Which of the following expressions is correct for the rate of reaction given below?
` 5 Br^(-) (aq) +BrO_(3)^(-) (aq) + 6H^(+) (aq) to 3Br_2(aq) + 2H_2 O(l)`

Options

A. `(Delta [Br^(-)])/(Delta t)=5 ( Delta [H^(+)])/( Delta t)`
B. `(Delta [Br^(-)])/(Delta t)=6/5 ( Delta [H^(+)])/( Delta t)`
C. `(Delta [Br^(-)])/(Delta t)=5/6 ( Delta [H^(+)])/( Delta t)`
D. `(Delta [Br^(-)])/(Delta t)=6 ( Delta [H^(+)])/( Delta t)`

Explanation

To determine the correct expression for the rate of the given reaction:

**Reaction:**
\[ 5 \text{Br}^- (aq) + \text{BrO}_3^- (aq) + 6 \text{H}^+ (aq) \rightarrow 3 \text{Br}_2 (aq) + 2 \text{H}_2\text{O} (l) \]

**Step 1: Write the Rate Expression**
The rate of a reaction can be expressed in terms of the change in concentration of reactants and products over time. The general form is:

\[ R = -\frac{1}{n} \frac{\Delta [A]}{\Delta t} \]

where \( n \) is the stoichiometric coefficient of the reactant \( A \).

**Step 2: Identify Stoichiometric Coefficients**
From the balanced equation, we have:
- For \( \text{Br}^- \): coefficient = 5
- For \( \text{BrO}_3^- \): coefficient = 1
- For \( \text{H}^+ \): coefficient = 6
- For \( \text{Br}_2 \): coefficient = 3
- For \( \text{H}_2\text{O} \): coefficient = 2

**Step 3: Write the Rate in Terms of Each Reactant and Product**
Using the stoichiometric coefficients, we can write the rate expressions for each species:

1. For \( \text{Br}^- \):
\[ R = -\frac{1}{5} \frac{\Delta [\text{Br}^-]}{\Delta t} \]

2. For \( \text{BrO}_3^- \):
\[ R = -\frac{1}{1} \frac{\Delta [\text{BrO}_3^-]}{\Delta t} \]

3. For \( \text{H}^+ \):
\[ R = -\frac{1}{6} \frac{\Delta [\text{H}^+]}{\Delta t} \]

4. For \( \text{Br}_2 \):
\[ R = \frac{1}{3} \frac{\Delta [\text{Br}_2]}{\Delta t} \]

5. For \( \text{H}_2\text{O} \):
\[ R = \frac{1}{2} \frac{\Delta [\text{H}_2\text{O}]}{\Delta t} \]

**Step 4: Equate the Expressions**
All these expressions represent the same rate \( R \), so we can set them equal to each other.

For example, equating the rate expressions for \( \text{Br}^- \) and \( \text{H}^+ \):
\[ -\frac{1}{5} \frac{\Delta [\text{Br}^-]}{\Delta t} = -\frac{1}{6} \frac{\Delta [\text{H}^+]}{\Delta t} \]

From this, we can derive:
\[ \frac{\Delta [\text{Br}^-]}{\Delta [\text{H}^+]} = \frac{5}{6} \]

**Step 5: Check Each Expression**
Now, we can check the expressions given in the question against our derived relationships.

- The expression for \( \text{Br}^- \) in terms of \( \text{H}^+ \) should be:
\[ \Delta [\text{Br}^-] = \frac{5}{6} \Delta [\text{H}^+] \]

Thus, the correct expression for \( \Delta [\text{H}^+] \) in terms of \( \Delta [\text{Br}^-] \) would be:
\[ \Delta [\text{H}^+] = \frac{6}{5} \Delta [\text{Br}^-] \]

**Final Conclusion: Identify the Correct Expression**
After analyzing the expressions, we can determine which one is correct based on the stoichiometric relationships derived from the rate expressions.