Problem 14 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

For a first order reaction \(A \rightarrow P\), the temperature ( \(T\) ) dependent rate constant \((k)\) was found to follow the equation \(\log k=-(2000)\) \(\frac{1}{T}+6.0 .\) The pre-exponential factor \(A\) and the activation energy \(E_{a}\), respectively, are (a) \(1.0 \times 10^{6} \mathrm{~s}^{-1}\) and \(9.2 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (b) \(6.0 \mathrm{~s}^{-1}\) and \(16.6 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (c) \(1.0 \times 10^{6} \mathrm{~s}^{-1}\) and \(16.6 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (d) \(1.0 \times 10^{6} \mathrm{~s}^{-1}\) and \(38.3 \mathrm{~kJ} \mathrm{~mol}^{-1}\)

Step-by-Step Solution

Answer

Option (d) is correct: \(1.0 \times 10^6 \, \text{s}^{-1}\) and \(38.3 \, \text{kJ/mol}\).

Step 1: Understanding the Given Equation

We are given the equation \( \log k = -\frac{2000}{T} + 6.0 \). This is an expression of the Arrhenius equation in the logarithmic form, where \( k \) is the rate constant, \( T \) is the temperature in Kelvin, \( A \) is the pre-exponential factor, and \( E_a \) is the activation energy.

Step 2: Identifying the Arrhenius Formula

The Arrhenius equation is given by \( k = A e^{-E_a/(RT)} \). Its logarithmic form is \( \log k = \log A - \frac{E_a}{2.303RT} \). We need to compare this form with the given equation \( \log k = -\frac{2000}{T} + 6.0 \).

Step 3: Matching Parameters

In the given equation, \( -\frac{2000}{T} \) represents \(-\frac{E_a}{2.303RT}\), from which we equate \( \frac{E_a}{2.303R} = 2000 \). Also, \( \log A = 6.0 \).

Step 4: Calculate the Activation Energy \(E_a\)

Using the equation \( \frac{E_a}{2.303R} = 2000 \), we solve for \( E_a \):Substituting \( R = 8.314 \, \text{J/mol K} \):\[ E_a = 2000 \times 2.303 \times 8.314 \].\[ E_a = 38351.364 \, \text{J}\, \approx 38.3 \, \text{kJ/mol} \].

Step 5: Calculate the Pre-exponential Factor \( A \)

Given \( \log A = 6.0 \), it follows that\[ A = 10^6 = 1.0 \times 10^6 \, \text{s}^{-1} \].

Step 6: Conclusion

The pre-exponential factor \( A \) and the activation energy \( E_a \) are found to be \( 1.0 \times 10^6 \, \text{s}^{-1} \) and \( 38.3 \, \text{kJ/mol} \) respectively. This matches with option (d).

Key Concepts

First Order Reaction

In the world of chemistry, a **first order reaction** is one where the rate of reaction is directly proportional to the concentration of a single reactant. This means that if you double the concentration of the reactant, the rate of the reaction also doubles. First order reactions are common and can be represented by the general reaction:

• \( A \rightarrow P \)

In this notation, **A** is the reactant that changes into a product **P** over time. The rate constant, often denoted as **k**, is crucial as it determines how fast this transformation happens. If a reaction has a higher rate constant, it proceeds more quickly.
To explore the kinetics of first order reactions, scientists often employ the Arrhenius equation, which helps in understanding how temperature affects the rate constant. Keep reading to find out how this connects to other critical concepts like activation energy and pre-exponential factor.

Activation Energy

The **activation energy** is a core concept for understanding chemical reactions. It's the minimum energy that reacting species must have for a reaction to occur. Imagine it as the barrier that reactants have to overcome to transform into products. In essence, it's a hurdle that the reactants face.
Within the Arrhenius equation, activation energy is represented as **\(E_{a}\)**. In the context of a reaction's speed, reactions with lower activation energies proceed faster at a given temperature compared to those with higher activation energies. This is because fewer reactant molecules exceed the energy threshold required for the reaction to proceed.The formula from the Arrhenius equation used to determine activation energy is:

• \( k = A e^{-E_{a}/(RT)} \)

In our exercise, we calculated \(E_{a}\) using a relation derived from the equation, resulting in approximately \( 38.3 \, \text{kJ/mol} \). This energy quantifies how much kinetic energy is needed for reactants to turn into products, a crucial step in the kinetics of any reaction.

Pre-exponential Factor

The **pre-exponential factor**, also known as the frequency factor, is a parameter of the Arrhenius equation that indicates the number of collisions resulting in a reaction per unit time assuming that every collision leads to a reaction. It is denoted by **A**. This factor is unique to each reaction and heavily depends on the nature of the reactants involved.
The pre-exponential factor is not influenced by temperature but can be affected by the physical state of the reactants and the surface interactions. For example, in our original exercise, **A** was determined to be \( 1.0 \times 10^6 \, \text{ s}^{-1} \). Such magnitude suggests a significant frequency of successful collisions leading to product formation.
Understanding the pre-exponential factor helps chemists predict how variables like surface area and catalyst presence can alter reaction rates besides activation energy. This is essential for practical applications such as designing chemical reactors and optimizing industrial chemical processes.