J. Phys. Chem. A 2011, 115, 507–513
507
Consistency between Kinetics and Thermodynamics: General Scaling Conditions for Reaction Rates of Nonlinear Chemical Systems without Constraints Far from Equilibrium Marcel O. Vlad,†,‡,§ Vlad T. Popa,| and John Ross*,‡ Department of Chemistry, Stanford UniVersity, Stanford California 94305-5080, United States, Institute of Mathematical Statistics and Applied Mathematics, Casa Academiei Romane, Calea 13 Septembrie 13, Bucharest, 050711 Romania, and Institute of Physical Chemistry, Romanian Academy, Splaiul Independentei 202, Bucharest, 060021 Romania ReceiVed: July 27, 2010; ReVised Manuscript ReceiVed: October 8, 2010
We examine the problem of consistency between the kinetic and thermodynamic descriptions of reaction networks. We focus on reaction networks with linearly dependent (but generally kinetically independent) reactions for which only some of the stoichiometric vectors attached to the different reactions are linearly independent. We show that for elementary reactions without constraints preventing the system from approaching equilibrium there are general scaling relations for nonequilibrium rates, one for each linearly dependent reaction. These scaling relations express the ratios of the forward and backward rates of the linearly dependent reactions in terms of products of the ratios of the forward and backward rates of the linearly independent reactions raised to different scaling powers; the scaling powers are elements of the transformation matrix, which relates the linearly dependent stoichiometric vectors to the linearly independent stoichiometric vectors. These relations are valid for any network of elementary reactions without constraints, linear or nonlinear kinetics, far from equilibrium or close to equilibrium. We show that similar scaling relations for the reaction routes exist for networks of nonelementary reactions described by the Horiuti-Temkin theory of reaction routes where the linear dependence of the mechanistic (elementary) reactions is transferred to the overall (route) reactions. However, in this case, the scaling conditions are valid only at the steady state. General relationships between reaction rates of the two levels of description are presented. These relationships are illustrated for a specific complex reaction: radical chlorination of ethylene. 1. Introduction There has been an interest for many decades in the relations of thermodynamic quantities (equilibrium constants) to kinetic quantities (rate coefficients). For single elementary reactions this relation, i.e. the equilibrium constant equals the ratio of the rate coefficient in the forward reaction to that in the reverse reaction, has been known since Boltzmann, Gibbs and Arrhenius. For complex reactions, which consist of many elementary reactions, the pioneering contribution to this subject was made by Wegscheider1 in 1902. This study did not receive much attention for many years because scientific efforts in chemistry and biology were focused on simple, mostly elementary, reactions. The situation changed as the experimental analysis of complex reaction systems was made possible by substantial advances in methods of measuring concentrations of chemical species in time, for many species simultaneously, and by advances in theoretical approaches to complex systems. These advances kindled renewed interest in this subject.2 In this work we examine the consistency between kinetics and thermodynamics.3,4 In section 2 we derive the main result of new scaling conditions for the nonequilibrium rates of a chemical system, which obeys mass action kinetics and evolves without constraints toward equilibrium. In section 3 we show the relation of these scaling conditions to the Wegscheider * To whom correspondence should be addressed. E-mail: john.ross@ stanford.edu. † Deceased, April 2009. ‡ Stanford University. § Casa Academiei Romane. | Romanian Academy.
cyclicity conditions. In section 4 we extend our analysis to systems of nonelementary reactions described by the HoriutiTemkin5-7 theory of reaction routes and find that similar scaling relations exist, but subject to steady state constraints. In section 5 the derived general relations are illustrated for a reasonably simple example: the radical ethylene chlorination.
