Graph Theoretical Analysis on the Kinetic Rate Equations of Linear

Article pubs.acs.org/JPCA

Graph Theoretical Analysis on the Kinetic Rate Equations of Linear Chain and Cyclic Reaction Networks Somnath Karmakar and Bholanath Mandal* Department of Chemistry, The University of Burdwan, Burdwan 713104, India ABSTRACT: 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.

1. INTRODUCTION Graph theory1−7 is an immensely important tool in the scientific community for having of its scope to deal with any problem whether it is an abstract or real in an easy and conceivable way. In chemical kinetics graphs have been found to be used for a long time. Christiansen8 first used plane diagram for the kinetic analysis of straight chain reactions and two years later King et al.9,10 put forward the schematic method for deriving the rate laws for enzyme-catalyzed reactions. Graphical notations that could be cast as directed graphs have been used by Temkin11 to derive kinetic rate equations for steady state chemical reactions. Balaban12,13 introduced “reaction graphs” to represent isomerization reactions for the first time where the vertices and the edges of the graph were used to describe the isomers and ismerization steps respectively and subsequently with his colleagues made important contributions to this field. Two recent review articles14,15 that give a brief outline on reactions graphs may be mentioned in this context. Glass16 developed topological and combinatorial techniques to classify nonlinear chemical kinetics networks and subsequently utilized to study such systems. Mishra et al.17 used a Fortran 77 program to study some interstellar reactions through graph theoretical method and proposed plausible mechanisms for the reactions thereof. Segal et al.18 proposed several models for irreversible inactivation of catalase and obtained kinetic equations under quasi-steady state approximation with the use of graphs. Very recently graphs have been used by the Segal19−21 for the kinetic analysis of multiroute reactions and for the kinetic description of catalytic and chain reactions. Most of the graph theoretical works on chemical kinetics were devoted to the derivation of reaction rates and subsequently the concentrations of the species on the basis of the quasi-steady state approximation. The solutions for the kinetic rate equations of nonstationary reaction networks have immense importance because of their general validity for any instant of time; for instance they could be reduced to stationary © 2014 American Chemical Society

state solutions under long time approximation. Very recently a graph theoretical method has been developed for solving the coupled kinetic rate equations and subsequently been illustrated with some well-known reactions.22 In the present work we have dealt with multisteps coupled nonstationary reaction networks of linear chains and cycles those are very much common in chemistry and physics. Graph theoretical expressions for the decay constants and hence the concentrations of the species in such networks have been derived that are found to be helpful to predict the course of the reaction for a given network along with other consequences thereof.

2. DECAY CONSTANTS AND THE CONCENTRATION VECTORS OF REACTION NETWORKS To solve the kinetic rate equations (i.e., decay constants and hence the concentration vectors) for any reaction network it is required first to represent the reaction network by a graph called reaction network graph; such a graph is generally a weighted graph and is obtained by translating the rate constant matrix of the reaction network into the corresponding diagram (weighted graph) with the help of some graph theoretical rules.22 Thereafter graph theoretical operations are subsequently performed on it to obtain the desired properties. Decay constants are the functions of rate constants of a particular reaction network that indicate how the concentrations of the chemical species involved in the reaction network change with time. The concentration vector whose elements are the concentrations of the chemical species in the reaction is the exponential function of the decay constants, i.e., concentration of each species is found to be expressed as linear combination of the exponentials of the respective decay constant Received: May 13, 2014 Revised: August 6, 2014 Published: August 13, 2014 7672

dx.doi.org/10.1021/jp504722q | J. Phys. Chem. A 2014, 118, 7672−7682

The Journal of Physical Chemistry A

Article

multiplied by time. Networks of linear chains and cycles are studied for the solutions of their respective kinetic rate laws with the consequences therein and are given in the ongoing subsections. 2.1. Linear Chain Reaction Networks. There a large variety of reaction having such features, for example, condensation polymerization reaction,23−25 radioactive decay series,26,27 etc. are well-known. Let us consider a condensation polymerization reaction as shown in Scheme 1 below, where M is a monomer and Aj is a j-mer obtained in the (j − 1)th step of polymerization process. Scheme 1

Figure 1. Reaction kinetic graphs for linear chain reaction network (G1), three member cyclic reaction network with all irreversible steps (G2) and three member cyclic reaction network with all reversible steps (G3).

Comparing eq 3 with eq 1 we have the concentration vector polynomials22,30 As usual the concentration of the monomer is very high in a polymerization process so [M] is assumed to be fixed through out the polymerization process and hence the order of the reaction is assumed to be pseudo-first order with the rate constant in the jth step, kj′ = kj[M]. In a radioactive series26,27 a radioactive element undergoes disintegration until the ultimate product (daughter element) are stable isotopes of lead. In general a radioactive disintegration series can be represented by Scheme 2 as follows.

C1j = λj(λj + k 2′)(λj + k 3′)...(λj + kn′− 1)⎫ ⎪ ⎪ C2j = k1′λj(λj + k 3′)...(λj + kn′− 1) ⎪ ⎪ C3j = k1′k 2′λj(λj + k4′)...(λj + kn′− 1) ⎬ . ⎪ . ⎪ . ⎪ ⎪ Cnj = k1′k 2′k 3′...kn′− 1 ⎭

Scheme 2

and hence the concentration vector ⎛ [A1] ⎞ ⎜ ⎟ ⎜[A 2]⎟ ⎜ . ⎟ ⎜ . ⎟= ⎜ . ⎟ ⎜ ⎟ ⎝[A n]⎠

This scheme results in the same reaction graph as is given in the condensation polymerization reaction,23−25 the only difference is that the rate constant in the jth step is kj instead of k′j . The reaction graph G1 corresponding to the Schemes 1 and 2 can be written down as discussed in ref 22. and is given in Figure 1. The characteristic polynomial (CP)4−6 of the graph G1 can be obtained as P(G1; λ) = λ(λ + k1′)(λ + k 2′)(λ + k 3′)...(λ + kn′− 1) 22,28−30

and equating eq 1 to zero the decay constants

λ = 0, −k1′, −k 2′ , −k 3′ , ..., − kn′− 1 , ...

