The possible mechanism for the reaction :
`2" H"_(2)+2" NO"to"N"_(2)+2" H"_(2)"O"` is (i)`2"NO"iff"N"_(2)"O"_(2)" (ii) ""N"_(2)"O"+"H"_(2)"O"("slow")" (iii) ""N"_(2)"O"+"H"_(2)to"N"_(2)+"H"_(2)"O"("fast")`
What is (i) the rate law for the reaction
(ii) the order of the reaction ?

Explanation

To determine the rate law and the order of the reaction based on the provided mechanism, we can follow these steps:

### Step 1: Identify the Rate-Determining Step
The rate-determining step is the slowest step in the reaction mechanism. According to the provided mechanism:
1. \( 2 \text{NO} \rightarrow \text{N}_2\text{O}_2 \) (fast)
2. \( \text{N}_2\text{O} + \text{H}_2\text{O} \rightarrow \text{N}_2 + 2 \text{H}_2\text{O} \) (slow)
3. \( \text{N}_2\text{O} + \text{H}_2 \rightarrow \text{N}_2 + \text{H}_2\text{O} \) (fast)

The slow step is the second one: \( \text{N}_2\text{O} + \text{H}_2\text{O} \).

### Step 2: Write the Rate Law
The rate law is derived from the rate-determining step. The rate of the reaction is proportional to the concentrations of the reactants involved in this step.

For the slow step:
\[ \text{Rate} = k[\text{N}_2\text{O}][\text{H}_2\text{O}] \]

### Step 3: Substitute for Intermediate
Since \( \text{N}_2\text{O} \) is an intermediate and not a reactant in the overall reaction, we need to express it in terms of the reactants. From the first step, we know that \( 2 \text{NO} \) produces \( \text{N}_2\text{O} \), so we can express \( [\text{N}_2\text{O}] \) in terms of \( [\text{NO}] \).

From the first step:
\[ [\text{N}_2\text{O}] = \frac{1}{2}[\text{NO}]^2 \]

Substituting this into the rate law gives:
\[ \text{Rate} = k \left(\frac{1}{2}[\text{NO}]^2\right)[\text{H}_2\text{O}] \]

### Step 4: Final Rate Law Expression
Thus, the rate law can be simplified to:
\[ \text{Rate} = k'[\text{NO}]^2[\text{H}_2\text{O}] \]
where \( k' = \frac{k}{2} \).

### Step 5: 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. In this case:
- The order with respect to \( \text{NO} \) is 2.
- The order with respect to \( \text{H}_2\text{O} \) is 1.

Therefore, the total order of the reaction is:
\[ 2 + 1 = 3 \]

### Summary of Results
(i) The rate law for the reaction is:
\[ \text{Rate} = k'[\text{NO}]^2[\text{H}_2\text{O}] \]

(ii) The order of the reaction is:
\[ 3 \]