In the following reaction, how is the rate of appearance of the underlined Product related to the rate of disappearance of the underlined reactant ?
`BrO_(3)^(ɵ)(aq) + 5underline(Br)^(ɵ)(aq) + 6H^(o+) (aq) rarr 3underline(Br_(2))(l) + 3H_(2)O(l)`

Options

A. `-(d[BrO_(3)^(-)])/(dt)=(d[Br_(2)])/(dt)`
B. `-(1)/(3)(d[BrO_(3)^(-)])/(dt)=(d[Br_(2)])/(dt)`
C. `-(d[BrO_(3)^(-)])/(dt)=(1)/(3)(d[Br_(2)])/(dt)`
D. None of these

Answer

To determine the relationship between the rate of appearance of the product \( \text{Br}_2 \) and the rate of disappearance of the reactant \( \text{BrO}_3^- \) in the reaction:

\[ \text{BrO}_3^-(aq) + 5\text{Br}^-(aq) + 6\text{H}^+(aq) \rightarrow 3\text{Br}_2(l) + 3\text{H}_2O(l) \]

we can use the concept of reaction rates.

### Step 1: Write the Rate Expressions
The rate of a reaction can be expressed in terms of the change in concentration of reactants and products over time. For the given reaction, we can write:

\[ -\frac{d[\text{BrO}_3^-]}{dt} = \frac{1}{5} \frac{d[\text{Br}^-]}{dt} = \frac{1}{6} \frac{d[\text{H}^+]}{dt} = \frac{1}{3} \frac{d[\text{Br}_2]}{dt} \]

### Step 2: Relate the Rates
From the rate expressions, we can relate the rate of disappearance of \( \text{BrO}_3^- \) to the rate of appearance of \( \text{Br}_2 \):

\[ -\frac{d[\text{BrO}_3^-]}{dt} = \frac{1}{3} \frac{d[\text{Br}_2]}{dt} \]

### Step 3: Rearranging the Equation
To express the rate of appearance of \( \text{Br}_2 \) in terms of the rate of disappearance of \( \text{BrO}_3^- \), we can rearrange the equation:

\[ \frac{d[\text{Br}_2]}{dt} = -3 \frac{d[\text{BrO}_3^-]}{dt} \]

### Final Answer
Thus, the relationship between the rate of appearance of \( \text{Br}_2 \) and the rate of disappearance of \( \text{BrO}_3^- \) is given by:

\[ \frac{d[\text{Br}_2]}{dt} = -3 \frac{d[\text{BrO}_3^-]}{dt} \]