⎛ C1j ⎞ ⎜ ⎟ ⎜C2j ⎟ n ∑ cj⎜⎜ . ⎟⎟e λjt . j=1 ⎜ . ⎟ ⎜⎜ ⎟⎟ ⎝ Cnj ⎠

⎛ λj(λj + k 2′)(λj + k 3′)...(λj + kn′− 1)⎞ ⎜ ⎟ ⎜ ⎟ ′ ′ ′ λ λ + λ + k k k ( )...( ) n j n−1 1 j j 3 ⎜ ⎟ λjt . = ∑ cj⎜ ⎟e . j=1 ⎜ ⎟ . ⎜ ⎟ ⎜ ⎟ k1′k 2′k 3′...kn′− 1 ⎝ ⎠

(1)

are found to be (2)

Now from the graph G1 shown in Figure 1 the concentration vector polynomials22,30 can be written down using path deleting procedure as ⎫ ⎪ C2j = k1′Pn − 2(G1; λ) ⎪ ⎪ C3j = k1′k 2′Pn − 3(G1; λ)⎪ ⎬ . ⎪ . ⎪ . ⎪ Cnj = k1′k 2′k 3′...kn′− 1 ⎪ ⎭

(4)

22,30

⎛ [A1] ⎞ ⎜ ⎟ ⎜[A 2]⎟ ⎜ . ⎟ ⎜ . ⎟= ⎜ . ⎟ ⎜ ⎟ ⎝[A n]⎠

C1j = Pn − 1(G1; λ)

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⎛ C1j ⎞ ⎜ ⎟ ⎜C2j ⎟ n ⎜ ⎟ c ∑ j⎜ .. ⎟e λjt j=1 ⎜ . ⎟ ⎜⎜ ⎟⎟ ⎝ Cnj ⎠

(5)

n−1 ⎛ ⎞ ⎜ λj(∏ (λj + kl′)) ⎟ ⎜ ⎟ l=2 ⎜ ⎟ n−1 ⎜ ⎟ ⎜ k1′λj ∏ (λj + kl′) ⎟ n ⎟ λt ⎜ l=3 = ∑ cj⎜ ⎟e j n−1 ⎟ j=1 ⎜ ⎜ k1′k 2′λj ∏ (λj + kl′)⎟ l=4 ⎜ ⎟ . ⎜ ⎟ . ⎜ ⎟ ⎜ ⎟ ′ ′ ′ ′ k k k k ... ⎝ ⎠ n−1 1 2 3



Article

Thus, the concentration of p-mer at an instant of time t can be expressed as n

[A p] =

n

n−1

∑ cj(λj + (1 − λj)δnp)(

∏

n

j=1

j=2

(9)

(λj + kl′))e ;

n

l=p+1

(kl′ − k′j − 1))e

−k′j − 1t

n

l=p+1

= − c 2k1′(

∏

(kl′ − k1′))e

; p=1

l=p+1 n p−1 ⎫ [A p] = − ∑ cj(k′j − 1 − (1 + k′j − 1)δnp)(∏ kl′) ⎪ ⎪ l=1 j=1 ⎪ n−1 ⎪ ( ∏ (kl′ − k′j − 1))e−k′j − 1t ⎪ l=p+1 ⎪ ⎬; 2 ≤ p ≤ n p+1 p−1 ⎪ ′ ′ ′ = − ∑ cj(kj − 1 − (1 + kj − 1)δnp)(∏ kl )⎪ ⎪ l=1 j=1 n−1 ⎪ −k′j − 1t ⎪ ( ∏ (kl′ − k′j − 1))e ⎪ l=p+1 ⎭

j=2

l=1 ≠j−1

n

p−1

j=2

n−1

j=1

l=1

(7)

j=1

}

j=2

n−1

∏

(kl′ − k′j − 1))−1(1 − e−k′j − 1t )}

l=1 ≠j−1

Since the decay constants for such networks of linear chains are real and negative, the intermediate(s) would proceed to the respective steady state(s) monotonically resulting steady decrease and increase in concentration(s) of the reactant(s) and product(s) respectively with time. 2.2. Cyclic Reaction Networks. In this section, we deal with two types of such reaction networks involving three species interconverting to one another in a cyclic process. For the first type forward reactions are allowed but the backreactions are forbidden while for the second case the reverse reactions are also allowed. 2.2.1. Three-Member Cyclic Reaction Having No BackReactions. Such a reaction involving interconversion three species, A, B and D, is shown by Scheme 3 below, and the

n−1

c 2k1′(∏ (kl′ − k1′))

l=3

j=2

j − 1t

(12)

l=2

∑ cjk′j− 1(∏ (kl′ − k′j− 1)) = 0;

n

n

∑ cje−k′

22,28,29

= − A 0 ; c 2 = − A 0 /(k1′ ∏ (kl′ − k1′))

p = 2,

l=1

j=1

l=1

l=2

n−1

n−1

= A 0(∏ kl′){∑ (k′j − 1

n−1

n

(11)

n

n−1

c 2k1′(∏ (kl′ − k1′))

l=2

l=1 ≠j−1

l=1

[A n] = (∏ kl′) ∑ cje−k′j − 1t = (∏ kl′){c1 +

n−1

∑ cjk′j − 1(∏ (kl′ − k′j − 1)) = −A 0 ;

p

1≤p≤n−1

n−1

p = 1,

p

[A p] = A 0 ∑ (∏ kl′)( ∏ (kl′ − k′j − 1))−1e−k′j−1t ;

The values of the coefficients, cj s, can be calculated from the initial condition that is by considering [A1] = A0 and [Ap] = 0 for all p ≠ 1 at time t = 0. Keeping this in mind we can evaluate the cj s with 2 ≤ j ≤ n for any given p (2 ≤ p ≤ n− 1) from eq 7 as follows. n

(10)

[A p] = A 0 ∑ ( ∏ (kl′ − k′j − 1))−1e−k′j−1t ; p = 1

n−1 −k1′t

(kl′ − k′j − 1))−1

Thus, putting cj values given in eqs 1 and 10 as may be applicable into eq 7, we have

n−1

∏

∏ l=1 ≠j−1

j=2

Introducing λj = −k′j − 1 in eq 6 we have j=1

n−1

c1 = A 0 ∑ (k′j − 1 (6)

