The reaction $$ 2 \mathrm{NO}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{NO}_{2}(g) $$ exhibits the rate law $$ \text {Rate} =k[\mathrm{NO}]^{2}\left[\mathrm{O}_{2}\right] $$ Which of the following mechanisms is consistent with this rate law? $$ \begin{array}{l}{\text { a. } \mathrm{NO}+\mathrm{O}_{2} \longrightarrow \mathrm{NO}_{2}+\mathrm{O}} \\ {\mathrm{O}+\mathrm{NO} \longrightarrow \mathrm{NO}_{2}} \\ {\text { b. } \mathrm{NO}+\mathrm{O}_{2} \rightleftharpoons \mathrm{NO}_{3}} \\ {\mathrm{NO}_{3}+\mathrm{NO} \longrightarrow 2 \mathrm{NO}_{2}}\end{array} $$ $$ \begin{array}{l}{\text { c. } 2 \mathrm{NO} \longrightarrow \mathrm{N}_{2} \mathrm{O}_{2}} \\ {\mathrm{N}_{2} \mathrm{O}_{2}+\mathrm{O}_{2} \longrightarrow \mathrm{N}_{2} \mathrm{O}_{4}} \\ {\mathrm{N}_{2} \mathrm{O}_{4} \longrightarrow 2 \mathrm{NO}_{2}} \\ {\text { d. } 2 \mathrm{NO} \rightleftharpoons \mathrm{N}_{2} \mathrm{O}_{2}} \\\ {\mathrm{N}_{2} \mathrm{O}_{2} \longrightarrow \mathrm{NO}_{2}+\mathrm{O}} \\\ {\mathrm{O}+\mathrm{NO} \longrightarrow \mathrm{NO}_{2}}\end{array} $$

The rate-determining step is the slowest step in the mechanism, and it determines the speed of the overall reaction. For each mechanism, find the rate-determining step and the rate law associated with it.

Now, we will compare each rate law associated with the rate-determining step of each mechanism, and see which one matches the given rate law: \(\text{Rate} =k[\mathrm{NO}]^{2}[\mathrm{O}_{2}].\) For mechanism a: a1. NO + O2 → NO2 + O a2. O + NO → NO2 The slowest step (rate-determining step) in this mechanism might be a1 or a2. If it's a1, the rate law would be: \(\text{Rate} =k[\mathrm{NO}][\mathrm{O}_{2}]\), which does not match the given rate law. For mechanism b: b1. NO + O2 ⇌ NO3 (equilibrium step) b2. NO3 + NO → 2 NO2 The slowest step (rate-determining step) in this mechanism is b2. In this case, the rate law would be: \(\text{Rate} =k[\mathrm{NO}][\mathrm{NO}_{3}]\), and because of the equilibrium in step b1, \([\mathrm{NO}_{3}]\) is proportional to \([\mathrm{NO}][\mathrm{O}_{2}]\). Therefore, the rate law matches the given rate law. For mechanism c: c1. 2 NO → N2O2 (rate-limiting step) c2. N2O2 + O2 → N2O4 c3. N2O4 → 2 NO2 The slowest step (rate-determining step) in this mechanism is c1. The rate law associated with this step would be: \(\text{Rate} =k[\mathrm{NO}]^{2}\), which does not match the given rate law. For mechanism d: d1. 2 NO ⇌ N2O2 (equilibrium step) d2. N2O2 → NO2 + O d3. O + NO → NO2 The slowest step (rate-determining step) in this mechanism could be d2 or d3. If it's d2, the rate law would be: \(\text{Rate} =k[\mathrm{N}_{2}\mathrm{O}_{2}]\), which does not match the given rate law.

Based on the comparison in Step 2, we find that only mechanism b has a rate-determining step whose rate law is consistent with the given rate law. So, the correct answer is mechanism b.