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 sec and 20 sec through \(1^{\text {t }}\) 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 \(0.5 \, \mathrm{mol}^{-1} \, \mathrm{dm}^{3}\) (Answer: a).

Step 1: Understanding Half-Life for First Order

For a first-order reaction, the half-life \( t_{1/2} \) is given by \( t_{1/2} = \frac{0.693}{k_1} \). Here, \( t_{1/2} = 40 \) seconds. Thus, \( k_1 = \frac{0.693}{40} \) s\(^{-1}\). Calculate \( k_1 \): \( k_1 = 0.017325 \) s\(^{-1}\).

Step 2: Understanding Half-Life for Zero Order

For a zero-order reaction, the half-life is \( t_{1/2} = \frac{[A]_0}{2k_0} \). Given \( t_{1/2} = 20 \) seconds and \( [A]_0 = 1.386 \, \mathrm{mol \, dm}^{-3} \). Solve for \( k_0 \): \( k_0 = \frac{[A]_0}{2 imes 20} = \frac{1.386}{40} = 0.03465 \, \mathrm{mol \, dm}^{-3} \, s^{-1} \).

Step 3: Calculate Ratio of Rate Constants

Using the computed rate constants, find the ratio \( \frac{k_1}{k_0} \): \( \frac{k_1}{k_0} = \frac{0.017325}{0.03465} = 0.5 \).

Key Concepts

Rate Constants

The rate constant, often denoted as \( k \), is a crucial part of understanding chemical kinetics. It captures the speed at which a reaction progresses.
These constants vary significantly based on the reaction order and the conditions under which the reaction occurs. They are derived from the rate law, which expresses the reaction rate as a function of the concentration of the reactants.

• For first-order reactions, the rate constant \( k_1 \) has units of \( ext{time}^{-1} \), such as \( ext{s}^{-1} \). This signifies that the rate is directly proportional to the concentration of a single reactant.
• For zero-order reactions, the rate constant \( k_0 \) has units of \( ext{concentration} \, ext{time}^{-1} \), for example, \( ext{mol} \, ext{dm}^{-3} \text{s}^{-1} \). In this scenario, the rate is constant and does not depend on the concentration of any reactants.

Understanding these constants helps predict how quickly reactions will occur under given conditions, allowing chemists to manipulate and control reaction speeds effectively.

First-Order Reaction

A first-order reaction is one where the rate depends linearly on the concentration of only one reactant. This implies that if the concentration of the reactant doubles, the rate of reaction also doubles, demonstrating a direct correlation.The behavior of first-order reactions is characterized by a mathematical expression of their rate:\[ ext{Rate} = k_1[A] \]One key feature of first-order reactions is their predictable half-life, the time it takes for half of the reactant to be converted into products. The half-life equation for a first-order reaction is given by:\[ t_{1/2} = \frac{0.693}{k_1} \]This highlights that the half-life is constant and independent of initial concentration, making first-order reactions particularly straightforward to analyze and predict. Their constant half-life allows easy calculation of how quickly a reactant decreases in concentration over time.

Zero-Order Reaction

In contrast to first-order reactions, zero-order reactions have a rate that is independent of the concentration of the reactants. This means the reaction proceeds at a constant rate until one of the reactants is depleted.For zero-order reactions, the rate is expressed as:\[ ext{Rate} = k_0 \]Since the reaction rate remains constant regardless of concentration changes, the concentration of the reactant decreases linearly over time. The half-life for a zero-order process depends on the initial concentration of the reactant \( [A]_0 \) and is given by:\[ t_{1/2} = \frac{[A]_0}{2k_0} \]This formula indicates that the half-life changes with the initial concentration, unlike a first-order reaction. Zero-order kinetics are often observed under saturated conditions where catalysts or reactants are in excess, leading to a constant rate as dictated by the rate constant \( k_0 \).