Problem 61 - A Complete Resource Book in Chemistry for JEE Main

Question

The half-life of a radioactive isotope is 3 hours. If the initial mass of the isotope was \(300 \mathrm{~g}\), the mass which remained undecayed in 18 hours would be (a) \(4.68 \mathrm{~g}\) (b) \(2.34 \mathrm{~g}\) (c) \(1.17 \mathrm{~g}\) (d) \(9.36 \mathrm{~g}\)

Step-by-Step Solution

Answer

(a) 4.68 g

Step 1: Understanding Half-Life Concept

The half-life of a radioactive isotope is the time required for half of its mass to decay. In this case, it is given that the half-life is 3 hours.

Step 2: Determine Number of Half-Lives Passed

To find out how many half-lives have passed in 18 hours, divide the total time by the half-life duration: \(\frac{18}{3} = 6\). So, 6 half-lives have passed in 18 hours.

Step 3: Calculate Remaining Mass Using Half-Life Formula

The mass remaining after a certain number of half-lives \(n\) can be calculated using the formula: \( \text{Remaining mass} = \text{Initial mass} \times \left(\frac{1}{2}\right)^n \). Substituting the values: \( 300 \times \left(\frac{1}{2}\right)^6 \).

Step 4: Compute the Remaining Mass

Calculate \( \left(\frac{1}{2}\right)^6 \), which is \(\frac{1}{64}\). Multiply this by the initial mass \(300\mathrm{~g}\): \(300 \times \frac{1}{64} = \frac{300}{64} = 4.6875\mathrm{~g}\).

Step 5: Round to the Nearest Given Option

Round 4.6875 to the nearest given option. The closest option is \(4.68\mathrm{~g}\).

Key Concepts

Radioactive Decay

Radioactive decay is a natural process where unstable atomic nuclei lose energy by emitting radiation. This results in the transformation of an element into a different isotope or a completely different element. A key characteristic of radioactive decay is that it's a random process at the individual atom level, but predictable in terms of amount over time for a large quantity of atoms. Understanding this concept is crucial because it underpins the calculations we do in half-life problems.

Initial Mass

The initial mass is the starting quantity of a radioactive substance before any decay has occurred. In half-life problems, it serves as the baseline for determining the amount of substance left after various intervals. In our example problem, the initial mass is given as 300 grams. This value is essential because it allows us to calculate how much of the substance remains after a certain number of half-lives have passed.

Remaining Mass Calculation

Calculating the remaining mass involves using the half-life formula, which predicts how much of a substance will remain after a certain number of half-lives. The formula is as follows: \[ \text{Remaining mass} = \text{Initial mass} \times \left(\frac{1}{2}\right)^n \] Here, \(n\) is the number of half-lives that have occurred. This formula helps us understand how substances decay exponentially over time. In our example, starting with 300 grams and passing through 6 half-lives, we use the formula: \[ 300 \times \left(\frac{1}{2}\right)^6 = 4.6875 \text{ grams} \] This shows us that after 18 hours, only a small fraction of the initial mass remains.

Number of Half-Lives

To determine the number of half-lives that have passed, you divide the total time elapsed by the duration of one half-life. In the given problem, each half-life lasts 3 hours. To find out how many have passed in 18 hours, you compute: \[ \frac{18}{3} = 6 \] This tells us that 6 half-lives have elapsed. By understanding how many half-lives a sample has undergone, we can predict its remaining mass using the half-life formula outlined earlier.