The forward reaction rate for the nitric oxide-oxygen reaction
`2NO+O_(2) rarr 2NO_(2)` has the rate law as:
`Rate = k[NO]^(2)[O_(2)]`.
If the mechanism is assumed to be:
`2NO+O overset(k_(eq))hArr` , (rapid equilibration)
`N_(2)O_(2) + O_(2) overset(k_(2))rarr 2NO_(2)` (slow step),
then which of the following is (are) correct?
(I) Rate constant `= k_(eq)k_(2)` , (II) `[N_(2)O_(2)] = k_(eq)[NO]^(2)`
(III) `[N_(2)O_(2)] = k_(eq)[NO]` , (IV) Rate constant `= k_(2)`
The correct option is

Options

A. I, II
B. III, IV
C. I, III
D. None of these

Explanation

To solve the problem, we will analyze the given reaction mechanism and the rate law provided. The reaction is:

\[ 2NO + O_2 \rightarrow 2NO_2 \]

with the rate law:

\[ \text{Rate} = k[NO]^2[O_2] \]

The proposed mechanism is:

1. \( 2NO + O \overset{k_{eq}}{\rightleftharpoons} N_2O_2 \) (rapid equilibrium)
2. \( N_2O_2 + O_2 \overset{k_2}{\rightarrow} 2NO_2 \) (slow step)

### Step 1: Understanding the Equilibrium Step
The first step is an equilibrium step, which means that we can express the equilibrium constant \( k_{eq} \) as:

\[ k_{eq} = \frac{[N_2O_2]}{[NO]^2} \]

From this, we can rearrange to find the concentration of \( N_2O_2 \):

\[ [N_2O_2] = k_{eq} [NO]^2 \]

### Step 2: Rate of the Slow Step
The second step is the slow step and is the rate-determining step. The rate of this step can be expressed as:

\[ \text{Rate} = k_2 [N_2O_2][O_2] \]

### Step 3: Substituting for \( [N_2O_2] \)
Now, we substitute the expression for \( [N_2O_2] \) from Step 1 into the rate expression from Step 2:

\[ \text{Rate} = k_2 (k_{eq} [NO]^2)[O_2] \]

This simplifies to:

\[ \text{Rate} = k_2 k_{eq} [NO]^2 [O_2] \]

### Step 4: Identifying the Overall Rate Constant
From the derived rate expression, we can identify the overall rate constant \( k \):

\[ k = k_2 k_{eq} \]

### Step 5: Evaluating the Options
Now we can evaluate the options given in the question:

1. **(I) Rate constant \( = k_{eq} k_2 \)**: This is correct as derived above.
2. **(II) \( [N_2O_2] = k_{eq} [NO]^2 \)**: This is also correct as derived from the equilibrium expression.
3. **(III) \( [N_2O_2] = k_{eq} [NO] \)**: This is incorrect; it should be \( [N_2O_2] = k_{eq} [NO]^2 \).
4. **(IV) Rate constant \( = k_2 \)**: This is incorrect; the rate constant is \( k_2 k_{eq} \).

### Conclusion
The correct options are (I) and (II).