2. Scaling Conditions for Systems of Elementary Reactions We consider a system of elementary reactions represented in the following general form:
M
∑
u)1
kw+
+ νuw Au h
M
∑ νuw- Au, w ) 1, ..., S
kw- u)1
(1)
+ and νuw are the stoichiometric coefficients of the where νuw forward and backward reactions 1 respectively, u ) 1, ..., M and w ) 1, ..., S are labels referring to the chemical species and to the elementary reactions, respectively and kw ( are rate coefficients. As formulated, eq 1 allows for the presence of autocatalytic steps such as A + X f 2A + Y. We denote by rw( the forward and backward reaction rates of the reactions w ) 1, ..., S and assume the validity of the ideal form of the massaction law
10.1021/jp107020k 2011 American Chemical Society Published on Web 12/23/2010
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rw( ) kw(
M
∏
Vlad et al. Ξ
(
[Au]νuw, w ) 1, ..., S
(2)
u)1
(Ξ+γ) fM )
∑ f(δ) M θδγ, with θδγ ) (ΘΞ;Ζ)δγ ;
δ)1
γ ) 1, ..., Z ) S - Ξ (8) where [Au] is the concentration of the chemical species, Au, u ) 1, ..., M. We assume the reactions 1 to be elementary, so that the formal (stoichiometric) representation contains all information8 for writing down the kinetic equations 2. Although chemically independent, meaning that the occurrence of any step does not influence the occurrence of remaining ones, these equations are not necessarily stoichiometrically independent. In terms of stoichiometry, we can express the chemical reactions 2 in a simplified, reduced form: M
∑ fuwAu ) 0, w ) 1, ..., S
(3)
u)1
where
We have Z ) S - Ξ linearly dependent vectors f(Ξ+1) ... f(S) M M (1) expressed in terms of the Ξ linearly independent vectors fM ... f(Ξ) M . The expansion coefficients θδγ are the elements of a transformation matrix ΘΞΖ that can be determined in different ways; linear algebra packages such as Matlab or Octave have standard procedures for computing ΘΞΖ, related to the problem of singular value decomposition. Analytical methods are also available: for example, a method based on structural characteristics of the stoichiometric matrix,9 and a more recent one4 based on building a Gramian matrix by starting out from the linearly independent (Ξ) stoichiometric vectors f(1) M ... fM . For each elementary reaction w ) 1, ..., S we introduce a reaction extent ξw defined by the differential equations:
dξw ) + fuw ) νuw - νuw , w ) 1, ..., S; u ) 1, ..., M
f(w) M
()
and a reaction affinity
(5)
which describes the stoichiometric properties of that reaction. The reactions 1 and 3 need not be all linearly independent, that is, some of the stoichiometric vectors f(w) M , w ) 1, ..., S can be linear combinations of the remaining others. We denote by Ξ the number of linearly independent stoichiometric vectors. The number Ξ can be calculated by introducing a stoichiometric matrix (1) (S) FM;S ) (f M ...f M ) ) (fuw), w ) 1, ..., S; u ) 1, ..., M
(6) The number Ξ of linearly independent vectors is given by the rank of the stoichiometric matrix FMS:
Ξ ) rankFM;S e min(M, S)
(9)
(4)
are net stoichiometric coefficients. To each of the S reactions 1 and 3, we can attach a column stoichiometric vector,
f1w . ) . , w ) 1, ..., S . fMw
dwnu , u ) 1, ..., M; w ) 1, ..., S fuw
(7)
Through a convenient relabeling of the chemical species and of the reactions, we can always make sure that the first Ξ (Ξ) stoichiometric vectors f(1) M ... fM , w ) 1, ..., Ξ are linearly independent and that the remaining Z ) S - Ξ vectors, ... f(S) f(Ξ+1) M M , w ) Ξ + 1, ..., S are linear combinations of the first ones. We consider the case where Ξ < min (M,S), a condition needed for the linearly dependent vectors to exist, and express ... f(S) the linearly dependent stoichiometric vectors f(Ξ+1) M M as linear (Ξ) combinations of the Ξ linearly independent ones, f(1) M ... fM .
