Two linear independent reactions

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.,... 

If just two linear independent reactions take place in a system, the absorbance differences for three different wavelengths (A = 1, 2, 3) are given according to eq. (5.65)... 

EDQ-diagrams become linear if either just two linear independent reactions take place or if in more than two linear independent partial reactions the rank of the matrix Q reduces to two by chance. For a district discrimina-... 

Fig. 5.14. Schematic E-diagram for two linear independent reactions. The figure visualises eq.

In the system used in example 1 there holds, that N = 5, M = 3, H = 3. According to relation (2.13) R = 2 and a maximum of two linearly independent reactions can take place in the system. 

Investigation of the temperature dependence of the degree of conversion in a system, in which a minimum of two linearly independent reactions are taking place, is rendered complicated by the fact, that the relationship involved depends on the ratios of heats... 

The rank of the matrix is H = 4, i.e. two linearly independent reactions will take place in the system. 

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. 

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. 

To appreciate the special role of a conical intersection as a transition point between the excited and the ground state in a photochemical reaction, it is useful to draw an analogy with a transition state associated with the barrier in a potential energy surface in a thermally activated reaction (Figure 6.6). In the latter, one characterizes the transition state with a single vector that corresponds to the reaction path through the saddle point. The transition structure is a minimum in all coordinates except the one that corresponds to the reaction path. In contrast, a conical intersection provides two possible linearly independent reaction path directions. 

In the case of mechanisms whose elementary steps incorporate one intermediate to the left and right of the reaction equality (called by Temkin linear mechanisms ), each edge in the cyclic graph stands for an elementary step of the reaction mechanism, i.e. for a pair of mutually reversed elementary reactions. Each vertex of the kinetic graph corresponds to a certain intermediate while the linearly independent reaction routes are represented by graph cycles. For example, the mechanism of the water vapour methane conversion over Ni incorporates two independent routes, five intermediates, and six steps it is depicted by kinetic graph 1. 

The above reactions represent two possible independent reactions. Other initial choices for two out of the five stoichiometric coefficients may generate different reactions, but these new reactions can always be generated by linear combinations of the two reactions given above. 

The backward reactions cause a reduced reaction scheme. In addition, the third step of the reaction can be cancelled and one obtains two linear independent degrees of advancement... 

Both the matrices K and Kj. have to be regular. They are called Jacobi matrices. In general the Jacobi matrix for two linear independent steps of a reaction (s = 2) has the following form ... 

Thermal reactions, which take place in closed systems, do not show complex eigenvalues. In contrast, in the case of quasilinear photoreactions, such solutions cannot be excluded in principle. Complex eigenvalues always appear in pairs. In the following, a general approach to solve this problem is given for two linear independent steps of reaction (j = 2). The two eigenvalues are... 

It is difficult to define general rules. However, one is able to simulate K-diagrams using a desk-top computer for assumed mechanisms and to analyse the curves for points of inflection. The principle approach was discussed in Section 2.3.2. Points of inflection are allowed in K- or X-diagrams only, if more than two linear independent steps of reaction s > 2) take place. 

In the case of two linear independent partial reactions, one can set up the following theorems for linear transformations ... 

Another example is a mechanism containing two linear independent steps, the latter including a back reaction A —> B C... 

More than two linear independent steps of reaction... 

Depending on the reproducibility, the two linear independent steps of reaction of the consecutive mechanism (existing in reality) appear as a uniform reaction. 

B and D form an addition reaction with the excited states A or A", respectively, in two linear independent steps of the reaction. In this case the correlation between the degrees of advancement is equivalent to that in the case of analogous thermal reactions treated in Section 2.4.2.1 as independent parallel reactions. If the reaction takes place according to this mechanism, the relationship will differ from that in the example given above. 

Both the photo-parallel reaction and the consecutive photoreaction, being physically sensitised, form two linear independent steps of the reaction. Their mechanisms and the reduced reaction scheme are listed in Table 3.5. The overall reaction schemes are derived in Appendix 6.6.2. 

The amount of light absorbed is substituted according to eq. (1.36) and the local average according to eq. (3.32) is formed. The result is an explicit expression for the change of the degree of advancement x of the Arth partial reaction step with time. This relationship can be used to set up the differential equations for the concentrations a, of the reactants A, according to eq. (2.5). This procedure is discussed in Section 2.2.1 and its subsections. The results for photoreactions are compared to the thermal Jacobi matrices in Section 2.2.1.4. There the results for the two linear independent steps of a consecutive photoreaction... 

If all the KDQ-diagrams are either linear or degenerate, only two linear independent partial reactions take place. This graphical method can be expanded to more complex mechanisms as discussed in the following example. 

If the photochemical reactions are superimposed by thermal reactions, in some cases additional information on the nature of the partial reactions can be obtained. This can be demonstrated in the case of the photoisomerisation of fra 5-azoxybenzene (A) to ort/io-hydroxy-azoxybenzene (C) [145]. In methanolic solution this reaction does not proceed uniformly, cis-Azoxybenzene (B) is formed as an intermediate. Using EDQ-diagrams explained in the next section one can demonstrate that just two linear independent partial reactions take place. The reaction can either follow the mechanism ps,... 

For two linear independent partial reactions the vector E is given by... 

The assumption of two linear independent partial reactions was controlled by an EDQ-diagram for both reactions. These are plotted in combination in Fig. 5.24. The photoreaction of the tra 5-azobenzene was evaluated according to the method of formal integration using the following equations ... 

Between the three compounds A, B, C two linear independent quasi-linear photoreactions are assumed to take place. A list of such reaction is given in Table 5.3. 

Survey on possible reaction with two linear independent steps... 

Since the reaction involves two linear independent steps, two wavelengths of observation have to be used. One has to solve two coupled linear differential equations by formal integration ... 

Taking the data as used for Fig. 4.18 the following linear relationship is obtained demonstrating that the reaction is represented by two linear independent steps. Therefore any of the mechanisms given in Table 2.5 can be used. On the other hand chemical information requires a consecutive reac-... 

It is customary to separate all reactions in water into two large groups linearly independent and dependent. With the linearly independent reactions belong reactions with participation of the basis components, and with linearly dependent ones - reactions between secondary components. Independent reactions are actually reactions of the formation of secondary components from the basis ones. 

A case is frequently encountered, in which the chemical conversion in a system is described by more than three linearly independent reactions, but yet the resulting set of equilibrium relations can be simplified to such a degree that one of the above-described procedures can be applied. In principle, two kinds of systems may be involved ... 

It has been shown in section 4.3.3, that this is a complicated problem in the case where a single reaction is taking place in a system. Where two or more linearly independent reactions are proceeding in a system, the influence of composition of the initial mixture cannot be elucidated beforehand. 

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