For a reaction, `2NO + 2H_(2) to N_(2) + 2H_(2) O`, the possible mechanism is
`NO hArr N_(2)O_(2)`
`N_(2)O_(2) + H_(2) overset("Slow")(to) N_(2)O + H_(2) O`
`N_(2) O + H_(2) O overset("fast")(to) N_(2) + H_(2) O`
What is the rate law and order of the reaction?

Options

A. Rate = `[N_(2)O_(2)]`, order = 1
B. Rate = `[N_(2)O_(2)][H_(2)]`, order = 2
C. Rate = `[N_(2)O_(2)]^(2)` order = 2
D. Rate = `[N_(2)O_(2)]^(2) [H_(2)]`, order = 3

Explanation

To determine the rate law and order of the reaction given the mechanism, we will follow these steps:

### Step 1: Write the Mechanism
The reaction mechanism provided is:
1. \( \text{NO} \rightleftharpoons \text{N}_2\text{O}_2 \) (Equilibrium)
2. \( \text{N}_2\text{O}_2 + \text{H}_2 \overset{\text{Slow}}{\rightarrow} \text{N}_2\text{O} + \text{H}_2\text{O} \)
3. \( \text{N}_2\text{O} + \text{H}_2\text{O} \overset{\text{Fast}}{\rightarrow} \text{N}_2 + \text{H}_2\text{O} \)

### Step 2: Identify the Rate-Determining Step
The rate law is determined by the slowest step of the reaction mechanism, which is the second step:
\[ \text{N}_2\text{O}_2 + \text{H}_2 \rightarrow \text{N}_2\text{O} + \text{H}_2\text{O} \]

### Step 3: Write the Rate Law for the Slow Step
For the slow step, the rate law can be expressed as:
\[ \text{Rate} = k_2 [\text{N}_2\text{O}_2][\text{H}_2] \]
where \( k_2 \) is the rate constant for this step.

### Step 4: Identify the Intermediate
In the rate law, \( \text{N}_2\text{O}_2 \) is an intermediate, which means we need to express its concentration in terms of the reactants.

### Step 5: Use the Equilibrium Expression
From the first step, we can write the equilibrium expression:
\[ K = \frac{[\text{N}_2\text{O}_2]}{[\text{NO}]^2} \]
Rearranging gives:
\[ [\text{N}_2\text{O}_2] = K [\text{NO}]^2 \]

### Step 6: Substitute the Intermediate into the Rate Law
Now substitute \( [\text{N}_2\text{O}_2] \) into the rate law:
\[ \text{Rate} = k_2 (K [\text{NO}]^2) [\text{H}_2] \]
Let \( k' = k_2 K \), then:
\[ \text{Rate} = k' [\text{NO}]^2 [\text{H}_2] \]

### Step 7: Determine the Order of the Reaction
The overall order of the reaction is the sum of the powers of the concentration terms in the rate law:
- The order with respect to \( [\text{NO}] \) is 2.
- The order with respect to \( [\text{H}_2] \) is 1.

Thus, the total order of the reaction is:
\[ \text{Order} = 2 + 1 = 3 \]

### Final Answer
The rate law is:
\[ \text{Rate} = k' [\text{NO}]^2 [\text{H}_2] \]
The order of the reaction is 3.

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