Question

The relation between the mean life time \( \tau \) and the half life time \( T_{1/2} \) of a radioactive substance is:

• A. \( T_{1/2} = \tau \log_e 2 \)
• B. \( T_{1/2} = \tau \log_{10} 2 \)
• C. \( T_{1/2} = \tau \)
• D. \( T_{1/2} = 2 \tau \log_e 2 \)

Hint:

Remember the basic relationships between the mean lifetime \( \tau \), decay constant \( \lambda \), and half-life \( T_{1/2} \).

Answer 1

The Correct Option is A

The relationship between the mean life time \( \tau \) and the half-life time \( T_{1/2} \) for a radioactive substance is given by the following formula: \[ T_{1/2} = \tau \log_e 2 \] This formula can be derived from the basic concepts of radioactive decay. The mean lifetime \( \tau \) is related to the decay constant \( \lambda \) by \( \tau = \frac{1}{\lambda} \), and similarly, the half-life \( T_{1/2} \) is related to \( \lambda \) by \( T_{1/2} = \frac{\ln 2}{\lambda} \). Therefore, combining these two gives the final result: \[ T_{1/2} = \tau \log_e 2 \]