Reaction Mechanism Graphs

Later, in 1970s and 1980s, Evstigneev et al. (1978, 1979, 1981) systematically analyzed this equation applying methods of graph theory They found a variety of its interesting structural properties regarding the link between kinetics of the complex reaction and structure of the reaction mechanism. 

A general method of application of (51) for the elimination of unknown concentrations with the help of a graph of a reaction mechanism is described elsewhere (27). 

King and Altman [5] and Temkin developed a method to represent a reaction mechanism as a graph. Its nodes are intermediates and its edges are steps. Reaction directions are marked by arrows on the edges. 

It is interesting that the product of the characteristic roots is the sum of the whole of the trees in the graph for this reaction mechanism [see eqns. (60) and (64)]. 

Fig. 1. Simple examples for bipartite graphs of reaction mechanisms. , Reaction nodes O, substance nodes.

In a certain sense, the simplest class of reaction mechanism is that whose bipartite graphs do not contain cycles, i.e. are acyclic. The dynamic behaviour of the corresponding reactions is always extremely simple [7]. An example for such a mechanism can be Ax - A2 - A3 - . . . - A [see Fig. 2(a)]. The contribution of acyclic mechanisms to the kinetics of catalytic reactions is not of importance. The mechanisms of catalytic reactions always contain cycles and these cycles are oriented, the directions of all the arrows being matched [the end of the ith arrow is the beginning of the... 

If all the elementary reactions are monomolecular, i.e. can be written as Ax —> Aj, it is more convenient to represent reaction mechanisms in a different way, namely nodes correspond to substances, edges are elementary reactions, and edge directions are the directions of reaction processes. As usual, this graph is simpler than the bipartite graph. For example, for the system of three isomers Al A2 and A3 we obtain... 

The graph for simple reaction mechanism of the liquid-phase hydrogenation A + H2 = AH2 is similar. 

It can also be interpreted in terms of the bipartite graph for the reaction mechanism (see Sect. 1.3). 

Let us examine the properties of eqn. (152) under the assumption of oriented connectivity. Let us fix some co-invariant simplex D0 zt 0, , 2,- = C > 0. Da has a unique steady state z°. Vector z° is positive since, due to the connectivity of the reaction digraph, no steady-state points exist on the boundary Da. Indeed, if we assume the opposite (some components z° are zero), we obtain kJt for such i and j as 2° 0 and z° = 0. But from this it follows that, moving along the direction of arrows in the graph of the reaction mechanism, we cannot get from the substances for which 2° 0 to those for which z° = 0, and this is contrary to oriented connectivity (the arrows in the reaction graph correspond, naturally, to the elementary reactions with non-zero rate constants). 

There is no doubt that studies for the establishment of new classes of mechanisms possessing an unique and stable steady state are essential and promising. On the other hand, it is of interest to construct a criterion for uniqueness and multiplicity that would permit us to analyze any reaction mechanism. An important contribution here has been made by Ivanova [5]. Using the Clark approach [59], she has formulated sufficiently general conditions for the uniqueness of steady states in a balance polyhedron in terms of the graph theory. In accordance with ref. 5 we will present a brief summary of these results. As before, we proceed from the validity of the law of mass action and its analog, the law of acting surfaces. Let us also assume that a linear law of conservation is unique (the law of conservation of the amount of catalyst). 

On the basis of the structure for a bipartite graph of the reaction mechanism, it is possible to formulate a sufficient condition (174) for the uniqueness of a steady state. Applying it to concrete reactions, it is possible to establish the parametric areas for which either a unique steady state exists or there is a multiplicity of such states. 

Fig. 3. Graphs of linear catalytic reaction mechanisms, (a), (b), (c), One-route (d), (e) two-route (f) multi-route mechanisms.

Though the reaction mechanism here is more complex than in the previous example and the kinetic equation also has non-Arrhenius parameters, it is possible to determine all the reaction rate constants. The fact is that there is a sufficient quantity of the Arrhenius complexes. In this case it appears that all "mixed complexes, i.e. complexes containing parameters of both direct and inverse reactions, are independent. Here these complexes evidently corresponding to the mixed spanning trees of the graph are coefficients for various concentration characteristics. It is this fact that permitted us to obtain the convenient eqns. (82). 

If Eq. (2.9) is appropriate, a graph of log A/Aq) vs. t should yield a straight line with a slope equal to —kc. However, based on this result alone, it is tenuous to conclude that Eq. (2.5) is the only possible interpretation of the data and that a straight-line graph indicates a first-order reaction. One can make these conclusions only if no other reaction mechanisms result in such a graphical relationship. 

Graphs of log (1 - A/Y0) vs. t are commonly used to test the validity of Eq. (2.10). However, Eq. (2.11), like Eq. (2.8), shows more complex behavior than simple graphical methods reveal. Thus, one should be cautious about making definitive statements concerning rate constants and particularly mechanisms, based solely on data according to integrated equations like those in Eqs. (2.9) and (2.10) unless other reaction mechanisms have been ruled out. 

Reaction mechanism graphs are particularly useful in comparing several different reaction mechanisms that yield the same overall reaction, and also in considering the relative locations of various steps. This is illustrated through several examples from molten carbonate and proton-exchange membrane (PEM) fuel cell. 

As a first example of the use of reaction mechanism graphs, consider the electrochemistry of molten carbonate fuel cell (MCFC) cathodes. These cathodes are typically nickel-oxide porous electrodes with pores partially filled with a molten carbonate electrolyte. Oxygen and carbon dioxide are fed into the cathode through the vacant portions of the pores. The overall cathodic reaction is 02 + 2C02 + 4e / 2C03=. This overall reaction can be achieved through a number of reaction mechanisms two such mechanisms are the peroxide mechanism and the superoxide-peroxide mechanism, and these are considered next. 

The relationship between the five peroxide mechanism reaction steps can be seen in the reaction mechanism graph in Figure 4. As defined above, each step occurs at one of the five nodes, and the directed edges give the forward direction for the mechanism. Current-carriers for the overall mechanism are in boxes, while carbonate ions that continue from one cycle to the next are circled. Dashed vertical lines represent interfaces between phases. Nodes on the gas-electrolyte interface represent reaction steps occurring at that interface nodes attached to the electrolyte-solid interface represent reaction steps occurring at sites on the surface of the solid phase. The location of each reaction on this reaction mechanism graph follows the description of the... 

Figure 4. Reaction mechanism graph for the peroxide mechanism reactions occur at dots, arrows show the forward direction for reactions, dashed lines separate phases, current-carriers are in boxes, and carbonate ions which continue from one cycle to the next are circled.

The reaction mechanism graph for this mechanism is given in Figure 5 again all of the steps except s3 (recombination) occur at an interface. 

Figure 5. Superoxide-peroxide reaction mechanism graph notation and symbols as in Figure 4.

---