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A unimolecular reaction (Eq. 1.1: A ➔ P) follows the first‐order rate law as shown in Equation 1.9

(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

(1.10)

Integrating Equation 1.10 on both sides and applying the boundary condition t = 0, [A] = [A]₀ (initial concentration), we have

(1.11)

From Equation 1.11, we have ln[A] – ln[A]₀ = −kt

Therefore,

(1.12)

Equation 1.12 is the integrated rate law for a unimolecular reaction.

The half‐life (t_1/2) of reactant A (the time required for conversion of one‐half of the reactant to the product, i.e., when t = t_1/2, [A] = ½ [A]₀) can be solved from Equation 1.12 as follows:

Therefore,

(1.13)

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.