Simple pathways and networks

For convenience, a distinction is made between "simple" and "non-simple" pathways or networks. A pathway, network, or any portion of one of these is called simple if it meets both of the following criteria ... 

To be sure, a simple pathway or network, or any portion of these, may be of any size and topology and contain any number of steps with higher molecularities as long as the co-reactants or co-products are not themselves intermediates ... 

For reactions with non-simple pathways or networks, the formulas and procedures described so far are not valid. Any step involving two or more molecules of intermediates as reactants destroys the linearity of mathematics, and any intermediate that builds up to higher than trace concentrations makes the Bodenstein approximation inapplicable. Such non-simple reactions are quite common. Among them are some of the kinetically most interesting combustion reactions, detonations, periodic reactions, and reactions with chaotic behavior. However, a discussion of more than only the most primitive types of non-simple reactions is beyond the scope of this book. The reader interested in more than this is referred to another recent volume in this series [1], in which such problems are specifically addressed. 

Note of caution As stated at the outset, all rules in this section are for simple pathways and do not necessarily apply to other types of networks. 

In this fashion, the set of rate equations of any simple pathway (unless it is part of a network) can be reduced to a single rate equation and the algebraic equations expressing the stoichiometry. To illustrate how much work can be saved in this way, let us return to the Gillespie-Ingold mechanism of nitration of aromatics, for which a repeated application of the Bodenstein approximation provided a rate equation in Example 4.4 in Section 4.3. 

The general formula for simple pathways, eqns 6.4 to 6.6, is also applicable to any (linear) simple segment between two nodes in a network or between a node and a network end member [8], In this way ... 

Application of the general formula for simple pathways allows any simple network to be reduced to one with only single, pseudo-first order steps between adjacent nodes or end members and adjacent nodes. 

If only a small minority of reaction steps are non-simple, much benefit can be had by breaking the pathway or network down into "piecewise simple" portions and then applying the methods described in the preceding sections to these [8], To this end the pathway or network is cut at the offending intermediates or steps, as will now be shown. 

All arrows represent multistep, irreversible pathways, and aldehyde and possibly aldol build up to higher than trace concentrations. The network is non-simple for that reason and also because the aldehyde, an intermediate, acts in addition as a co-reactant in the pathway to the aldol. 

The concept of "simplicity" of a pathway or network is introduced. For a pathway to be "simple," all its intermediates must be and remain at trace level, and no step may involve two or more molecules of intermediates as reactants. The first condition ensures... 

The rules deduced in this subsection are exclusively for simple pathways. A pathway or network is "simple" if all its intermediates are and remain at trace level and if no step involves two or more molecules of intermediates as reactants (see definition in Section 6.1). 

The procedure of arriving at a probable mechanism via an empirical rate equation, as described in the previous section, is mainly useful for elucidation of (linear) pathways. If the reaction has a branched network of any degree of complexity, it becomes difficult or impossible to attribute observed reaction orders unambiguously to their real causes. While the rate equations of a postulated network must eventually be checked against experimental observations, a handier tool in the early stages of network elucidation are the yield-ratio equations (see Section 6.4.3). This approach relies on the fact that the rules for simple pathways also hold for simple linear segments between network nodes and end products. 

If a pathway or network turns out to be non-simple, a good strategy is to try to break it up into piecewise simple portions that can be studied independently. Whether and how this can be done depends on the reaction at hand. The job is easiest if the portions are irreversible, so that none of them feeds back into a preceding one, and if the non-trace intermediates can be synthesized. 

As has already become apparent, some of the rules governing the relationships between network properties and kinetic behavior, derived in Section 7.3.1 for noncatalytic simple pathways, can no longer be relied upon in catalysis, except if the free catalyst is the macs. A re-examination therefore is called for. 

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