Problem 97 - A Complete Resource Book in Chemistry for JEE Main

Question

The half-life of a chemical reaction at a particular concentration is \(50 \mathrm{~min}\), when the concentration of reactants is doubled, the half-life becomes \(100 \mathrm{~min}\). Find the order. (a) zero (b) first (c) second (d) third

Step-by-Step Solution

Answer

The order of the reaction is second, corresponding to option (c).

Step 1: Understand reaction order concept

In chemical kinetics, the order of a reaction refers to the power to which the concentration of a reactant is raised in the rate equation. Half-life, which is the time needed for a reactant concentration to reduce to half its initial value, depends differently on concentrations based on the order of the reaction.

Step 2: Analyze given half-life data

We are given two half-life times: 50 minutes and 100 minutes. Initially, the half-life is 50 minutes, and when the concentration of reactants is doubled, the half-life becomes 100 minutes. This change in half-life duration with a change in concentration will help determine the order of the reaction.

Step 3: Apply half-life relation for order determination

For a zero-order reaction, half-life is inversely proportional to initial concentration; for a first-order reaction, half-life is constant; for a second-order reaction, half-life is directly proportional to the initial concentration. For this problem, the half-life doubles when the concentration is doubled, which indicates a second-order reaction where \[ t_{1/2} \propto \frac{1}{[A]}\].

Step 4: Conclude the reaction order

The doubling of the half-life when the concentration is doubled is consistent with the behavior of a second-order reaction. This verifies that the reaction's half-life directly depends on the initial concentration, confirming the order of the reaction is second.

Key Concepts

Reaction Order

Reaction order is a fundamental concept in chemical kinetics, helping us understand how a reaction's rate depends on the concentration of reactants. It reflects the sum of powers to which the concentration terms are raised in the rate equation. For example, in the rate equation \[ rate = k[A]^m[B]^n \]"m" and "n" represent the reaction orders with respect to reactants A and B.

• A reaction order of zero means that the rate is independent of the concentration of the reactants.
• A first-order reaction means the reaction rate is directly proportional to the concentration of one reactant.
• The second-order reaction implies that the rate is proportional to the square of the concentration of one reactant or to the product of the concentrations of two reactants.

Determining reaction order helps predict how changes in concentration will affect reaction rates, crucial for designing industrial processes and laboratory experiments. In our exercise, when the concentration of reactants doubles, the half-life doubles, pointing towards a second-order reaction.

Half-Life

The half-life of a chemical reaction is the time it takes for half of the reactant to be consumed. It provides insight into the speed of a reaction and is particularly useful for understanding kinetic behaviors. The relationship between half-life and concentration varies depending on the order of the reaction. - **Zero-order reactions** have a half-life that decreases with decreasing concentration, specifically, it's inversely proportional to the initial concentration. - **First-order reactions** feature a constant half-life that remains unchanged regardless of concentration. This characteristic makes isotopic decay processes fall under first-order reactions. - **Second-order reactions** have a half-life that is inversely proportional to the initial concentration, so as the concentration increases, the half-life decreases, and vice versa. In the exercise, as the concentration of the reactant is doubled and the half-life doubles, this unique relationship confirms a second-order reaction.

Second-Order Reaction

Second-order reactions are characterized by a reaction rate proportional to either the square of the concentration of a single reactant or the product of concentrations of two different reactants. The rate law for a simple second-order reaction can be expressed as:\[rate = k[A]^2\]Or, in the case involving two reactants, it may look like:\[rate = k[A][B]\]The half-life equation for a second-order reaction is:\[t_{1/2} = \frac{1}{k[A]_0}\]Here, \([A]_0\) is the initial concentration, and "k" is the rate constant. This relationship implies that as the reactant concentration increases, the half-life decreases, which explains why doubling the concentration also doubles the half-life in our exercise.In a laboratory or industrial setting, understanding second-order reactions helps in adjusting concentrations to control reaction times and ensure optimal product yields.