The relation between half-life T of a radioactive sample and its mean life `tau`is:

Options

A. `T=0.693tau`
B. `tau=0.693T`
C. `tau=T`
D. `tau=2.718T`

Answer

To find the relation between the half-life \( T \) of a radioactive sample and its mean life \( \tau \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Half-Life**:
- The half-life \( T \) of a radioactive substance is the time required for half of the radioactive atoms in a sample to decay. If we start with \( N_0 \) atoms, after time \( T \), we will have \( \frac{N_0}{2} \) atoms remaining.

2. **Decay Constant**:
- The decay constant \( \lambda \) is a probability rate at which a single atom decays. It is related to the half-life by the formula:
\[ T = \frac{\ln(2)}{\lambda} \]
- Here, \( \ln(2) \) is approximately 0.693.

3. **Mean Life**:
- The mean life \( \tau \) of a radioactive sample is the average time that a single atom will exist before it decays. The mean life is given by:
\[ \tau = \frac{1}{\lambda} \]

4. **Relating Half-Life and Mean Life**:
- We can substitute the expression for \( \lambda \) from the mean life into the half-life formula:
\[ T = \frac{\ln(2)}{\lambda} = \ln(2) \cdot \tau \]
- Therefore, we have:
\[ T = 0.693 \cdot \tau \]

5. **Final Relation**:
- The final relation between the half-life \( T \) and the mean life \( \tau \) is:
\[ T = 0.693 \tau \]

### Conclusion:
The relation between the half-life \( T \) of a radioactive sample and its mean life \( \tau \) is given by:
\[ T = 0.693 \tau \]