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 for the reaction mechanism given, we will follow these steps:

### Step 1: Identify the Rate Determining Step (RDS)
The slow step in the mechanism is the rate determining step. The overall reaction is:
\[ 2NO + O_2 \rightarrow 2NO_2 \]
The slow step is:
\[ N_2O_2 + O_2 \overset{k_2}{\rightarrow} 2NO_2 \]

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

### Step 3: Analyze the Fast Step
The fast step is:
\[ NO + NO \overset{k_1}{\rightleftharpoons} N_2O_2 \]
For this step, we can write the rate expressions for the forward and reverse reactions:
- Forward: \( \text{Rate}_{\text{forward}} = k_1 [NO]^2 \)
- Reverse: \( \text{Rate}_{\text{reverse}} = k_{-1} [N_2O_2] \)

Since the fast step is reversible and at equilibrium, we can set the rates equal:
\[ k_1 [NO]^2 = k_{-1} [N_2O_2] \]

### Step 4: Solve for [N2O2]
Rearranging the equation gives us:
\[ [N_2O_2] = \frac{k_1}{k_{-1}} [NO]^2 \]

### Step 5: Substitute [N2O2] into the Rate Expression
Now, substitute the expression for [N2O2] into the rate expression from Step 2:
\[ \text{Rate} = k_2 \left(\frac{k_1}{k_{-1}} [NO]^2\right)[O_2] \]

### Step 6: Simplify the Rate Expression
This simplifies to:
\[ \text{Rate} = \frac{k_2 k_1}{k_{-1}} [NO]^2 [O_2] \]

### Step 7: Identify the Overall Rate Constant
The overall rate constant \( k \) for the reaction can be defined as:
\[ k = \frac{k_2 k_1}{k_{-1}} \]

### Conclusion
Thus, the rate constant for the reaction \( 2NO + O_2 \rightarrow 2NO_2 \) is:
\[ k = \frac{k_2 k_1}{k_{-1}} \]