Fast pre-equilibrium Arrhenius intermediates

When the first step of the mechanism, step (13.11), is a fast eqnilibrium, the intermediate X is in equilibrium with the reactants, and corresponds to Arrhenius s concept of an equilibrium intermediate. This step leads to 

In this expression, the concentrations of C and S, [C] and [S], are their equilibrium concentrations and not their initial concentrations. For initial concentrations, [C](, and [Slg, we can write 

The intermediate X can be eliminated from this expression using eq. (13.4). 

It is often more useful to consider the relative concentrations of the catalyst and substrate and obtain limiting cases for the rate law, rather than carry ont aU the algebra with the quadratic equation. Because [X] cannot be larger than [C]q, we can obtain a first limiting case when the initial concentration of the substrate is relatively high, [S]g [C]q, and, consequently, [S]q-[X] [S]q. Under this approximation eq. (13.4) becomes 

We have seen this behaviour before in Chapter 10 in single-substrate reactions on the surface of solids, and will discuss it again in Chapter 14 in the context of enzyme catalysis. For both cases of surface reactions and enzyme catalysis, the species Y and W do not exist and eq. (13.8) becomes 

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