Question

(a) \(\mathrm{CO}+\mathrm{NO}_{2} \longrightarrow \mathrm{CO}_{2}+\mathrm{NO}\) (b) \(2 \mathrm{NO}_{2} \rightleftharpoons \mathrm{N}_{2} \mathrm{O}_{4}\) \(\quad \mathrm{~N}_{2} \mathrm{O}_{4}+2 \mathrm{CO} \longrightarrow 2 \mathrm{CO}_{2}+2 \mathrm{NO}\) (c) \(2 \mathrm{NO}_{2} \longrightarrow \mathrm{NO}_{3}+\mathrm{NO}\) \(\mathrm{NO}_{3}+\mathrm{CO} \longrightarrow \mathrm{NO}_{2}+\mathrm{CO}_{2}\) (d) \(2 \mathrm{NO}_{2} \longrightarrow 2 \mathrm{NO}+\mathrm{O}_{2}\) \(\mathrm{O}_{2}+2 \mathrm{CO} \longrightarrow 2 \mathrm{CO}_{2}\)At low temperatures, the rate law for the reaction $$ \mathrm{CO}(g)+\mathrm{NO}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g)+\mathrm{NO}(g) $$ is as follows: rate \(=(\) constant \()\left[\mathrm{NO}_{2}\right]^{2} .\) Which of the following mechanisms is consistent with the rate law?

Short answer

Answer: None of the mechanisms (a), (b), (c), and (d) is consistent with the rate law for the reaction between CO and NO₂ at low temperatures.

Step-by-step solution

Analyze Mechanism (a)

Mechanism (a) is given as: CO + NO₂ → CO₂ + NO As this is a single-step or elementary reaction, we can write the rate law directly from the reaction as: rate = k[CO][NO₂] This rate law does not match the given rate law (rate = (constant)[NO₂]²), so mechanism (a) is not consistent with the rate law.

Analyze Mechanism (b)

Mechanism (b) is given as: 2 NO₂ ⇌ N₂O₄ (Step 1) N₂O₄ + 2 CO → 2 CO₂ + 2 NO (Step 2) For Step 1, the rate law is given as: rate₁ = k₁[NO₂]² - k₂[N₂O₄] For Step 2, the rate law is given as: rate₂ = k₃[N₂O₄][CO]² However, the overall rate law should depend only on [NO₂], and in this case, it depends on [CO] as well. Mechanism (b) is not consistent with the given rate law (rate = (constant)[NO₂]²).

Analyze Mechanism (c)

Mechanism (c) is given as: 2 NO₂ → NO₃ + NO (Step 1) NO₃ + CO → NO₂ + CO₂ (Step 2) For Step 1, the rate law is given as: rate₁ = k₁[NO₂]² For Step 2, the rate law is given as: rate₂ = k₂[NO₃][CO] The overall rate law would depend on [NO₂], [CO], and [NO₃], and we must compare it with the given rate law (rate = (constant)[NO₂]²). Even though Step 1 follows the given rate law, the overall rate law involves other species as well, so mechanism (c) is not consistent with the given rate law.

Analyze Mechanism (d)

Mechanism (d) is given as: 2 NO₂ → 2 NO + O₂ (Step 1) O₂ + 2 CO → 2 CO₂ (Step 2) For Step 1, the rate law is given as: rate₁ = k₁[NO₂]² For Step 2, the rate law is given as: rate₂ = k₂[O₂][CO]² As for Mechanism (c), even though Step 1 follows the given rate law, the overall rate law involves other species. So mechanism (d) is also not consistent with the given rate law (rate = (constant)[NO₂]²).

Conclusion

None of the mechanisms (a), (b), (c), and (d) match the given rate law (rate = (constant)[NO₂]²). Hence, none of these mechanisms is consistent with the rate law for the reaction between CO and NO₂ at low temperatures.