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. \( \lambda_A = \lambda_B \ln 2 \)

Answer 1

The Correct Option is D

To solve the problem, let's begin by understanding the concepts of half-life and average life (mean life) for radioactive elements:

• Half-life (\( t_{1/2} \)) is the time required for half of the radioactive atoms present in a sample to decay.
• Mean life (\( \tau \)) is the average lifetime of all the atoms in a radioactive sample.
• The relationship between mean life and half-life is given by:

\(\tau = \frac{t_{1/2}}{\ln 2}\)

where \(\ln 2 \approx 0.693\).

In this question, we know:

• The half-life of element A, \( t_{1/2, A} \), is equal to the mean life of element B, \( \tau_B \).
• The decay constant \( \lambda \) is related to half-life as \(\lambda = \frac{\ln 2}{t_{1/2}}\).
• The decay constant \( \lambda \) is related to mean life as \(\lambda = \frac{1}{\tau}\).

Given the half-life of A is equal to the mean life of B, we can write:

\(\frac{\ln 2}{\lambda_A} = \frac{1}{\lambda_B}\)

Solving this equation for the relationship between \( \lambda_A \) and \( \lambda_B \):

• Rearrange the equation: \(\lambda_B = \lambda_A \ln 2\)
• Thus, the correct relation is: \(\lambda_A = \lambda_B \ln 2\)

Therefore, the correct answer is:

\(\lambda_A = \lambda_B \ln 2\), which matches the third option.