[A p] = − ∑ cjk′j − 1(

j=2

λjt

2≤p≤n

n

n−1

1 ( ∏ (kl′ − k′j − 1))−1 = 0 k′j − 1 l = 1

n−1

∑ cj(λj + (1 − λj)δnp)(∏ kl′)( ∏ l=1

n

≠j−1

p−1

j=1

n

∑ cj = c1 + ∑ cj = c1 − A 0 ∑

(λj + kl′))e λjt ; p = 1

l=p+1

j=1

[A p] =

The coefficient c1 is to be calculated after having all the coefficients as follows

l=3

n−1

+ c3k 2′(∏ (kl′ − k 2′)) = 0 l=3

n−1

−A 0 /(k 2′ − k1′) + c3k 2′(∏ (kl′ − k 2′)) = 0

Scheme 3

l=3

n−1

c3 = A 0 /((k 2′ − k1′)k 2′ ∏ (kl′ − k 2′)) l=3 n−1

= −A 0 /(k 2′ ∏ (kl′ − k 2′)) l=1 ≠2

corresponding reaction kinetic graph (G2) for the reaction is given in Figure 1. The characteristic polynomial4−6,22 of this graph G2 is shown to be given as

Proceeding step by step in this way one may arrive at the general expression of cj (for all j in the range, 2 ≤ j ≤ n) as n−1

cj = −A 0(k′j − 1

∏ l=1 ≠j−1

P(G2 ; λ) = (λ + k1′)(λ + k 2′)(λ + k 3′) − k1′k 2′k 3′

(kl′ − k′j − 1))−1

= λ{λ 2 + λ(k1′ + k 2′ + k 3′) + (k1′k 2′ + k 2′k 3′ + k 3′k1′)} (13)

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with the decay constants,22,28,29

Thus, for such Scheme 3, on insertion of the expressions of the decay constants (λj s) into eq 16, it reduces to the following expression of the concentration vector at any instant of time

λ1 = 0, λ 2 = {− (k1′ + k 2′ + k 3′) −

(k1′ − k 2′ − k 3′)2 − 4k 2′k 3′ }/2 = λ−

λ3 = {− (k1′ + k 2′ + k 3′) +

(k1′ − k 2′ − k 3′)2 − 4k 2′k 3′ }/2 = λ+

⎛(λ− + k 2′)(λ− + k 3′)⎞ ⎛ k 2′k 3′ ⎞ ⎛[A]⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ [B] ⎟ = c1⎜ k1′k 3′ + k 2′k 3′ ⎟ + c 2⎜ k1′(λ− + k 3′) + k 2′k 3′ ⎟e λ−t ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜ ⎟ ⎝[D]⎠ ⎝ k1′k 2′ + k 2′k 3′ ⎠ ⎝ k1′k 2′ + k 3′(λ− + k 2′) ⎠

(14)

Similarly, the concentration vector polynomials22,30 are, C1j = (λj + k 2′)(λj + k 3′) ⎫ ⎪ ⎪ C2j = k1′(λj + k 3′) + k 2′k 3′ ⎬ ⎪ C2j = k1′k 2′ + k 3′(λj + k 2′) ⎪ ⎭

Thus, concentration vector written down as

22,30

⎛ C1j ⎞ ⎛[A]⎞ 3 ⎜ ⎟ ⎜ ⎟ ⎜ [B] ⎟ = ∑ cj⎜C2j ⎟e λjt = ⎜ ⎟ j = 1 ⎜⎜ ⎟⎟ ⎝[D]⎠ ⎝ C 3j ⎠

⎛(λ+ + k 2′)(λ+ + k 3′)⎞ ⎜ ⎟ + c3⎜ k1′(λ+ + k 3′) + k 2′k 3′ ⎟e λ+t ⎜⎜ ⎟⎟ ⎝ k1′k 2′ + k 3′(λ+ + k 2′)⎠

(15)

for such a Scheme can be

Now putting the values of the coefficients (cj s) from eq 19 into eq 20, one can have the expression of the concentration vector for such reaction scheme at any instant of time, say at t. Let us consider few cases: If at the start of reaction i.e., at t = 0, [A] = A0, [B] = 0 and [D] = 0. Then from eq 19, we have

⎛ (λj + k 2′)(λj + k 3′) ⎞ ⎜ ⎟ 3 ⎜ ⎟ λjt ′ ′ ′ ′ + + λ ( ) k k k k ∑ cj⎜ 1 j 3 2 3 e ⎟ j=1 ⎜ ⎟ ′ ′ ′ ′ + λ + ( ) k k k k 3 j 2 ⎠ ⎝ 1 2 (16)

⎫ ⎪ ⎪ ⎪ ⎪ A0 c2 = ⎬ λ−(λ− − λ+) ⎪ ⎪ A0 ⎪ c3 = λ+(λ+ − λ−) ⎪ ⎭

Now the calculation of the concentrations of the species rests on the evaluation the coefficients, cj s that can be accomplished from the initial condition, that is, [A] = A0, [B] = B0 and [D] = D0 at t = 0. Under this condition eq 16 gets reduced to

c1 =

⎛(λ− + k 2′)(λ− + k 3′)⎞ ⎛ k 2′k 3′ ⎞ ⎛ A0⎞ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ B0 ⎟ = c1⎜ k1′k 3′ + k 2′k 3′ ⎟ + c 2⎜ k1′(λ− + k 3′) + k 2′k 3′ ⎟ ⎟⎟ ⎜⎜ ⎜⎜ ⎟⎟ ⎜ ⎟ ⎝ D0 ⎠ ⎝ k1′k 2′ + k 2′k 3′ ⎠ ⎝ k1′k 2′ + k 3′(λ− + k 2′) ⎠ ⎛(λ+ + k 2′)(λ+ + k 3′)⎞ ⎜ ⎟ + c3⎜ k1′(λ+ + k 3′) + k 2′k 3′ ⎟ ⎜⎜ ⎟⎟ ⎝ k1′k 2′ + k 3′(λ+ + k 2′)⎠

(21)

⎞ ⎛ ⎧ k ′k ′ (λ + k 2′)(λ− + k 3′) λ−t ⎟ ⎜ A 0⎨ 2 3 + − e λ−(λ− − λ+) ⎟ ⎜ ⎩ λ+λ− ⎟ ⎜ ⎫ (λ + k 2′)(λ+ + k 3′) λ+t ⎟ ⎜ + + e ⎬ ⎟ ⎜ λ+(λ+ − λ−) ⎭ ⎟ ⎜ (k1′k 3′ + k 2′k 3′) k1′(λ− + k 3′) + k 2′k 3′ λ−t ⎟ ⎛[A]⎞ ⎜ A ⎧ ⎨ + e ⎟ ⎜ ⎟ ⎜ 0⎩ λ+λ− λ−(λ− − λ+) ⎟ ⎜ ⎜ [B] ⎟ = ⎫ ⎟ ⎜ ′ ′ ′ ′ + + λ k ( k ) k k ⎜ ⎟ 1 + 3 2 3 λ+t ⎬ + e ⎟ ⎜ ⎝[D]⎠ λ+(λ+ − λ−) ⎭ ⎟ ⎜ ⎟ ⎜ k1′k 2′ + k 3′(λ− + k 2′) λ−t ⎟ ⎜ ⎧ (k1′k 2′ + k 2′k 3′) + e ⎟ ⎜ A 0⎨ λ+λ− λ−(λ− − λ+) ⎟ ⎜ ⎩ ⎟ ⎜ k1′k 2′ + k 3′(λ+ + k 2′) λ+t ⎫ e ⎬ ⎟ ⎜ + λ+(λ+ − λ−) ⎭ ⎠ ⎝

