The mechanism of the reaction
`2NO + O_(2) rarr 2NO_(2)` is
`NO + NO underset(k_(-1))overset(k_(1))hArr N_(2)O_(2) ("fast")`
`N_(2)O_(2) + O_(2) overset(k_(2))rarr 2NO_(2) (slow)`
The rate constant of the reaction is

Options

A. `k_(2)`
B. `k_(2)k_(1)(k_(-1))`
C. `k_(2)k_(1)`
D. `k_(2)((k_(1))/(k_(-1)))`

Explanation

To find the rate constant of the reaction given the mechanism, we will follow these steps:

### Step 1: Identify the rate-determining step
The mechanism provided consists of two steps:
1. \( NO + NO \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} N_2O_2 \) (fast step)
2. \( N_2O_2 + O_2 \overset{k_2}{\rightarrow} 2NO_2 \) (slow step)

The slow step is the rate-determining step, which means the rate of the overall reaction is governed by this step.

### Step 2: Write the rate expression for the slow step
The rate of the slow step can be expressed as:
\[ \text{Rate} = k_2 [N_2O_2][O_2] \]

### Step 3: Express the concentration of \( N_2O_2 \) in terms of \( NO \)
From the fast step, we can derive the concentration of \( N_2O_2 \) using the equilibrium expression:
\[ \frac{k_1}{k_{-1}} = \frac{[N_2O_2]}{[NO]^2} \]
Rearranging gives:
\[ [N_2O_2] = \frac{k_1}{k_{-1}} [NO]^2 \]

### Step 4: Substitute \( [N_2O_2] \) into the rate expression
Now, substitute the expression for \( [N_2O_2] \) into the rate expression from Step 2:
\[ \text{Rate} = k_2 \left(\frac{k_1}{k_{-1}} [NO]^2\right) [O_2] \]
This simplifies to:
\[ \text{Rate} = \frac{k_2 k_1}{k_{-1}} [NO]^2 [O_2] \]

### Step 5: Identify the overall rate constant
The overall rate constant \( k \) for the reaction can be expressed as:
\[ k = \frac{k_2 k_1}{k_{-1}} \]

### Final Answer
Thus, the rate constant for the given reaction is:
\[ k = \frac{k_2 k_1}{k_{-1}} \]

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