Problem 73 - A Complete Resource Book in Chemistry for JEE Main

Question

In the following reaction, how is the rate of appear ance of the underlined product related to the rate of disappearance of the underlined reactant? \(\mathrm{BrO}_{3}^{-}(\mathrm{aq})+5 \mathrm{Br}^{-}(\mathrm{aq})+6 \mathrm{H}^{+}(\mathrm{aq}) \longrightarrow 3 \mathrm{Br}_{2}(1)\) \(+3 \mathrm{H}_{2} \mathrm{O}(\mathrm{l})\) (a) \(\frac{\mathrm{d}\left[\mathrm{Br}_{2}\right]}{\mathrm{dt}}=-\frac{5}{3} \frac{\mathrm{d}\left[\mathrm{Br}^{-}\right]}{\mathrm{dt}}\) (b) \(\frac{\mathrm{d}\left[\mathrm{Br}_{2}\right]}{\mathrm{dt}}=-\frac{\mathrm{d}[\mathrm{Br}]}{\mathrm{dt}}\) (c) \(\frac{\mathrm{d}\left[\mathrm{Br}_{2}\right]}{\mathrm{dt}}=\frac{\mathrm{d}[\mathrm{Br}-]}{\mathrm{dt}}\) (d) \(\frac{\mathrm{d}\left[\mathrm{Br}_{2}\right]}{\mathrm{dt}}=-\frac{3}{5} \frac{\mathrm{d}\left[\mathrm{Br}^{-}\right]}{\mathrm{dt}}\)

Step-by-Step Solution

Answer

The correct relationship is (d) \( \frac{d[\text{Br}_2]}{dt} = -\frac{3}{5} \frac{d[\text{Br}^-]}{dt} \).

Step 1: Understanding Reaction Stoichiometry

The reaction is \( \text{BrO}_3^- (aq) + 5 \text{Br}^- (aq) + 6 \text{H}^+ (aq) \rightarrow 3 \text{Br}_2(l) + 3 \text{H}_2O (l) \). Underline the reactant \( \text{Br}^- \) and the product \( \text{Br}_2 \). The stoichiometric relationship is crucial to relate rates of disappearance and appearance.

Step 2: Expressing Rate Relationships

Write the rate of disappearance and appearance in terms of the stoichiometry of the reaction:\[ -\frac{1}{5} \frac{d[\text{Br}^-]}{dt} = \frac{1}{3} \frac{d[\text{Br}_2]}{dt} \]This expression comes from comparing the coefficients of \( \text{Br}^- \) and \( \text{Br}_2 \) in the balanced equation.

Step 3: Simplifying the Equation

Rearrange the equation from Step 2 to express the relationship directly between \( \frac{d[\text{Br}_2]}{dt} \) and \( \frac{d[\text{Br}^-]}{dt} \):\[ \frac{d[\text{Br}_2]}{dt} = -\frac{3}{5} \frac{d[\text{Br}^-]}{dt} \]This simplifies to a negative sign indicating \( \text{Br}^- \) disappears as \( \text{Br}_2 \) forms.

Step 4: Selecting the Correct Option

From the given options, match the simplified relationship found in Step 3 with the choices provided. The correct expression is option (d):\[ \frac{d[\text{Br}_2]}{dt} = -\frac{3}{5} \frac{d[\text{Br}^-]}{dt} \]

Key Concepts

Stoichiometry

Stoichiometry is the cornerstone of understanding chemical reactions, as it defines the quantitative relationships between reactants and products in a balanced equation. In a stoichiometric calculation, we use the coefficients from the chemical equation to relate the amounts of each substance present. This concept is critical for predicting how much of a reactant will be consumed or how much of a product will be formed in a reaction.

For instance, in the given reaction \[\mathrm{BrO}_{3}^{-} + 5 \mathrm{Br}^{-} + 6 \mathrm{H}^{+} \rightarrow 3 \mathrm{Br}_{2} + 3 \mathrm{H}_{2} \mathrm{O},\]we observe that 5 moles of \( \mathrm{Br}^{-} \) are needed to produce 3 moles of \( \mathrm{Br}_{2} \). This forms the basis to derive the rate of formation or disappearance of these substances, meaning if \( \mathrm{Br}^{-} \) is disappearing, \( \mathrm{Br}_{2} \) must be forming based on the stoichiometric values. Understanding stoichiometry helps in knowing the proportions and ensuring the balanced nature of reactions.

Rate of Reaction

The rate of a chemical reaction refers to how quickly or slowly reactants are converted into products. Reaction rates can be expressed in terms of the change in concentration of reactants or products per unit time. This can be represented with the formula:\[\text{Rate} = \frac{\Delta [\text{Product}]}{\Delta t} = -\frac{\Delta [\text{Reactant}]}{\Delta t},\]where \(\Delta[\text{Product}]\) and \(\Delta[\text{Reactant}]\) are the changes in concentration over time \(\Delta t\).

To connect stoichiometry and reaction rate, we use stoichiometric coefficients to scale these rates appropriately. The negative sign indicates that as the reactant \( \mathrm{Br}^{-} \) is being consumed, the product \( \mathrm{Br}_{2} \) is forming. Using stoichiometric coefficients, the rate can be related as such: \[\frac{d[\text{Br}_2]}{dt} = -\frac{3}{5} \frac{d[\text{Br}^-]}{dt},\]where the \( -\frac{3}{5} \) reflects the stoichiometric relationship from the balanced equation.

Chemical Equations

Chemical equations are the written representation of chemical reactions, detailing the substances involved, their physical states, and the stoichiometric relationships between them. They provide a clear and concise way to document the transformation of reactants into products.

• Reactants are the starting substances in a chemical reaction, shown on the left-hand side of the equation.
• Products are the substances formed as a result of the reaction, shown on the right-hand side.
• Stoichiometric coefficients indicate the proportional amounts of each substance involved.

The provided equation \[\mathrm{BrO}_{3}^{-} + 5 \mathrm{Br}^{-} + 6 \mathrm{H}^{+} \rightarrow 3 \mathrm{Br}_{2} + 3 \mathrm{H}_{2} \mathrm{O}\]informs us that 1 mole of \(\mathrm{BrO}_{3}^{-}\) reacts with 5 moles of \(\mathrm{Br}^{-}\) to yield 3 moles of \(\mathrm{Br}_{2}\) and water. By writing these symbols and coefficients, chemical equations simply and powerfully describe the condition and outcome of chemical reactions.