(such as radicals and carbocations) possess high energies and are unstable and highly reactive. Therefore, the formation of such an intermediate is usually relatively slow (with a high Ea), while the subsequent transformation the intermediate experiences is relatively fast (with a low Ea). Mathematically, k1 is much smaller than k2 (k1 ≪ k2). As a result, the concentration of the reactive intermediate (such as Y in Reaction 1.28) remains at a low level and it essentially does not change in the course of the overall reaction. This is referred to as the steady‐state approximation. The changes in concentrations of the reactant, intermediate, and product over time for Reaction 1.28 are illustrated in Figure 1.2. It shows that the intermediate Y in Reaction 1.28 remains in a steady‐state (an essentially constant low concentration) in the course of the overall reaction, formulated as
FIGURE 1.2 The changes in concentrations of the reactant (X), intermediate (Y), and product (Z) over time for Reaction 1.28. The intermediate Y is shown to remain in a steady‐state (d[Y]/dt = 0) in the course of the overall reaction.
In general, the steady‐state approximation is applicable to all types of reaction intermediates in organic chemistry. Equation 1.32 is the mathematical form of the steady‐state assumption.
With the help of the steady‐state approximation, the dependence of concentrations of all the substances in Reaction 1.28 on time can be obtained readily [2].
Integration of Equation 1.29 leads to Equation 1.33 (c.f. Eqs 1.9–1.12).
[X]0 is the initial concentration of X.
From Equation 1.30 (rate equation for Y) and Equation 1.32 (steady‐state assumption for Y), we have
Substituting Equation 1.33 for Equation 1.34, we have
(1.35)
On the basis of the stoichiometry for Reaction 1.28, the initial concentration of the reactant X can be formulated as
Therefore,
(1.36)
Combination of Equations gives Equation 1.37.
Equations show the dependence of concentrations of all the substances in Reaction 1.28 on time [2].
Substituting Equation 1.34 (derived from the steady‐state approximation for Y) for Equation 1.31 gives Equation 1.38.
Comparing Equations 1.29 and 1.38 indicates that rate for consumption of the reactant X is approximately equal to rate for the formation of the product Z (Eq. 1.39).
1.4.3 Rate‐Laws for Stepwise Reactions
Let us use the following consecutive reaction (Reaction 1.40) that involves both reversible and irreversible elementary processes to demonstrate the general procedure for obtaining rate laws for stepwise reactions [3]:
The rate (r) for the overall reaction can be expressed as an increase in concentration of the product (Z) per unit time (Eq. 1.41):
Since Z is produced only from the k2 step which is a unimolecular process, the rate equation for Z is first order in Y.
The steady‐state assumption is applied to the intermediate Y, and its rate
