Question

For the reaction \(2 \mathrm{NH}_{3} \rightarrow \mathrm{N}_{2}+3 \mathrm{H}_{2}\) it is found that \(-\frac{\mathrm{dNH}_{3}}{\mathrm{dt}}=\mathrm{K}_{1}\left(\mathrm{NH}_{3}\right)\) \(\frac{\mathrm{d} \mathrm{N}_{2}}{\mathrm{dt}}=\mathrm{K}_{2}\left[\mathrm{NH}_{3}\right]\) \(\frac{\mathrm{dH}_{2}}{\mathrm{dt}}=\mathrm{K}_{3}\left[\mathrm{NH}_{3}\right]\) the correct relation between \(\mathrm{K}_{1}, \mathrm{~K}_{2}\) and \(\mathrm{K}_{3}\) can be given as ? a. \(3 \mathrm{~K}_{1}=2 \mathrm{~K}_{2}=6 \mathrm{~K}_{3}\) b. \(6 \mathrm{~K}_{1}=3 \mathrm{~K}_{2}=2 \mathrm{~K}_{3}\) c. \(\mathrm{K}_{1}=\mathrm{K}_{2}=\mathrm{K}_{3}\) d. \(2 \mathrm{~K}_{1}=3 \mathrm{~K}_{2}=6 \mathrm{~K}_{3}\)

Short answer

The correct relation is option d: \(2 \mathrm{K}_{1}=3 \mathrm{K}_{2}=6 \mathrm{K}_{3}\).

Step-by-step solution

Understand the Reaction Stoichiometry

The given reaction is \(2 \mathrm{NH}_{3} \rightarrow \mathrm{N}_{2} + 3 \mathrm{H}_{2}\). This means 2 moles of \(\mathrm{NH}_{3}\) produce 1 mole of \(\mathrm{N}_{2}\) and 3 moles of \(\mathrm{H}_{2}\).

Determine the Rate Expressions

According to the problem, the rate of disappearance of \(\mathrm{NH}_{3}\) is given by \(-\frac{\mathrm{dNH}_{3}}{\mathrm{dt}}=\mathrm{K}_{1}[\mathrm{NH}_{3}]\). The rates of formation of \(\mathrm{N}_{2}\) and \(\mathrm{H}_{2}\) are \(\frac{\mathrm{d} \mathrm{N}_{2}}{\mathrm{dt}}=\mathrm{K}_{2}[\mathrm{NH}_{3}]\) and \(\frac{\mathrm{dH}_{2}}{\mathrm{dt}}=\mathrm{K}_{3}[\mathrm{NH}_{3}]\) respectively.

Relate Rates with Stoichiometry

From stoichiometry, we have the relations:- \(\frac{1}{2} \times \text{Rate of } \mathrm{NH}_{3} = \text{Rate of } \mathrm{N}_{2}\)- \(\frac{3}{2} \times \text{Rate of } \mathrm{NH}_{3} = \text{Rate of } \mathrm{H}_{2}\)

Establish Relationships Between Rates and Constants

By these stoichiometric ratios:\[ \mathrm{K}_{2} [\mathrm{NH}_{3}] = \frac{1}{2} \mathrm{K}_{1} [\mathrm{NH}_{3}] \Rightarrow \mathrm{K}_{2} = \frac{1}{2} \mathrm{K}_{1} \]\[ \mathrm{K}_{3} [\mathrm{NH}_{3}] = \frac{3}{2} \mathrm{K}_{1} [\mathrm{NH}_{3}] \Rightarrow \mathrm{K}_{3} = \frac{3}{2} \mathrm{K}_{1} \]

Compare with Given Options

Based on the relations, we have \( \mathrm{K}_{1} = 2 \mathrm{K}_{2} \) and \( \mathrm{K}_{1} = \frac{2}{3} \mathrm{K}_{3} \). Converting into a common factor, the option \(2 \mathrm{K}_{1}=3 \mathrm{K}_{2}=6 \mathrm{K}_{3}\) matches these conditions.

Key concepts

Rate Expressions

In reaction kinetics, a rate expression describes how the concentration of a reactant or product changes with time. These expressions help us understand how fast a reaction proceeds. For the reaction \(2\ \mathrm{NH}_{3} \rightarrow \mathrm{N}_{2} + 3\ \mathrm{H}_{2}\), the rate expressions are unique to each species involved.

The rate of disappearance of \(\mathrm{NH}_{3}\) is given by the expression \(-\frac{\mathrm{dNH}_{3}}{\mathrm{dt}}=\mathrm{K}_{1}[\mathrm{NH}_{3}]\). Here, the negative sign indicates that \(\mathrm{NH}_{3}\) is being consumed. Meanwhile, the formation rates of \(\mathrm{N}_{2}\) and \(\mathrm{H}_{2}\) are represented as \(\frac{\mathrm{d} \mathrm{N}_{2}}{\mathrm{dt}}=\mathrm{K}_{2}[\mathrm{NH}_{3}]\) and \(\frac{\mathrm{dH}_{2}}{\mathrm{dt}}=\mathrm{K}_{3}[\mathrm{NH}_{3}]\) respectively.

Each reactant or product in a reaction can exhibit a different rate expression based on the stoichiometry and nature of the reaction. Understanding these expressions is crucial to predicting and controlling reaction rates.

Stoichiometry

Stoichiometry is the study of the quantitative relationships in chemical reactions. It helps determine the ratios in which reactants combine and products form. This concept is integral in relation to reaction rates.

In the given reaction \(2\ \mathrm{NH}_{3} \rightarrow \mathrm{N}_{2} + 3\ \mathrm{H}_{2}\), stoichiometry informs us that 2 moles of \(\mathrm{NH}_{3}\) will produce 1 mole of \(\mathrm{N}_{2}\) and 3 moles of \(\mathrm{H}_{2}\). This allows us to understand how the rates of disappearance and formation relate to each other. By using stoichiometry, we can establish expressions like

• \(\mathrm{Rate\ of\ } \mathrm{N}_{2} = \frac{1}{2} \times \mathrm{Rate\ of\ } \mathrm{NH}_{3}\)
• \(\mathrm{Rate\ of\ } \mathrm{H}_{2} = \frac{3}{2} \times \mathrm{Rate\ of\ } \mathrm{NH}_{3}\)

Stoichiometry helps us link these rates to their respective rate constants, providing a full picture of the reaction's kinetics.

Rate Constants

Rate constants, denoted as \(K\), are specific to the reaction conditions and play a crucial role in the rate expression. They provide the proportionality factor that links the rate of reaction to the concentration of reactants.

In our scenario, the constants \(\mathrm{K}_{1}\), \(\mathrm{K}_{2}\), and \(\mathrm{K}_{3}\) appear in the rate expressions for \(\mathrm{NH}_{3}\), \(\mathrm{N}_{2}\), and \(\mathrm{H}_{2}\) respectively. By analyzing the stoichiometry and rate expressions, we can derive relationships between them:

• \(\mathrm{K}_{2} = \frac{1}{2} \mathrm{K}_{1}\)
• \(\mathrm{K}_{3} = \frac{3}{2} \mathrm{K}_{1}\)

These relations help us solve the problem by finding common factors or ratios among \(\mathrm{K}_{1}, \mathrm{K}_{2},\) and \(\mathrm{K}_{3}\). The conditions and relationships derived from these constants offer insights into the dynamics and efficiency of a chemical reaction.