Reaction linear independent, number

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. 

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) ... 

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. 

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... 

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. 

First, we note that the number of photons absorbed rather than the number of incident photons has to be taken into account. Second, integrations over extended time periods most likely bear substantial errors because the intensity of the source may fluctuate or drift. As a consequence of this, the only exact measure for the efficiency of a photochemical reaction is the true differential quantum yield, which needs to be determined for each step of the reaction. Similar to thermal reactions, photochemical reactions may be complex. Accordingly, the only correct measure is the so-called partial (true differential photochemical) quantum yield, which is defined for each linearly independent step of the reaction. 

However, there is one fly in the ointment here It may not be possible to determine the rate laws for each of the reactions. In this case it may be necessary to work with the minimum number of reactions and hope that a rate law can be found for each reaction. That is, you need to find the number of linearly independent reactions m your reaction set. In Example 6-8 just discussed, there are four reactions given [(E6-8.5) through (E6-8.8)]. However, only three of these reactions are independent, as the fourth can be formed from a linear combinafion of the other three, Tcchttiques for determining the number of independent reactions are given by Aris. ... 

We shall assume that the reactions are independent, i.e., that none of them can be expressed as a linear combination of the others. More precisely, we say that there exists no set of to numbers Xi,. . . X, except the trivial set Xj = X2 =. . . = X = 0, such that... 

Consider two reactions of the type A B and B 5 C. If the third process A C is not feasible the number of elementary reactions is the same as the number of linearly independent reaction equations the reactions are said to be uncoupled. If A 4= C represents a feasible reaction the three processes are said to be coupled generally, coupling occurs whenever there is a redundancy in the number of reaction steps. 

Let us just consider Eq. (2.4a)-(2.4c) and (2.5). Are they a set of independent equations If the answer (yes) is not obvious to you from the way in which the equations were formulated, you should review Appendix L to ascertain how to determine in a formal way whether the equations in a set of equations are independent. In general, but not always, in the absence of reactions, the number of independent equations equals the number of chemical compounds present, and with reactions occurring, the number of independent equations equals the number of atomic species present. But you may say the equations as posed are not linear because a mass fraction, an- unknown, multiplies W. also unknown. However, Eg. (2.5) can be substi-tuted into Eq. (2.4d), the latter solved directly for W, and W substituted into eachof Eq. (2.4a)-(2.4c). The result is three linear equations. Can you show that they are independent ... 

The maximum number of linearly independent chemical reactions. Mr, in a complex reaction system is given by ... 

The linearly independent chemical reactions create the stoichiometric (thermodynamic) basis for the reaction system. All other chemical processes in the system can be expressed as a linear function of these basic reactions. The stoichiometric basis is determined by the stoichiometry of the process, the number and compK>sition of... 

Contemporary chemical kinetics and the theory of reaction mechanisms are characterized not only by increased complexity of the mechanisms (hypotheses of mechanisms) but also by the considerable number of hypotheses (the possible mechanisms describing each reaction). Cases are known where the mechanism of formation of a certain product in a complicated multiroute mechanism incorporates completely different sequences of elementary steps and intermediates" even in the case of reactions that have one linearly independent stoichiometric equation The greater mechanistic complexity and high number of hypotheses raise the issue of the formalization and automation of the procedure adopted for the generation of hypotheses. 

In what follows we discuss these criteria and present the main results. The number of vertices in a KG, JV, is not important for the classification, but it is convenient to introduce this criterion into the coding procedure immediately after the notation for the number of routes. The number of KG edges, E (the mechanism s elementary steps), is not regarded as a criterion because it is determined uniquely by the Horiuti rule M = E — J, where J is the number of linearly independent intermediates. In the case of non-catalytic reactions the number N includes the vertex with the so-called zero reagent. 

Once a set of independent reactions is selected, write each of the dependent reactions as a linear combination of the independent reactions. The total number of reactions is the sum of the independent and dependent reactions ... 

According to the Horiuti-Temkin theorem [9,10], the number of linearly independent RRs is equal to I = p - ranka = p - q. Now, let Ju Ji be the corresponding rates (or, the RR fluxes) along the arbitrarily selected set of linearly independent RRs, namely RRu RR2,..., RRi. Then, within the QSS approximation the following relation between RR fluxes and the rates of individual elementary reaction steps is valid... 

Since rank a = 8, the surface intermediates are linearly independent and, hence, a direct RR involves no more that 9+l = 8 + l= 9 elementary reactions. It is further observed that 5i, sz, 53, and S5 (adsorption and desorption steps) should be involved in all full RRs. That is, it is not possible to obtain the OR by omitting these 4 elementary reactions. The remaining 9-4 = 5 elementary reactions involved in a RR need to be selected from among S4, sy, ss, s% s lo, S12, sw, Sis and sn. Thus, the total number of RRs does not exceed the number of ways 5... 

Equation 13 l-22b suggests that the logarithm of the equilibrium constant should be a linear function of the reciprocal of the absolute temperature if the heat of reaction is independent of temperature and, presumably, an almost linear function of l/T even if Arxnff° is a function of temperature. (Compare this behavior with that of the vapor pressure of a pure substance in Sec. 7.7, especially Eq. 7.7-6.) Consequently, it is common practice to plot the logarithm of the equilibrium constant versus the reciprocal of temperature. Figure 13.-1-2 gives the equilibrium constants for a number of reactions as a function of temperature plotted in this way. (Can you identify those reactions that are endothermic and those that are exothermic )... 

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 and a set of such routes is called the basis of routes. Although the basis of routes can be chosen in different ways the number of basis routes for a given reaction mechanism is determined in a unique way, being the dimension of the space of the routes. 

The issue of independence centers on the question of whether or not we can express any reaction in the network as a linear combination of the other reactions. If we can, then the set of reactions is not independent. It is not necessary to eliminate extra reactions and work with the smallest set, but it is sometimes preferable. In any case, the concept is important and is examined further. Before making any of these statements precise, we explore the question of whether or not the three reactions listed in Reactions 2.16 are independent. Can we express the first reaction as a linear combination of the second and third reactions By linear combination we mean multiplying a reaction by a number and adding it to the other reactions. It is clear from inspection that the first reaction is the sum of the second and third reactions, so the set of three reactions is not independent... 

---