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 question regarding the fraction of a radioactive element that disintegrates during its mean life, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Mean Life**: The mean life (T_mean) of a radioactive element is related to the decay constant (λ) by the formula:
\[ T_{\text{mean}} = \frac{1}{\lambda} \]

2. **Determine the Remaining Nuclei**: The number of nuclei remaining (N_t) after time t is given by the equation:
\[ N_t = N_0 \cdot e^{-\lambda t} \]
where \(N_0\) is the initial number of nuclei.

3. **Substitute Mean Life into the Equation**: We want to find the number of remaining nuclei at the mean life (t = T_mean). Substituting \(T_{\text{mean}} = \frac{1}{\lambda}\) into the equation gives:
\[ N_t = N_0 \cdot e^{-\lambda \cdot T_{\text{mean}}} = N_0 \cdot e^{-1} \]

4. **Calculate the Number of Disintegrated Nuclei**: The number of nuclei that have disintegrated (N_d) is the initial number minus the remaining number:
\[ N_d = N_0 - N_t = N_0 - \left(N_0 \cdot e^{-1}\right) = N_0 \left(1 - e^{-1}\right) \]

5. **Find the Fraction of Disintegrated Nuclei**: The fraction of disintegrated nuclei (f) is given by the ratio of disintegrated nuclei to the initial number of nuclei:
\[ f = \frac{N_d}{N_0} = \frac{N_0 \left(1 - e^{-1}\right)}{N_0} = 1 - e^{-1} \]

6. **Express the Fraction in Terms of e**: The fraction of disintegrated nuclei can also be expressed as:
\[ f = \frac{e - 1}{e} \]

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