Aw ) -
∂G , w ) 1, ..., S ∂ξw
(10)
where nu, u ) 1, ..., M are the numbers of moles of the different chemical species present in the system, dwnu is the variation of nu due to the occurrence of the wth reaction and G is the Gibbs free energy of the system. For each elementary reaction w there is a relation between the ratio of the forward and backward reaction rates rw+/rw- and the corresponding reaction affinity, Aw10
rw+ /rw- ) exp(Aw /RT), w ) 1, ..., S
(11)
where R is the universal gas constant and T is the absolute temperature of the system. The affinities Aw can be expressed as Aw ) -
( )
M M ∂nu ∂G ∂G ))fuwµu, w ) 1, ..., S ∂ξw ∂ξw ∂nu u)1 u)1
∑
∑
(12)
where µu ) ∂G/∂nu is the chemical potential of the species u. We recall that for an ideal system µu ) µu0 + RT ln[Au] and use the equilibrium state as a reference state. Under these circumstances, eq 12 leads to
( ) M
∏ [Au]feq
uw
Aw ) RT ln
u)1 M
∏ [Au]
) RT ΨMf(w) M
(13)
fuw
u)1
where ΨM is the line vector
ΨM ) [ln([A1]eq /[A1]), ..., ln([AM]eq /[AM])]
(14)
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We split eq 13 into two groups by separating the equations attached to the linearly independent reactions from the equations attached to the linearly dependent reactions:
Aδ ) RT ΨMf(δ) M , δ ) 1, ..., Ξ
(15)
AΞ+γ ) RT ΨMf(Ξ+γ) , γ ) 1, ..., S - Ξ M
(16)
We insert eq 8 into eq 16 and combine the result with eq 15 to obtain Ξ
AΞ+γ ) RT ΨMf(Ξ+γ) ) RT M
∑ ΨMf(δ) M θδγ )
δ)1
Equations 21 are the general form of Wegscheider cyclicity conditions1 for an arbitrary network of elementary reactions; various forms of these conditions were derived by Vlad and Segal in 1979,11 revised in 1984,12 and later by Schuster and Schuster in 198913 based on the explicit condition of detailed balance. However, the above derivation of eq 21 applies only to (ideal) mass action systems and, being based on the general relationship (eq 19), does not contain any explicit assumption concerning detailed balance. 4. Scaling Conditions for Reaction Routes In this section we extend the scaling condition (eq 19) for systems of nonelementary reactions, described by the HoriutiTemkin5-7 theory of reaction routes. We consider a complex chemical process with M stable species and J active intermediates involved in S elementary steps
Ξ
RT
∑ Aδθδγ, γ ) 1, ..., S - Ξ
(17)
that is, the affinities of the linearly dependent reactions are linear combinations of the affinities of the linearly independent reactions. From eqs 11 and 17 it follows that for each of the Z ) S - Ξ linearly dependent reactions we have + rΞ+γ rΞ+γ
( ∑ ) ∏ [ ( )] Ξ
) exp
Ξ
Aδθδγ ) RT δ)1
exp
δ)1
Aδ RT
θδγ
,
γ ) 1, ..., S - Ξ (18)
Now we use eq 11 once again, this time for the linearly independent reactions, and insert eq 11 applied for w ) 1, ..., Ξ into eq 18, resulting in + rΞ+γ rΞ+γ
Ξ
)
∏ δ)1
() rδ+
θδγ
, γ ) 1, ..., S - Ξ
rδ-
M
J
u)1
u)1
kw+
M
J
u)1
u)1
∑ Ruw+ Au + ∑ βuw+ Xu ak ∑ Ruw- Au + ∑ βuw- Xu;
δ)1
w ) 1, ..., S (22) M ( J ( with 3 g ∑u)1 Ruw + ∑u)1 βuw Xu g 1, where Au are stable ( ( g 0, βuw g 0 are their species, Xu are active intermediates, Ruw corresponding stoichiometric coefficients and kw+ and kw- are forward and backward rate coefficients, respectively. At the steady state, where concentrations of both stable and active species are constant, there is no physical (or rather chemical) constraint imposed by the occurrence of any of the above steps on the occurrence of any of the remaining ones. In other words, all steps in eq 22 are physically (chemically) independent. On the other hand, the mechanistic scheme (eq 22) may contain some steps that result from linear combinations of other steps: these are linearly (stoichiometrically) dependent steps, which means that some of the stoichiometric vectors
