Problem 68 - A Complete Resource Book in Chemistry for JEE Main

Question

The activation energy of a reaction is \(9 \mathrm{kcal} /\) mole. The increase in the rate constant when its temperature is raised from 295 to 300 is (a) \(14.9 \%\) (b) \(28.9 \%\) (c) \(78.9 \%\) (d) \(82.9 \%\)

Step-by-Step Solution

Answer

The rate constant increase is closest to (b) 28.9\%.

Step 1: Understand the Arrhenius Equation

The Arrhenius equation relates the rate constant of a reaction to its activation energy and temperature. It is given by \[ k = Ae^{-Ea/(RT)} \]where \( k \) is the rate constant, \( A \) is the pre-exponential factor, \( Ea \) is the activation energy, \( R \) is the universal gas constant \( \approx 1.987 \text{ cal/(mol K)} \), and \( T \) is the temperature in Kelvin.

Step 2: Set Up the Equation for Two Temperatures

To find the change in the rate constant due to a temperature change, we use the ratio:\[\frac{k_2}{k_1} = \frac{Ae^{-Ea/(R T_2)}}{Ae^{-Ea/(R T_1)}} = e^{-Ea/(R T_2) + Ea/(R T_1) }\]where \( T_1 = 295 \text{ K} \) and \( T_2 = 300 \text{ K} \).

Step 3: Plug in the Known Values

Substitute the given activation energy \( Ea = 9 \text{ kcal/mol} \) and convert it to calories (since \( R \) is in calories) \( Ea = 9000 \text{ cal/mol} \). Calculate\[ \frac{k_2}{k_1} = e^{-9000/(1.987 \times 300) + 9000/(1.987 \times 295)} \]

Step 4: Simplify the Expression and Calculate

First, calculate the terms inside the exponent:\[ \frac{9000}{1.987 \times 300} \approx 15.104 \]\[ \frac{9000}{1.987 \times 295} \approx 15.319 \]This results in:\[\frac{k_2}{k_1} = e^{15.319 - 15.104} = e^{0.215}\]

Step 5: Calculate the Ratio and Percentage Increase

Compute the exponential:\[ e^{0.215} \approx 1.239 \]This gives: \( k_2 = 1.239 k_1 \), so the rate constant increases by \( 0.239 k_1 \), or \( 23.9\% \).

Step 6: Choose the Nearest Answer

Among the given options, the increase closely matches option (b) \( 28.9\% \).

Key Concepts

Activation Energy

Activation energy is a crucial concept in chemical reactions. It represents the minimum energy required for a reaction to occur. Think of it as a barrier that reactants must overcome for a successful transformation into products.
Activation energy is often denoted by the symbol \( Ea \). In the context of a chemical reaction, the higher the activation energy, the more energy is needed to start the reaction.
Consider activation energy in the context of molecules. Molecules must collide with enough energy to break old bonds and form new ones. The activation energy is the threshold that these molecules need to cross.

• High activation energy means that few molecules have enough energy to react, leading to slower reactions.
• Low activation energy suggests easier and faster reactions, as more molecules can meet the energy requirement.

Understanding the role of activation energy helps in controlling and optimizing reactions in various chemical processes.

Rate Constant

The rate constant \( k \) is a vital parameter in the study of chemical kinetics. It connects the concentration of the reactants to the reaction rate, through rate laws. The Arrhenius equation expresses the rate constant with:
\[ k = Ae^{-Ea/(RT)} \]
In this equation:

• \( A \) is the pre-exponential factor or frequency factor, indicating how often molecules collide with the correct orientation.
• \( Ea \) is the activation energy, the barrier needed to start the reaction.
• \( R \) is the universal gas constant.
• \( T \) is the temperature in Kelvin.

The rate constant is not constant in the typical sense; it changes with temperature via the Arrhenius equation. As temperature increases, the rate constant often increases, meaning the reaction becomes faster. Understanding the rate constant's role is key to predicting reaction behavior.

Temperature Dependence

Temperature plays a pivotal role in chemical kinetics by influencing the rate of reactions. The Arrhenius equation explicitly shows this dependency by featuring temperature in its exponential term. As the temperature rises, molecules move more energetically, increasing the chances of successful collisions.
This increased molecular activity lowers the barrier set by activation energy, as more molecules can surpass it. Consequently, the rate constant \( k \) grows with temperature.

• An increase in temperature generally boosts reaction rates because of higher energy collisions.
• The increase in the rate constant due to temperature changes follows the exponential trend defined in the Arrhenius equation.

For example, a slight increase from 295 K to 300 K can significantly impact the rate by increasing the rate constant, as seen in the provided exercise. This highlights how delicate temperature changes can affect chemical kinetics.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the speed or rate of a reaction and how different conditions affect this rate. By understanding kinetics, we can delve into the complexities of reactions and determine ways to control them effectively.
Key factors affecting reaction rates include:

• Concentration of reactants
• Temperature
• Presence of a catalyst
• Surface area

One of the fundamental tools in chemical kinetics is the Arrhenius equation, which correlates the rate constants of reactions with temperature and activation energy. Understanding these factors allows chemists to manipulate conditions for desired reaction rates, which is essential in industrial applications, environmental processes, and everyday life. By studying kinetics, chemists can also predict the time required for reactions to reach significant completion, balance reaction efficiency, and energy consumption.