Question

At some instant, a radioactive sample \(S_1\) having an activity \(5\ \mu\mathrm{Ci}\) has twice the number of nuclei as another sample \(S_2\) which has an activity of \(10\ \mu\mathrm{Ci}\). The half lives of \(S_1\) and \(S_2\) are :

• A. 20 years and 5 years, respectively
• B. 20 years and 10 years, respectively
• C. 5 years and 20 years, respectively
• D. 10 years and 20 years, respectively

Hint:

Activity \(A = \lambda N\) and half-life \(T_{1/2} = \frac{\ln 2}{\lambda}\), so \(T_{1/2} \propto N/A\).

Answer 1

The Correct Option is A

\(A_1 = \lambda_1 N_1 = 5\) and \(A_2 = \lambda_2 N_2 = 10\) with \(N_1 = 2N_2\).
Divide: \(\frac{\lambda_1 N_1}{\lambda_2 N_2} = \frac{5}{10}\), so \(\frac{\lambda_1}{\lambda_2}\times 2 = \frac12\) and \(\frac{\lambda_1}{\lambda_2} = \frac14\).
Half-life \(T = \frac{\ln 2}{\lambda}\), so \(\frac{T_1}{T_2} = \frac{\lambda_2}{\lambda_1} = 4\).
Only option (1) has \(T_1 : T_2 = 20 : 5 = 4\).
Answer: (1) 20 years and 5 years, respectively