(19) (w) fM+J )
Eq 19 is the main result of this paper. It is a set of scaling conditions for the nonequilibrium rates of a chemical system obeying mass-action law kinetics which evolves, without constraints, toward equilibrium. It has been obtained based on consistency requirements between chemical kinetics and nonequilibrium thermodynamics. We emphasize that eq 19 was derived without explicit use of the principle of detailed balance. 3. from Scaling Conditions to Wegscheider Relations for Systems of Elementary Reactions
w
( ) a(w) M
b(w) J
, w ) 1, ..., S
(23)
that is, the vectors of the net stoichiometric coefficients, with + a(w) M ) (auw)u)1, ..., M ; auw ) Ruw - Ruw ; w ) 1, ..., S (24)
+ b(w) J ) (buw)u)1, ..., J ; buw ) βuw - βuw ; w ) 1, ..., S
(25)
We insert eq 2 into eq 19 and group the concentrationdependent terms together, resulting in from which, by taking
may be linearly dependent. It follows that the rank Ξ of the stoichiometric matrix, (1) (S) w)1, ..., S FM+J;S ) (fM+J ...fM+J ) ) (fuw)u)1, ..., M+J )
into account that from eq 8 it follows that fu(Ξ+γ) ) ∑δΞ ) 1fuδθδγ, we come to + kΞ+γ kΞ+γ
Ξ
)
∏ δ)1
() kδ+
kδ-
( ) aM;S bJ;S
(26)
obeys the inequality
θδγ
, γ ) 1, ..., S - Ξ
(21)
rankFM+J;S ) Ξ < min(M + J, S)
(27)
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where aM;S and bJ;S are partitions of the stoichiometric matrix FM+J;S corresponding to stable species and active intermediates, respectively. Obviously, eqs 5-7 are similar to 23, 26, and 27. The only difference consists in the evidencing of the active intermediates in the latter set. According to Horiuti,5,6 starting from the mechanistic reaction 22 we can obtain overall reactions (routes) by multiplying each of eq 22 by a stoichiometric number σw, summing over w and requiring the cancellation of the stoichiometric coefficients of the active intermediates, Xu, in the resulting overall reaction. This condition leads to an undetermined homogeneous system of linear equations in the stoichiometric numbers. There are P infinite sets of independent solutions of this homogeneous system that determine the number of reaction routes (overall reactions) necessary for the kinetic description of the system in terms of the stable species Au. Such a set is termed algebraically “a fundamental system of solutions”14 and chemically “a route basis”.7 According to the Horiuti-Nakamura theorem,6 which is a kinetic analogue of Gibbs phase rule, the number P of reaction routes is given by P ) S - I, where I is the number of independent intermediates in the Horiuti sense, that is
I ) rankbJ;S
(28)
where, according to eq 26, (S) w)1, ..., S bJ;S ) (b(1) J ... bJ ) ) (buw)u)1, ..., J
(29)
Equation 33 was advanced by Horiuti and Nakamura6 in 1957 and express net step rates as weighted (by stoichiometric numbers) sums of their occurrence along different routes. As clearly stated later by Temkin,7 these relationships, which he termed “step steady state conditions”, as opposed to “intermediate steady state conditions” (originally advanced by Bodenstein15), are valid only at the steady state, when the reaction rate along a certain route has a physical meaning. Although formulated for net reaction rates, these relationships can be applied for unidirectional rates, as follows:
