Question

The half-life of a radioactive element A is the same as the mean-life of another radioactive element B. Initially both substances have the same number of atoms, then :

• A. A and B decay at the same rate always.
• B. A and B decay at the same rate initially.
• C. A will decay at a faster rate than B.
• D. B will decay at a faster rate than A.

Hint:

Decay rate is \(\lambda N\); compare the decay constants using \(T_{1/2} = \frac{0.693}{\lambda}\) and \(\tau = \frac{1}{\lambda}\).

Answer 1

The Correct Option is D

For A: \(T_{1/2}(A) = \frac{0.693}{\lambda_A}\). For B: mean life \(\tau_B = \frac{1}{\lambda_B}\).
Given \(T_{1/2}(A) = \tau_B\): \(\frac{0.693}{\lambda_A} = \frac{1}{\lambda_B}\), so \(\lambda_B = \frac{\lambda_A}{0.693} = 1.44\,\lambda_A\).
Both start with the same number of atoms \(N_0\): initial rates \(R_A = \lambda_A N_0\), \(R_B = \lambda_B N_0\).
\(\frac{R_B}{R_A} = 1.44 \gt 1\), so B decays at a faster rate than A.
Answer: (4) B will decay at a faster rate than A.