For the reactions \(5Br^- (aq) + BrO_3(aq) + 6H^+ (aq) \longrightarrow 3Br_2 (aq) + 3H_2O(l)\)
The rate of the reaction is:

one-fifth of rate of disappearance of \(Br^-\) ions

The correct answer is option 1. one-fifth of rate of disappearance of \(Br^-\) ions.

Let us break down the problem in detail by examining the relationship between the reaction rate and the rates of disappearance or formation of different species in the reaction:

\(5 \text{Br}^- (aq) + \text{BrO}_3^- (aq) + 6 \text{H}^+ (aq) \longrightarrow 3 \text{Br}_2 (aq) + 3 \text{H}_2\text{O} (l) \)

The rate of a chemical reaction can be expressed in terms of the rates of change of concentration of the reactants and products. For the given reaction, the rate \( r \) of the reaction can be related to the rates at which the concentrations of reactants are decreasing and the products are increasing.

Rate of Disappearance of Reactants

Disappearance of \( \text{Br}^- \):

From the balanced equation, 5 moles of \( \text{Br}^- \) are consumed for every reaction. The rate of disappearance of \( \text{Br}^- \) can be expressed as:

\(\text{Rate of disappearance of } \text{Br}^- = -\frac{1}{5} \frac{d[\text{Br}^-]}{dt} \)

Here, the negative sign indicates that \( \text{Br}^- \) is decreasing over time. The factor of \(\frac{1}{5}\) comes from the stoichiometric coefficient of \( \text{Br}^- \) in the reaction equation.

Relating the Rate of Reaction to the Rate of Disappearance of \( \text{Br}^- \)

The rate of the reaction \( r \) is related to the rate of disappearance of \( \text{Br}^- \) by the stoichiometric coefficients. Since 5 moles of \( \text{Br}^- \) disappear for every reaction, the rate of reaction is

\(r = -\frac{1}{5} \frac{d[\text{Br}^-]}{dt} \)

Thus, the rate of the reaction \( r \) is one-fifth of the rate of disappearance of \( \text{Br}^- \) ions.

Verification with Other Species

Let us verify the rate of the reaction with respect to other species for completeness:

2. Formation of \( \text{Br}_2 \): From the balanced equation, 3 moles of \( \text{Br}_2 \) are formed. The rate of formation of \( \text{Br}_2 \) is:

\(\text{Rate of formation of } \text{Br}_2 = \frac{1}{3} \frac{d[\text{Br}_2]}{dt} \)

Thus, the rate of reaction is:

\(r = \frac{1}{3} \frac{d[\text{Br}_2]}{dt} \)

3. Disappearance of \( \text{BrO}_3^- \): From the balanced equation, 1 mole of \( \text{BrO}_3^- \) is consumed. The rate of disappearance of \( \text{BrO}_3^- \) is:

\(\text{Rate of disappearance of } \text{BrO}_3^- = -\frac{d[\text{BrO}_3^-]}{dt} \)

Thus, the rate of the reaction is:

\(r = -\frac{d[\text{BrO}_3^-]}{dt}\)

4. Formation of \( \text{H}_2\text{O} \):

From the balanced equation, 3 moles of \( \text{H}_2\text{O} \) are formed. The rate of formation of \( \text{H}_2\text{O} \) is:

\(\text{Rate of formation of } \text{H}_2\text{O} = \frac{1}{3} \frac{d[\text{H}_2\text{O}]}{dt} \)

Thus, the rate of reaction is:

\(r = \frac{1}{3} \frac{d[\text{H}_2\text{O}]}{dt}\)

Conclusion

Given the stoichiometric coefficients in the balanced equation, the rate of the reaction \( r \) is directly proportional to the rate of disappearance of \( \text{Br}^- \) and is one-fifth of that rate. This matches option (1). Other species, while useful for cross-verification, do not change this specific ratio for \( \text{Br}^- \).