Reaction : - `3ClO^(-) to ClO_3^(-) +2Cl^(-) ` occurs in following two steps
`(i) 2ClO^(-) rightarrow ClO_(2)^(-) +Cl^(-) ` ( slow step )
`(ii) ClO_(2) ^(-) +ClO^(-) overset(K_(2)) to ClO_(3)^(-) +Cl^(-) ` ( fast step)
then the rate of given reaction - ______.

Options

A. `K_(1) [ClO^(-) ]^(2)`
B. `K_(1) [ClO^(-)]`
C. `K_(2) [ClO_(2)^(-) ][ClO^(-)]`
D. `K_(2) [ClO^(-)]^(3)`

Answer

To determine the rate of the reaction given by the equation:

\[ 3ClO^{-} \rightarrow ClO_{3}^{-} + 2Cl^{-} \]

which occurs in two steps, we will analyze the slow and fast steps of the reaction mechanism.

### Step 1: Identify the Slow Step
The first step of the reaction is:

\[ 2ClO^{-} \rightarrow ClO_{2}^{-} + Cl^{-} \] (slow step)

Since this is the slow step, it is the rate-determining step (RDS) of the overall reaction. The rate of the overall reaction will depend on this step.

### Step 2: Write the Rate Law for the Slow Step
For the slow step, the rate can be expressed in terms of the concentration of the reactants involved in that step. The rate law for this elementary reaction is:

\[ \text{Rate} = k_1 [ClO^{-}]^2 \]

where \( k_1 \) is the rate constant for the slow step and \([ClO^{-}]\) is the concentration of the \( ClO^{-} \) ion.

### Step 3: Consider the Fast Step
The second step of the reaction is:

\[ ClO_{2}^{-} + ClO^{-} \overset{k_2}{\rightarrow} ClO_{3}^{-} + Cl^{-} \] (fast step)

Since this step is fast and follows the slow step, it does not directly affect the rate law of the overall reaction. However, it is important to note that the concentration of \( ClO_{2}^{-} \) produced in the slow step will be involved in the fast step.

### Step 4: Determine the Rate of the Overall Reaction
Since the slow step is the rate-determining step, we can conclude that the rate of the overall reaction is governed by the rate of the slow step:

\[ \text{Rate} = k_1 [ClO^{-}]^2 \]

This means that the rate of the overall reaction is proportional to the square of the concentration of \( ClO^{-} \).

### Final Answer
Thus, the rate of the given reaction is:

\[ \text{Rate} = k_1 [ClO^{-}]^2 \]