A Procedure for Finding Every Direct Mechanism

A procedure, which was introduced by Happel and Sellers (1) for finding all direct mechanisms for a given reaction will be demonstrated here from the standpoint of how to apply it in practice. We demonstrate it by applying it to an arbitrary S-step chemical system, as defined in Section II, to find all the direct mechanisms for the general overall reaction (14) derived in Section III. 

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 

The same result can be achieved by considering columns C at a time in the following smaller C x S matrix  

Let us find every direct mechanism for a given overall reaction r. Assume r to be of the general form given in Eq. (14) and of multiplicity R (R = Q - H), which means that an expression for it contains R parameters. Any mechanism for r is of the general form given in Eq. (13) and depends not only on the R parameters in its reaction, but on C additional parameters, where C is the dimension of the space of all cycles (R + C = S — H). Therefore, to determine a unique mechanism for r, we need to determine C parameters, and they can be chosen to make it a direct mechanism by the following reasoning  

Case 1. If C = 0, there are no cycles in the system. For any reaction r in the system there is one mechanism, and it is direct. This applies to Example 1 in Section V and Example 6 in Section VI. 

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