Network elucidation

Pseudo-orders can also be established by semi-batch experiments in which one or several reactants are replenished to keep their concentration or concentrations constant. In this way, the reaction order or orders with respect to the other reactant or reactants can be established. The order with respect to a replenished reactant can then be found by comparison of the rates obtained with different concentrations of that reactant. This method is particularly convenient for gas-liquid reactions such as homogeneous oxidation, halogenation, hydrogenation, hydroformylation, and hydrocyanation. Here, the gaseous reactant or reactants can be admitted to the reactor on demand as they are consumed, or by bubbling an excess of the gas at constant pressure through the reactor. Some later examples of network elucidation are of cases in which this method was used. 

In essence, the application of any one of the three principal tools allows one or several rate equations to be replaced by algebraic equations which, in turn, can be used to eliminate concentrations of intermediates from the set. In the best of all worlds, mathematics can be reduced to a single rate equation and simple algebraic relationships between the concentrations of the reactants and products. More often, several simultaneous rate equations remain, but the reduced set is nevertheless much easier to handle for network elucidation and more convenient for modeling. 

The presentation is geared toward recognition of symptoms of typical step combinations in observable kinetic behavior as an essential facet of network elucidation. 

The principal application of yield ratio equations is in network elucidation, to be discussed in Section 7.3.2. An additional example will be given in that context. Further examples for establishment of yield ratio equations and their application in network elucidation can be found in Temkin s book on reaction networks [24], although not under that name. Also, in mathematical modeling, simple algebraic yield ratio equations can sometimes be substituted for rate equations, which may be differential (see Section 11.2). 

The reduction in mathematical complexity of non-simple networks is mainly of value in modeling. For network elucidation, a better approach usually is to study as many portions of the network separately, say, by experiments that use synthesized intermediates as starting materials or by appropriate lumping, as the following example will illustrate (see also Section 7.3.3). 

Network elucidation starts with the identification of the participants in the reaction reactants, products, known intermediates, and possibly catalysts and any other species that affect the rate. From there on approaches differ. 

For mathematical convenience and economy of effort, rate equations in network elucidation and modeling are best written in terms of the minimum necessary number of constant "phenomenological" coefficients, which may be combinations of rate coefficients of elementary steps. This not only simplifies algebra and increases clarity, but also lightens the experimental burden fewer coefficients, fewer experiments to determine them and their temperature dependences. 

One-plus rate equations play a key role in network elucidation. Perhaps the most difficult step in that endeavor is the translation of a mathematical description of experimental results into a correct network of elementary reaction steps. The observed behavior can usually be fitted quite well by a traditional power law with empirical, fractional exponents, at least within a limited range of conditions. This has indeed been standard procedure in times past. However, such equations are highly unlikely to result from a combinations of elementary steps. Their acceptance may be expedient, but as far as network elucidation is concerned they are a dead... 

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. 

In practice, many reaction systems involve non-trace intermediates, but no obvious non-simple reactions of intermediates. A good strategy in such situations is to cut the overall reaction network into portions at the non-trace intermediate or intermediates (see Section 6.5), then reduce the portions as described for simple networks in Section 6.4.1. Network reduction makes it unnecessary to keep track of trace intermediates (except those reacting in a non-simple manner) and so obviates much of the hard work Trace intermediates are the more troublesome ones in network elucidation because they are difficult or impossible to detect, identify, analyze for, or synthesize, tasks that usually do not pose problems with intermediates that rise above trace level. Often, the network portions will turn out to be "piecewise simple" (see Section 6.5). If not, further cutting at additional nonsimple steps is called for when these become apparent. 

Piecewise simple portions feeding into others may, of course, be reversible. This complicates network elucidation significantly. Often, however, such back-reactions can be blocked by an additive, the omission of a catalyst or co-catalyst, or some other experimental stratagem. Alternatively, the intermediate produced by the portion can be trapped in some fashion this allows at least the forward reaction through the portion to be studied without interference. 

If many or even a majority of the steps are non-simple, the network reduction methods described here are of little use in network elucidation. This is typically the case in hydrocarbon pyrolysis and combustion, where reactions of free radicals with one another are common. Fortunately, an extensive data base of rate coefficients and activation energies of reaction steps of species in this field of chemistry has been compiled over the years and can be of help in network elucidation [12-16]. 

In addition to the simplification of mathematics and the criteria and guidelines described so far, many other techniques can be brought to bear in network elucidation. Excellent literature on these is available, and only a brief overview will be given here. For quick orientation the reader is referred to a comprehensive review with copious references [38],... 

The general formula for the rate in simple pathways, derived in Section 6.3, can be used for deducing a large number of rules that relate observable kinetic behavior, such as reaction orders, to properties the network may have or definitely cannot have. (Catalytic reactions require qualifications see Section 8.6.) These rules greatly facilitate network elucidation Pathways or networks that include a feature producing behavior contrary to observation can be ruled out by whole groups rather than one at a time. 

Criteria and guidelines useful in network elucidation and supplementing the rules derived in this chapter include considerations of steric effects, molecularities of postulated reaction steps, and thermodynamic constraints as well as Tolman s 16- or 18-electron rule for reactions involving transition-metal complexes and the Woodward-Hoffmann exclusion rules based on the principle of conservation of molecular orbital symmetry. Auxiliary techniques that can be brought to bear include, among others, determinations of isomer distribution, isotope techniques, and spectrophotometry. 

To arrive at rate equations of catalysis by a single species that is present almost exclusively as free catalyst, the formalism developed for noncatalytic reactions can be used with the catalyst appearing as both a reactant and a product. In fact, many of the examples used in earlier chapters to illustrate the deduction and application of rate equations were from catalytic reactions of this type. Thus, all the rules derived in Chapter 7 for network elucidation remain valid in such cases. 

A prerequisite for fundamental mathematical modeling is that the reaction network or networks have been established. This will be taken for granted here (for network elucidation, see Chapter 7). 

Regarding the second criterion, the proposed mechanism must withstand scrutiny from all angles such as stereochemistry, thermodynamics, molecular-orbital theory, experience with analogous chemistry, etc. Such matters have been discussed in the context of network elucidation in Section 7.4. 

---