Nitric oxide NO reacts with oxygen to given nitrogen dioxide `2NO(g)+O_2(g) to 2NO_2 (g)` and the mechanism is:
`NO+O_2 hArr NO_3 (fast), NO_3+NO hArr^(k_1) NO_2+NO_2` (slow). The rate of the reaction will be:

Options

A. `k[NO][O_2]`
B. `k[NO_3][NO]`
C. `k[NO0^2[O_2]`
D. `(k[NO][O_2])/([NO_2])`

Answer

To determine the rate of the reaction between nitric oxide (NO) and oxygen (O2) to form nitrogen dioxide (NO2), we need to analyze the given reaction mechanism step by step.

### Step 1: Identify the Reaction and Mechanism
The overall reaction is:
\[ 2NO(g) + O_2(g) \rightarrow 2NO_2(g) \]

The mechanism provided is:
1. \( NO + O_2 \rightleftharpoons NO_3 \) (fast)
2. \( NO_3 + NO \rightarrow 2NO_2 \) (slow)

### Step 2: Determine the Rate-Determining Step
In a reaction mechanism, the slowest step is often referred to as the rate-determining step (RDS). This is because the rate of the overall reaction cannot exceed the rate of this slow step. Here, the second step is the slow step.

### Step 3: Write the Rate Law for the Rate-Determining Step
Since the second step is the rate-determining step, we can write the rate law based on this step. The rate law is generally expressed as:
\[ \text{Rate} = k[\text{Reactants}] \]
For the slow step:
\[ NO_3 + NO \rightarrow 2NO_2 \]
The rate law can be written as:
\[ \text{Rate} = k_1 [NO_3][NO] \]

### Step 4: Express Concentration of Intermediate
Since \( NO_3 \) is an intermediate and not present in the overall balanced equation, we need to express its concentration in terms of the reactants. From the fast step, we can derive the equilibrium expression:
\[ K = \frac{[NO_3]}{[NO][O_2]} \]
Thus, we can express \( [NO_3] \) as:
\[ [NO_3] = K[NO][O_2] \]

### Step 5: Substitute Back into the Rate Law
Now, substituting \( [NO_3] \) back into the rate law:
\[ \text{Rate} = k_1 [NO_3][NO] \]
Substituting for \( [NO_3] \):
\[ \text{Rate} = k_1 (K[NO][O_2])[NO] \]
\[ \text{Rate} = k' [NO]^2 [O_2] \]
Where \( k' = k_1 K \) is the new rate constant.

### Final Rate Expression
Thus, the rate of the reaction can be expressed as:
\[ \text{Rate} = k' [NO]^2 [O_2] \]

### Conclusion
The rate of the reaction is proportional to the square of the concentration of nitric oxide and the concentration of oxygen.

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