The `t_(0.5)` of a radioactive element is related to its average life by the expression

Options

A. `T_12=1.44 T_(av)`
B. `T_12=1.36 T_(av)`
C. `T_12=0.693 T_(av)`
D. `T_12= (T_(av))/0.693`

Answer

To solve the problem of how the half-life \( t_{0.5} \) of a radioactive element is related to its average life, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Average Life**:
The average life \( t_{avg} \) of a radioactive element is defined as the average time a nucleus exists before it decays. Mathematically, it can be expressed as:
\[ t_{avg} = \frac{\int_0^\infty t \cdot \frac{dN}{dt} dt}{N_0} \]
where \( N_0 \) is the initial number of nuclei and \( \frac{dN}{dt} \) is the rate of change of the number of nuclei.

**Hint**: Average life is the integral of time weighted by the rate of decay.

2. **Decay Constant Relation**:
The rate of disintegration is proportional to the number of nuclei present, which can be expressed as:
\[ \frac{dN}{dt} = -\lambda N \]
where \( \lambda \) is the decay constant.

**Hint**: Remember that the decay rate is linked to the number of remaining nuclei.

3. **Integrating the Decay Equation**:
Rearranging and integrating gives:
\[ N(t) = N_0 e^{-\lambda t} \]
This shows how the number of nuclei decreases over time.

**Hint**: The exponential decay function is key to understanding radioactive decay.

4. **Finding Average Life**:
To find \( t_{avg} \), we can substitute \( N(t) \) into the average life equation. After performing the integration and simplifications, we find:
\[ t_{avg} = \frac{1}{\lambda} \]

**Hint**: The average life is inversely proportional to the decay constant.

5. **Relating Half-Life to Average Life**:
The half-life \( t_{0.5} \) is defined as the time taken for half of the radioactive nuclei to decay. It can be derived from the decay equation:
\[ N_0 e^{-\lambda t_{0.5}} = \frac{N_0}{2} \]
Taking the natural logarithm of both sides gives:
\[ -\lambda t_{0.5} = \ln\left(\frac{1}{2}\right) = -\ln(2) \]
Thus, we can express the half-life as:
\[ t_{0.5} = \frac{\ln(2)}{\lambda} \]

**Hint**: The half-life is derived from the decay equation using logarithms.

6. **Final Relation**:
Now, we can relate \( t_{0.5} \) and \( t_{avg} \):
\[ t_{0.5} = \ln(2) \cdot t_{avg} \]
Since \( t_{avg} = \frac{1}{\lambda} \), we can see that:
\[ t_{0.5} = \ln(2) \cdot \frac{1}{\lambda} \]

**Hint**: The relationship between half-life and average life involves the natural logarithm of 2.

### Conclusion:
The relationship between the half-life \( t_{0.5} \) and the average life \( t_{avg} \) of a radioactive element is given by:
\[ t_{0.5} = \ln(2) \cdot t_{avg} \]
Thus, the correct answer is option C: \( t_{0.5} = \ln(2) \cdot t_{avg} \).