rw( )
rw( ) kw(
J
∏ [Au] ∏ [Xu] ( Ruw
u)1
( βuw
, w ) 1, ..., S
(30)
u)1
We denote by σ1ς, ..., σSς the stoichiometric numbers attached to the elementary steps 1, ..., S contributing to the reaction route ς, where ς ) 1, ..., P. Each reaction route ς can be represented by an overall stoichiometric reaction:
∑ σwςr˜(ς , w ) 1, ..., S
(34)
ς)1
Equation 34 express the distribution of elementary reactions rates along the basic routes of a complex chemical process on a chosen route basis. In 1979, Temkin7 has thoroughly stressed the fact that the route basis is not unique, as linear combinations of basic routes yield a valid (chemically possible) reaction route. Because reaction rates along different routes are well-defined quantities at the steady state, the rate of the whole process is a vector in the route space, whose components are the route rates. For any complex reaction mechanism, although the choice of basic routes is arbitrary, their number, P, which is the dimension of the route vector space, is a unique parameter of the mechanism. The search of equations similar to eq 19, but pertaining to route rates, starts from a general relationship derived by Temkin7 for single-route reactions comprising S elementary steps:
is the partition of the stoichiometric matrix which corresponds to the active intermediates. The forward and backward rates of the elementary reactions 22 are given by the ideal form of the mass-action law: M
P
r˜+ ) r˜-
S
r+
∏ r-w
(35)
w)1 w
For multiroute systems eq 35 applies for each route; thus each route can be viewed as a complex reaction with a single overall stoichiometric equation, with one “run”7 along a certain route comprising all elementary steps pertaining to the route in question taken σwς (their corresponding stoichiometric numbers) times:
r˜ς+
S
)
r˜ς-
∏ w)1
() rw+
σwς
rw-
, ς ) 1, ..., P
(36)
M
∑ ˜fuςAu ) 0, ς ) 1, ..., P
(31)
u)1
where route stoichiometric coefficients, ˜f uς, are given by S
˜fuς )
∑
(Ruw
-
+ Ruw )σwς
˜ P ) ΛSΣS;P Λ
S
∑ fuwσwς; ς ) 1, ..., P;
)
w)1
w)1
u ) 1, ..., M (32) The reaction rates r˜ς attached to the different routes ς ) 1, ..., P are related to the rates rw, w ) 1, ..., S of the elementary steps 22 through the relations
rw )
rw+
-
rw-
P
)
As written, eq 36 ensure that steps not involved in a certain route are not taken into account as their stoichiometric numbers are zero. Taking the logarithms and making use of vector-matrix notation in eq 36 leads to
∑ σwςr˜ς, w ) 1, ..., S
ς)1
(33)
(37)
which contains the line-vectors
(
˜ P ) ln Λ
r˜+ 1
r˜P+
r˜1
r˜P-
, ..., ln -
) (
; ΛS ) ln
r+ 1
r+ S
r1
rS
, ..., ln -
)
(38)
and the stoichiometric numbers matrix (that will be termed “Horiuti matrix”), the columns of which are route stoichiometric numbers (termed “Horiuti vectors”):
Reaction Rates of Nonlinear Chemical Systems ς)1, ..., P (1) (P) ΣS;P ) (σwς)w)1, ..., S ) (σS , ..., σS )
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(39)
For multiroute systems with linearly dependent mechanistic stoichiometric equations a general theorem16 states that (1) linearly dependent mechanistic equations always result in linearly dependent overall equations and (2) the number of linearly dependent elementary reactions equals the number or linearly dependent overall ones. Writing eq 32 in matrix form we get
( )
F˜ F˜M+J;P ) M;P ) FM+J;SΣS;P 0J;P
˜ Z ) ) ΛS(ΣS;Q|Σ¯ S;Z ) ) ˜ Q|Λ ˜ P ) (Λ Λ ˜ QεQ;Z ) (46) ˜ Q|Λ (ΛSΣS;Q|ΛSΣS;QεQ;Z ) ) (Λ Thus, the Z line vectors, pertaining to logarithms of route rate ratios of linear dependent routes, are linear combinations of the Q similar quantities of linear independent ones:
