giving product molecules. Common examples of bimolecular reactions are thermal decomposition of hydrogen iodide (HI) to elemental iodine (I2) and hydrogen (H2) (Reaction 1.6), the SN2 reaction of hydroxide with bromomethane (Reaction 1.7), and Diels–Alder reaction of 1,3‐butadiene and ethylene (Reaction 1.8).
Almost all the concerted processes in organic reactions are either unimolecular or bimolecular steps.
1.4 KINETICS
1.4.1 Rate‐Laws for Elementary (Concerted) Reactions
For elementary reactions, the reaction orders are consistent with the molecularity. A unimolecular reaction is the first‐order in the reactant and a bimolecular reaction has a second‐order rate law.
Unimolecular reactions
A unimolecular reaction (Eq. 1.1: A ➔ P) follows the first‐order rate law as shown in Equation 1.9
where k is the rate constant (with the typical unit of s−1) for the reaction, and it is independent of the concentration of the reactant. The rate constant is the quantitative measure of how fast the reaction proceeds at a certain temperature.
Rearranging Equation 1.9 leads to
Integrating Equation 1.10 on both sides and applying the boundary condition t = 0, [A] = [A]0 (initial concentration), we have
From Equation 1.11, we have ln[A] – ln[A]0 = −kt
Therefore,
Equation 1.12 is the integrated rate law for a unimolecular reaction.
The half‐life (t1/2) of reactant A (the time required for conversion of one‐half of the reactant to the product, i.e., when t = t1/2, [A] = ½ [A]0) can be solved from Equation 1.12 as follows:
Therefore,
Equation 1.13 shows that the half‐life of a substance that undergoes first‐order decay is inversely proportional to the rate constant and independent of the initial concentration.
Bimolecular reactions
A bimolecular reaction that involves two reactant molecules of the same compound (Eq. 1.4: 2A ➔ P) follows the second‐order rate law as shown below:
where k is the rate constant (with the typical unit of M−1s−1) for the reaction.
Rearranging Equation 1.14 leads to
Integrating Equation 1.15 on both sides and applying the boundary condition t = 0, [A] = [A]0 (initial concentration), we have
From Equation 1.16, we have
Equation 1.17 is the integrated rate law for a bimolecular reaction involving two molecules from the same compound.
A bimolecular reaction that involves two reactant molecules of different compounds (Eq. 1.5: A + B ➔ P) also follows the second‐order rate law (first‐order in each of the reactants) as shown in Equation 1.18.
Assume
