Problem 154 - A Complete Resource Book in Chemistry for JEE Main

Question

The rate constant, the activation energy and Arrhenius parameter of a chemical reaction at \(300 \mathrm{~K}\) are \(\mathrm{K}, \mathrm{Ea}\) and \(\mathrm{A}\) respectively. The value of rate constant at \(\mathrm{T} \rightarrow\) \(\infty\) is (a) \(\mathrm{A}\) (b) \(\mathrm{Ea}\) (c) \(\mathrm{Ea} \mathrm{x} \mathrm{A}\) (d) \(\mathrm{A}-\mathrm{Ea}\)

Step-by-Step Solution

Answer

The rate constant at \( T \to \infty \) is \( A \).

Step 1: Understanding Arrhenius Equation

The Arrhenius equation is given by \( k = A \exp\left( -\frac{E_a}{RT} \right) \) where \( k \) is the rate constant, \( A \) is the pre-exponential factor or Arrhenius parameter, \( E_a \) is the activation energy, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

Step 2: Analyzing Conditions Approach

We need to find the value of the rate constant \( k \) as the temperature \( T \) approaches infinity. As \( T \to \infty \), the term \( \frac{E_a}{RT} \to 0 \). This implies that \( \exp\left( -\frac{E_a}{RT} \right) \to \exp(0) \).

Step 3: Exponential Function Simplification

Since \( \exp(0) = 1 \), the Arrhenius equation simplifies to \( k = A \times 1 = A \). Thus, as \( T \) approaches infinity, the rate constant \( k \) will approach \( A \).

Key Concepts

Rate Constant

The rate constant, denoted as \( k \), is a crucial factor in determining the speed of a chemical reaction. It is a proportionality constant in the Arrhenius equation that relates to the rate of reaction with the concentration of reactants. Rate constants can vary greatly with temperature, underlining their dependence on the thermal environment of the reaction. They encapsulate how quickly a reaction can proceed under given conditions. In simpler terms, if you think of a balloon filled with gas, the rate constant is akin to how quickly you'd expect the balloon's gas to leak out, given its material properties and the environment. Key points about the rate constant:

• Depends on temperature but not on concentration or pressure.
• Specific for every reaction under given conditions and necessary for calculating reaction rates.
• Changes with temperature as described by the Arrhenius equation.

Understanding the rate constant helps chemists predict how a reaction behaves and adjust parameters to control the reaction pace.

Activation Energy

Activation energy, represented by \( E_a \), is the minimum energy required for a chemical reaction to proceed. It can be visualized as the energy barrier that reactants must overcome to transform into products. High activation energy means the reaction will be slower and require more energy, while lower activation energy speeds up the reaction.Imagine pushing a rock up a hill: the higher the hill (activation energy), the more effort it takes to get it rolling down the other side. Important aspects of activation energy include:

• It indicates the sensitivity of the reaction rate to temperature changes.
• Higher values typically correspond to slower reactions unless external energy is supplied.
• By lowering activation energy, catalysts make reactions proceed faster without being consumed themselves.

Understanding activation energy is fundamental for processes like enzyme activity, where the activation energy is lowered, enabling life-sustaining reactions in biological systems.

Pre-exponential Factor

The pre-exponential factor, often denoted as \( A \), is a vital component of the Arrhenius equation. Also known as the frequency factor, it reflects the number of collisions resulting in a reaction per unit time, assuming the correct orientation and sufficient energy. This factor essentially sets the upper limit to the rate constant \( k \) at infinite temperature, as seen in the solution for the provided exercise. At very high temperatures, where the exponential term approaches unity, \( A \) remains as the sole influencing factor.Consider \( A \) as the readiness of a ballroom dance where dancers (molecules) are always prepared to move if the conditions are right. Critical points about the pre-exponential factor:

• Depends on factors like the frequency of collisions and the probability that collisions are successful.
• It is inherent to the molecular structure and orientation of reacting species.
• Its value is derived from empirical data and varies widely across different reactions.

Comprehending \( A \) helps chemists understand reaction mechanics on a molecular level, offering insights into reaction feasibility and control under various conditions.