that at a given time t, the molar concentration of the product P is x. Therefore, the molar concentrations of reactants A and B are [A] = [A]0 − x and [B] = [B]0 − x, respectively. [A]0 and [B]0 are initial concentrations of reactants A and B, respectively.
From Equation 1.18, we have
If the quantities of the two reactants A and B are in stoichiometric ratio ([A] 0 = [B]0), Equation 1.19 becomes
Rearranging Equation 1.20 leads to Equation 1.21.
Integrating Equation 1.21 on both sides and applying the boundary condition t = 0, x = 0, we have
From Equation 1.22, we have
Since [A] = [A]0 − x, Equation 1.23 becomes
If the reactants A and B have different initial concentrations, Equation 1.19 becomes
Integrating Equation 1.24 on both sides and applying the boundary condition t = 0, x = 0, we have
From Equation 1.25, we have
Since [A] = [A]0 − x and [B] = [B]0 − x, Equation 1.26 becomes
Equation 1.27 represents the integrated rate law for a bimolecular reaction involving two different reactant molecules with different initial concentrations.
If one of the reactants (such as B) in Equation 1.5 (the bimolecular reaction: A + B ➔ P) is in large excess (typically 10–20‐folds, i.e., [B]0/[A]0 = 10–20), the change in molar concentration of reactant B in the course of the reaction can be neglected ([B] ~ [B]0) [2]. The rate law (Eq. 1.18) becomes
Let k′ = k[B]0 (the observed rate constant). We have
The reaction becomes pseudo first order. The integrated rate law is
1.4.2 Reactive Intermediates and the Steady‐State Assumption
First, let us consider a reaction that consists of two consecutive irreversible unimolecular processes as shown in Reaction 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]:
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
