Question

Which of the following statement about the Arrhenius equation is/are correct? a. On raising temperature, rate constant of the reaction of greater activation energy increases less rapidly than that of the reaction of smaller activation energy. b. The term \(\mathrm{e}^{-E a / \mathrm{RT}}\) represents the fraction of the molecules having energy in excess of threshold value. c. The pre-exponential factor becomes equal to the rate constant of the reaction at extremely high temperature. d. When the activation energy of the reaction is zero, the rate becomes independent of temperature

Short answer

Statements (b), (c), and (d) are correct.

Step-by-step solution

Understanding Arrhenius Equation

The Arrhenius equation is expressed as \( k = A \cdot e^{-\frac{E_a}{RT}} \), where \( k \) is the rate constant, \( A \) is the pre-exponential factor, \( E_a \) is the activation energy, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin. This equation describes how the rate constant \( k \) changes with temperature and activation energy.

Analyzing Statement (a)

Statement (a) is about the relationship of activation energy and temperature. According to the Arrhenius equation, reactions with higher activation energy \( E_a \) tend to have a smaller rate constant at lower temperatures. As temperature increases, the rate constant for these reactions increases faster compared to those with lower activation energy because the exponential term \( \mathrm{e}^{-E_a/RT} \) is more affected by \( E_a \). Thus, statement (a) is incorrect.

Analyzing Statement (b)

In the Arrhenius equation, the term \( \mathrm{e}^{-E_a/RT} \) indeed represents the fraction of molecules that have kinetic energy greater than or equal to the activation energy required for the reaction. Hence, statement (b) is correct.

Analyzing Statement (c)

At extremely high temperatures, the factor \( \mathrm{e}^{-E_a/RT} \) approaches 1, assuming \( E_a \) is finite, thus making \( k \approx A \). Therefore, the pre-exponential factor \( A \) does approximate \( k \), meaning statement (c) is correct.

Analyzing Statement (d)

When activation energy \( E_a \) is zero, the Arrhenius equation simplifies to \( k = A \), indicating that the rate constant, and thus the reaction rate, is completely independent of temperature. Thus, statement (d) is correct.

Key concepts

Activation Energy

Activation energy is a key concept when discussing reaction kinetics. It determines the minimum amount of energy required for a chemical reaction to occur. Imagine activation energy as a barrier that reactant molecules need to overcome to transform into products.
It's what keeps a reaction from happening instantaneously at all temperatures. If the energy of colliding molecules exceeds this activation energy, a reaction can take place. Otherwise, the molecules simply collide without reacting.

• Lower activation energy means that a larger fraction of molecules have enough energy to surpass this barrier and react.
• A higher activation energy means fewer molecules can react without additional energy input.

This concept explains why some reactions occur rapidly while others are slow unless heated or otherwise energized.

Rate Constant

The rate constant, denoted as \( k \), is an important factor in the Arrhenius equation and determines the speed of a reaction. It is influenced by several factors, including temperature and activation energy.
The Arrhenius equation demonstrates how \( k \) varies with temperature \( T \) and activation energy \( E_a \). The equation is expressed as \( k = A \cdot e^{-\frac{E_a}{RT}} \).

• \( k \) increases with an increase in temperature, accelerating the reaction.
• If \( E_a \) is high, \( k \) will be smaller at lower temperatures, indicating a slower reaction.

Rate constants are essential for predicting how quickly reactions proceed under different conditions.

Temperature Dependence

Temperature plays a crucial role in the rate of chemical reactions. As per the Arrhenius equation, the rate constant \( k \) increases as temperature \( T \) rises. This is because a higher temperature boosts the kinetic energy of molecules.
More molecules possess enough energy to surpass the activation energy, leading to a higher frequency of successful collisions.

• The relationship is exponential, meaning even small changes in temperature can significantly alter reaction rates.
• The term \( e^{-\frac{E_a}{RT}} \) in the equation illustrates the exponential increase in fraction of molecules gaining requisite energy with increasing temperature.

Therefore, reactions generally speed up as the temperature increases, a principle widely used in chemical processes and industries.

Pre-exponential Factor

In the Arrhenius equation, the pre-exponential factor \( A \) represents the frequency of collisions with proper orientation between reacting molecules. It's sometimes referred to as the "frequency factor" and dictates how often reactions occur if there were no energy barriers.

• \( A \) signifies how many reactant collisions could lead to a reaction, assuming all are successful.
• It incorporates factors like collision frequency and proper molecular orientation, which are necessary for a reaction.

At extremely high temperatures, the equation suggests \( k \approx A \), as the exponential factor \( e^{-\frac{E_a}{RT}} \) approaches 1. This occurs because the activation energy barrier is insignificant relative to thermal energy. This part of the equation gives insights into the intrinsic kinetics of the reaction beyond just energy considerations.