(17)

c1k 2′k 3′ + c 2(λ− + k 2′)(λ− + k 3′) + c3(λ+ + k 2′)(λ+ + k 3′) = A 0

(18)

Solving eq 18 we have ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ y(λ+ + k 2′ + k 3′) − xk1′ ⎤⎛ yD − zB ⎞⎬ yA 0 − xB0 0 0 ⎪ ⎥⎜ ⎟ c2 = −⎢ yλ−(λ− − λ+) ⎣ yλ−(λ− − λ+) ⎦⎝ yk 3′ − zk1′ ⎠⎪ ⎪ ⎡ y(λ− + k 2′ + k 3′) − xk1′ ⎤⎛ yD − zB ⎞ ⎪ yA 0 − xB0 0 ⎪ ⎥⎜ 0 ⎟ c3 = −⎢ yλ+(λ+ − λ−) ⎣ yλ+(λ+ − λ−) ⎦⎝ yk 3′ − zk1′ ⎠ ⎪ ⎭

A0 λ+λ−

and hence eq 20 takes the form

Thus, the equations relating the coefficients, c1, c2 and c3 are

⎫ ⎪ ⎪ ⎪ c1(k1′k 3′ + k 2′k 3′) + c 2(k1′(λ− + k 3′) + k 2′k 3′) ⎪ ⎬ + c3(k1′(λ+ + k 3′) + k 2′k 3′) = B0 ⎪ ⎪ c1(k1′k 2′ + k 2′k 3′) + c 2(k1′k 2′ + k 3′(λ− + k 2′))⎪ ⎪ + c3(k1′k 2′ + k 3′(λ+ + k 2′)) = D0 ⎭

(20)

(22)

yA 0 + (λ+λ− − x)B0 yλ+λ− y(λ+ + λ− + k 2′ + k 3′) + k1′(λ+λ− − x) ⎛ yD0 − zB0 ⎞ ⎜ ⎟ − yλ+λ− ⎝ yk 3′ − zk1′ ⎠

c1 =

Here, from eq 22, we can see that how do the concentrations of the species, A, B, and D, change with time depend on the nature of the decay constants λ+ and λ−. Here, from eq 13, we can see that both the decay constants, λ+ and λ−, are negative. Now let us analyze the nature of the decay constants λ+ and λ− given in eq 14 whether they are real or imaginary or under what condition(s) they would be so. Equation 14 can be rearranged for λ± as follows.

(19)

where x = k2′ k3′ , y = k1′ k3′ + k2′ k3′ , and z = k1′ k2′ + k2′ k3′ . 7675



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Scheme 4

λ± = {−(k1′ + k 2′ + k 3′) ± (k1′ + k 2′ + k 3′)2 − 4(k1′k 2′ + k 2′k 3′ + k 3′k1′) }/2 = {−(kl′ + km′ + kn′) ±

(kl′ − km′ − kn′)2 − 4km′ kn′ }/2 (23)

where the subscripts l, m, n have the values 1, 2, 3 with the condition that l ≠ m ≠ n. Equation 23 indicates that the reactions corresponding to Scheme 3 would proceed monotonically to the steady states provided (i) (k′l − k′m − k′n)2 ≥ 4k′mk′n or (ii) anyone of the rate constants (kl′, km′ , and kn′) becomes zero since only in that cases the decay constants are real and negative. The requirement of condition (ii) is the change of reaction Scheme 3 to Scheme 1; thus the only condition for such a cyclic system for approaching steady states monotonically is condition (i). Except these two cases reactions for such a Scheme oscillate in between steady states,29,31−33 i.e., the concentrations of A, B, and D would not vary monotonically rather they would oscillate while approaching to the steady states for each of the reactants. Let us consider few cases where such situation may occur: Case 1: If k′l = k′m = k′n, λ0 = 0 ⎛3 + i λ− = − ⎜ ⎝ 2 ⎛3 − i λ+ = −⎜ ⎝ 2

⎫ ⎪ 3⎞ ⎪ ⎟kl′⎪ ⎠ ⎬; i = ⎪ 3⎞ ⎪ ⎟kl′ ⎪ ⎠ ⎭

in Figure 2. Thus, the reaction kinetic graph (G3) for Scheme 4 can be obtained from the superposition of two graphs one with clockwise and other with anticlockwise direction of flow as is illustrated in Figure 2. The characteristic polynomial4−6 of G3 is shown to be given as P(G3; λ) = (λ + k1′ + k −′ 3)[(λ + k 2′ + k −′ 1)(λ + k 3′ + k −′ 2) − k 2′k −′ 2] − k1′k −′ 1(λ + k 3′ + k −′ 2) − k 3′k −′ 3(λ + k 2′ + k −′ 1) − k1′k 2′k 3′ − k −′ 1k −′ 2k −′ 3 = λ[λ 2 + qλ + r ] (24)

where q = (k1′ + k 2′ + k 3′ + k −′ 1 + k −′ 2 + k −′ 3) ⎫ ⎪ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ r = (k1k 2 + k 3k1 + k 2k 3 + k 3k −1 + k1k −2 ⎬ ⎪ + k 2′k −′ 3 + k −′ 1k −′ 2 + k −′ 2k −′ 3 + k −′ 3k −′ 1) ⎪ ⎭

−1

Thus, the decay constants

Case 2: If k′l ≠ k′m ≠ k′n but k′l = k′m = k′n ⎫ ⎪ ⎪ λ− = −(kl′ + i km′ kn′ )⎬; i = ⎪ λ+ = −(kl′ − i km′ kn′ ) ⎪ ⎭

