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: \[ T_{1/2} = \tau \log_e 2 \]

Answer 2

Step 1: Understand the definitions.
- Mean life time (\( \tau \)) is the average time a nucleus exists before decaying.
- Half life time (\( T_{1/2} \)) is the time required for half of the radioactive nuclei to decay.

Step 2: Mathematical relationship between \( \tau \) and decay constant \( \lambda \).
\[ \tau = \frac{1}{\lambda}, \quad T_{1/2} = \frac{\ln 2}{\lambda} \]

Step 3: Derive the relation between \( \tau \) and \( T_{1/2} \).
Substitute \( \lambda = \frac{1}{\tau} \) into the expression for \( T_{1/2} \):
\[ T_{1/2} = \ln 2 \times \tau = \tau \log_e 2 \]

Step 4: Conclusion.
The correct relation is: \( \boxed{T_{1/2} = \tau \log_e 2} \)