Problem 30 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

According to the Arrhenius equation, (a) a high activation energy usually implies a fast reaction. (b) rate constant increases with increase in temperature. This is due to a greater number of collisions whose energy exceeds the activation energy. (c) higher the magnitude of activation energy, stronger is the temperature dependence of the rate constant. (d) the pre-exponential factor is a measure of the rate at which collisions occur, irrespective of their energy.

Step-by-Step Solution

Answer

Only statements (b), (c), and (d) are true.

Step 1: Understanding the Arrhenius Equation

The Arrhenius equation is given by \( k = Ae^{-\frac{E_a}{RT}} \), where \( k \) is the rate constant, \( A \) is the pre-exponential factor, \( E_a \) is the activation energy, and \( R \) is the gas constant. Temperature \( T \) is in Kelvin. This equation shows how \( k \) changes with \( T \) and \( E_a \).

Step-by-step Evaluation of Statements

Evaluate each statement based on the Arrhenius equation. Compare the statements (a) to (d) with the mathematical form and implication of the equation.

Step 3: Analyzing Statement (a): High Activation Energy and Reaction Speed

According to the Arrhenius equation, high activation energy \( E_a \) implies a lower rate constant \( k \) when temperature \( T \) is constant, because more energy is required for reactants to reach the transition state. This results in a slower reaction, thus (a) is false.

Step 4: Analyzing Statement (b): Effect of Temperature on Rate Constant

As per the Arrhenius equation, as temperature \( T \) increases, \( \frac{E_a}{RT} \) decreases, thus \( e^{-\frac{E_a}{RT}} \) increases, leading to an increase in the rate constant \( k \). This directly supports statement (b), which is true.

Step 5: Analyzing Statement (c): Activation Energy and Temperature Dependence

From the Arrhenius equation, a higher activation energy suggests that the rate constant \( k \) will be more sensitive to changes in temperature. This is because \( E_a \) affects \( e^{-\frac{E_a}{RT}} \) significantly, thus making statement (c) true.

Step 6: Analyzing Statement (d): Pre-exponential Factor

The pre-exponential factor \( A \) in the Arrhenius equation represents the frequency of collisions. However, it assumes that all collisions could potentially lead to reactions, regardless of the energy involved. Thus, (d) is true.

Key Concepts

Activation Energy

Activation energy, denoted as \( E_a \), is a fundamental concept in chemical reactions. It represents the minimum energy required for a chemical reaction to occur. Often considered as an energy barrier that reactants must overcome to transform into products, activation energy can be visualized as the hill that reactants need to climb. The Arrhenius equation reveals that reactions with high activation energies tend to proceed slower at a given temperature because fewer molecules have the necessary energy to surpass this barrier.
Consequently, as activation energy increases, it typically implies a slower reaction unless temperatures are increased significantly to compensate for the larger energy demand.

Temperature Dependence of Rate Constant

The rate constant \( k \) of a reaction depends heavily on temperature, which is a crucial insight provided by the Arrhenius equation. As temperature rises, the fraction \( \frac{E_a}{RT} \) decreases, leading to an increase in the exponential component \( e^{-rac{E_a}{RT}} \), and thus, the rate constant \( k \) increases. This means that even a small increase in temperature can lead to a significant rise in the rate constant, and consequently, the speed of the reaction.
This dependence is why many reactions can proceed more rapidly at higher temperatures, as more reactants possess sufficient energy to overcome the activation energy barrier.

Pre-exponential Factor

The pre-exponential factor \( A \) in the Arrhenius equation is crucial for understanding reaction kinetics. It primarily accounts for the frequency of collisions between reactant molecules that could lead to a reaction. While the barrier of activation energy limits which collisions are successful, the pre-exponential factor assumes all collisions have the potential to result in a reaction if the energy condition is met.
Therefore, \( A \) serves as a measure of the inherent rate of collisions in a system, disregarding their energy. It reflects the probability that molecules will collide with the correct orientation and remain critical in the calculation of the rate constant.

Rate Constant

The rate constant \( k \) is a vital parameter in reaction kinetics that determines how fast a reaction proceeds. Calculated using the Arrhenius equation, \( k \) is influenced by various factors, including temperature and activation energy.

• At constant temperature, higher activation energy leads to a lower rate constant, signifying a slower reaction.
• Conversely, increasing temperature boosts \( k \), accelerating the reaction.

This factor is essential in predicting the speed of a reaction and helps chemists understand and control reaction conditions in practical applications.

Reaction Kinetics

Reaction kinetics is the study of how and why chemical reactions occur at particular rates. The Arrhenius equation is a cornerstone of this field, offering insight into the microscopic dynamics of reactions. Understanding the relationship between temperature, activation energy, and the rate constant allows chemists to manipulate conditions to favor desired reaction rates.

• It helps in designing and optimizing industrial processes.
• Offers insights into the feasibility and efficiency of reactions under varying environmental conditions.

Overall, reaction kinetics is indispensable for scientists to control and predict how fast reactions take place, making it a fundamental topic in both theoretical and applied chemistry.