Problem 67 - A Complete Resource Book in Chemistry for JEE Main

Question

A radioactive substance having a half-life of 3 days was received in 12 days. It was found that there was only \(3 \mathrm{~g}\) of the isotope in the container. The initial weight of the isotope when packed was (a) \(12 \mathrm{~g}\) (b) \(24 \mathrm{~g}\) (c) \(48 \mathrm{~g}\) (d) \(96 \mathrm{~g}\)

Step-by-Step Solution

Answer

The initial weight of the isotope was 48 grams (option c).

Step 1: Determine the Decay Formula for Half-Life

The decay of a radioactive substance is given by the formula \( N(t) = N_0 \cdot \left( \frac{1}{2} \right)^{\frac{t}{T}} \), where \( N(t) \) is the amount remaining, \( N_0 \) is the initial amount, \( t \) is the time elapsed, and \( T \) is the half-life of the substance.

Step 2: Identify Known Values

From the exercise, the half-life \( T = 3 \) days, time elapsed \( t = 12 \) days, and the remaining amount \( N(t) = 3 \) grams. We need to find \( N_0 \), the initial amount.

Step 3: Set Up the Equation

Using the decay formula, substitute the known values: \( 3 = N_0 \cdot \left( \frac{1}{2} \right)^{\frac{12}{3}} \). Simplifying the exponent gives \( 3 = N_0 \cdot \left( \frac{1}{2} \right)^4 \).

Step 4: Solve for Initial Amount \( N_0 \)

The expression \( \left( \frac{1}{2} \right)^4 \) equals \( \frac{1}{16} \). Thus, \( 3 = N_0 \cdot \frac{1}{16} \). To isolate \( N_0 \), multiply both sides by 16: \( N_0 = 3 \times 16 \).

Step 5: Calculate \( N_0 \) and Choose the Correct Answer

Calculate \( N_0 = 48 \) grams. Comparing with the options, the initial weight of the isotope was 48 grams, option (c).

Key Concepts

Half-Life Calculation

The concept of half-life is a fundamental principle in radioactive decay measurements. It describes the time required for a quantity of a radioactive substance to reduce to half its initial amount. In the context of our exercise, the half-life of the substance is given as 3 days. This means that every 3 days, the remaining amount of the isotope will be halved.
Understanding half-life helps predict how long it will take for a radioactive material to decay to a safer, less radioactive level.

• If a radioactive substance starts with a certain number of atoms, in one half-life, that number will be reduced by 50%.
• This does not mean the substance is half gone, just that half of its radioactive activity has decayed.

When calculating decay over multiple half-lives, we multiply the time period by the number of half-lives elapsed to find the total decay period.

Exponential Decay Formula

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. In radioactive decay, this is captured mathematically by the exponential decay formula: \[N(t) = N_0 \cdot \left( \frac{1}{2} \right)^{\frac{t}{T}}\]where:

• \(N(t)\) is the remaining quantity of the substance after time \(t\).
• \(N_0\) is the initial quantity.
• \(T\) is the half-life of the substance.

This formula is central to solving half-life problems because it relates all relevant variables: the past, present, and decay process. In our exercise, this formula helps us determine how much of the isotope remains after it has undergone decay for 12 days.
Understanding the exponential decay formula equips you to calculate the amount of substance remaining at any given time.

Initial Quantity of Isotope

Determining the initial quantity of an isotope is critical in many scientific applications, from radiometric dating to medical diagnostics. The initial quantity, often denoted as \(N_0\), refers to the amount of the substance before any decay has occurred.
This value can be found using the exponential decay formula by rearranging it to solve for \(N_0\) if the remaining amount and elapsed time are known.

• In the problem, we find that after 12 days, only 3 grams remain.
• Using the formula \(3 = N_0 \cdot \left( \frac{1}{2} \right)^{\frac{12}{3}}\), we determine \(N_0\).
• Upon solving, we find that the initial quantity was 48 grams.

Knowing the initial quantity is crucial for tracking substances' decay over time and ensures that mathematical models or scientific predictions are based on accurate, initial starting conditions.