The reaction \( {A}_2 + {B}_2 \to 2{AB} \) follows the mechanism:
\[
{A}_2 \xrightarrow{k_1} {A} + {A} \ ({fast}) \quad
{A} + {B}_2 \xrightarrow{k_2} {AB} + {B} \ ({slow}) \quad
{A} + {B} \to {AB} \ ({fast})
\]
The overall order of the reaction is:
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The overall order of a reaction is determined by the rate-determining step (slow step). If there is equilibrium in one of the steps, the concentration of that reactant can be substituted with the equilibrium expression.
The rate law for the overall reaction is dictated by the slowest step in the mechanism: \[ \text{Rate} = k_2 [{A}][{B}_2] \]. Given that the rapid step, \( {A}_2 \to {A} + {A} \), is at equilibrium, the equilibrium constant \( k_1 \) can be employed to express \( [{A}] \) as a function of \( [{A}_2] \): \[ [{A}] = \sqrt{k_1 [{A}_2]} \]. Upon substitution into the rate law: \[ \text{Rate} = k_2 \sqrt{k_1 [{A}_2]} [{B}_2] \]. Consequently, the overall rate law is: \[ \text{Rate} = k [{A}_2]^{1/2} [{B}_2]^1 \]. The total order of the reaction is \( 1/2 + 1 = 1.5 \).
