Graph Theoretical Solutions for the Coupled Kinetic Rate Equations

Karmakar, S.; Mandal, Bholanath

doi:10.1021/jp4109865

Cited by 9 publications

(29 citation statements)

References 18 publications

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“…The concentration vector polynomials ( C r (λ j ); r = 1, 2, 3,…, n ) for the chemical species obtained by the path deletion procedure developed by Kassman 29,35 have their respective values for each of the decay constants (λ j ). These polynomials have resemblances to the eigenvector polynomials 7,9,29,35 in graph theoretical molecular orbital theory 28–35 where the eigenvectors are obtained by substituting the variables in the polynomials by the eigenvalues. The concentration vector (i.e., the concentrations of the species involved in the reaction arranged in a column matrix form) is obtained as a linear combination 4,7–9 of the exponential functions (

e^{λ_{j} t}

) of the time product of decay constant multiplied by concentration vector polynomial C r (λ j ) of the concerned decay constant 7–9 as

(\begin{matrix} [normalI] \\ [normalA] \\ [A_{1}] \\ [A_{- 1}] \\ [A_{2}] \\ [A_{- 2}] \\ . \\ . \\ . \\ [A_{n}] \\ [A_{- n}] \end{matrix}) = \sum_{j = 1}^{n + 2} c_{j} ((), \begin{matrix} C_{1} (λ_{j}) \\ C_{2} (λ_{j}) \\ C_{3} (λ_{j}) \\ C_{4} (λ_{j}) \\ C_{5} (λ_{j}) \\ C_{6} (λ_{j}) \\ . \\ . \\ . \\ C_{n} \end{matrix})

…”

Section: Methodsmentioning

confidence: 99%

“…Following the above rules, the RKG for this RS is constructed and is shown in Figure 1. With the use of standard graph theoretical procedure, the characteristic polynomial (CP) 7–9,28–30 of the RKG is obtained and that on equating to zero gives the eigenvalues, 7–9,28–30,31–34 commonly known as the decay constants 7–9 for the RS. The concentration vector polynomials ( C r (λ j ); r = 1, 2, 3,…, n ) for the chemical species obtained by the path deletion procedure developed by Kassman 29,35 have their respective values for each of the decay constants (λ j ).…”

Section: Methodsmentioning

confidence: 99%

“…Equating this Equation (4) to zero, we obtain the roots of the CP of the graph, that is, the eigenvalues or the decay constants 7–9 of the corresponding reaction scheme (RS 1) as

λ = 0, 0, - k_{0}^{'}, - ({k^{'}}_{1} + {k^{'}}_{- 1}) .

…”

Section: Methodsmentioning

confidence: 99%

“…As it is well known that a reaction kinetic graph (RKG) 7,8 can be constructed from a reaction scheme (RS) where the vertices are the different chemical species (reactant(s), active center(s), intermediate(s), by‐product(s), product(s), etc.) and the edges are the different elementary steps of conversion in the chemical reaction network 7–10 . Such an RKG is generally a weighted as well as a directed graph.…”

Section: Introductionmentioning

confidence: 99%

“…Very recently, graph theory has been used 7–9 in solving coupled kinetic rate equations and explored several consequences thereof: like the finding conditions for monotonic or oscillatory pathways for time evolution of concentrations of the species, the condition of microscopic reversibility for a cyclic reaction involving three reversible paths, 8,9 etc. Simple algebraic solutions to the kinetic problems of triangle, quadrangle, and pentangle reactions have also been recently explored 25 …”

Section: Introductionmentioning

confidence: 99%

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Graph theoretical exploration for the solutions of the kinetics rate equations of nonchain complex reaction networks

Mondal

Mandal

2021

Int J of Chemical Kinetics

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Nonchain complex reactions involving active center, intermediate(s), by‐products, and the final product(s) are explored with the use of graph theory for solving their rate equations. Three such reaction schemes in a varying number of intermediates are solved for the concentrations of the species involved at any instant of time (t). Time derivatives of these solutions result in the reaction rates of the species concerned. The time functions of rates and concentrations for three successive reaction schemes are then used to generalize the respective solutions for reaction scheme having n intermediates. The rate equations are approximated in three different time regions and are found that: (i) at time t = 0, the rate of reaction of each of the intermediate(s), by‐products, and final product is zero; (ii) at t < < < τ (average life‐time of the active center and intermediate(s)), the reaction rate of an mth intermediate, an mth by‐product, or the final product bears proportionality relationship with tm; and (iii) at t > > > τ, the reaction rate of each of the intermediates is zero whereas that for each by‐product or final product is constant and is expressed from an initial rate multiplied by the product of the probabilities of reaction of the intermediate(s) concerned.

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e^{λ_{j} t}

) of the time product of decay constant multiplied by concentration vector polynomial C r (λ j ) of the concerned decay constant 7–9 as

(\begin{matrix} [normalI] \\ [normalA] \\ [A_{1}] \\ [A_{- 1}] \\ [A_{2}] \\ [A_{- 2}] \\ . \\ . \\ . \\ [A_{n}] \\ [A_{- n}] \end{matrix}) = \sum_{j = 1}^{n + 2} c_{j} ((), \begin{matrix} C_{1} (λ_{j}) \\ C_{2} (λ_{j}) \\ C_{3} (λ_{j}) \\ C_{4} (λ_{j}) \\ C_{5} (λ_{j}) \\ C_{6} (λ_{j}) \\ . \\ . \\ . \\ C_{n} \end{matrix})

…”

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

“…Equating this Equation (4) to zero, we obtain the roots of the CP of the graph, that is, the eigenvalues or the decay constants 7–9 of the corresponding reaction scheme (RS 1) as

λ = 0, 0, - k_{0}^{'}, - ({k^{'}}_{1} + {k^{'}}_{- 1}) .

…”

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Graph theoretical exploration for the solutions of the kinetics rate equations of nonchain complex reaction networks

Mondal

Mandal

2021

Int J of Chemical Kinetics

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Reaction kinetic graphs of chain reactions: Solutions for their rate equations

Mondal

Mandal

2021

Chemical Physics Letters

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Graph Theoretical Analysis on the Kinetic Rate Equations of Linear Chain and Cyclic Reaction Networks

Karmakar

Mandal

2014

J. Phys. Chem. A

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Graph theoretical solutions for kinetic rate equations of some reaction networks involving linear chains and cycles have been derived; condensation polymerization and long chain of radioactive decay come under the purview of the former whereas the interconversion of the species in cycles under the later. The reactions for the linear chains considered here proceed monotonically to the steady states with time whereas the cycle with all irreversible steps has been found to have either periodic or monotonic time evaluation of concentrations depending on the values of rate constants of the involved paths. In case of a cyclic reaction having all reversible paths, the condition for the microscopic reversibility has been derived on the basis of the assumption that the decay constants obtained for this case are all real.

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Graph Theoretical Solutions for the Coupled Kinetic Rate Equations

Abstract: A graph theoretical procedure for solving multistep coupled kinetic rate equations and thereby obtaining the concentrations of the species involved in the reaction has been developed. The method so developed has been illustrated with some well-known reaction schemes.

Cited by 9 publications

References 18 publications

Graph theoretical exploration for the solutions of the kinetics rate equations of nonchain complex reaction networks

Graph theoretical exploration for the solutions of the kinetics rate equations of nonchain complex reaction networks

Reaction kinetic graphs of chain reactions: Solutions for their rate equations

Graph Theoretical Analysis on the Kinetic Rate Equations of Linear Chain and Cyclic Reaction Networks

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