The correct relation between `t_(av)`=average life and `t_(1//2)=` half life for a radioactive nuclei.

Options

A. `T_(1//2)=("In"2)/(lambda)`
B. `T_(1//2)"In" 2 =lambda`
C. `T_(1//2)=1/(lambda)`
D. `(lambda+T_(1//2))="In"2`

Answer

To find the correct relation between the average life (\( t_{av} \)) and the half-life (\( t_{1/2} \)) of a radioactive nucleus, we can start by using the definitions of average life and half-life.

### Step-by-Step Solution:

1. **Definitions**:
- The average life (\( t_{av} \)) of a radioactive nucleus is defined as the expected time before a nucleus decays. It is given by the formula:
\[ t_{av} = \frac{1}{\lambda} \]
where \( \lambda \) is the decay constant.

- The half-life (\( t_{1/2} \)) is the time taken for half of the radioactive nuclei to decay. It is given by the formula:
\[ t_{1/2} = \frac{\ln 2}{\lambda} \]

2. **Finding the Relation**:
- To find the relation between \( t_{av} \) and \( t_{1/2} \), we can express both in terms of \( \lambda \):
\[ t_{av} = \frac{1}{\lambda} \]
\[ t_{1/2} = \frac{\ln 2}{\lambda} \]

3. **Dividing the Two Equations**:
- Now, we can divide the average life by the half-life:
\[ \frac{t_{av}}{t_{1/2}} = \frac{\frac{1}{\lambda}}{\frac{\ln 2}{\lambda}} = \frac{1}{\ln 2} \]

4. **Rearranging the Equation**:
- From the above relation, we can rearrange it to express \( t_{av} \) in terms of \( t_{1/2} \):
\[ t_{av} = \frac{t_{1/2}}{\ln 2} \]

5. **Conclusion**:
- Thus, the correct relation between the average life and half-life for a radioactive nucleus is:
\[ t_{av} = \frac{t_{1/2}}{\ln 2} \]

### Final Answer:
The correct relation is:
\[ t_{av} = \frac{t_{1/2}}{\ln 2} \]