λ1 = 0

λ0 = 0

λ 2 = [−q −

−1

λ3 = [ − q +

Case 3: If k′l = k′m ≠ k′n

(25)

are

⎫ ⎪ ⎪ 2 q − 4r ]/2 = λ− ⎬ ⎪ q2 − 4r ]/2 = λ+ ⎪ ⎭

(26)

Since the rate constants are always positive, then from eq 25 one can see that that q and r will be positive quantities. Again eq 24 shows that λ+ + λ− = −q and λ+λ− = r, and hence, the decay constants (λ+ and λ−) given by eq 26 are negative. The values of λ+ and λ− may be real or imaginary depending on the individual magnitudes of q and r. To illustrate this let us consider the following specific cases: (a) For a reaction with all rate constants having the same value, i.e., one unit each, then λ+ and λ− are found from eqs 24 and 26 to be real, having values −3.0 unit each. (b) For a reaction of having the values of rate constants k′j (j = 1, 2, 3) = 1.0 unit and k′‑j (j = 1, 2, 3) = 0.5 unit, the decay constants, λ+ and λ−, calculated with the use of eqs 24 and 25 are found to be imaginary with values [−4.5 + i(0.75)1/2]/2 and [−4.5 − i(0.75)1/2]/2 respectively. Similarly as discussed in subsection 2.2.1 here the concentration vector polynomials22,30 are,

⎫ λ0 = 0 ⎪ ⎪ λ− = −{(2kl′ + km′ ) + km′ (km′ − 4kl′) }/2 ⎬ ⎪ λ+ = −{(2kl′ + km′ ) − km′ (km′ − 4kl′) }/2 ⎪ ⎭

In this case (a), if km′ /kl′ < 4 ⎫ ⎪ ⎪ λ− = −{(2kl′ + km′ ) + i km′ (km′ − 4kl′) }/2 ⎬; i = ⎪ λ+ = −{(2kl′ + km′ ) − i km′ (km′ − 4kl′) }/2 ⎪ ⎭

22,28,29

λ0 = 0

−1

the decay constants are imaginary, but if (b) km′ /kl′ ≥ 4, the decay constants are all real and negative. 2.2.2. Three-Member Cyclic Reaction Having BackReactions in All Three Steps. Such a reaction34 network involving the interconversion of three reactants, A, B and D, is shown in Scheme 4 below and the reaction kinetic graph (G3) corresponding to such a Scheme is given in Figure 1. The reaction Scheme 4 is the superposition of two reaction Schemes; one is Scheme 3 with flow in clockwise direction and the other is the same but the flow is in anticlockwise direction as shown

C1j = (λj + k 2′ + k −′ 1)(λj + k 3′ + k −′ 2)⎫ ⎪ ⎪ − k 2′k −′ 2 ⎪ ⎬ C2j = k1′(λj + k 3′ + k −′ 2) + k 2′k 3′ ⎪ ⎪ C3j = k1′k 2′ + k 3′(λj + k 2′ + k −′ 1) ⎪ ⎭ 7676

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Figure 2. Illustration of superposition of two reaction kinetic graphs of three member irreversible cycles (one showing flow in a clockwise (G2) and the other showing flow in an anticlockwise direction (G−2)) for obtaining the graph G3.

⎛ P(0)Q (0) − k 2′k −′ 2 ⎞ ⎛ P(λ−)Q (λ−) − k 2′k −′ 2 ⎞ ⎛ A0⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ B0 ⎟ = c1⎜ k1′Q (0) + k 2′k 3′ ⎟ + c 2⎜ k1′Q (λ−) + k 2′k 3′ ⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜ ⎟ ⎝ D0 ⎠ ⎝ k 3′P(0) + k1′k 2′ ⎠ ⎝ k 3′P(λ−) + k1′k 2′ ⎠

Thus, concentration vector22,30 for such a Scheme can be written down as ⎛[A]⎞ ⎜ ⎟ ⎜ [B] ⎟ = ⎜ ⎟ ⎝[D]⎠

⎛ P(λj)Q (λj) − k 2′k −′ 2 ⎞ ⎟ ⎜ ∑ cj⎜⎜ k1′Q (λj) + k 2′k 3′ ⎟⎟e λjt j=1 ⎜ ⎟ ⎝ k 3′P(λj) + k1′k 2′ ⎠ 3

⎛ P(λ+)Q (λ+) − k 2′k −′ 2 ⎞ ⎜ ⎟ + c3⎜ k1′Q (λ+) + k 2′k 3′ ⎟ ⎜⎜ ⎟⎟ ⎝ k 3′P(λ+) + k1′k 2′ ⎠

⎛ P(λ−)Q (λ−) − k 2′k −′ 2 ⎞ ⎛ P(0)Q (0) − k 2′k −′ 2 ⎞ ⎟ ⎜ ⎟ ⎜ = c1⎜ k1′Q (0) + k 2′k 3′ ⎟ + c 2⎜ k1′Q (λ−) + k 2′k 3′ ⎟ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎝ k 3′P(λ−) + k1′k 2′ ⎠ ⎝ k 3′P(0) + k1′k 2′ ⎠ ⎛ P(λ+)Q (λ+) − k 2′k −′ 2 ⎞ ⎟ ⎜ λ−t e + c3⎜ k1′Q (λ+) + k 2′k 3′ ⎟e λ+t ⎟⎟ ⎜⎜ ⎝ k 3′P(λ+) + k1′k 2′ ⎠

(30)

Equation 30 turns to A 0 = c1R1(0) + c 2R1(λ−) + c3R1(λ+) ⎫ ⎪ ⎪ B0 = c1R 2(0) + c 2R 2(λ−) + c3R 2(λ+)⎬ ⎪ D0 = c1R3(0) + c 2R3(λ−) + c3R3(λ+) ⎪ ⎭

(31)

with the Rs having values as follows R1(λj) = (P(λj)Q (λj) − k 2′k −′ 2)⎫ ⎪ ⎪ ′ ′ ′ R 2(λj) = (k1Q (λj) + k 2k 3) ⎬ ⎪ R3(λj) = (k 3′P(λj) + k1′k 2′) ⎪ ⎭

(28)

where P(λj) = (λj + k 2′ + k −′ 1) ⎫ ⎪ ⎬ Q (λj) = (λj + k 3′ + k −′ 2)⎪ ⎭

The solutions of eq 31 have been found by using a little bit of

If the initial concentrations of the reactants are that is, at t = 0, [A] = A0, [B] = B0 and [D] = D0, then eq 28 becomes

c3 =

(32)

(29)

algebra, and the values of cj s are as follows.

