Nitric oxide (NO) reacts with oxygen to produce nitrogen dioxide
`2NO_((g))+O_2(g)rarr2N_(2(g))`
If the mechanism of reaction is
`NO+O_2overset(K)hArrNO_3` (fast)
`NO_3+NOoverset(K_1)rarrNO_2+NO_2` (slow)
then rate law is

Options

A. Rate `=K'[NO][O_2]`
B. Rate `=K'[NO][O_2]^2`
C. Rate `=K'[NO]^2[O_2]`
D. Rate `=K'[NO]^3[O_2]`

Answer

To determine the rate law for the reaction of nitric oxide (NO) with oxygen (O₂) to produce nitrogen dioxide (NO₂), we will analyze the provided reaction mechanism step by step.

### Step 1: Identify the Reaction Steps
The reaction mechanism is given as:
1. \( \text{NO} + \text{O}_2 \overset{K}{\rightleftharpoons} \text{NO}_3 \) (fast, equilibrium step)
2. \( \text{NO}_3 + \text{NO} \overset{K_1}{\rightarrow} 2\text{NO}_2 \) (slow step)

### Step 2: Determine the Rate Law from the Slow Step
The rate of the overall reaction is determined by the slowest step in the mechanism. Therefore, the rate law can initially be written based on the slow step:
\[ \text{Rate} = K_1 [\text{NO}_3][\text{NO}] \]

### Step 3: Identify the Intermediate
In this case, \(\text{NO}_3\) is an intermediate, which means it does not appear in the overall balanced equation. To express the rate law without the intermediate, we need to eliminate \([\text{NO}_3]\) from the equation.

### Step 4: Use the Equilibrium Expression
From the first step, which is an equilibrium step, we can write the equilibrium constant \(K\):
\[ K = \frac{[\text{NO}_3]}{[\text{NO}][\text{O}_2]} \]
From this, we can express \([\text{NO}_3]\) in terms of the other concentrations:
\[ [\text{NO}_3] = K [\text{NO}][\text{O}_2] \]

### Step 5: Substitute the Intermediate into the Rate Law
Now, substitute \([\text{NO}_3]\) back into the rate law:
\[ \text{Rate} = K_1 (K [\text{NO}][\text{O}_2]) [\text{NO}] \]
This simplifies to:
\[ \text{Rate} = K_1 K [\text{NO}]^2 [\text{O}_2] \]

### Step 6: Define the Rate Constant
Let \(k' = K_1 K\) (a new constant that combines both constants). Thus, the final rate law can be expressed as:
\[ \text{Rate} = k' [\text{NO}]^2 [\text{O}_2] \]

### Final Answer
The rate law for the reaction is:
\[ \text{Rate} = k' [\text{NO}]^2 [\text{O}_2] \]