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First, let us consider a reaction that consists of two consecutive irreversible unimolecular processes as shown in Reaction 1.28.

(1.28)

X is the reactant. Z is the product. Y is a reactive intermediate. k₁ and k₂ are rate constants for the two unimolecular processes. In order to determine the way in which the concentrations of the substances change over time, the rate equation for each of the substances is written down as follows (Eq. ) [2]:

(1.29)

(1.30)

(1.31)

Equation 1.30 shows the net rate of increase in the intermediate Y, which is equal to the rate of its formation (k₁[X]) minus the rate of its disappearance (k₂[Y]). Equation 1.31 shows the rate of formation of the product Z. Since Z is produced only from the k₂ step which is a unimolecular process, the rate equation for Z is first order in Y.

It is tedious to obtain the accurate solutions of the above simultaneous differential equations. Appropriate approximations may be employed to ease the situation [2].

In most of the stepwise organic reactions, the intermediates (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 E_a), while the subsequent transformation the intermediate experiences is relatively fast (with a low E_a). 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

(1.32)

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).

(1.33)

[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

(1.34)

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.

(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.

(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).

(1.39)