(40) ln
where the middle term accounts for the condition of intermediates cancellation in the overall stoichiometric equations. If we denote by Q the rank of the overall stoichiometric coefficients matrix,
rankF˜M+J;P ) rankF˜M;P ) Q; Q < P
Taking into account eqs 44 and 45, eq 37 written for the stoichiometric route basis reads
+ r˜Q+γ r˜Q+γ
( )
Q
)
∑
r˜β+
ln
β)1
r˜β-
εβγ, γ ) 1, ..., Z ) P - Q
(47)
or expressed in a form similar to eq 19: + r˜Q+γ
(41)
r˜Q+γ
Q
)
∏ β)1
() r˜β+
εβγ
r˜β-
, γ ) 1, ..., Z ) P - Q ) S - Ξ
(48)
the above-mentioned theorem15 gives
S-Ξ)P-Q)Z
(42)
With 42 taken into account, a convenient labeling of chemical species and reactions affords the following partitioning in 40:
Equation 48 represents the overall (macroscopic) expression of the mechanistic (microscopic) scaling conditions (eq 19). There is the notable difference that eq 48 is only valid for steady state systems. In particular, if we consider a stoichiometric route basis with Z empty routes on the right-hand side of eq 44 and with eq 37 taken into account we get S
∏
w)1
We examine a special case of the Horiuti matrix, that is, one in which the last Z Horiuti vectors result in cancellation of all chemical species, both stable and reactive, in the overall equations. This is possible because the last Z mechanistic stoichiometric vectors are linearly dependent. Through linear combinations of basic Horiuti vectors one can obtain what Temkin termed “a stoichiometric basis of routes”,7 resulting in Q normal overall stoichiometric equations and Z equations of the type 0 ) 0, which Temkin called “empty routes”. This is not an algebraic artifact. An empty route has a clear chemical meaning: it is a cycle, a subset of mechanistic steps coupled in such a way that the start and end species are the same. j S;P be a route basis where we group on the In general, let Σ right-hand side of the Horiuti matrix the Z Horiuti vectors that result in linearly dependent routes, while the first Q vectors yield linearly independent ones:
¯ S;P ) (σS(1), ..., σS(Q) σS(Q+1), ..., σS(P-Q) ) ) (ΣS;Q|Σ¯ S;Z ) Σ (44) These Z vectors can be obtained by linear combinations of the remaining Q vectors, as all possible routes are generated as linear combinations of members of some basic set: Q
σS(Q+γ) )
∑ σS(β)εβγ, γ ) 1, ..., ZSΣ¯ S;Z ) ΣS;QεQ;Z
β)1
(45)
() rw+
σwγ
Q
)
rw-
∏ β)1
() r˜β+
r˜β-
εβγ
, γ ) 1, ..., Z ) P - Q ) S - Ξ
(49)
which links elementary reaction (mechanistic) rates and overall (route) rates conditions for certain cycles (empty routes) within a Horiuti-Temkin complex chemical system at steady state. Via mass action expressions of the elementary steps, such conditions result in Wegscheider-type relationships between rate coefficients, as shown in eq 21. Such steady-state relationships are “weaker” (less restrictive) than equilibrium detailed balance ones. This is similar to the case of equilibrium “complex balancing”, studied by means of a different approach by Horn and Jackson17 and fully developed by Feinberg.18 For a stoichiometric route basis eq 43 becomes
with
0M+J;Z )
(
)
FΞ;ΞΣ¯ Ξ;Z + FΞ;ZΣ¯ Z;Z ) FM+J-Ξ;ΞΣ¯ Ξ;Z + FM+J-Ξ;ZΣ¯ Z;Z FM+J;ΞΣ¯ Ξ;Z + FM+J;ZΣ¯ Z;Z
(51)
j Z;Z is not singular (which can be always From 51, provided that Σ achieved through a proper labeling of species and steps), we get
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¯ ΞZ ;Θ ¯ ΞZ ) -Σ¯ Ξ;ZΣ¯ Z;Z-1 FM+J;Z ) FM+J;ΞΘ
Vlad et al.
