`3BrO^(-) to BrO_(3)^(-) + 3Br^(-)`
If `-(d[BrO^(-)])/(dt) = k_(1) [BrO^(-)]^(2), +(d[BrO_(3)^(-)])/(dt) =k_(2)[BrO^(-)]^(2) + (d[Br^(-)])/(dt) =k_(3)[BrO^(-)]^(2)`, the correct relation between `k_(1),k_(2)` and `k_(3)` is

Options

A. `3k_(1) = k_(2) = 2k_(3)`
B. `k_(1) = 3k_(2) = 1.5 k_(3)`
C. `k_(1) = k_(2)= k_(3)`
D. `2k_(1) = 3k_(2) = k_(3)`

Explanation

To solve the problem, we need to analyze the given reaction and the rates of change of the concentrations of the species involved. The reaction is:

\[ 3 \text{BrO}^- \rightarrow \text{BrO}_3^- + 3 \text{Br}^- \]

We are given the following rate expressions:

1. \(-\frac{d[\text{BrO}^-]}{dt} = k_1 [\text{BrO}^-]^2\)
2. \(\frac{d[\text{BrO}_3^-]}{dt} = k_2 [\text{BrO}^-]^2\)
3. \(\frac{d[\text{Br}^-]}{dt} = k_3 [\text{BrO}^-]^2\)

### Step 1: Write the rate of the reaction in terms of stoichiometric coefficients.

The rate of the reaction can be expressed in terms of the change in concentration of the reactants and products:

\[ \text{Rate} = -\frac{1}{3} \frac{d[\text{BrO}^-]}{dt} = \frac{1}{1} \frac{d[\text{BrO}_3^-]}{dt} = \frac{1}{2} \frac{d[\text{Br}^-]}{dt} \]

### Step 2: Substitute the rate expressions into the stoichiometric rate equation.

From the first expression, we have:

\[ \text{Rate} = -\frac{1}{3} \left( k_1 [\text{BrO}^-]^2 \right) \]

From the second expression, we have:

\[ \text{Rate} = k_2 [\text{BrO}^-]^2 \]

From the third expression, we have:

\[ \text{Rate} = \frac{1}{2} \left( k_3 [\text{BrO}^-]^2 \right) \]

### Step 3: Set the rates equal to each other.

Since all expressions represent the same rate, we can set them equal to each other:

\[ -\frac{1}{3} k_1 [\text{BrO}^-]^2 = k_2 [\text{BrO}^-]^2 = \frac{1}{2} k_3 [\text{BrO}^-]^2 \]

### Step 4: Eliminate the common factor \([\text{BrO}^-]^2\).

Assuming \([\text{BrO}^-]^2 \neq 0\), we can divide through by \([\text{BrO}^-]^2\):

\[ -\frac{1}{3} k_1 = k_2 = \frac{1}{2} k_3 \]

### Step 5: Express \(k_1\), \(k_2\), and \(k_3\) in terms of each other.

From the first part, we can express \(k_1\) in terms of \(k_2\):

\[ k_1 = -3k_2 \]

From the second part, we can express \(k_3\) in terms of \(k_2\):

\[ k_3 = 2k_2 \]

### Step 6: Final relation between \(k_1\), \(k_2\), and \(k_3\).

Now substituting \(k_3\) back into the expression for \(k_1\):

\[ k_1 = -3k_2 \quad \text{and} \quad k_3 = 2k_2 \]

Thus, we can summarize the relations as:

\[ k_1 = -3k_2 \quad \text{and} \quad k_2 = \frac{1}{3}k_1 \quad \text{and} \quad k_3 = 2k_2 \]

### Conclusion:

The correct relation between \(k_1\), \(k_2\), and \(k_3\) is:

\[ k_1 = 3k_2 \quad \text{and} \quad k_3 = \frac{3}{2}k_2 \]