Question

For a reaction \(\mathrm{X} \rightarrow \mathrm{Y}\), the rate law is \(\mathrm{R}=\mathrm{k}[\mathrm{X}]^{2}\), which of the following statements are correct? a. \(\mathrm{K}\) remains constant in the reaction at all temperatures. b. The unit of rate constant Litre \(\mathrm{mole}^{-1} \mathrm{~s}^{-1}\). c. The reaction follows first order kinetics. d. The \(t_{1 / 2}\) of the reaction depends upon initial concentration of reactant.

Short answer

Correct statements: (b) and (d).

Step-by-step solution

Analyze Rate Law Equation

The rate law is given as \( R = k[X]^2 \). This indicates the reaction is second order with respect to \( X \), as the exponent on \( [X] \) is 2.

Determine Units of Rate Constant

For a second-order reaction with rate law \( R = k[X]^2 \), the rate \( R \) has units of \( ext{mol L}^{-1} ext{s}^{-1} \) and concentration \( [X] \) has units of \( ext{mol L}^{-1} \). Thus, \( k \) must have units of \( ext{L mol}^{-1} ext{s}^{-1} \) to ensure the rate units match.

Assess Behavior of Rate Constant \( k \)

The rate constant \( k \) is only constant at a fixed temperature. It changes with temperature, so statement (a) is incorrect.

Evaluate Reaction Order

The reaction is second order overall (due to \( [X]^2 \)), not first order. Therefore, statement (c) is incorrect.

Consider Half-Life Dependence

For second-order reactions, the half-life \( t_{1/2} \) is given by \( t_{1/2} = \frac{1}{k[X]_0} \), meaning it depends on the initial concentration \( [X]_0 \). Hence, statement (d) is correct.

Key concepts

Second-order reactions

In chemistry, understanding reaction orders is crucial to predict how substances behave during a reaction. A second-order reaction is one where the rate of the reaction depends on the concentration of one reactant raised to the second power or two reactants each raised to the first power.
For the given reaction \( \mathrm{X} \rightarrow \mathrm{Y} \), the rate law \( R = k[X]^2 \) clearly indicates it's a second-order reaction with respect to \( X \). The "2" in the exponent signifies that doubling the concentration of \( X \) results in a fourfold increase in the reaction rate.
Second-order reactions can have various implications, such as:

• Altered reaction dynamics, meaning they can proceed faster or slower based on changes in concentration.
• An essential role in determining the lifespan of reactants in any reaction.
• Special considerations in designing reactors and processes in industrial applications.

Understanding these dynamics allows chemists to manipulate conditions to achieve desired outcomes efficiently.

Rate constant units

The rate constant \( k \) is a crucial part of the rate law equation in chemical kinetics. Its units vary depending on the overall order of the reaction. For a second-order reaction like \( \mathrm{R} = k[X]^2 \), identifying the correct units is essential for the calculations to make sense.
To ensure unit consistency, consider:

• The rate \( R \) has units of \( \text{mol L}^{-1} \text{s}^{-1} \), because it represents a change in concentration over time.
• The concentration \([X]\) is typically in \( \text{mol L}^{-1} \).

Thus, for the rate \( R \) units to be balanced, \( k \) must have units of \( \text{L mol}^{-1} \text{s}^{-1} \). These units, \( \text{L mol}^{-1} \text{s}^{-1} \), specifically indicate a second-order reaction, ensuring everything mathematically checks out. This is not constant across all reactions and understanding unit derivation is important for solving kinetics problems efficiently.

Half-life dependency

Second-order reactions present unique characteristics regarding half-life dependency. Unlike first-order reactions where half-life is constant, the half-life of a second-order reaction depends on the initial concentration of the reactant. This can significantly impact reaction monitoring and control in practical settings.
For the rate law \( R = k[X]^2 \), the half-life \( t_{1/2} \) for a second-order reaction is expressed as \( t_{1/2} = \frac{1}{k[X]_0} \).
From this formula, it's clear that:

• The half-life is inversely proportional to the initial concentration \([X]_0\).
• As \([X]_0\) increases, \( t_{1/2} \) decreases, meaning the reactant concentration halves more quickly.

Therefore, in practical scenarios, altering the initial concentration \([X]_0\) is a common technique to control the timescales of a second-order reaction, making this half-life dependency a crucial factor in designing chemical processes.