Question

Two radioactive elements A and B initially have the same number of atoms. The half-life of A is the same as the average life of B. If \( \lambda_A \) and \( \lambda_B \) are the decay constants of A and B respectively, then choose the correct relation from the given options:

• A. \( \lambda_A = 2\lambda_B \)
• B. \( \lambda_A = \lambda_B \)
• C. \( \lambda_A \ln 2 = \lambda_B \)
• D. \( \lambda_A = \lambda_B \ln 2 \)

Hint:

When comparing the half-life and average life, use the relationship between the decay constant and the half-life to derive the necessary formula.

Answer 1

The Correct Option is D

We are given that the half-life of A is the same as the average life of B. The average life \( \tau \) and half-life \( T \) are related to the decay constant \( \lambda \) by the equations: \[ T = \frac{\ln 2}{\lambda} \quad \text{and} \quad \tau = \frac{1}{\lambda} \] Since the half-life of A is the same as the average life of B, we have: \[ \lambda_A = \lambda_B \ln 2 \] Hence, the correct relation is \( \lambda_A = \lambda_B \ln 2 \).