Problem 94 - A Complete Resource Book in Chemistry for JEE Main

Question

The activation energies of two reactions with rate constants \(k\), and \(k_{2}\), are \(E_{a 1}\) and \(E_{a 2}\) respectively. If \(E_{a 1}\) \(<\mathrm{E}_{\mathrm{a} 2}\), when the temperature is increased from \(\mathrm{T}\), to \(\mathrm{T}_{2}\) the rate constants are \(k_{1}^{\prime}\) and \(k_{2}^{\prime} .\) Which one of the following statements is correct? (a) \(\mathbf{k}_{1}^{\prime}=\mathrm{k}_{2}\) (b) \(\frac{\mathrm{k}_{1}}{\mathrm{k}_{1}}<\frac{\mathrm{k}_{2}}{\mathrm{k}_{2}}\) (c) \(\frac{\mathrm{k}_{1}}{\mathrm{k}_{1}}>\frac{\mathrm{k}_{2}}{\mathrm{k}_{2}}\) (d) \(\frac{k_{1}}{k_{1}^{\prime}}=\frac{k_{2}}{k_{2}}\)

Step-by-Step Solution

Answer

(b) \( \frac{k_{1}'}{k_{1}} < \frac{k_{2}'}{k_{2}} \).

Step 1: Understand the Problem

We have two reactions with different activation energies, E_{a1} and E_{a2}, and rate constants k_{1} and k_{2}. We want to find out the relationship between the new rate constants k_{1}' and k_{2}' when temperature is increased, given that E_{a1} < E_{a2}.

Step 2: Recall Arrhenius Equation

The Arrhenius equation relates the rate constant k with the activation energy E_{a}, temperature T, and a pre-exponential factor A. It is given by:\( k = A e^{-\frac{E_{a}}{RT}} \), where R is the universal gas constant. The rate constant increases with an increase in temperature, T.

Step 3: Compare Temperature Sensitivity

A lower activation energy, E_{a1}, indicates the reaction is less sensitive to temperature changes. The reaction with higher activation energy, E_{a2}, will have a more significant change in rate constant for the same temperature increase. Thus, the rate constant k_{1} may not increase as much as k_{2} when T changes to T_{2}.

Step 4: Evaluate Options

Option (b): \( \frac{k_{1}'}{k_{1}} < \frac{k_{2}'}{k_{2}} \) seems correct because if E_{a1} < E_{a2}, the sensitivity of k_{1} to temperature change is less compared to k_{2}. Option (c): \( \frac{k_{1}'}{k_{1}} > \frac{k_{2}'}{k_{2}} \) contradicts this logic, and options (a) and (d) do not logically follow given the change in rate constants.

Key Concepts

Arrhenius Equation

The Arrhenius equation is a fundamental concept in chemistry that describes how the rate constant of a reaction depends on temperature and activation energy. It is formulated as:\[ k = A e^{-\frac{E_a}{RT}} \]Where:

• \( k \) is the rate constant.
• \( A \) is the pre-exponential factor, often interpreted as the frequency of molecules colliding with the correct orientation.
• \( E_a \) is the activation energy, representing the minimum energy required for the reaction to occur.
• \( R \) is the universal gas constant, valued at approximately 8.314 J/(mol K).
• \( T \) is the temperature in Kelvin.

​This equation shows that, as temperature increases, the exponential factor \( e^{-\frac{E_a}{RT}} \) increases, leading to a higher rate constant \( k \). Consequently, reactions tend to speed up at higher temperatures, but the extent of this acceleration is crucially dependent on the activation energy \( E_a \). Thus, understanding the Arrhenius equation is pivotal for comparing how different reactions are temperature-sensitive.

Rate Constants

Rate constants are integral to understanding the speed or rate of a chemical reaction and are represented as \( k \) in the Arrhenius equation. They embody how often reactions occur in a given period and are influenced by several factors like temperature, pressure, and the presence of catalysts.For two reactions with rate constants \( k_1 \) and \( k_2 \), their changes under temperature variations reflect differences in their activation energies. If a reaction has a higher activation energy \( E_{a2} \), its rate constant \( k_2 \) is more affected by temperature changes compared to a reaction with lower activation energy \( E_{a1} \). This means:

• For \( E_{a1} < E_{a2} \), increasing temperature leads to more substantial increases in \( k_2 \) than in \( k_1 \).
• The relationship \( \frac{k_1'}{k_1} < \frac{k_2'}{k_2} \) becomes valid under these conditions, indicating that although both rate constants increase with temperature, \( k_2 \) increases relatively more.

By examining rate constants and activation energies, one can predict and subsequently control reaction rates, crucial for industrial processes and theoretical chemistry.

Temperature Sensitivity

Temperature sensitivity in chemical reactions refers to how much the rate of reaction changes with temperature. It is largely dictated by the activation energy \( E_a \) of the reaction. When a reaction has a low activation energy, its rate constant \( k \) is less sensitive to temperature increases. Conversely:

• A high activation energy signifies greater temperature sensitivity, meaning that with the same temperature rise, the rate constant of such a reaction would increase more radically.

This concept is important in predicting and controlling reaction speeds. For example:

• Chemical kinetics can be tuned by adjusting temperatures to favor or impede certain reactions in processes like enzyme catalysis or chemical manufacturing.
• In our context, with \( E_{a1} < E_{a2} \), reaction 2's rate constant \( k_2 \) becomes more pronounced as temperature increases, resulting in a greater relative increase compared to reaction 1's \( k_1 \).

By understanding temperature sensitivity, chemists can effectively design controlled experiments and optimize reaction conditions for various practical applications.