledHoriuti numbers σ, and then adding the results.While the relationbetweenσandNint is σ:Nint¼0 (15) Horiuti number allowsus to distinguish the short-lived intermediate and long-lived compo- nents, i.e., to eliminate the intermediates using an RREF of the stoichiometric matrix S, the intermediatesmust be listed first, not last. Then the rows inwhich all intermediates vanish provideabasis for theoverall reactions [2]. ThenumbersofkeycomponentsNkcaregivenby theequation Nkc¼Nc�rankMð Þ (16) and thenumber of key components equals the number of key reactions.Also, the number of keycomponents+numberofnonkeyreactions=numberof reactions In Figure 2, their curves represent two different solution curves of their respective reaction routes lyingatdifferentphasespace, i.e., one lies in2Dwhile thesecond lieswithin3D. Now the question arises, if a complex reaction adopts different completion routes before giving the product, then how can one relate (or distinguish) such available routes andwhy theyare important tobemeasured? For this, the reactionrouteNrrof thesystemcanbemeasuredas Complex Reactions and Dynamics http://dx.doi.org/10.5772/intechopen.70502 9