During mean life of a radioactive element, the fraction that disintegrates is

Options

A. e
B. `(e-1)/e`
C. `1/e`
D. `e/(e-1)`

Answer

To solve the problem of determining the fraction of a radioactive element that disintegrates during its mean life, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Mean Life**:
The mean life (or average life) of a radioactive element is denoted as \( T_{avg} \) and is given by the formula:
\[ T_{avg} = \frac{1}{\lambda} \]
where \( \lambda \) is the decay constant.

2. **Initial and Remaining Nuclei**:
Let \( N_0 \) be the initial number of nuclei at time \( t = 0 \), and \( N_t \) be the number of remaining nuclei after time \( t \). The relationship between these quantities is given by:
\[ N_t = N_0 e^{-\lambda t} \]

3. **Finding the Fraction that Disintegrates**:
The fraction of the radioactive element that has disintegrated by time \( t \) can be expressed as:
\[ \text{Fraction disintegrated} = \frac{N_0 - N_t}{N_0} \]
Substituting \( N_t \) into the equation:
\[ \text{Fraction disintegrated} = \frac{N_0 - N_0 e^{-\lambda t}}{N_0} \]
This simplifies to:
\[ \text{Fraction disintegrated} = 1 - e^{-\lambda t} \]

4. **Substituting Mean Life into the Equation**:
Now, we substitute \( t = T_{avg} = \frac{1}{\lambda} \) into the fraction:
\[ \text{Fraction disintegrated} = 1 - e^{-\lambda \cdot \frac{1}{\lambda}} = 1 - e^{-1} \]

5. **Final Calculation**:
We can express \( e^{-1} \) as \( \frac{1}{e} \):
\[ \text{Fraction disintegrated} = 1 - \frac{1}{e} \]
To combine these terms, we can write:
\[ \text{Fraction disintegrated} = \frac{e - 1}{e} \]

6. **Conclusion**:
Therefore, the fraction of the radioactive element that disintegrates during its mean life is:
\[ \frac{e - 1}{e} \]

### Final Answer:
The fraction that disintegrates during the mean life of a radioactive element is \( \frac{e - 1}{e} \).