Question

The half-life of a radioactive nucleus is 5 years. The fraction of the original sample that would decay in 15 years is:

• A. \(\frac{1}{8}\) of initial value
• B. \(\frac{7}{8}\) of initial value
• C. \(\frac{1}{4}\) of initial value
• D. \(\frac{3}{4}\) of initial value

Answer 1

The Correct Option is A

Understanding the Problem

We are given a radioactive substance with a half-life of 5 years. We need to find the fraction of the original sample that decays after 15 years.

Solution

1. Radioactive Decay Formula:

The number of remaining nuclei after time \(t\) is given by:

\( N = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \)

where:

• \( N_0 \) is the initial number of nuclei
• \( N \) is the remaining number of nuclei
• \( t \) is the elapsed time
• \( T_{1/2} \) is the half-life

2. Substitute Values:

Given \( T_{1/2} = 5 \, \text{years} \) and \( t = 15 \, \text{years} \), we have:

\( N = N_0 \left( \frac{1}{2} \right)^{\frac{15}{5}} = N_0 \left( \frac{1}{2} \right)^3 \)

3. Calculate Remaining Fraction:

\( N = N_0 \left( \frac{1}{8} \right) = \frac{N_0}{8} \)

This means that \(\frac{1}{8}\) of the original sample remains.

4. Calculate Decayed Fraction:

The fraction that decayed is the difference between the initial fraction (1) and the remaining fraction (\(\frac{1}{8}\)):

\( \text{Fraction decayed} = 1 - \frac{1}{8} = \frac{7}{8} \)

Final Answer

The fraction of the original sample that decays is \(\frac{7}{8}\).

Answer 2

The correct answer is (A) : \(\frac{1}{8}\) of initial value
A substance has a half-life of 5 years.
The number of half life periods in 15 years\(=\frac{15}{5}=3\)
The relation used is :
\(A_t=\frac{A_∘}{2^n}\)
Here, n is number of half life periods.
\(A_{15}\ \ years=\frac{1}{2^3}=\frac{1}{8}\)
The amount of A left after 15 years is \(\frac{1}{8}\) of initial value.