Problem 153 - A Complete Resource Book in Chemistry for JEE Main

Question

A redox reaction is carried out at \(127^{\circ} \mathrm{C}\). If the same reaction is carried out in presence of a catalyst at the same temperature, the rate of reaction is doubled. To what extent is the energy barrier lowered by the catalyst? [Use \(\mathrm{R}=2 \mathrm{cal} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\) and \(\left.\log 2=0.301\right]\) (a) \(455 \mathrm{cal}\) (b) \(231 \mathrm{cal}\) (c) \(693 \mathrm{cal}\) (d) \(554 \mathrm{cal}\)

Step-by-Step Solution

Answer

The energy barrier is lowered by 554 cal with the catalyst. Option (d).

Step 1: Identify Variables

The temperature given is \(127^{\circ} \mathrm{C}\). Convert this temperature to Kelvin:\[ T = 127 + 273 = 400 \; \text{K} \] The rate of reaction doubles in the presence of a catalyst, and we have constants \( R = 2 \; \text{cal} \; \text{mol}^{-1} \; \text{K}^{-1} \) and \( \log 2 = 0.301 \).

Step 2: Use Arrhenius Equation

The Arrhenius equation for the rate constant \( k \) is:\[ k = A \cdot e^{-E_a/RT} \]where \( E_a \) is the activation energy, and \( A \) is the pre-exponential factor. Under the presence of a catalyst, the rate constant is doubled, so:\[ 2k = A \cdot e^{-E_a'/RT} \]with \( E_a' \) representing the reduced activation energy.

Step 3: Relate Original and Catalysed Reaction Rates

Divide the equation of the catalyzed reaction by the original:\[ \frac{2k}{k} = \frac{A \cdot e^{-E_a'/RT}}{A \cdot e^{-E_a/RT}} \]Simplifying, we have:\[ 2 = e^{(E_a - E_a')/RT} \]

Step 4: Apply Logarithms

Take the natural logarithm of both sides:\[ \ln(2) = \frac{E_a - E_a'}{RT} \]Substitute the known values of \( R \) and \( T \) and \( \log 2 \):\[ 0.301 \times 2.303 = \frac{E_a - E_a'}{2 \times 400} \]

Step 5: Solve for Energy Barrier Difference

First, calculate \( 0.301 \times 2.303 \):\[ 0.301 \times 2.303 \approx 0.693 \]Rearrange to find the difference in energy barrier:\[ E_a - E_a' = 0.693 \times 2 \times 400 \approx 554 \; \text{cal} \]

Step 6: Verification with Options

Compare the calculated energy barrier lowering to the given options. The reduction corresponds to option (d), which is \( 554 \; \text{cal} \).

Key Concepts

Arrhenius Equation

The Arrhenius Equation is a fundamental formula used in chemistry to describe how reaction rates increase with temperature. It is given by:

• \[ k = A imes e^{-E_a/RT} \]

where:

• \( k \) is the rate constant of a reaction.
• \( A \) is the pre-exponential factor, reflecting frequency of collisions and orientations.
• \( E_a \) is the activation energy needed for the reaction.
• \( R \) is the universal gas constant.
• \( T \) is the temperature in Kelvin.

The equation reveals that as the temperature increases, or the activation energy decreases, the rate constant \( k \) becomes larger, indicating a faster reaction rate. The exponential factor \( e^{-E_a/RT} \) explains the sensitivity of the reaction rate to temperature changes and activation energy adjustments.

Catalyst Effect

Catalysts are substances that increase the rate of a reaction without undergoing permanent chemical changes themselves. They achieve this by lowering the activation energy \( E_a \), making the reaction pathway more favorable.

• Catalysts do not alter the reactants or products in a chemical equation.
• They provide an alternative pathway with a lower activation energy barrier.
• A catalyst makes it easier for reactant molecules to reach the transition state.

In our problem, the presence of a catalyst doubles the reaction rate by lowering the activation energy. This is evident from the relation:

• \( 2 = e^{(E_a - E_a')/RT} \)

The catalyst effect is very useful in industry to enhance efficiency and energy use, allowing reactions to occur at lower temperatures and saving resources.

Reaction Kinetics

Reaction kinetics is the study of rates at which chemical processes occur and the factors affecting them. It helps us understand and control the speed of reactions efficiently. Factors influencing reaction rates include:

• Nature of reactants and products.
• Concentration of reactants.
• Temperature and pressure conditions.
• Catalyst presence, which changes the activation energy.

Using the Arrhenius equation, we can precisely predict how a change in conditions, like adding a catalyst, impacts the reaction rate. Kinetics provides invaluable insights into mechanism details - which steps are rate-limiting and how interventions can alter them. By doubling the reaction rate, the catalyst demonstrates its profound effect on reaction kinetics.

Temperature Conversion

Temperature conversion is crucial in chemistry as reactions depend heavily on temperature conditions. To apply the Arrhenius equation, temperatures must be in Kelvin because the Kelvin scale is an absolute temperature scale crucial for gas law calculations.

• To convert Celsius to Kelvin, add 273 to the Celsius temperature.
• For example, given \(127^{\circ} \mathrm{C}\), convert it to Kelvin:
  • \( T = 127 + 273 = 400 \; \text{K} \)

The use of Kelvin ensures that we accurately compute temperature-dependent properties and align with thermodynamic principles. Proper temperature conversion is essential to precisely determine reaction rates using the Arrhenius equation.