Chemical feedback branched-chain ignition

Examples of such empirical rate laws for the H2 + O2 reaction are found in the literature [1-4]  

Here the coefficients a-e are the individual (empirical) reaction orders with respect to the species A-E, respectively. The overall reaction order would then be given by a + p + x + e, but this may not be a particularly significant quantity. Other possible (and indeed quite common) forms for the empirical rate law involve rational polynomial terms such as [ l] /(l + [C] ) in which case the concept of an overall order, or of an order with respect to C, is not appropriate. 

Empirical rate laws are particularly useful for introducing the idea of feedback. For this, we can proceed by plotting a graph of how the overall reaction rate varies with the degree of advancement of the reaction. The latter can be simply represented by the extent of reaction described above, i.e., by the ratio of how much of a selected reactant has been used up with respect to its initial concentration. In terms of species A in our general reaction, which for convenience we can simplify here to 

The form of the dependence of / on will be determined by the order of the reaction with respect to A and B. Provided the reaction does indeed follow the overall stoichiometry in the above equation (in this particular case we have ru = vb = -1) then, once we know the initial concentrations of A and B, the concentration of B at any time can be determined uniquely if the concentration of A is known, i.e., these concentrations are not independent, and so the rate can be formally expressed as a function solely of one concentration or extent of reaction variable. 

The curves in Fig. 5.1(b) with n + I art non-linear but do share the feature that the rate is highest at the beginning of the reaction ( = 0) and the rate decreases monotonically in each case as the extent of reaction increases. This feature is characteristic of deceleratory reactions. As the number of elementary steps and intermediate species (particularly if these involve reactive species such as radicals) increases, so the possibility of more interesting shapes for the reaction rate curve increases. Two such interesting shapes are illustrated in Fig. 5.1(c). These are characteristic of reactions that display an acceleratory phase at low extents of reaction before attaining a maximum prior to a final deceleratory phase at high extents at the end of the reaction as the state of chemical equilibrium is 

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