Question

If half life of a radio-active nuclide A is equal to average life of another radio-active nuclide B. Find the ratio of decay constant of A to that of B.

• A. In 2:1
• B. 1: In 2
• C. 2: In 2
• D. In 2:2

Answer 1

The Correct Option is A

To solve this problem, we need to understand the relation between the half-life and average life (mean life) of a radioactive nuclide and how they relate to the decay constant. Let's break it down step-by-step:

1. Definitions and Formulas:
   • The half-life (\( t_{1/2} \)) of a radioactive nuclide is the time required for half of the radioactive atoms to decay. It is related to the decay constant (\( \lambda \)) as: \(t_{1/2} = \frac{\ln 2}{\lambda}\)
   • The average life (mean life, \( \tau \)) of a radioactive nuclide is the average time each atom will survive before it decays. It is related to the decay constant as: \(\tau = \frac{1}{\lambda}\)
2. Given Condition:
   • It is given that the half-life of nuclide \( A \) is equal to the average life of nuclide \( B\): \(t_{1/2}^A = \tau^B\)
   • Substituting the formulas for half-life and average life, we get: \(\frac{\ln 2}{\lambda_A} = \frac{1}{\lambda_B}\)
3. Finding the Ratio of Decay Constants:
   • Rearrange the formula to find the relation between \( \lambda_A \) and \( \lambda_B \): \(\lambda_A = \lambda_B \ln 2\)
   • Hence, the ratio of decay constant of A to that of B is: \(\frac{\lambda_A}{\lambda_B} = \ln 2\)
4. Conclusion:
   • The ratio of the decay constant of nuclide \( A \) to that of nuclide \( B \) is \(\ln 2 : 1\).
   • The correct answer is: In 2:1.