Problem 109 - A Complete Resource Book in Chemistry for JEE Main

Question

The rate constant of a reaction at temperature 200 is 10 times less than the rate constant at \(400 \mathrm{~K}\). What is the activation energy \(\left(\mathrm{E}_{\alpha}\right)\) of the reaction? \((\mathrm{R}=\) gas constant) (a) \(1842.4 \mathrm{R}\) (b) \(921.2 \mathrm{R}\) (c) \(460.0 \mathrm{R}\) (d) \(230.3 \mathrm{R}\)

Step-by-Step Solution

Answer

The activation energy \( E_a \) is \( 921.2 R \) (Option b).

Step 1: Use the Arrhenius Equation

The rate constant is defined by the Arrhenius equation: \[ k = A e^{-E_{a}/(RT)} \] where \( k \) is the rate constant, \( A \) is the pre-exponential factor, \( E_{a} \) is the activation energy, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.

Step 2: Set Up the Relationship Between Rate Constants

Given that the rate constant at 200 K is 10 times less than at 400 K, write the relationship as:\[ \frac{k_{200}}{k_{400}} = \frac{1}{10} \]

Step 3: Express Rate Constants Using Arrhenius Equation

Substitute the Arrhenius equation for each rate constant:\[ \frac{A e^{-E_{a}/(R \cdot 200)}}{A e^{-E_{a}/(R \cdot 400)}} = \frac{1}{10} \]

Step 4: Simplify the Expression

Cancel \( A \) from both sides of the equation, then simplify:\[ e^{-E_{a}/(R \cdot 200) + E_{a}/(R \cdot 400)} = \frac{1}{10} \] Simplify the exponent:\[ e^{-E_{a}(1/(R \cdot 200) - 1/(R \cdot 400))} = \frac{1}{10} \]

Step 5: Calculate the Exponent

Calculate the difference in the exponents:\[ \frac{1}{200} - \frac{1}{400} = \frac{2 - 1}{400} = \frac{1}{400} \] So the equation becomes:\[ e^{-E_{a}/(R \cdot 400)} = \frac{1}{10} \]

Step 6: Take the Natural Logarithm of Both Sides

Apply the natural logarithm:\[ -\frac{E_{a}}{R \cdot 400} = \ln\left(\frac{1}{10}\right) \] \[ \frac{E_{a}}{R \cdot 400} = -\ln(10) \] Substitute the natural logarithm of 10 (approximately 2.302):\[ \frac{E_{a}}{R \cdot 400} = 2.302 \]

Step 7: Solve for Activation Energy

Multiply both sides by \( R \cdot 400 \) to solve for the activation energy:\[ E_{a} = 2.302 \times R \times 400 \] Calculate the activation energy value:\[ E_{a} = 921.2 R \]

Key Concepts

Arrhenius Equation

The Arrhenius Equation is a fundamental formula in the field of chemical kinetics and it establishes a relationship between the rate constant of a reaction and its temperature. This equation is presented as:\[ k = A e^{-E_{a}/(RT)} \]Here,

• \( k \) represents the rate constant, which is crucial as it dictates how fast a reaction occurs.
• \( A \) is the pre-exponential factor, also known as the frequency factor, indicating how often molecules collide in the right orientation.
• \( E_{a} \) is the activation energy, the energy barrier that needs to be surpassed for a reaction to proceed.
• \( R \) is the universal gas constant, providing a link to energy based on temperature.
• \( T \) is the absolute temperature measured in Kelvin.

The exponential factor \( e^{-E_{a}/(RT)} \) indicates how likely the reactants are to overcome the activation energy. A smaller activation energy or higher temperature results in effectively larger rate constants, implying faster reactions.

Rate Constant

The rate constant \( k \) is an integral component in the study of reaction kinetics. It quantifies the speed of a reaction at a given temperature. In essence, the rate constant provides the proportional factor when calculating the rate of chemical reactions, offering clarity on how concentration changes over time.

• The rate constant is sensitive to temperature changes, as seen in many chemical processes.
• Each reaction has its own unique rate constant, dependent on the specific conditions and nature of the reactants.
• Larger values of \( k \) typically imply a faster reaction, as more molecules surpass the activation energy barrier.

Since the rate constant is directly linked to the Arrhenius equation, modifications in factors like temperature will significantly influence its value. This aspect is fundamental in industries where reaction rates need optimization for efficiency.

Temperature Effects in Reactions

Temperature plays a pivotal role in influencing the rate of chemical reactions. As per the Arrhenius principle, raising the temperature generally increases the rate of a reaction. This effect of temperature on reaction rates can be explained by a few key principles:

• With increased temperature, molecules move faster, leading to more frequent and energetic collisions. This rise in activity raises the probability that collisions will have enough energy to overcome the activation energy.
• Even a small increase in temperature can result in a significant increase in reaction rates, highlighting the sensitivity of rates to thermal fluctuations.
• For the Arrhenius equation, as temperature \( T \) increases, the exponential factor \( e^{-E_{a}/(RT)} \) also increases. This rise results in a larger rate constant \( k \), accelerating the reaction's rate.

Overall, temperature is a controllable factor that can be manipulated to optimize reaction speeds, significantly impacting both laboratory research and industrial applications where efficiency is crucial.