Nitric oxide reacts with hydrogen to give nitrogen and water `(2" NO"+2" H"_(2)to"N"_(2)+2" H"_(2)"O")`. The kinetics of this reaction is explained by the following steps :
(i) `2 "NO"+"H"_(2)to"N"_(2)+"H"_(2)"O"_(2)("slow")" ""(ii) ""H"_(2) "O"_(2)+"H"_(2)to2" H"_(2)"O"("fast")`
What is the predicted rate law ?

Answer

To determine the predicted rate law for the reaction between nitric oxide (NO) and hydrogen (H₂) that produces nitrogen (N₂) and water (H₂O), we need to analyze the provided reaction steps, focusing particularly on the slow step, which is the rate-determining step.

### Step-by-Step Solution:

1. **Identify the Reaction Steps**:
- The first step is the slow step:
\[ 2 \text{NO} + \text{H}_2 \rightarrow \text{N}_2 + \text{H}_2\text{O}_2 \quad (\text{slow}) \]
- The second step is the fast step:
\[ \text{H}_2\text{O}_2 + \text{H}_2 \rightarrow 2 \text{H}_2\text{O} \quad (\text{fast}) \]

2. **Determine the Rate-Determining Step**:
- The rate of a reaction is determined by the slowest step, which is the first step in this case.

3. **Write the Rate Law**:
- The rate law is expressed in terms of the concentrations of the reactants involved in the rate-determining step. For the first step, we have:
- 2 moles of NO and 1 mole of H₂.
- Therefore, the rate law can be written as:
\[ \text{Rate} = k [\text{NO}]^2 [\text{H}_2]^1 \]
- This simplifies to:
\[ \text{Rate} = k [\text{NO}]^2 [\text{H}_2] \]

4. **Determine the Overall Order of the Reaction**:
- The overall order of the reaction is the sum of the powers of the concentrations in the rate law:
- Order with respect to NO = 2
- Order with respect to H₂ = 1
- Therefore, the overall order of the reaction is:
\[ 2 + 1 = 3 \]

### Final Rate Law:
The predicted rate law for the reaction is:
\[ \text{Rate} = k [\text{NO}]^2 [\text{H}_2] \]