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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
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book Advanced Chemical Kinetics"
Advanced Chemical Kinetics
- Title
- Advanced Chemical Kinetics
- Author
- Muhammad Akhyar Farrukh
- Editor
- InTech
- Location
- Rijeka
- Date
- 2018
- Language
- English
- License
- CC BY 4.0
- ISBN
- 978-953-51-3816-7
- Size
- 18.0 x 26.0 cm
- Pages
- 226
- Keywords
- Engineering and Technology, Chemistry, Physical Chemistry, Chemical Kinetics
- Categories
- Naturwissenschaften Chemie