Linearly independent reactions

Reciprocally, any set of m chemical reactions linearly independent in a system of n chemical species corresponds to a subtorus T of T". 

Note that the matrix of stoichiometric coefficients devotes a row to each of the N components and a column to each of the M reactions. We require the reactions to be independent. A set of reactions is independent if no member of the set can be obtained by adding or subtracting multiples of the other members. A set will be independent if every reaction contains one species not present in the other reactions. The student of linear algebra will understand that the rank of v must equal M. 

The components must be linearly independent of one another. In other words, we should not be able to write a balanced reaction to form one component in terms of the others. 

For a complex system, determination of the stoichiometry of a reacting system in the form of the maximum number (R) of linearly independent chemical equations is described in Examples 1-3 and 14. This can be a useful preliminary step in a kinetics study once all the reactants and products are known. It tells us the minimum number (usually) of species to be analyzed for, and enables us to obtain corresponding information about the remaining species. We can thus use it to construct a stoichiometric table corresponding to that for a simple system in Example 2-4. Since the set of equations is not unique, the individual chemical equations do not necessarily represent reactions, and the stoichiometric model does not provide a reaction network without further information obtained from kinetics. 

Note that this constraint implies that the (/) columns of Y are orthogonal to the (E) rows of A. Thus, since each column of Y represents one elementary reaction, the maximum number of linearly independent elementary reactions is equal to K — E, i.e., N-f = rank(Y) < K — E. Formost chemical kinetic schemes, Ny = K — E however, this need not be the case. 

It can be straightforwardly verified that indeed NK = 0. Each feasible steady-state flux v° can thus be decomposed into the contributions of two linearly independent column vectors, corresponding to either net ATP production (k ) or a branching flux at the level of triosephosphates (k2). See Fig. 5 for a comparison. An additional analysis of the nullspace in the context of large-scale reaction networks is given in Section V. 

Introduction of stoichiometric number concept and linear transformation of the "conventional" QSSA equations (16) to the equivalent system (20) was essentially the major (and, possibly, only) result of theory of steady reactions developed independently by J. Horiuti in 1950s and M. I. Temkin in 1960s. 

The elementary reactions in Eqs. (1) are not necessarily linearly independent, and, accordingly, let Q denote the maximum number of them in a linearly independent subset. This means that the set of all linear combinations of them defines a 0-dimensional vector space, called the reaction space. In matrix language 0 is the rank of the S x A matrix (2) of stoichiometric coefficients which appear in the elementary reactions (1) ... 

There are special cases where the direct mechanisms are linearly independent and constitute a basis. If all the direct mechanisms for a particular reaction r are disjoint, in the sense of containing no steps in common, then they are obviously linearly independent, or if there is only one direct mechanism for r, it is independent. This last case suggests a way of finding all the direct mechanisms in a chemical system. If we can find a subsystem which contains at most one mechanism m for any reaction r, then m is direct. In other words, m is a direct mechanism if S — Q in the chemical system, consisting just of the steps in m. 

In a chemical system with S steps and a maximum of Q linearly independent elementary reactions every set of Q steps whose reactions are linearly independent constitutes a cycle-free subsystem. It is apparent that, if R(Sj),..., R(sq) are linearly independent, then no cycle can be formed with the steps... 

These quantities are preserved like atoms in the given reactions and hence are called reaction invariants (ref. 16). In this example we found 4 linearly independent reaction invariants. It does not mean, however, that the species Ml, 2,. .. M6 are built necessarily from 4 atoms. In fact, introducing the species M = CH4, 2 — O2, 3 = M4 = 5 = 2 9 and M — 2 2... 

We consider a chemical system consisting of the following species methane (CH4), water (H20), carbon monoxid (CO), carbon dioxide (CO2), and hydrogen (H2). There are two linearly independent reactions among these species, e.g.,... 

It is shown below (Chapter 4) that its solution is also important for the bimolecular stage of a reaction defining a stationary recombination profile. The second linearly independent solution of this equation could be expressed through y(r) as... 

Such a definition can, evidently, be extended to any number of routes. It is clear that if A(1), A(2), A<3) are routes of a given reaction, then any linear combination of these routes will also be a route of the reaction (i.e., will produce the cancellation of intermediates). Obviously, any number of such combinations can be formed. Speaking in terms of linear algebra, the reaction routes form a vector space. If, in a set of reaction routes, none can be represented as a linear combination of others, then the routes of this set are linearly independent. A set of linearly independent reaction routes such that any route of the reaction is a linear combination of these routes of the set will be called the basis of routes. It follows from the theorems of linear algebra that although the basis of routes can be chosen in different ways, the number of basis routes for a given reaction mechanism is determined uniquely, being the dimension of the space of the routes. Any set of routes is a basis if the routes of the set are linearly independent and if their number is equal to the dimension of the space of routes. 