⎧ ⎫ ⎪(A 0 R 2(0) − B0 R1(0))(R3(0)R 2(λ −) − R 2(0)R3(λ −))− ⎪ ⎨ ⎬ ⎪ ⎪ ⎩(B0 R3(0) − D0R 2(0))(R 2(0)R1(λ−) − R1(0)R 2(λ−)) ⎭ ⎧ ⎫ ⎪(R 2(0)R1(λ+) − R1(0)R 2(λ+))(R3(0)R 2(λ −) − R 2(0)R3(λ −))− ⎪ ⎨ ⎬ ⎪ ⎪ ⎩(R 2(0)R1(λ−) − R1(0)R 2(λ−))(R3(0)R 2(λ+) − R 2(0)R3(λ+)) ⎭

7677

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c2 =

Article

⎧ ⎫ ⎪(A 0 R 2(0) − B0 R1(0))(R3(0)R 2(λ+) − R 2(0)R3(λ+))− ⎪ ⎨ ⎬ ⎪ ⎪ ⎩(B0 R3(0) − D0R 2(0))(R3(0)R 2(λ+) − R 2(0)R3(λ+)) ⎭ ⎧(R 2(0)R1(λ−) − R1(0)R 2(λ−))(R3(0)R 2(λ+) − R 2(0)R3(λ+))−⎫ ⎪ ⎨ ⎬ ⎪ ⎪ ⎩(R3(0)R 2(λ−) − R 2(0)R3(λ−))(R 2(0)R1(λ+) − R1(0)R 2(λ+)) ⎭ ⎪

⎡ ⎧ ⎫ ⎤ ⎪(A 0 R 2(0) − B0 R1(0))(R3(0)R 2(λ+) − R 2(0)R3(λ+))−⎪ ⎢ ⎨ ⎬ ⎥ R1(λ−)⎪ ⎪ ⎥ ⎢ ⎩(B0 R3(0) − D0R 2(0))(R3(0)R 2(λ+) − R 2(0)R3(λ+)) ⎭ ⎥ ⎢A − ⎧(R 2(0)R1(λ−) − R1(0)R 2(λ−))(R3(0)R 2(λ+) − R 2(0)R3(λ+))−⎪ ⎫⎥ ⎢ 0 ⎪ ⎢ ⎨ ⎬⎥ ⎪ ⎪ ⎢ ⎩(R3(0)R 2(λ−) − R 2(0)R3(λ−))(R 2(0)R1(λ+) − R1(0)R 2(λ+)) ⎭ ⎥ 1 ⎥ c1 = ×⎢ ⎥ R1(0) ⎢ ⎧ ⎫ A R B R R R R R λ λ − − − ( (0) (0))( (0) ( ) (0) ( )) ⎪ ⎪ 0 2 0 1 3 2 − 2 3 − ⎢ ⎥ ⎨ ⎬ R1(λ+)⎪ ⎪ ⎢ ⎩(B0 R3(0) − D0R 2(0))(R 2(0)R1(λ−) − R1(0)R 2(λ−)) ⎭ ⎥ ⎢− ⎥ ⎧(R 2(0)R1(λ+) − R1(0)R 2(λ+))(R3(0)R 2(λ−) − R 2(0)R3(λ−))−⎪ ⎫ ⎥ ⎢ ⎪ ⎢ ⎨ ⎬ ⎥ ⎪ ⎢⎣ ⎪ ⎩(R3(0)R 2(λ+) − R 2(0)R3(λ+))(R 2(0)R1(λ−) − R1(0)R 2(λ−)) ⎭ ⎥⎦

Thus, putting these values of coefficients (cj; j = 1, 2, 3) that depend on the initial concentrations of the reactants and the rate constants given in eq 30, the reaction concentration vector22,30 can be calculated.

c3 =

(35)

If at the starting of the reaction, i.e., at t = 0, [A] = A0, [B] = 0, and [D] = 0,then the expressions of the coefficients (cj; j = 1, 2, 3) reduce to the following.

⎡ ⎤ ⎡ R1(λ−){R3(0)R 2(λ+) − R 2(0)R3(λ+)}−⎤ ⎢ ⎥ ⎢ ⎥ R 2(0) ⎥ ⎢⎣ R1(λ+){R3(0)R 2(λ−) − R 2(0)R3(λ−)} ⎥⎦ A0 ⎢ ⎢1 − ⎥ c1 = ⎧ ⎫⎥ R1(0) ⎢ ⎪(R 2(0)R1(λ −) − R1(0)R 2(λ −))(R3(0)R 2(λ+) − R 2(0)R3(λ+))− ⎪ ⎢ ⎨ ⎬⎥ ⎪ ⎪ ⎢⎣ ⎩(R3(0)R 2(λ−) − R 2(0)R3(λ−))(R 2(0)R1(λ+) − R1(0)R 2(λ+)) ⎭ ⎥⎦

c2 =

(34)

(36)

A 0R 2(0)(R3(0)R 2(λ+) − R 2(0)R3(λ+)) ⎧ ⎫ ⎪(R 2(0)R1(λ −) − R1(0)R 2(λ −))(R3(0)R 2(λ+) − R 2(0)R3(λ+))− ⎪ ⎨ ⎬ ⎪ ⎪ ⎩(R3(0)R 2(λ−) − R 2(0)R3(λ−))(R 2(0)R1(λ+) − R1(0)R 2(λ+)) ⎭

(37)

A 0R 2(0)(R3(0)R 2(λ−) − R 2(0)R3(λ−)) ⎧ ⎫ ⎪(R 2(0)R1(λ+) − R1(0)R 2(λ+))(R3(0)R 2(λ −) − R 2(0)R3(λ −))− ⎪ ⎨ ⎬ ⎪ ⎪ ⎩(R 2(0)R1(λ−) − R1(0)R 2(λ−))(R3(0)R 2(λ+) − R 2(0)R3(λ+)) ⎭

(38)

⎫ ⎪ ⎪ ⎪ ⎪ A0 c2 = ⎬ λ−(λ− − λ+) ⎪ ⎪ A0 ⎪ c3 = λ+(λ+ − λ−) ⎪ ⎭ c1 =

Substituting the values of Rk(λj) obtained from the respective values of P(λj), Q(λj) into the eqs 36 − 38 results in the following values of the coefficients 7678

A0 λ+λ−

(39)



