Problem 21 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

A hydrogenation reaction is carried out at \(500 \mathrm{~K}\). If same reaction is carried out in the presence of a catalyst at the same rate, the temperature required is \(400 \mathrm{~K}\). Calculate the activation energy of the reaction if the catalyst lowers the activation barrier by \(20 \mathrm{~kJ} \mathrm{~mol}^{-1}\).

Step-by-Step Solution

Answer

Activation energy without catalyst is 60 kJ/mol.

Step 1: Understand the Arrhenius Equation

The Arrhenius equation is important for understanding reaction rates. It is given by \( k = A e^{-E_a/(RT)} \), where \( k \) is the rate constant, \( A \) is the pre-exponential factor, \( E_a \) is the activation energy, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

Step 2: Establish the Relation with and without Catalyst

For reactions carried out at the same rate with and without a catalyst, the ratios of the rate constants \( k_1 \) and \( k_2 \) at two different temperatures \( T_1 \) and \( T_2 \) can be related as follows: \[ \frac{k_2}{k_1} = \frac{A e^{-E_a'/RT_2}}{A e^{-E_a/RT_1}} \] Where \( E_a' \) is the activation energy with the catalyst.

Step 3: Simplify and Apply Given Information

Given that the catalyst lowers the activation energy by 20 kJ/mol and both reactions proceed at the same rate (\( k_1 = k_2 \)), we have: \[ E_a' = E_a - 20 \text{ kJ/mol} \].

Key Concepts

Arrhenius equation

The Arrhenius equation plays a crucial role in understanding how temperature affects chemical reaction rates. It is formulated as \( k = A e^{-E_a/(RT)} \). In this equation:

• \( k \) is the rate constant, which tells us how fast a reaction occurs.
• \( A \) is the pre-exponential factor, representing the frequency of collisions having the correct orientation.
• \( E_a \) is the activation energy, the minimal energy barrier that must be overcome for a reaction to proceed.
• \( R \) is the universal gas constant, with a value of \( 8.314 \, \text{J/mol}\cdot\text{K} \).
• \( T \) signifies the temperature in Kelvin.

The exponential factor \( e^{-E_a/(RT)} \) shows how the probability of molecules having sufficient energy to react depends on temperature. Lower activation energy or higher temperature increases the rate constant \( k \), thus speeding up the reaction. Understanding this relationship helps predict how reactions will fare under different temperatures.

catalyst effects

Catalysts are substances that speed up chemical reactions without being consumed in the process. Their main role is to lower the activation energy needed for a reaction to occur. In the hydrogenation reaction example, the catalyst reduces the activation barrier by 20 kJ/mol, which significantly impacts the rate at which the reaction proceeds.

• By reducing \( E_a \), catalysts provide an alternative pathway for the reaction, making it easier for reactants to convert into products.
• Catalysts do not alter the temperature or equilibrium position of a reaction but enable it to reach equilibrium faster.

In our example, the reaction originally carried out at 500 K can proceed at 400 K with the catalyst, illustrating the dramatic effect a catalyst can have on reaction conditions. This is a practical application of catalysts benefiting many industrial processes, saving energy and improving efficiency.

reaction rates

Reaction rates describe how quickly a reactant is consumed or a product is formed in a chemical reaction. The rate is influenced by several factors, including temperature, concentration of reactants, and the presence of a catalyst.

Analyzing reaction rates:

• **Temperature:** Higher temperatures increase molecular movement, leading to more frequent and energetic collisions between particles.
• **Concentration:** More reactants present often result in more collisions and an increased likelihood of reaction.
• **Catalyst:** As discussed, it reduces the activation energy, enabling the reaction to proceed faster even at lower temperatures.

The Arrhenius equation again supports this, where \( k \), the rate constant, predictably changes with the conditions mentioned.

Reaction rates dictate how efficiently a reaction yields products, which informs decisions in chemical manufacturing and synthesis. By controlling these factors, chemists can optimize reactions for desired outcomes efficiently.