Problem 87 - A Complete Resource Book in Chemistry for JEE Main

Question

The rate constant, the activation energy and the Arrhenius parameter of a chemical reaction at \(25^{\circ} \mathrm{C}\) are \(3.0 \times 10^{-4} \mathrm{~s}^{-1}, 104.4 \mathrm{~kJ} \mathrm{~mol}^{-1}\) and \(6 \times 10^{14} \mathrm{~s}^{-1}\) respectively. The value of the rate constant as \(\mathrm{T} \longrightarrow \infty\) is (a) \(2.0 \times 10^{18} \mathrm{~s}^{-1}\) (b) \(6.0 \times 10^{14} \mathrm{~s}^{-1}\) (c) infinity (d) \(3.6 \times 10^{30} \mathrm{~s}^{-1}\)

Step-by-Step Solution

Answer

As temperature approaches infinity, the rate constant is approximately \( 6 \times 10^{14} \mathrm{~s}^{-1} \), matching option (b).

Step 1: Understanding the Formula

The rate constant of a chemical reaction can be described by the Arrhenius equation: \( k = Ae^{-\frac{E_a}{RT}} \), where \( k \) is the rate constant, \( A \) is the Arrhenius parameter (also known as the frequency factor), \( E_a \) is the activation energy, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

Step 2: Analyzing as Temperature Approaches Infinity

As temperature \( T \) approaches infinity, the exponent \( -\frac{E_a}{RT} \) in the Arrhenius equation approaches zero. When \( e^x \) approaches \( e^0 \), it simplifies to 1. Therefore, the expression simplifies to \( k \approx A \times e^0 = A \). This implies that at infinite temperature, the rate constant \( k \) will be approximately equal to the Arrhenius parameter \( A \).

Step 3: Finding the Correct Answer Choice

Since \( A = 6 \times 10^{14} \mathrm{~s}^{-1} \), the rate constant at infinite temperature will be \( 6 \times 10^{14} \mathrm{~s}^{-1} \). This matches option (b) in the provided answer choices.

Key Concepts

Arrhenius Equation

The Arrhenius Equation is a cornerstone of chemical kinetics. It connects the rate constant of a reaction to temperature and activation energy. This equation is written as:

• \[ k = A e^{-\frac{E_a}{RT}} \]

Here, "k" represents the rate constant, defining how fast a reaction proceeds. "A" is the Arrhenius parameter, also known as the frequency factor. It considers how often molecules collide in the correct orientation for a reaction.
The term \( e^{-\frac{E_a}{RT}} \) involves the activation energy \( E_a \) and the temperature \( T \). "R" is the universal gas constant. This part of the equation shows how higher temperatures can overcome energy barriers more easily, thus accelerating reactions. Utilizing the Arrhenius equation allows us to understand the dynamics of molecular transformations at different temperatures. For example, as temperature increases, the exponential term suggests that more molecules have the required energy to overcome activation barriers.

Activation Energy

Activation energy \( E_a \) is the minimal energy required for a reaction to occur. It represents the barrier that reactants must overcome to form products. Reactions with low activation energy happen more easily compared to those with high activation energy.
Consider, if you're pushing a boulder up a hill. The effort you need to put in to get the boulder over the hill and down the other side is akin to activation energy. Molecules at lower activation energies find it easier to collide successfully and transform.
Activation energy is crucial in determining the rate of reaction. By using a catalyst, reactions can be sped up because catalysts lower activation energy, allowing more molecules to react per unit time. Catalysts provide alternate pathways with lower energy requirements, thus making reactions happen more smoothly.

Temperature Dependency of Reactions

Temperature greatly influences how quickly reactions occur. According to the Arrhenius Equation, as temperature increases, the rate constant "k" typically rises as well. This is because higher temperatures provide molecules with more kinetic energy.
Think of heat as providing the extra boost that molecules need to overcome activation energy barriers. There's a greater probability for successful collisions when molecules move faster, increasing the rate at which products form.
In practical terms, this is why we observe that reactions often happen more rapidly at elevated temperatures. For example, food spoils faster in warm conditions compared to refrigeration. The temperature dependency of reactions is an essential consideration in industries like pharmaceuticals, where reaction rates can affect production timelines and product stability. Understanding how temperature affects reaction rates allows chemists to manipulate conditions to achieve desired outcomes efficiently.