Kinetics graph

Remember that when plotting kinetic graphs the concentration at t = 0 is also a valid data point. 

Figure 8.22 Kinetic graph for a reversible first-order reaction with the axes for an integrated rate equation ln([A], — A fcq ) (as 3/ ) against time (as V). The gradient is —5.26 x 10 3 min 1...

As an example of the construction of kinetic graphs, consider the following set of reaction steps that yield the electrochemical oxidation of hydrogen ... 

Callaghan and Datta (2005)11 discuss this reaction in another context, taking H-S and HyS as the intermediates. For this choice of intermediates, a kinetic graph can be constructed as in Figure 2. The tradition for kinetic graphs is to use directed edges only when... 

Figure 2. Kinetic graph for the electrochemical oxidation of hydrogen nodes represent individual intermediates, an edge connects two nodes exactly when the two corresponding intermediates are the only intermediates on opposite sides of the corresponding reaction step. If no intermediate is on one side of a given step, an edge corresponding to that step connects to the 0-species node.

Mathematical model of growth of such heterogeneous population in accordance with kinetic graph (Figure 7) is reflected by the following system of equations ... 

Figure 7. Kinetic graph of ecotoxicological model of tumor growth of equations system (37) solution.

Construction of the kinetic graph and evaluation of the weight of each of its arcs. 

In the following, a kinetic graph should always be understood to be associated with its corresponding linear reaction mechanism and vice versa a linear reaction mechanism should always be associated with its corresponding reaction graph. 

The kinetic graph shown in Fig. 1.1 corresponds to mechanism (41 based on the isomorphism between the reaction mechanism and the graph. Here vertex 1 corresponds to the free site Z on the catalyst surface, vertex 2 corresponds to the ISC cis-1,2-ChH,6 j Z, and vertex 3 corresponds to the ISC trans-1, 2-CaH,6-. Z. All steps in the mechanism (4) are reversible. For every elementary reaction of the mechanism in the kinetic graph separate arcs appear and for every elementary step in the reaction graph two arcs with the opposite directions appear. 

Let us construct the kinetic graph for given mechanism in such a way that for every elementary step from the mechanism (which ma be reversible or nonreversible) there corresponds an edge. This kinetic graph is nonoriented. Comparing equations (1) and (17), and taking into account the isomorphism between a graph and a n echanism, it is evident that the number of independent... 

The total number of cycles in the kinetic graph is given by ... 

This means that every kinetic graph is equivalent to a steady state system (11). Evaluation of the kinetic equation by the methods of graph theory is equivalent to solving system (11). 

For noncatalytic reactions a null vertex is added to the kinetic graph, while for catalytic reactions in corresponding kinetic graphs no null vertex appears because the role of the null vertex is that of a free active site on the catalyst s surface. 

The analogy with electric circuits made earlier is certainly not an obvious one. and in the works of Vorkenshtein and Gordshtein (2,3), a requirement exists that the kinetic graph for noncataly tic reactions contain a nuU vertex with a concentration equal to 1. For catalytic reactions the role of the null vertex is that of a free catalytic surface. 

The application of the methods of graph theory to experimental data from kinetics studies of heterogeneous catalytic reactions,is strongly restricted by the difficulties in calculating the base determinants of the vertices in kinetic graphs. While for a simple graph it is possible to find the base determinants by hand, for a complex mechanism this is often a difficult task. 

Several algorithms (34,35) for solving this problem have been proposed, but they proved to be inefficient. Three algoriths (6.39), proposed by the author for the efficient calculation of the base determinants of kinetic graphs, are now presented. 

Classification and coding principles of undirected kinetic graphs. ... 

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 following intermediates are included Z is a reaction site on the nickel surface and is assumed to be bivalent CH2. CHOH, CO, and O are hemosorped radicals. In the same listing order these intermediates correspond respectively to vertices 1,3,4,5, and 2 in the kinetic graph. 

The use of cyclic graphs proved to be very fruitful for the deduction of kinetic equations and in the analysis of kinetic data for linear mechanisms. It should also be mentioned that Temkin s kinetic graphs provide a unique approach to both catalytic and non-catalytic reactions. In the latter case, one of the graph vertices contains a zero intermediate of concentration 1 not included in the kinetic equations. Thus, the use of kinetic graphs for non-catalytic reactions is justified only for mechanisms with at least one intermediate. 

Basic Classification and Coding Principles of Kinetic Graphs... 

Following on from the Introduction, the problem associated with the classification of linear mechanisms, when taking into account only structural information, is reduced to the classiHcation of kinetic graphs. Such a classification, which provides a unique coding of the graphs, must fulfill certain specific requirements ... 

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