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same rate. Applying this concept to macroscopic systems at local equilibrium leads to the principle of detailed balance.35,36 This principle along with the linear phenomenological relations among the thermodynamic forces and fluxes led Onsager to derive the famous Onsager reciprocity relation37−41 in 1931. The principle of microscopic reversibility35,36 that differentiates equilibrium from the steady state can excellently be depicted by the Schemes 3 and 4. If there exist a steady flow of A, B, and D in the cycle at molecular level (Scheme 3) maintaining the constancy of individual concentrations then all the conditions of equilibrium imposed by the first and second laws of thermodynamics would be satisfied. Onsager rightly pointed out that the Chemists are accustomed to impose one additional restriction for a system to achieve equilibrium requires each individual reaction must balance itself.42 This is the principle of detailed balance warded in a very simpler manner and is a special case of microscopic reversibility.35,36 The system represented by Scheme 3 has no provision to maintain the restriction of detail balancing of individual chemical reaction thus would not be in equilibrium in true sense; whereas the provision for detail balancing results in true equilibrium in Scheme 4. Now let us see how the expressions of decay constants lead us to the condition of microscopic reversibility for the reaction given in Scheme 4. The expressions of the decay constants22,28,29 given in eq 26 is written as

Here it is noticed that the expressions of the coefficients (cj; j = 1, 2, 3) given in eqs 21 and 39 are the same except the values of the decay constants. Thus, under this situation the concentration vector22,30 would look like ⎛ ⎞ ⎛ P(0)Q (0) − k 2′k −′ 2 ⎞ ⎜ ⎟ ⎜ ⎟ A0 ⎜ A 0 ⎜ k ′Q (0) + k ′k ′ ⎟ + ⎟ 2 3 ⎜ λ+λ− ⎜ 1 ⎟ − λ λ λ ( ) ⎟ − − + ⎜ ⎟ ⎜ ⎟ ⎝ k 3′P(0) + k1′k 2′ ⎠ ⎜ ⎟ ⎛ ⎞ ⎟ ⎛[A]⎞ ⎜ ⎜ P(λ−)Q (λ−) − k 2′k −′ 2 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ λ−t ⎟ ⎟ ⎜ [B] ⎟ = ⎜ ⎜ k1′Q (λ−) + k 2′k 3′ ⎟e ⎟ ⎜ ⎟ ⎜ ⎜ k ′P(λ ) + k ′k ′ ⎟ ⎠ 3 − 1 2 ⎝[D]⎠ ⎜ ⎝ ⎟ ⎜ ⎟ ⎛ P(λ+)Q (λ+) − k 2′k −′ 2 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A0 λ t ⎜ k1′Q (λ+) + k 2′k 3′ ⎟e + ⎟ ⎜+ ⎟⎟ ⎜⎜ λ+(λ+ − λ−) ⎜⎜ ⎟⎟ ′ ′ ′ + λ ( ) k P k k ⎝ ⎠ 3 + 1 2 ⎝ ⎠ (40) 35,36

The principle of microscopic reversibility is an important principle in irreversible or nonequilibrium thermodynamics whose sole property is the time reversal invariance of the mechanical equation of motion of individual particles in the system. For a system at equilibrium any molecular process and the reverse of that process will be taking place on average at the

⎫ ⎪ ⎫ ⎪ ⎧−(k1′ + k 2′ + k 3′ + k −′ 1 + k −′ 2 + k −′ 3)± ⎪ ⎬ ⎪ ⎬/2 ⎪ λ± = ⎨ ⎪ (k1′ − k 2′ − k 3′ + k −′ 1 − k −′ 2 + k −′ 3)2 + 4(k 3′ − k −′ 1)(k −′ 3 − k 2′) ⎪ ⎪ ⎭ ⎭ ⎩ λ1 = 0

(41)

that can be rearranged to the expression with subscripts l,m,n having values 1,2,3 such that l ≠ m ≠ n as ⎫ ⎪ ⎧− (kl′ + km′ + kn′ + k −′ l + k −′ m + k −′ n)± ⎫ ⎪ ⎪ ⎪ ⎬; l , m , n → 1, 2, 3 ⎬/2 ⎪ λ± = ⎨ ⎪ (k ′ − k ′ − k ′ + k ′ − k ′ + k ′ )2 − 4(k ′ − k ′ )(k ′ − k ′ ) ⎪ ⎪ ⎩ l m n n m −l −m −n −l −n ⎭ ⎭ λ1 = 0

(42)

This equation can be expressed in general form as ⎫ ⎪ ⎧ 3 ⎪ ⎪ ⎪ λ± = ⎨−∑ (k′j + k −′ j) ⎪ ⎪ j=1 ⎬ ⎩ ⎪ ⎫ 3 ⎪ ⎪ 2 νj ν−j νl νl − 4 νm νm − 4 ± (∑ ( −1) k′j + (− 1) k −′ j) − 4(( −1) kl′ + ( −1) kl′− 4)(( −1) km′ + ( −1) km′ − 4) ⎬/2 ⎪ ⎪ ⎪ j=1 ⎭ ⎭ λ1 = 0

(43)

3

where m can have any values among 1,2,3 except l such that both νl and νm have the same parity i.e. either odd or even parity. Here νl and νl−4 and νm and νm−4 have different parity. For this case the decay constants λ± would be real only when

(∑ ( −1)νj k′j + ( −1)ν−j k −′ j)2 ≥ 4(( −1)νl kl′ + ( −1)νl−4 kl′− 4) j=1

× (( −1)νm km′ + ( −1)νm−4 km′ − 4) 7679



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If k′l = k′l‑4 and/or k′m = k′m‑4, for such a Scheme the decay constants, ⎫ ⎪ ⎪ ⎪ λ± = {∑ ( −k′j ± ( −1)νj k′j) ⎬ j=1 ⎪ 3 ± (∑ −k −′ j ± ( −1)ν−j k −′ j)}/2 ⎪ ⎪ j=1 ⎭

below. The reaction kinetic graph G4 for such a Scheme is given in Figure 3. The characteristic polynomial of this graph has been shown to be given by

λ1 = 0

n

n

3

P(G2 ; λ) =

∏ (λ + k′j) − ∏ k′j j

j

n ⎧ ⎫ 1 n−1 n−2 ⎪λ + (∑ k′j)λ + (∑ k′jkl′)λ n − 3 ⎪ 2! j ≠ l ⎪ ⎪ j n ⎪ ⎪ 1 ⎪ ⎪ ′jkl′km′ )λ n − 4 + ∑ ( k ⎬ = λ⎨ 3! j ≠ l ≠ m ⎪ ⎪ ⎪ ⎪ n 1 ⎪ + ... + ( ∑ k′jkl′km′ ···) ⎪ ⎪ ⎪ (n − 1)! j ≠ l ≠ m ≠··· ⎩ ⎭ n

