Question

The rate constant, the activation energy and the Arrhenius parameter of a chemical reaction at \(25^{\circ} \mathrm{C}\) are \(3.0 \times 10^{4} \mathrm{~s}^{-1}, 104.4 \mathrm{~kJ} \mathrm{~mol}^{-1}\) and \(6 \times 10^{14} \mathrm{~s}^{-1}\) respectively. The value of the rate constant as \(\mathrm{T} \longrightarrow \infty\) is (a) \(2.0 \times 10^{18} \mathrm{~s}^{-1}\) (b) \(6.0 \times 10^{14} \mathrm{~s}^{-1}\) (c) infinity (d) \(3.6 \times 10^{30} \mathrm{~s}^{-1}\)

Short answer

(b) \(6.0 \times 10^{14} \mathrm{~s}^{-1}\)

Step-by-step solution

Understand the Problem

We need to determine the value of the rate constant, \(k\), as the temperature \(T\) approaches infinity. We know that the rate constant \(k\) is given by the Arrhenius equation.

Write the Arrhenius Equation

The Arrhenius equation is given by: \[ k = A \cdot e^{-\frac{E_a}{RT}} \] where \(A\) is the Arrhenius parameter (frequency factor), \(E_a\) is the activation energy, \(R\) is the gas constant, and \(T\) is the temperature in Kelvin.

Analyze Asymptotic Behavior

When \(T \rightarrow \infty\), the term \(\frac{E_a}{RT}\) becomes very small, approaching zero. Thus, \(e^{-\frac{E_a}{RT}}\) tends towards 1 as \(T\) becomes very large.

Evaluate Limit as T Approaches Infinity

Given the asymptotic behavior, the rate constant \(k\) approaches \(A\) because \(e^{-\frac{E_a}{RT}} \approx 1\) for \(T \rightarrow \infty\). Thus, the rate constant \(k\) tends towards the value of \(A\).

Identify the Arrhenius Parameter

The Arrhenius parameter \(A\) is given as \(6 \times 10^{14} \text{s}^{-1}\). This is the value that \(k\) approaches as \(T \rightarrow \infty\).

Select the Correct Answer

The value of the rate constant as \(T \rightarrow \infty\) is \(6.0 \times 10^{14} \text{s}^{-1}\). Therefore, the correct answer is option (b).

Key concepts

Understanding Rate Constant

The rate constant, often denoted as \( k \), is a critical factor in the Arrhenius equation that helps determine the speed of a chemical reaction. It essentially tells us how quickly a reaction proceeds under specific conditions. Unlike the reaction rate, which changes with concentration, the rate constant remains fixed at a given temperature, providing a more consistent measure.
- The rate constant is specific to particular reactions and varies with temperature changes.- Its dimensions can vary based on the reaction order, like \( ext{s}^{-1} \) for first-order reactions.
Incorporated into the Arrhenius equation, the rate constant shows a strong dependence on temperature. With increasing temperature, typically, the rate constant increases, signifying an increase in reaction speed. This relationship is crucial for predicting how a reaction will behave under different thermal conditions.

Activation Energy: The Reaction Threshold

Activation energy, denoted as \( E_a \), represents the minimum energy required for a reaction to occur. Think of it as the energy barrier that the reactants must overcome to transform into products.
- It is typically measured in kilojoules per mole (kJ/mol) and can assess whether a reaction is likely to proceed at a given temperature.- Higher activation energies imply that more energy is needed, often meaning slower reactions unless significant temperature increases are applied.
In the context of the Arrhenius equation, \( E_a \) plays a vital role in determining how temperature affects the rate constant \( k \). An exponential factor, \( e^{-\frac{E_a}{RT}} \), indicates how small changes in \( E_a \) or temperature \( T \) could lead to substantial changes in the rate constant, thereby altering the reaction rate significantly.

Asymptotic Behavior in Reaction Rates

Asymptotic behavior is a concept used to describe how functions behave as the input approaches a boundary—even infinity. In the realm of chemical kinetics, understanding the asymptotic behavior is crucial when evaluating how a rate constant, \( k \), changes with temperature.
- The Arrhenius equation expresses this asymptotic nature as temperature \( T \) rises to very high values.- The term \( \frac{E_a}{RT} \) becomes negligible since \( R \), the universal gas constant, along with \( T \) (temperature in Kelvin), ensure the fraction decreases.
As a result, the exponential factor \( e^{-\frac{E_a}{RT}} \) in the Arrhenius equation approaches 1. This implies the rate constant \( k \) asymptotically approaches the frequency factor \( A \), especially as \( T \) goes towards incredibly large numbers, essentially infinite temperatures, demonstrating the reaction's temperature dependency decline.