Question

A radioactive sample has half-life of 3 years. The time required for the activity of the sample to reduce to \(\frac{1}{5}\)th of its initial value is about

• A. 7 years
• B. 15 years
• C. 5 years
• D. 10 years

Answer 1

The Correct Option is A

Given: 

• Half-life of the radioactive sample, \( T = 3 \) years
• Final activity is \( \frac{1}{5} \) of the initial activity

Decay Formula:

\[ N = N_0 \times \left( \frac{1}{2} \right)^{t/T} \]

Substituting values:

\[ \frac{1}{5} = \left( \frac{1}{2} \right)^{t/3} \]

Taking natural logarithm on both sides:

\[ \ln \left( \frac{1}{5} \right) = \frac{t}{3} \ln \left( \frac{1}{2} \right) \]

Using values:

\[ \ln(1/5) = -\ln 5 \approx -1.609 \]

\[ \ln(1/2) = -\ln 2 \approx -0.693 \]

Solving for \( t \):

\[ t = \frac{1.609 \times 3}{0.693} \]

\[ t \approx \frac{4.827}{0.693} \approx 7 \text{ years} \]

Answer: The time required is 7 years (Option A).