(52)
Equation 52 is practically the matrix formulation of eq 8. It offers an expression of the transformation matrix of mechanistic stoichiometric vectors in terms of the Horiuti-Temkin approach, that is, in terms of partitions of the Horiuti matrix corresponding to the empty routes of the stoichiometric route basis. We thus found an alternative method to calculate the exponents in the general relationship 19. 5. An Example: Radical Chlorination of Ethylene The radical addition of chlorine to ethylene is a classical example of a complex reaction involving a reasonable amount of species and steps. The basic mechanistic scheme,19 where the Horiuti vectors are displayed on the right-hand side and their corresponding overall equations are shown at the lower side, reads
where the last stoichiometric vector is the linearly dependent one and the exposed partition is similar with that of the lefthand side of eq 43, while in eq 42 we have Z ) 1. The stoichiometric matrix partitions are
( ()
)
FΞ;Ξ ) F4;4
-1 0 -1 0 0 -1 0 0 ) , (rankF4;4 ) 4); 0 0 1 0 0 0 0 1
FΞ;Z ) F4;1
0 0 ) 1 0
FM+J-Ξ;Ξ ) F2;4 ) FM+J-Ξ;Z ) F2;1 )
( ( )
)
2 -1 1 0 ; 0 1 -1 -2 -1 -1
(56)
As mentioned above, the linearly dependent stoichiometric vector is easily obtained as: It is a mechanism involving four stable chemical species and two active species (chlorine and chloroethyl radicals), connected via five kinetically (chemically) independent simple steps. The term “simple step” or “simple stage” refers, according to Temkin’s terminology,7 to the ensemble of opposed (forward/ reverse) elementary reactions. The appropriate labeling is given in eq 54:
(3) (1) f(5) 6 ) f6 - f6
(57)
In this case, eq 8 reads 4
(
(Ξ+γ) fM ) f(4+1) ) f(5) 6 6 )
∑ f(δ) 6 θδ1 ) F6;4θ4 )
)( ) ( )
δ)1
-1 0 -1 0 0 0 -1 0 0 -1 0 0 0 0 1 0 1 ) 1 0 0 0 1 0 0 2 -1 1 0 -1 0 1 -1 -2 -1
(58)
Obtaining of the θ matrix is straightforward in this case of a “simple complex reaction”, where eq 19 becomes One may easily notice that step (5) may be obtained by algebraically subtracting step (1) from step (3). The mechanistic scheme, eq 53, thus contains one linearly (stoichiometrically) dependent step. The same is true for the overall representation of this complex process, given by its reaction routes. In this case, the linear dependence of overall stoichiometric equations, given in the lower side of eq 53, is even more evident. Routes σ(1) and σ(3) result in the same overall reaction, although along different mechanisms: for σ(1) the main product, 1,2-dichloroethane, is the result of a chain propagation reaction, while for σ(3) it is formed via a termination one. The stoichiometric matrix of the above mechanism reads:
+ r4+1 r4+1
)
r+ 5 r5
4
)
∏ δ)1
( ) ( ) ( )( )( ) ()() rδ+
rδ-
θδ1
)
r+ 1
r1
-1
r+ 2
r2
0
r+ 3
1
r3
r+ 1
r1
r+ 4
0
r4
-1
r+ 3
r3
)
1
(59)
With the application of either mass action rate laws in 59 or the actual value of θ4 in eq 21 we obtain the Wegscheider cyclicity conditions for this complex reaction:
Reaction Rates of Nonlinear Chemical Systems
k+ 5 k5
4
∏
)
δ)1
J. Phys. Chem. A, Vol. 115, No. 4, 2011 513
() ()() kδ+
k+ 1
θδ1
)
kδ-
-1
k1
k+ 3
1
(60)
k3
Finding the empty route for this example is also straightforward. One may readily notice from 53 that σ(e) ) σ(3) - σ(1), where superscript “e” stands for “empty”. Replacing σ(3) by σ(e) in eq 53 we obtain the stoichiometric route basis7 representation of this complex reaction:
The empty route evidenced in eq 61 illustrates some terms extensively used throughout this contribution. “0 ) 0” expresses algebraically the zero overall chemical transformation along the empty route. However, mechanistic chemical motion is present along this route and its forward and reverse rates are preserved: the difference consists in the fact that both forward sense, comprising elementary reactions (1)-(-3)-(5), and the reverse sense, with reactions (-1)-(3)-(-5), both result in zero chemical change of the stable species. In spite of this, the net rate along the empty route is still a valid concept, meaning that chemical motion in the forward sense could be faster than the one pertaining to the reverse sense. The existence of the empty routes is a consequence of the existence of linearly (stoichiometrically) dependent elementary reactions. This dependence does not affect the occurrence of these reactions, that is, their chemical (kinetic) independence. Again, one readily obtains the actual forms of eq 48 and 49 pertaining to the analyzed mechanism:
r˜+ e r˜e
3
)
∏ β)1
( ) ( ) ( )( ) r˜β+
εβ1
r˜β-
)
r˜+ 1
-1
r˜1
r˜+ 2
0
r˜2
r˜+ 3
1
(62)
r˜3
and 5
∏
w)1
( ) ( )( ) ( ) rw+
rw-
σwe
)
r+ 1
r1
1
r+ 3
r3
-1
r+ 5
r5
1
)
r˜+ e r˜e
)
()() r˜+ 1
r˜1
-1
r˜+ 3
1
r˜3 (63)
where eq 62 has been taken into account in the last term of eq 63, resulting in the thought for relationship linking route rates ration to the mechanistic rate ratios.
