The decomposition of \(H_2O_2\) in alkaline medium, in presence of \(I^-\) ion occurs as follows:

\(2H_2O_2 \overset{I^-, \text{alk. medium}}{\longrightarrow}2H_2O + O_2\)

If the reaction occurs in the following two steps:

\(\text{Step 1: }H_2O_2 + I^- \underset{slow}{\longrightarrow} H_2O + IO^-\)

\(\text{Step 2: }H_2O_2 + IO^- \underset{fast}{\longrightarrow} H_2O + I^- + O_2\)

The correct rate law expression for the reaction is:

\(r = k[H_2O_2][I^-]\)

The correct answer is option 2. \(r = k[H_2O_2][I^-]\).

To determine the correct rate law expression for the given reaction, we need to analyze the steps of the reaction and the rate-determining step.

Given Reaction Steps

Step 1 (Slow Step): 

\(H_2O_2 + I^- \underset{slow}{\longrightarrow} H_2O + IO^-\)

Step 2 (Fast Step):

\(H_2O_2 + IO^- \underset{fast}{\longrightarrow} H_2O + I^- + O_2\)

Rate Law Determination

Identify the Rate-Determining Step: The rate-determining step is the slow step. The overall rate of the reaction depends on this slow step.

\(H_2O_2 + I^- \underset{slow}{\longrightarrow} H_2O + IO^-\)

Rate Law from Slow Step: For the slow step, the rate law is determined by the concentrations of the reactants in this step. The rate law expression for the slow step is:

\( r = k[\text{H}_2\text{O}_2][\text{I}^-] \)

Here, \(k\) is the rate constant, and \([\text{H}_2\text{O}_2]\) and \([\text{I}^-]\) are the concentrations of hydrogen peroxide and iodide ion, respectively.

Checking the Fast Step:

The fast step involves the reaction between \(\text{H}_2\text{O}_2\) and \(\text{IO}^-\), but since \(\text{IO}^-\) is a product of the slow step and not a reactant in the overall rate law, its concentration is not part of the rate law expression for the overall reaction rate.

Correct Rate Law Expression

Given that the rate law is based on the slow step, the correct rate law expression for the reaction is:

\( r = k[\text{H}_2\text{O}_2][\text{I}^-] \)

The correct rate law expression is: 2. \( r = k[\text{H}_2\text{O}_2][\text{I}^-] \)