Problem 180 - A Complete Resource Book in Chemistry for JEE Main

Question

For a reaction if effective rate constant \(\mathrm{k}\) ' is given by \(\mathrm{k}^{\prime}=\frac{2 \mathrm{k}_{2}}{\mathrm{k}_{3}}\left(\frac{\mathrm{k}_{1}}{\mathrm{k}_{5}}\right)^{1 / 5}\) where \(\mathrm{k}_{1}, \mathrm{k}_{2}, \mathrm{k}_{3}, \mathrm{k}_{5}\) are rate constants for different steps of the reaction. The effective frequency factor (a) \(\mathrm{A}^{\prime}=\frac{\mathrm{A}_{2}}{\mathrm{~A}_{3}}\left(\frac{\mathrm{A}_{1}}{\mathrm{~A}_{5}}\right)^{2}\) (b) \(\mathrm{A}^{\prime}=\frac{2 \mathrm{~A}_{2}}{\mathrm{~A}_{3}}\left(\frac{\mathrm{A}_{1}}{\mathrm{~A}_{5}}\right)^{1 / 5}\) (c) \(A^{\prime}=A_{1} \times A_{2} \times A_{3} \times A_{4} \times A_{5}\) (d) \(\mathrm{A}^{\prime}=\frac{\mathrm{A}_{1}}{\mathrm{~A}_{2}}\left(\frac{\mathrm{A}_{3}}{\mathrm{~A}_{5}}\right)^{1 / 5}\)

Step-by-Step Solution

Answer

The effective frequency factor is (b) \( A^{\prime} = \frac{2 A_{2}}{A_{3}}\left(\frac{A_{1}}{A_{5}}\right)^{1/5} \).

Step 1: Identify the Given Effective Rate Formula

The exercise provides the effective rate constant formula as \( k^{\prime} = \frac{2 k_{2}}{k_{3}}\left(\frac{k_{1}}{k_{5}}\right)^{1/5} \). This formula involves a proportional relationship between different reaction steps, which usually correlate with frequency factors under similar formulations.

Step 2: Understand the Concept of Frequency Factor

In chemical reactions, the frequency factor (or pre-exponential factor) describes the number of times reactants approach the activation barrier per unit time. It is generally represented by \( A \) in the Arrhenius equation. The frequency factors are related similarly to the rate constants in multi-step reactions.

Step 3: Compare Effective Rate and Frequency Factor Formulas

To match the frequency factor with the given effective rate formula structure, focus on option (b): \( A^{\prime} = \frac{2 A_{2}}{A_{3}}\left(\frac{A_{1}}{A_{5}}\right)^{1/5} \). This structure mirrors the rate constant formula, indicating the relationship between \( A \) values corresponding to those \( k \) values.

Step 4: Conclude with the Correct Answer

The correct choice must have a similar form to \( k^{\prime} = \frac{2 k_{2}}{k_{3}}\left(\frac{k_{1}}{k_{5}}\right)^{1/5} \). After comparing the options, the correct option that matches the format is (b): \( A^{\prime} = \frac{2 A_{2}}{A_{3}}\left(\frac{A_{1}}{A_{5}}\right)^{1/5} \).

Key Concepts

Reaction Kinetics

Understanding reaction kinetics is like understanding the speed limits of a chemical highway. This branch of chemistry explores how fast reactions occur, how they can be tuned, and what factors affect their rates. Typically, reaction kinetics involve looking at how the concentration of reactants or products changes over time.
For students delving into reaction kinetics, it's essential to know that reactions don't all go at the same speed. Some fire up instantly, while others take their sweet time converting reactants into products. In equations, this speed is expressed through \[ k = \text{rate constant} \] where \( k \) is a unique value for each specific reaction at a given temperature. It tells us how fast the reaction proceeds when all reactants are present at standard concentrations.

• The higher the rate constant \( k \), the faster the reaction speeds along.
• Temperature changes can beef up or dampen reaction speeds.
• Presence of catalysts can dramatically affect the value of \( k \).

So when students see complex formulas in kinetics, remember—it simply relates how various factors bump up or slow down the reaction speed.

Frequency Factor

Diving into the frequency factor is like figuring out how often cars approach a toll booth—from here, the traffic flow of a reaction can be further comprehended. In reaction kinetics, the frequency factor, often denoted as \( A \), is a critical component of the Arrhenius equation:\[ k = A e^{-E_a / RT} \] where \( E_a \) is the activation energy, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.
Understanding \( A \) tells us about the effectiveness of reactant collisions in a chemical process. Here’s what to remember:

• \( A \) indicates how frequently reactant particles collide with correct orientation to react.
• While \( E_a \) represents the energy hurdle to overcome, a large \( A \) suggests more paths for reactants to react.
• Frequency factors are key for assessing different steps in multi-step reactions, comparing them to see which step might be the bottleneck.

Readily related to the rate constant, the frequency factor excellently conveys reaction possibilities, painting a clearer picture of the kinetic energy landscape of reactions.

Chemical Reaction Steps

Chemical reactions often unfold in multiple steps, each a tiny event on the way to the final product. Understanding these steps is like tracing a route on a map—it helps predict how a journey unfolds, stop by stop.
In multi-step reactions, each step has its rate constant and frequency factor. The interaction and progression through these steps dictate the overall reaction path:

• Each step might have its own particular \( k \) and \( A \), contributing to the overall effective rate constant and frequency factor.
• The slowest step is often the rate-determining step, akin to the slow car in a convoy that sets the pace.
• Effective rate constants, such as \( k' \), provide an aggregate perspective on complex reactions.

Mastering the link between individual steps and their cumulative impact is vital. It allows chemists to fine-tune processes, maximize yields, or minimize unwanted byproducts. Understanding each step sharpens reaction insights, enhancing students’ comprehension of complex kinetics problems.