Problem 134 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following statements are correct about half-life period? (1) time required for \(99.9 \%\) completion of a reaction is 100 times the half-life period (2) time required for \(75 \%\) completion of a first-order reaction is double the half-life of the reaction (3) average life \(=1.44\) times the half-life for firstorder reaction (4) it is proportional to initial concentration for zeroth-order (a) 1,2 and 3 (b) 2,3 and 4 (c) 2 and 3 (d) 3 and 4

Step-by-Step Solution

Answer

The correct statements are 2, 3, and 4. Option (b) is correct.

Step 1: Understanding Half-Life

Half-life is the time required for half of the reactant in a reaction to be consumed. It is a measure often used to describe the kinetics of first-order reactions but can be related to other order reactions as well.

Step 2: Analyze Statement 1

Statement 1 claims that the time required for the completion of 99.9% of a reaction is 100 times the half-life. This is true for first-order reactions, as the formula for the fraction remaining can be exponential, and a 99.9% completion will be approximately 10 half-lives.

Step 3: Analyze Statement 2

For first-order reactions, the time for 75% completion can be expressed as: \[ t_{75 ext{%}} = rac{2}{k}\ln 4 = 2\times\frac{0.693}{k} = 2t_{1/2} \]. This indicates that the time for 75% completion is indeed double the half-life period, making this statement true for first-order reactions.

Step 4: Analyze Statement 3

The average life for a first-order reaction is indeed calculated as \( \tau = \frac{1}{k} \), where \( \tau = 1.44 \times t_{1/2} \) (as \( k = \frac{0.693}{t_{1/2}} \)). Thus, this statement is correct.

Step 5: Analyze Statement 4

For zeroth-order reactions, the rate of reaction is constant, so the half-life is given by \( t_{1/2} = \frac{[A]_0}{2k} \), where \([A]_0\) is the initial concentration. This directly shows that the half-life is proportional to the initial concentration, making this statement true.

Step 6: Choose Correct Statements

From the analysis, statements 2, 3, and 4 are true. Thus, the correct option containing these statements is (b) 2, 3, and 4.

Key Concepts

First-Order Reactions

In chemistry, first-order reactions are those reactions whose rate depends linearly on the concentration of a single reactant. This means that the rate of reaction is directly proportional to the concentration of one of the reactants. The mathematical expression for the rate of a first-order reaction can be written as: \[ Rate = k[A] \]where \( k \) is the rate constant, and \([A]\) is the concentration of the reactant. A key characteristic of first-order reactions is that they exhibit an exponential decay over time.
For such reactions, the half-life (\( t_{1/2} \)) is independent of the initial concentration and is constant over time. It is given by the formula:\[ t_{1/2} = \frac{0.693}{k} \]This indicates that as the reaction progresses, the time it takes for the concentration to reduce by half remains the same. Furthermore, other important time-related observations can be made:

• The time required for 75% completion is double the half-life period.
• The average life is equal to \( 1.44 \times t_{1/2} \).

Zeroth-Order Reactions

Zeroth-order reactions are distinct because their rate is constant and does not depend on the concentration of the reactants involved. This means that the rate at which the reactants are consumed is constant over time. The rate expression for a zeroth-order reaction can be represented as: \[ Rate = k \]In these reactions, the reactant decreases in a linear manner over time. The half-life of zeroth-order reactions is uniquely characterized by its dependence on the initial concentration. This relationship can be expressed through the formula:\[ t_{1/2} = \frac{[A]_0}{2k} \]Here, \([A]_0\) represents the initial concentration.

As a result, the half-life becomes shorter as the concentration increases. This is contrary to first-order reactions, where the half-life is constant regardless of concentration. In zeroth-order reactions, understanding these kinetics is crucial when adjusting the amount of reactants to manage reaction time efficiently.

Reaction Kinetics

Reaction kinetics is the study of the rates at which chemical processes occur and the factors influencing these rates. This field provides insights into the steps involved in chemical reactions and the conditions that affect their speed.

• Temperature: Increasing the temperature generally increases the reaction rate, as it provides more energy for reactant molecules to collide and react.
• Concentration: For most reactions, an increase in concentration will increase the rate because there are more molecules available to collide.
• Nature of Reactants: Different substances react at different rates. Some reactants inherently react faster than others.
• Catalysts: Catalysts speed up the reaction process by lowering the activation energy needed for the reaction to occur.

By analyzing reaction kinetics, chemists can determine the most efficient pathways and conditions for chemical processes. It also allows them to predict how long a reaction will take and how various factors might alter its speed. Understanding these dynamics is essential for both industrial applications and theoretical chemistry.