Complex reaction schemes and approximations

Many reactions with mechanisms like those discussed in the previous section have been investigated, i.e. reactions with relatively simple experimental rate laws which are congruent 

Fortunately, introduction of chemical approximations often leads to mathematical simplification of complex rate equations, which allows the reaction to be modelled. The following are the three most commonly used approximations. 

We shall use the representative mechanism in Equation 4.7 comprising three unimolecular steps to illustrate their application each elementary step is assigned a mechanistic rate constant  

Qualitatively, this mechanism involves the reversible formation of an intermediate (B) which partitions between return to starting material (A) and irreversible forward reaction to product (C). Because these are coupled first-order processes, the differential equations can be solved exactly and, after rather involved but standard mathematical manipulations, the analytical solutions for the dependences of the concentrations of A, B and C upon time (assuming no B or C is initially present) are shown in Equations 4.8 [6]  

The mechanism of Equation 4.7 is not especially complicated, yet the rigorous derivation of the rate equations is mathematically challenging, and the concentration-time expressions in Equations 4.8 are complex. It will be clear that when more unimolecular steps are involved in a mechanism, or if bimolecular elementary steps intervene, derivation of analytical solutions may become a formidable task. If the magnitudes of the elementary rate constants are similar, mathematical simplifications are not feasible, so the difficult rigorous methods have to be used. However, approximations become possible when the elementary rate constants are appreciably unequal in magnitude. This allows considerable mathematical simplification of the concentration-time relationships. Fortunately, the approximations are valid for many reactions of interest to organic chemists as we shall demonstrate. 

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