Problem 138 - A Complete Resource Book in Chemistry for JEE Main

Question

Half-life of a substance A, following first order kinetics is 5 days. Starting with \(100 \mathrm{~g}\) of \(\mathrm{A}\), amount left after 15 days is \(|\mathbf{2 0 0 2}|\) (a) \(25 \mathrm{~g}\) (b) \(50 \mathrm{~g}\) (c) \(12.5 \mathrm{~g}\) (d) \(6.25 \mathrm{~g}\)

Step-by-Step Solution

Answer

The amount of substance A left after 15 days is 12.5 grams (option c).

Step 1: Understand Half-Life and Initial Conditions

The half-life of a substance is the time required for half of the initial amount to decay. For this exercise, the half-life of substance A is given as 5 days, and the initial amount is 100 grams.

Step 2: Determine Number of Half-Lives Passed

Calculate the number of half-lives that have passed in 15 days. Since each half-life is 5 days, in 15 days, the substance goes through three half-lives: \( \text{Number of half-lives} = \frac{15 \text{ days}}{5 \text{ days/half-life}} = 3 \text{ half-lives} \).

Step 3: Apply Decay Formula

After each half-life, the substance amount is halved. Therefore, after 3 half-lives, the remaining amount is given by: \( \text{Remaining amount} = 100 \text{ grams} \times \left(\frac{1}{2}\right)^3 \).

Step 4: Calculate Remaining Amount

Perform the calculation from the previous step: \( 100 \times \left(\frac{1}{2}\right)^3 = 100 \times \frac{1}{8} = 12.5 \text{ grams} \).

Step 5: Select the Correct Answer

Comparing this result with the given options, the closest is option (c) 12.5 grams.

Key Concepts

Half-Life

The concept of half-life is central to understanding how substances decay over time. Half-life is defined as the time required for half of the original amount of a substance to transform or disappear due to decay processes. For instance, if you start with 100 grams of a substance, after one half-life, you would have 50 grams left. After another half-life, you would have 25 grams left, and so on.
Understanding half-life helps us predict how long it will take for a substance to reduce to a certain amount, which is crucial for areas like pharmacology and environmental sciences. Knowing the half-life allows scientists and engineers to calculate how quickly a substance will diminish, which can inform everything from medication dosages to waste management strategies.

Decay Formula

The decay formula is a mathematical expression used to calculate the remaining amount of a substance after a given number of half-lives. It is particularly useful for substances following first-order kinetics, where the rate of decay is proportional to the current amount.
The basic decay formula is:

• Amount remaining = Initial amount × ewline\((\frac{1}{2})^{\text{Number of half-lives}}\)

This formula uses the power of a fraction to determine how much of the substance remains after several cycles of halving. As each cycle occurs, the substance amount is continually divided by two. Using this formula, you can accurately determine how much of the substance remains at any given time period.

Half-Life Calculation

Calculating half-life involves determining the elapsed time in relation to the rate at which a substance decays. To find out how many half-lives have passed within a certain period, you divide the total time by the length of one half-life.
For example, if the half-life of a substance is 5 days and you want to know the remaining amount after 15 days, the calculation would be:

• Number of half-lives = Total time / Length of one half-life- In this example: Number of half-lives = ewline\(\frac{15 \text{ days}}{5 \text{ days/half-life}} = 3 \)

Such calculations are crucial to determine timelines for how substances behave over time, especially in chemical reactions and radiological assessments.

Chemical Kinetics

Chemical kinetics involves the study of reaction rates and how different conditions impact these rates. In the context of half-life and decay, it helps explain the speed and processes by which chemical reactions occur. In first-order kinetics, which many radioactive and chemical decay processes follow, the rate of the reaction is directly proportional to the concentration of one reactant.
The knowledge of chemical kinetics allows scientists to:

• Predict how quickly a substance will decay or transform under various conditions.
• Optimize conditions for reactions to occur at desirable rates.
• Formulate strategies to control these rates in industrial processes.

Understanding first-order kinetics is crucial when working with reactions that have half-lives, as it forms the foundation for calculations and predictions about how substances will behave over time.