Problem 21 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The half-life period of a first order reaction is 15 minutes. The amount of substance left after one hour will be: [Main Online April 9, 2014] (a) \(\frac{1}{4}\) of the original amount (b) \(\frac{1}{8}\) of the original amount (c) \(\frac{1}{16}\) of the original amount (d) \(\frac{1}{32}\) of the original amount

Step-by-Step Solution

Answer

The amount left after one hour is \( \frac{1}{16} \), option (c).

Step 1: Understanding Half-Life for First Order Reactions

For a first-order reaction, the half-life period \( t_{1/2} \) is the time required for the concentration of the substance to decrease to half of its initial value. The half-life \( t_{1/2} \) is defined as \( t_{1/2} = \frac{0.693}{k} \), where \( k \) is the rate constant.

Step 2: Calculate Rate Constant

Given the half-life period \( t_{1/2} = 15 \) minutes, we can solve for the rate constant \( k \):\[ k = \frac{0.693}{t_{1/2}} = \frac{0.693}{15} \space \text{min}^{-1}\]

Step 3: Determine Number of Half-Lives in One Hour

In one hour (60 minutes), we need to determine how many half-lives have passed:\[ \text{Number of half-lives} = \frac{60}{15} = 4\]

Step 4: Calculate Remaining Amount of Substance

For a first-order reaction, after \( n \) half-lives, the amount of substance left is \( \frac{1}{2^n} \) of the original amount. Here, \( n = 4 \):\[ \text{Remaining amount} = \frac{1}{2^4} = \frac{1}{16}\]

Step 5: Identify the Correct Answer Option

The amount of substance left after one hour is \( \frac{1}{16} \) of the original amount, which corresponds to option (c).

Key Concepts

First Order Reaction

In chemical kinetics, reactions are often classified into different types based on their reaction order. A first order reaction is a type where the reaction rate is directly proportional to the concentration of one reactant. This means that as the concentration of the reactant decreases, the rate at which the reaction occurs also slows down.

First order reactions are quite common in nature and laboratory settings. The rate law for a first order reaction can be expressed as:

\( ext{Rate} = k[A] \)

Where:

• \( ext{Rate} \) is the speed at which the reaction occurs.
• \( k \) is the reaction rate constant, which is a fixed value specific to each reaction.
• \( [A] \) is the concentration of the reactant.

Understanding how first order reactions work helps predict how long it will take for a reaction to reach a certain point, such as when a reactant is fully consumed.

Half-Life

The concept of half-life is crucial when discussing reactions, particularly first order reactions. Half-life, denoted as \( t_{1/2} \), is defined as the time required for the concentration of a reactant to reduce to half its initial amount. It is an indication of how quickly a reaction proceeds.

For first order reactions, the half-life is independent of the initial concentration. This means that regardless of how much reactant you start with, the time it takes for half of it to react will be the same. The formula for calculating half-life in first order reactions is:

\( t_{1/2} = \frac{0.693}{k} \)

Here, \( k \) is the reaction rate constant.

The half-life can be used to quickly determine how much of a substance remains after various periods of time by calculating how many half-lives have elapsed.

Reaction Rate Constant

The reaction rate constant, symbolized as \( k \), is a fundamental factor in chemical kinetics. It provides an indication of the speed of a reaction under specific conditions. For first order reactions, the rate constant is determined from the half-life using the equation:

\( k = \frac{0.693}{t_{1/2}} \)

The rate constant \( k \) has units of time\(^{-1}\), often noted as \( ext{min}^{-1} \) in cases where time is measured in minutes.

A larger value of \( k \) indicates a faster reaction, meaning the reactant is consumed more quickly. Conversely, a smaller \( k \) suggests a slower reaction.

Understanding how \( k \) interacts with the half-life and concentration allows researchers to manipulate reaction conditions to achieve desired results, such as increasing the speed of a reaction or prolonging a reactant's availability.