A linear independence of routes does not imply linear independence of the respective overall equations. For instance, as mentioned, the basic routes of the reaction (35) result in the same overall equation. A more complicated... 

The number of complexes minus that of connected components of the graph for their conversions equals the number of linearly independent reactions (stoichiometric vectors). A second Horn and Jackson condition for quasi-thermodynamic behaviour is the weak reversibility of the graph for complex conversions. This graph is called weakly reversible if any of its connected components contain a route to get from any node to any other moving in the direction of its arrows. For example, the scheme... 

Due to the fulfilment of this law of conservation, the number of linearly independent intermediates is not three but one fewer, i.e. it amounts to two. To the right of mechanism (1) we gave a column of numerals. Steps of the detailed mechanism must be multiplied by these numerals so that, after the subsequent addition of the equations, a stoichiometric equation for a complex reaction (a brutto equation) is obtained that contains no intermediates. The Japanese physical chemist Horiuti suggested that these numerals should be called "stoichiometric numerals. We believe this term is not too suitable, since it is often confused with stoichiometric coefficients, indicating the number of reactant molecules taking part in the reaction. In our opinion it would be more correct to call them Horiuti numerals. For our simplest mechanism, eqn. (1), these numerals amount to unity. 

Horiuti numbers v (S x P) is the route of a complex reaction. The rank of the matrix rint cannot be higher than (S - P) since, according to eqn. (19) there are P linearly independent rows of Tint. As usual, we have... 

From eqns. (86) one must choose a set of linearly independent equations and by using known methods find the reaction rate constants. 

In summary, it can be seen for the three-step reaction scheme of this example that the net rate of the overall reaction is controlled by three kinetic parameters, KTSi, that depend only on the properties of the transition states for the elementary steps relative to the reactants (and possibly the products) of the overall reaction. The reaction scheme is represented by six individual rate constants /c, and /c the product of which must give the equilibrium constant for the overall reaction. However, it is not necessary to determine values for five linearly independent rate constants to determine the rate of the overall reaction. We conclude that the maximum number of kinetic parameters needed to determine the net rate of overall reaction is equal to the number of transition states in the reaction scheme (equal to three in the current case) since each kinetic parameter is related to a quasi-equilibrium constant for the formation of each transition state from the reactants and/or products of the overall reaction. To calculate rates of heterogeneous catalytic reactions, an addition kinetic parameter is required for each surface species that is abundant on the catalyst surface. Specifically, the net rate of the overall reaction is determined by the intrinsic kinetic parameters Kf s as well as by the fraction of the surface sites, 0, available for formation of the transition states furthermore, the value of o. is determined by the extent of site blocking by abundant surface species. 

Brinkley (4 postulated C species at equilibrium, p species, referred to as "components," were selected to have linearly independent formula vectors, where p is the rank of the atom matrix, (mjk), and Yj is the formula vector for the jth species, [mj, mj2f...mjE]. Given the choice of components, the stoichiometric coefficients for an independent set of chemical reactions are computed ... 

First, we need to identify the state variables y describing the chemical equilibrium. Following the notation of Gorban and Karlin (2003), the conservation laws in chemical reactions introduce k linearly independent vectors blr b2,..., bk. The state variables describing the system at the chemical equilibrium are... 

Methanol synthesis from C02 (Equation [1]) and CO (Equation [2]) is mildly exothermic and results in volumetric contraction. Methanol steam reforming (MSR) refers to the inverse of reaction (1), and the inverse of reaction (2) is conventionally referred to as methanol decomposition - an undesired side reaction to MSR. The slightly endothermic reverse water-gas shift (rWGS) reaction (Equation [3]) occurs as a side reaction to methanol synthesis and MSR. According to Le Chatelier s principle, high pressures and low temperatures would favor methanol synthesis, whereas the opposite set of conditions would favor MSR and methanol decomposition. It should be noted that any two of the three reactions are linearly independent and therefore sufficient in describing the compositions of equilibrated mixtures. 

Indeed, catalysis of the water gas shift reaction is observed using (I) dissolved in methanol (50 ml) and water (25 ml) as catalyst in a 300 mL bomb at 100°C and with pressures of CO from 80-160 psi. Under these conditions the rate of reaction is independent of CO pressure and first order with respect to concentration of catalyst (I), giving a turnover rate of 3.6 0.6 (moles C02 or H2)/(mole catalyst)(hour). Good linear kinetics are observed for at least one day and solutions remain homogeneous... 

This example illustrates the fundamental principle that if one describes coupled reactions in terms of a set of linearly independent steps, then sufficiently close to equilibrium the reaction rates may be described in terms of phenomenological equations involving the chemical affinities as driving forces. 

As already discussed, at equilibrium only the reaction rates wr of the R linearly independent reactions can initially be assumed to vanish. To handle the correspondence between the linearly dependent and independent reactions we relate the A, and Ar affinities through linear equations of the form... 

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