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

### Step 1: Understand the Definitions
- **Half-life (\( T \))**: The time required for half of the radioactive nuclei in a sample to decay.
- **Mean life (\( \tau \))**: The average lifetime of a radioactive particle before it decays.

### Step 2: Use the Decay Constant
The mean life \( \tau \) is related to the decay constant \( \lambda \) by the formula:
\[ \tau = \frac{1}{\lambda} \]

### Step 3: Relate Half-life to Decay Constant
The half-life \( T \) is related to the decay constant \( \lambda \) by the formula:
\[ T = \frac{\ln(2)}{\lambda} \]
where \( \ln(2) \) is approximately \( 0.693 \).

### Step 4: Substitute for \( \lambda \)
From the half-life equation, we can express \( \lambda \) as:
\[ \lambda = \frac{\ln(2)}{T} \]

### Step 5: Substitute \( \lambda \) into Mean Life Equation
Now, substituting \( \lambda \) back into the mean life equation:
\[ \tau = \frac{1}{\lambda} = \frac{1}{\frac{\ln(2)}{T}} = \frac{T}{\ln(2)} \]

### Step 6: Final Relationship
Since \( \ln(2) \) is approximately \( 0.693 \), we can express the relationship as:
\[ \tau = \frac{T}{0.693} \]
or rearranging gives:
\[ T = 0.693 \cdot \tau \]

### Conclusion
Thus, the relationship between the half-life \( T \) and the mean life \( \tau \) of a radioactive sample is:
\[ T = 0.693 \cdot \tau \]