(44)

are always real and thus the reaction proceeds monotonically to the equilibrium state. Thus, the conditions that keep such a Scheme far from oscillation29,31−33 can be obtained from the ′ and by putting l = 1,2,3, we have k1′ = k‑3 ′, relations kl′ = kl‑4 ′ and k3′ = k‑1 ′ that altogether result in k1′ k2′ k3′ = k‑1 ′ k‑2 ′ k‑3 ′, k2′ = k‑2 the condition for the microscopic reversibility35−42 being the criterion for a Scheme to be devoid of oscillation.29,31−33 A nth Step Cyclic Reaction Having no Direct Feedback Step. Such a reaction is represented by Scheme 5 as shown

(45)

It is obvious for eq 45 that one of the decay constants is zero and the rest of them are difficult to solve for their analytical solutions. However, it is clear from eq 45 that rest of the decay

Figure 3. Illustration of superposition of two n-membered reaction kinetic graphs of n-membered irreversible cycles (one showing flow in clockwise (G4) and the other showing flow in anticlockwise direction (G−4) for obtaining the n-membered reaction kinetic graph G5 corresponding to n-membered reversible cyclic reaction network. 7680



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Scheme 5

in the decay constants for the reaction Scheme 6 when it is equated to zero and hence the concentration vectors.

constants are negative. They may be real or imaginary depending on the values of the rate constants. Let us consider few cases: Case 1 (n is either even or odd): If all the rate constants are same, i.e., k′1 = k′2 = k′3 = ... = k′n = k′j , then the above equation of CP is reduced to

3. CONCLUSION The reaction networks of linear chains and cycles considered in this study have immense importance in physics, chemistry, and their interfaces. The solutions for the kinetic rate equations of such networks have been derived graph theoretically that does not require any integration for having such solutions. There are a large variety of reaction-networks that may come under linear chains among those the long chain of radioactive decay and condensation polymerization have been treated in this study and their concentration vectors have been expressed as a function of time. Under the heading of cyclic networks, the cycle with three irreversible steps and/or three reversible steps have been derived graph theoretically. The solution for the former is found to be applicable to deduce the criteria for the cyclic reactions to have oscillatory behaviors while the solution for the later is found to be used to arrive at the principle of microscopic reversibility, the criterion for the system to be far from oscillation. This method has also been used for a cycle having n steps and thereby several conclusions for such systems have been made. Moreover the solutions obtained so far for a system are very general and can be reduced to any desired state results under proper approximations, e.g., stationary or steady state results can be obtained under long time approximation.

P(G ; λ) = (λ + k′j)n − k′j n

which on equating to zero we obtain the decay constants as follows.

(λ + k′j)n − k′j n = 0 λj = k′j(e 2πij / n − 1); j = 1, 2, 3, ..., n

(46)

■

Case 2 (even n): If k′1 = k′2 = k′3 = ... = k′n/2 = k′ and k(n/2 ′ + 1) = k(n/2 ′ + 2) = k(n/2 ′ + 3)... = ... = k′n = k″ then P(G2; λ) = (λ + k′)n/2 (λ + k″)n/2 − k′n/2k″n/2 and equating the CP to zero we have (λj + k′)(λj + k″) = k′k″e4πij/n, j = 1, 2, ..., n/2, that ultimately results in the decay constants in the following analytical forms

AUTHOR INFORMATION

Corresponding Author

*(B.M.) Telephone: +919332133069. Fax: +91-342-2567938/ 2530452. E-mail: [email protected]. Notes

(k ′ + k ″ ) 1 ± (k′ − k″)2 + 4k′k″e 4πij / n ; 2 2 n j = 1, 2, ..., (47) 2 Case 3 (even n): If there are two types of rate constants and they are in alternant fashion along the cycle, i.e., k′1 = k′3 = k′5 = ... = k′n‑1 = k′ and k′2 = k′4 = k′6 = ... = k′n = k″, then the decay constants are expressed by eq 47 as in case 2. It is clear from values of the decay constants in the above three cases that such reactions will not result stationary reactions rather lead to oscillation since all but few of the decay constants are imaginary. The reaction kinetic graph (G5) of a nth steps cyclic reaction with direct feedback in each step corresponding to Scheme 6

The authors declare no competing financial interest.

λj = −

■

ACKNOWLEDGMENTS The authors are thankful to the reviewers for their valuable comments. The University Grants Commission, New Delhi, is thankfully acknowledged for financial assistance extended through the CAS Program in the Department of Chemistry of the University of Burdwan.

■

REFERENCES

(1) Ore, O. Graphs and Their Uses; The L. W. Singer Company: New York, 1963. (2) Deo, N. Graph Theory with Applications to Engineering and Computer Science; Prentice-Hall of India Pvt. Ltd.: New Delhi, 1997. (3) Harary, F., Ed.: Graph Theory and Theoretical Physics; Academic Press: New York, 1967. (4) Trinajstić, N. Chemical Graph Theory; CRC Press: Boca Raton, FL, 1992. (5) Gutman, I. and Polansky, O. E. Mathematical Concepts in Organic Chemistry; Springer-Verlag: New York, 1986. (6) Dias, J. R. Molecular Orbital Calculations Using Chemical Graph Theory; Springer-Verlag: New York, 1993. (7) Balaban, A. T., Ed. Chemical Applications of Graph Theory; Academic Press: London, 1967. (8) Christiansen, A. J. The Elucidation of Reaction Mechanisms by the Method of Intermediates in Quasi-Stationary Concentrations. Adv. Catal. 1953, 5, 311−353. (9) King, E. I.; Altman, C. A Schematic Method of Deriving the Rate Laws for Enzyme Catalyzed Reactions. J. Phys. Chem. 1956, 60, 1375− 1378. (10) King, E. I. Unusual Kinetic Consequences of Certain Enzyme Catalysis Mechanisms. J. Phys. Chem. 1956, 60, 1378−1381.

Scheme 6

can be obtained from the superposition of the cyclic graph of G4 type one showing flow in clockwise and another in anticlockwise direction as has been illustrated in Figure 3. The CP of G5 can be obtained using graph theoretically that results 7681



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Graph Theoretical Analysis on the Kinetic Rate Equations of Linear

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