Problem 98 - 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 reaction is second-order.

Step 1: Understand the Problem

We are given that the half-life changes when the concentration is doubled. This indicates that the reaction is not first-order, as first-order reactions have a constant half-life regardless of concentration. Thus, we need to determine if the reaction is zero-order, second-order, or third-order.

Step 2: Analyze Half-life Relationship

The half-life of zero-order reactions is directly proportional to concentration. For second-order reactions, the half-life is inversely proportional to the concentration. For third-order reactions, the dependency is more complex, but it does not simply double when concentration is doubled.

Step 3: Compare Concentration and Half-life Changes

We note that doubling the concentration results in doubling the half-life from 50 min to 100 min. This suggests a specific relationship characteristic of reaction order.

Step 4: Determine Reaction Order

For a second-order reaction, the half-life \( t_{1/2} \) is inversely proportional to the initial concentration (i.e., \( t_{1/2} \propto \frac{1}{[A]} \)). Here, the half-life actually increases when the concentration is increased, which is consistent with second-order reactions.

Key Concepts

Understanding Half-life

Half-life is a fundamental concept in chemical kinetics. It refers to the time required for half of a given reactant concentration to be consumed during a chemical reaction. The half-life of a reaction can provide insight into how fast a reactant is depleted.
For example:

• In first-order reactions, the half-life (\( t_{1/2} \)) remains constant regardless of the initial concentration, because it depends only on the rate constant (\( k \)).
• In contrast, zero-order and second-order reactions have half-lives that depend on reactant concentration.

The half-life helps chemists determine the speed and progress of chemical processes without needing to monitor reactant concentrations constantly.

Second-order Reaction Characteristics

Second-order reactions are characterized by a reaction rate that is proportional to either the square of the concentration of one reactant or the product of concentrations of two different reactants. The rate law for a second-order reaction could be expressed as:\[ \text{Rate} = k [A]^2 \]or \[ \text{Rate} = k [A][B] \]where \( k \) is the rate constant, and \( [A] \) and \( [B] \) are the concentrations of the reactants.
In such reactions, the half-life is inversely proportional to the initial concentration of the reactant:\[ t_{1/2} = rac{1}{k[A]_0} \]This means that if the concentration of a reactant is doubled, the half-life doubles as well, which aligns perfectly with the behavior of second-order reactions. This property helps identify reaction order in experimental scenarios where reactant concentrations change.

Introduction to Chemical Kinetics

Chemical Kinetics is the branch of chemistry that studies the speed of chemical reactions and the factors affecting them. It primarily focuses on reaction rates, mechanisms, and the steps taken during the transformation of reactants to products. Understanding kinetics allows scientists to:

• Predict how a reaction progresses over time.
• Determine the reaction rate and its dependence on reactant concentrations, temperature, and catalysts.
• Identify the steps or mechanisms involved in a chemical process.

Knowing these elements provides the tools to control reactions, optimize conditions in industrial processes, or explore new reactions with desired properties. Kinetics is central to developing efficient industrial processes, creating stable drugs, and advancing chemical manufacturing. This field is essential for innovation and ensures that reactions can be scaled safely from the lab to large-scale production.