JEE Physics 2023 Question with Solution

Given: The half-life of the radioactive nucleus is $5 \, \text{years}$ and the given time is $15 \, \text{years}$.

Find: The fraction of the original sample that decays in $15 \, \text{years}$.

Step 1: Identify the number of half-lives elapsed.

$$\text{Number of half-lives} = \frac{15}{5} = 3$$

Step 2: Calculate the fraction of the substance remaining. After $n$ half-lives, the fraction remaining is

$$\left(\frac{1}{2}\right)^n$$

For $n = 3$,

$$\text{Fraction remaining} = \left(\frac{1}{2}\right)^3 = \frac{1}{8}$$

Step 3: Find the fraction that has decayed.

$$\text{Fraction decayed} = 1 - \text{Fraction remaining} = 1 - \frac{1}{8} = \frac{7}{8}$$

Step 4: Conclusion. Therefore, the fraction of the original sample that decays in $15 \, \text{years}$ is $\dfrac{7}{8}$. The correct option is D.