In the reaction, `"BrO"_(3)^(-)(aq)+5"Br"^(-)(aq)+6" H"^(+)to3" Br"_(2)(l)+3" H"_(2)O(l),`
the rate of appearance of bromine `("Br"_(2))` is related to the rate of disappearance of bromide ions as follows :

Options

A. `(d[Br_(2)])/(dt)=-(5)/(3)(d[Br^(-)])/(dt)`
B. `(d[Br_(2)])/(dt)=(5)/(3)(d[Br^(-)])/(dt)`
C. `(d[Br_(2)])/(dt)=(3)/(5)(d[Br^(-)])/(dt)`
D. `(d[Br_(2)])/(dt)=-(3)/(5)(d[Br^(-)])/(dt)`

Explanation

To solve the problem, we need to relate the rate of appearance of bromine \( \text{Br}_2 \) to the rate of disappearance of bromide ions \( \text{Br}^- \) in the given reaction:

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

### Step 1: Write the Rate Expressions
For the reaction, we can express the rates of disappearance of reactants and appearance of products as follows:

- The rate of disappearance of bromate \( \text{BrO}_3^- \):
\[ -\frac{d[\text{BrO}_3^-]}{dt} \]

- The rate of disappearance of bromide \( \text{Br}^- \):
\[ -\frac{d[\text{Br}^-]}{dt} \]

- The rate of disappearance of hydrogen ions \( \text{H}^+ \):
\[ -\frac{d[\text{H}^+]}{dt} \]

- The rate of appearance of bromine \( \text{Br}_2 \):
\[ \frac{d[\text{Br}_2]}{dt} \]

- The rate of appearance of water \( \text{H}_2O \):
\[ \frac{d[\text{H}_2O]}{dt} \]

### Step 2: Relate the Rates
From the stoichiometry of the balanced equation, we can relate these rates:

\[ -\frac{1}{1} \frac{d[\text{BrO}_3^-]}{dt} = -\frac{1}{5} \frac{d[\text{Br}^-]}{dt} = -\frac{1}{6} \frac{d[\text{H}^+]}{dt} = \frac{1}{3} \frac{d[\text{Br}_2]}{dt} = \frac{1}{3} \frac{d[\text{H}_2O]}{dt} \]

### Step 3: Focus on \( \text{Br}^- \) and \( \text{Br}_2 \)
To find the relationship between the rate of disappearance of bromide ions and the rate of appearance of bromine, we focus on the rates of \( \text{Br}^- \) and \( \text{Br}_2 \):

From the stoichiometry, we see that for every 5 moles of \( \text{Br}^- \) that disappear, 3 moles of \( \text{Br}_2 \) appear. Therefore, we can write:

\[ -\frac{d[\text{Br}^-]}{dt} = \frac{5}{3} \frac{d[\text{Br}_2]}{dt} \]

### Step 4: Rearranging the Equation
Rearranging the equation gives us:

\[ \frac{d[\text{Br}_2]}{dt} = -\frac{3}{5} \frac{d[\text{Br}^-]}{dt} \]

### Conclusion
Thus, the relationship between the rate of appearance of bromine \( \text{Br}_2 \) and the rate of disappearance of bromide ions \( \text{Br}^- \) is:

\[ -\frac{d[\text{Br}^-]}{dt} = \frac{5}{3} \frac{d[\text{Br}_2]}{dt} \]