6. Conclusions A general relationship is derived for nonequilibrated complex chemical systems, subject to the validity of mass action law, comprising an arbitrary number of chemical species involved in kinetically independent but stoichiometrically dependent elementary reactions. This relationship, given by eq 19 of section 2, links rates ratios of linear dependent reactions to rate ratios linear independent ones. Expressing of rates via mass action law in this general relation between rates result in a similar relation in rate coefficients, given in eq 21, that is a generalization of Wegscheider’s cyclicity condition. Unlike other derivations of such rate coefficients relations, the one presented in section 3 does not explicitly apply the detailed balance conditions. In section 4 the approach was extended for complex systems with stable and reactive chemical species, described by the Horiuti-Temkin reaction route theory. Scaling conditions similar to eq 19 were found to exist between route (overall) rates, as expressed by eq 48, but these are subject to the additional steady state requirements. The difference between chemical (kinetic) independence of elementary reactions and their stoichiometric (linear, algebraic) independence was outlined. The stoichiometric route basis representation (containing empty routes) was found to offer an additional relationship between the mechanistic (elementary reactions) representation and overall (route) representation of the chemical system, given by eq 49. New relations among rate coefficients of complex reaction systems are valuable additions to the theory of chemical kinetics, and can be used to determine rate coefficients not measured or not readily measurable. A reasonably simple multiroute reaction, the radical chlorine addition to ethylene, was used in section 5 for the illustration of the general results obtained in preceding sections. Acknowledgment. J.R. gratefully acknowledges the support from the National Science Foundation - CHE 0847073. References and Notes (1) Wegscheider, R. Z. Phys. Chem. 1902, 39, 257–303. (2) Ross, J.; Schreiber, I.; Vlad, M. O. Determination of Complex Reaction Mechanisms: Analysis of Chemical. Biological and Genetic Networks; Oxford University Press: New York, 2006. (3) Boudart, M. J. Phys. Chem. 1976, 80, 2869–2870. (4) Vlad, M. O.; Ross, J. Wiley Interdisciplinary ReViews: Systems Biology and Medicine; Wiley: New York, 2009. (5) Horiuti, J. Ann. N. Y. Acad. Sci. 1973, 213, 5–30. (6) Horiuti, J.; Nakamura, T. Z. Phys. Chem. N. F. 1957, 11, 358365; AdV. Catal. 1967, 17, 1-49. (7) Temkin, M. I. AdV. Catal. 1979, 28, 173-291; Int. Chem. Eng. 1971, 11, 709-717. (8) Boudart, M. Kinetics of Chemical Processes; Prentice Hall, Inc.: Englewood Cliffs, NJ, 1968. (9) Segal, E.; Vlad, M.; Popa, V. ReV. Roum. Chim. 1984, 29, 593– 596. (10) Boudart, M.; Djega-Mariadassou, G., Kinetics of Heterogeneous Catalytic Reactions; Princeton University Press: Princeton, NJ, 1984. (11) Vlad, M. O.; Segal, E. ReV. Roum. Chim. 1979, 24, 807–808. (12) Popa, V.; Segal, E.; Vlad, M. ReV. Roum. Chim. 1984, 29, 625– 629. (13) Schuster, S.; Schuster, R. J. Math. Chem. 1989, 3, 25–42. (14) Kurosh, G. Higher Algebra; Mir Publishers: Moscov, 1972. (15) Bodenstein, M. Z. Phys. Chem. 1913, 85, 329–397. (16) Popa, V. T.; Vlad, M. O.; Segal, E. Z. Naturforsch. 1989, 44A, 704–706. (17) Horn, F. J. M.; Jackson, R. Archs Ration. Mech. Analysis 1972, 47, 81–116. (18) Feinberg, M. Chem. Eng. Sci. 1989, 44, 1819–1827. (19) Emanuel, N.; Knorre, D. Cinetique Chimique; Mir: Moscou, 1975; p 198.
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