Problem 140 - A Complete Resource Book in Chemistry for JEE Main

Question

A certain reaction proceeds in a sequence of three elementary steps with the rate constants \(\mathrm{k}_{1}, \mathrm{k}_{2}\) and \(\mathrm{k}_{3} .\) If the observed rate constant of the expressed as \(\mathrm{k}\) (obs) \(=\mathrm{k}(\mathrm{obs})=\left[\frac{\mathrm{k}_{1}}{\mathrm{k}_{2}}\right]^{1 / 2} \mathrm{k}_{3}\), the observed energy of activa- tion of the reaction is (a) \(\frac{\mathrm{E}_{3}+\mathrm{E}_{1}}{2}\) (b) \(\frac{1}{2}\left[\frac{E_{1}}{E_{2}}\right]+E_{3}\) (c) \(\mathrm{E}_{3}+\frac{1}{2}\left[\mathrm{E}_{1}-\mathrm{E}_{2}\right]\) (d) \(\mathrm{E}_{3}\left[\frac{\mathrm{E}_{1}}{\mathrm{E}_{2}}\right]^{12}\)

Step-by-Step Solution

Answer

The correct answer is (c): \( E_3 + \frac{1}{2}(E_1 - E_2) \).

Step 1: Understanding the Expression for Observed Rate Constant

The given expression for the observed rate constant is \( k(\text{obs}) = \left[\frac{k_1}{k_2}\right]^{1/2} k_3 \). This expression combines the elementary rate constants using a square root function and multiplication, suggesting a relationship between these rate constants.

Step 2: Relating Activation Energy to Rate Constants

The rate constant \( k \) for a reaction step can be expressed using the Arrhenius equation: \( k = Ae^{-E/RT} \), where \( E \) is the activation energy. For \( k_1, k_2, \text{ and } k_3 \), these correspond to activation energies \( E_1, E_2, \text{ and } E_3 \) respectively.

Step 3: Derivation of the Combined Activation Energy

Using the Arrhenius equation, express each rate constant: \( k_1 = A_1e^{-E_1/RT} \), \( k_2 = A_2e^{-E_2/RT} \), \( k_3 = A_3e^{-E_3/RT} \). Substitute these into the expression for \( k(\text{obs}) \): \[ k(\text{obs}) = \left[\frac{A_1e^{-E_1/RT}}{A_2e^{-E_2/RT}}\right]^{1/2} A_3e^{-E_3/RT} \].

Step 4: Simplification of the Activation Energy Expression

Simplify the expression for \( k(\text{obs}) \): \[ k(\text{obs}) = \left[\frac{A_1}{A_2}\right]^{1/2} A_3 e^{-\frac{1}{2}(E_1 - E_2 + 2E_3)/RT} \]. This shows that the effective activation energy \( E_{\text{obs}} \) is \[ E_{\text{obs}} = \frac{1}{2}(E_1 - E_2) + E_3 \].

Step 5: Identifying the Correct Answer

Compare the simplified effective activation energy \( \frac{1}{2}(E_1 - E_2) + E_3 \) with the given options. The option that matches is (c): \( E_3 + \frac{1}{2}(E_1 - E_2) \).

Key Concepts

Rate Constants

In chemical kinetics, the rate constant is a crucial factor that provides insight into the speed of a reaction. Simply put, it determines how fast reactants convert into products. The rate constant, denoted by \( k \), is specific to a particular reaction and varies with temperature. For each elementary step in a reaction mechanism, there is a unique rate constant associated with that particular step. Understanding rate constants helps in determining the rate law, which is a mathematical expression that shows how the concentration of reactants affects the rate of reaction. The observed rate constant for a multi-step reaction often combines the rate constants of individual steps. As shown in the solution for our problem, the expression \( k(\text{obs}) = \left[\frac{k_1}{k_2}\right]^{1/2} k_3 \) merges the rate constants from three elementary steps into a single entity that represents the overall reaction speed.

• The rate constant is integral for calculating reaction rates.
• It varies with temperature and is unique to the reaction conditions.
• Combining rate constants can give a comprehensive view of multi-step reactions.

Activation Energy

Activation energy is the minimum energy needed for a reaction to proceed, an essential concept in understanding why different reactions occur at different rates. Represented typically by \( E \), this energy barrier must be overcome by the reactants to form products.In the context of our exercise, activation energies \( E_1, E_2, \) and \( E_3 \) refer to the minimum energy requirements for each elementary step within the multi-step reaction. The expression derived in the solution, \( E_{\text{obs}} = \frac{1}{2}(E_1 - E_2) + E_3 \), shows how these individual energies combine to give an overall effective activation energy for the complete reaction.Here are some key points about activation energy:

• It acts as an energy barrier that influences reaction rates.
• A higher activation energy means a slower reaction rate at a given temperature.
• It plays a significant role in determining the efficiency of elementary steps in a multi-step reaction.

Arrhenius Equation

The Arrhenius equation is a fundamental formula that illustrates the dependency of the rate constant on temperature and activation energy. The equation is given by:\[ k = A e^{-E/RT} \]In this equation, \( k \) is the rate constant, \( E \) is the activation energy, \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( A \) is the pre-exponential factor that relates to the frequency of collisions and the orientation of the reacting particles.Applying the Arrhenius equation helps reveal why reactions speed up as temperatures rise: the exponential term \( e^{-E/RT} \) becomes larger, thus increasing \( k \), the rate constant. In the problem’s context, knowing each elementary step's activation energy allows us to use the Arrhenius equation to find those step-wise rate constants \( k_1, k_2, \) and \( k_3 \), and eventually, the effective observed activation energy.

• The Arrhenius equation provides insight into how temperature affects the reaction rate.
• It connects key variables such as activation energy and the rate constant.
• This equation is pivotal in predicting how reactions respond to temperature changes.