Problem 8 - A Complete Resource Book in Chemistry for JEE Main

Question

Activation energy of a chemical reaction can be determined by (a) evaluating rate constant at standard temperature (b) evaluating velocities of reaction at two different temperatures (c) evaluating rate constants at two different tempera tures (d) changing concentration of reactants

Step-by-Step Solution

Answer

(c) evaluating rate constants at two different temperatures.

Step 1: Understanding the Arrhenius Equation

The Arrhenius equation is given by \( k = A e^{-\frac{E_a}{RT}} \), where \( k \) is the rate constant, \( A \) is the frequency factor, \( E_a \) is the activation energy, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin. The equation shows the relationship between the rate constant and the temperature.

Step 2: Evaluating Multiple Temperatures

To determine the activation energy \( E_a \), we evaluate rate constants at two different temperatures using the integrated form of the Arrhenius equation. By taking the natural logarithm on both sides of the Arrhenius equation, we get \( \ln k = \ln A - \frac{E_a}{R} \times \frac{1}{T} \).

Step 3: Using Two Rate Constants

To find \( E_a \), the equation can be modified to \( \ln\left(\frac{k_1}{k_2}\right) = \frac{E_a}{R} \left(\frac{1}{T_2} - \frac{1}{T_1}\right) \), where \( k_1 \) and \( k_2 \) are the rate constants at temperatures \( T_1 \) and \( T_2 \), respectively. This approach directly uses the differences in rate constants at two different temperatures for calculating \( E_a \).

Step 4: Application to the Options

Option (c) 'evaluating rate constants at two different temperatures' aligns with the method to determine activation energy using the Arrhenius equation. This method involves calculating \( E_a \) by comparing the rate constants from two temperatures, which is consistent with the formulation shown in Step 3.

Key Concepts

Arrhenius Equation

The Arrhenius Equation is a fundamental concept in chemical kinetics that helps us understand how the rate of a chemical reaction depends on temperature. It is expressed mathematically as \( k = A e^{-\frac{E_a}{RT}} \), where:

• \( k \) is the rate constant of the reaction.
• \( A \) is the frequency factor, which indicates the number of collisions that result in a reaction.
• \( E_a \) is the activation energy - the minimum energy needed for a reaction to occur.
• \( R \) is the gas constant (8.314 J/mol·K).
• \( T \) is the temperature in Kelvin.

This equation demonstrates that the rate of a reaction increases exponentially with an increase in temperature, assuming the activation energy and frequency factor remain constant. The Arrhenius equation is crucial for predicting how fast a reaction will proceed under different conditions. Applying this equation helps chemists and researchers calculate important parameters like the activation energy and understand kinetic behavior.

Rate Constant

The rate constant \( k \) is a crucial part of the Arrhenius equation and plays a vital role in chemical kinetics. It quantifies the speed of a chemical reaction at a given temperature and provides insights into how quickly reactants are converted into products.
The value of the rate constant is influenced by several factors:

• Temperature: As described by the Arrhenius equation, the rate constant increases with rising temperatures.
• Activation Energy: Lower activation energy results in a higher rate constant, indicating a faster reaction.
• Frequency Factor: This reflects the likelihood of reactant collisions leading to a reaction.

In practical terms, the rate constant allows researchers to predict reaction times and make informed decisions about reaction conditions, such as temperature adjustments to achieve desired reaction speeds. Evaluating rate constants at different temperatures can further help in determining the activation energy needed for the reaction.

Temperature Dependence

Temperature significantly influences the rate of chemical reactions, which is prominently reflected in the Arrhenius equation. As temperature increases, the molecules involved in a reaction have more kinetic energy. This increased energy results in a greater number of effective collisions, surpassing the activation energy threshold and thus speeding up the reaction.
Key points about temperature dependence include:

• An increase in temperature typically results in a higher rate constant \( k \), indicating a faster reaction.
• The temperature effect is more pronounced for reactions with higher activation energies, as even a slight temperature increase can substantially affect the rate.
• The Arrhenius equation mathematically captures this dependence by the exponential term \( e^{-\frac{E_a}{RT}} \); thus, it allows precise calculation of reaction rates at varying temperatures.

By understanding temperature dependence, chemists can manipulate conditions to optimize reaction rates, making this knowledge crucial in industrial and laboratory settings.

Chemical Kinetics

Chemical kinetics is the study of reaction rates and the factors affecting them. It helps us understand how reactions proceed over time, why some are faster than others, and how different conditions influence the speed and mechanisms of reactions.
Important aspects of chemical kinetics include:

• Reaction Rate: This describes how fast a reaction proceeds, typically measured in terms of concentration change over time.
• Mechanism: Kinetics can provide insights into the series of steps or pathways that a reaction follows from reactants to products.
• Rate Laws: These mathematical expressions relate the reaction rate to the concentrations of reactants. They help determine the order of a reaction.
• Influencing Factors: Temperature, reactant concentration, catalysts, and physical state all affect reaction rates.

Chemical kinetics is essential for designing chemical processes, developing new materials, and even understanding biological systems. By leveraging kinetic theories and applications, scientists can control and accelerate reactions efficiently in various fields.