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Step-by-Step Solution
Step 1: Understand the Proposed Mechanism
The reaction mechanism is proposed in two steps:
(i) $NO(g) + Br_2(g) \leftrightharpoons NOBr_2(g)$
(ii) $NOBr_2(g) + NO(g) \rightarrow 2\,NOBr(g)$
The second step (ii) is the rate-determining step, meaning it effectively controls the overall rate of the reaction.
Step 2: Write the Rate Law From the Rate-Determining Step
Because the second step is slow (rate determining), its rate law directly sets the rate of the overall reaction:
$ \text{Rate} = k \,[NOBr_2]\,[NO]$
Here, $[NOBr_2]$ appears as an intermediate, which must be eliminated from the final rate expression.
Step 3: Express the Intermediate Concentration
From the first equilibrium step, we have:
$NO(g) + Br_2(g) \leftrightharpoons NOBr_2(g)$
The equilibrium constant for this step is
$K_C = \frac{[NOBr_2]}{[NO] \,[Br_2]}$
Rearranging to find $[NOBr_2]$:
$[NOBr_2] = K_C\, [NO] \,[Br_2]$
Step 4: Substitute the Intermediate Concentration into the Rate Law
Substitute $[NOBr_2]$ from above into the rate law obtained in Step 2:
$ \text{Rate} = k \,[NOBr_2]\,[NO] = k \,\bigl(K_C [NO] [Br_2]\bigr)\,[NO]$
$ \text{Rate} = k\,K_C \,[NO]^2\,[Br_2]$
Step 5: Determine the Reaction Order with Respect to NO
From the final rate expression $ \text{Rate} = k\,K_C \,[NO]^2\,[Br_2]$, it is clear that the reaction is second order with respect to $NO$ (because of $[NO]^2$) and first order with respect to $Br_2$.
Step 6: Conclude the Overall Order with Respect to NO
The problem specifically asks for the order of the reaction with respect to $NO(g)$. We see that it is 2. Hence, the correct answer is 2.
