Problem 65 - A Complete Resource Book in Chemistry for JEE Main

Question

Under the same reactions conditions, initial concentration of \(1.386 \mathrm{~mol} \mathrm{dm}^{-3}\) of a substance becomes half in \(40 \mathrm{sec}\) and 20 sec through \(1^{\text {st }}\) order and zero order kinetics respectively. Ratio \(\left[\frac{\mathrm{K}_{1}}{\mathrm{~K}_{0}}\right]\) of the rate constants for first order \(\left(\mathrm{k}_{1}\right)\) and zero order \(\left(\mathrm{k}_{0}\right)\) of the reactions is (a) \(0.5 \mathrm{~mol}^{-1} \mathrm{dm}^{-3}\) (b) \(1 \mathrm{~mol} \mathrm{dm}^{-3}\) (c) \(1.5 \mathrm{~mol} \mathrm{dm}^{-3}\) (d) \(2 \mathrm{~mol}^{-1} \mathrm{dm}^{3}\)

Step-by-Step Solution

Answer

The ratio \( \frac{k_1}{k_0} \) is (a) 0.5 \( \mathrm{mol}^{-1} \mathrm{dm}^{3} \).

Step 1: Understanding First Order Kinetics

First order kinetics mean the rate of reaction is proportional to the concentration of one reactant. The half-life for a first-order reaction is given by the formula \( t_{1/2} = \frac{0.693}{k_1} \). We are given that the half-life \( t_{1/2} \) is 40 seconds.

Step 2: Calculate First Order Rate Constant \( k_1 \)

Substitute the given half-life into the first-order half-life equation: \( 40 = \frac{0.693}{k_1} \). Solving for \( k_1 \), we get \( k_1 = \frac{0.693}{40} = 0.017325 \, \text{sec}^{-1} \).

Step 3: Understanding Zero Order Kinetics

For zero-order kinetics, the rate of reaction is constant and does not depend on the concentration of the reactant. The half-life is given by \( t_{1/2} = \frac{[A]_0}{2k_0} \). Here, the half-life is 20 seconds and the initial concentration \( [A]_0 = 1.386 \, \text{mol} \, \text{dm}^{-3} \).

Step 4: Calculate Zero Order Rate Constant \( k_0 \)

Using the zero-order half-life equation: \( 20 = \frac{1.386}{2k_0} \). Solving for \( k_0 \), we find \( k_0 = \frac{1.386}{40} = 0.03465 \, \text{mol} \, \text{dm}^{-3} \, \text{sec}^{-1} \).

Step 5: Compute the Ratio \( \frac{k_1}{k_0} \)

Now that we have \( k_1 = 0.017325 \, \text{sec}^{-1} \) and \( k_0 = 0.03465 \, \text{mol} \, \text{dm}^{-3} \, \text{sec}^{-1} \), we calculate the ratio: \( \frac{k_1}{k_0} = \frac{0.017325}{0.03465} \).

Step 6: Simplify the Ratio

Compute \( \frac{0.017325}{0.03465} = 0.5 \, \text{mol}^{-1} \, \text{dm}^{3} \). This matches option (a).

Key Concepts

First-Order Kinetics

In first-order kinetics, the rate at which a reaction proceeds is directly proportional to the concentration of one reactant. This means that even as the reactant concentration changes, the relationship between the concentration and the rate remains linear. An important feature of first-order reactions is their characteristic half-life formula:

• \( t_{1/2} = \frac{0.693}{k_1} \), where
  • \( t_{1/2} \) is the half-life of the reaction, and
  • \( k_1 \) is the rate constant for the reaction.

This formula indicates that the half-life of a first-order reaction is constant and does not depend on the initial concentration of the reactant. It's particularly useful for calculating the rate constant when the half-life is known, as it simplifies mathematical manipulation by keeping the form consistent.

Zero-Order Kinetics

In contrast to first-order kinetics, zero-order kinetics describes a scenario where the rate of reaction is independent of the concentration of the reactant. This means the reaction proceeds at a constant rate regardless of how much of the reactant is left. The formula used to describe the half-life of a zero-order reaction is:

• \( t_{1/2} = \frac{[A]_0}{2k_0} \), where
  • \([A]_0\) represents the initial concentration of the reactant, and
  • \( k_0 \) is the rate constant for the zero-order reaction.

Because the half-life depends directly on the initial concentration, zero-order reactions typically become slower with time as the concentration decreases, making the reaction characteristics quite different from first-order reactions.

Rate Constant Calculation

The rate constants, often denoted as \( k_1 \) for first-order reactions and \( k_0 \) for zero-order reactions, are crucial for determining the speed of reactions under specific conditions. To calculate the rate constants:

• For first-order kinetics, you can rearrange the half-life formula: \[ k_1 = \frac{0.693}{t_{1/2}} \]
• For zero-order kinetics, solve the following equation for \( k_0 \): \[ k_0 = \frac{[A]_0}{2t_{1/2}} \]

These formulas are straightforward and emphasize the dependency (or lack thereof in zero-order kinetics) of the reaction rate on the concentration of the reactants. Understanding these equations allows one to predict how long a reaction will take to reach a certain point or to calculate the change in concentration over time.

Half-Life Calculation

The concept of half-life is a vital aspect of chemical kinetics, providing insight into how long it takes for a concentration to decrease by half. It is distinct for different orders of reactions:

• For first-order reactions, the half-life is a constant value and does not rely on the concentration of the reactant, allowing for the use in diverse applications such as pharmacokinetics and radioactive decay.
• For zero-order reactions, the half-life is directly proportional to the initial concentration, suggesting that higher starting concentrations lead to longer half-lives. This feature makes zero-order dynamics important in cases where a catalyst or saturated enzyme activity dominates.

The calculation and understanding of half-life provide a framework for modeling and predicting reaction behaviors in various scientific and industrial applications, emphasizing the importance of accurately determining rate constants and understanding reaction order.