Question

In Arrhenius equation, \(\mathrm{k}=\mathrm{A} \exp (-\mathrm{Ea} / \mathrm{RT})\). A may be regarded as the rate constant at a. Very high temperature b. Very low temperature c. High activation energy d. Zero activation energy

Short answer

A is the rate constant at very high temperature or zero activation energy (options a and d).

Step-by-step solution

Analyze the Arrhenius Equation

The Arrhenius equation is given by \( k = A \exp(-Ea/RT) \). Here, \( k \) is the rate constant, \( A \) is the pre-exponential factor, \( Ea \) is the activation energy, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.

Consider the Case of Very High Temperature

At very high temperatures, \( T \to \infty \). As the temperature increases, the term \( \exp(-Ea/RT) \) approaches 1 since \( Ea/RT \) becomes very small. Thus, \( k \approx A \).

Evaluate Very Low Temperature

At very low temperatures, \( T \to 0 \). Thus, \( Ea/RT \) is large, making \( \exp(-Ea/RT) \) approach zero. Therefore, \( k \) is much smaller than \( A \).

Check High Activation Energy

With a high activation energy, \( Ea \) is large, making \( Ea/RT \) large and \( \exp(-Ea/RT) \) small regardless of the temperature, leading to a small \( k \).

Consider Zero Activation Energy

If \( Ea = 0 \), then \( \exp(-Ea/RT) = \exp(0) = 1 \). In this case, \( k = A \) because the exponential term does not reduce the value of \( A \).

Determine the Correct Option

From the analysis, we see that \( A \) can be regarded as the rate constant at very high temperatures and when the activation energy is zero.

Key concepts

Activation Energy

Activation energy is a crucial concept in understanding how chemical reactions occur. It represents the minimum amount of energy needed for a reaction to proceed. This energy is required to overcome the energy barrier between reactants and products.
In the Arrhenius equation, the activation energy is denoted as \( Ea \). It significantly influences the rate of a reaction:

• A higher activation energy implies that fewer molecules have sufficient energy to react at a given temperature, thus resulting in a slower reaction rate.
• A lower activation energy means more molecules can react, leading to a faster reaction rate.

Using the concept of activation energy, we can predict whether a proposed reaction will need an external energy source, such as heat, to proceed.

Rate Constant

In chemical kinetics, the rate constant, denoted \( k \), is a fundamental parameter that indicates the speed of a reaction. It is dependent on factors like temperature and the presence of a catalyst. In the context of the Arrhenius equation, \( k \) can be expressed as:
\[ k = A \exp(-Ea/RT) \]
where \( A \) is the pre-exponential factor and reflects the frequency of collisions having the correct orientation, and \( \exp(-Ea/RT) \) represents the fraction of collisions that can overcome the activation energy barrier.

• If \( k \) is large, the reaction is fast.
• If \( k \) is small, the reaction is slow.

Understanding the rate constant helps predict how quickly a reaction can occur under specific conditions.

Temperature Dependence

Temperature plays a pivotal role in chemical kinetics and the Arrhenius equation illustrates how this happens. The rate of a reaction generally increases with temperature due to the increase in molecular energy and movement.
The Arrhenius equation, \( k = A \exp(-Ea/RT) \), explicitly shows temperature's impact:

• As temperature \( T \) increases, the exponential term \( \exp(-Ea/RT) \) increases (since \( Ea/RT \) becomes smaller), leading to an increase in the rate constant \( k \).
• This means that at higher temperatures, more molecules have sufficient energy to overcome the activation energy, resulting in a higher reaction rate.

Temperature is a key factor that scientists manipulate to control reaction rates, often increasing the temperature to accelerate reactions.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the speed, or rate, of reactions and the factors affecting them. It provides insights into reaction mechanisms and intermediate stages of reactions.
The Arrhenius equation is fundamental to chemical kinetics as it quantifies how reaction rates depend on temperature and activation energy. Critical aspects of chemical kinetics include:

• Understanding how different factors such as temperature, concentration, surface area, and catalysts affect reaction rates.
• Analyzing how reactions proceed over time and identifying different phases like initiation, propagation, and termination.
• Supporting the development of mathematical models to predict reaction behaviors under varied conditions.

Through chemical kinetics, researchers can design better chemical processes and optimize conditions for industrial reactions.