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

Options

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

Answer

To solve the problem of finding 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 (τ) of a radioactive element is defined as the average time a nucleus exists before it disintegrates. It is mathematically expressed as:
\[ \tau = \frac{1}{\lambda} \]
where \( \lambda \) is the disintegration constant.

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

3. **Calculate Remaining Nuclei After Mean Life**:
Substitute \( t \) with the mean life \( \tau \):
\[ N(\tau) = N_0 e^{-\lambda \tau} \]
Since \( \tau = \frac{1}{\lambda} \), we can substitute this into the equation:
\[ N(\tau) = N_0 e^{-1} \]
This simplifies to:
\[ N(\tau) = \frac{N_0}{e} \]

4. **Find the Number of Disintegrated Nuclei**:
The number of disintegrated nuclei is given by the initial number minus the remaining number:
\[ N_{\text{disintegrated}} = N_0 - N(\tau) = N_0 - \frac{N_0}{e} \]
This can be simplified to:
\[ N_{\text{disintegrated}} = N_0 \left(1 - \frac{1}{e}\right) \]

5. **Calculate the Fraction of Disintegrated Nuclei**:
The fraction of disintegrated nuclei is given by:
\[ \text{Fraction} = \frac{N_{\text{disintegrated}}}{N_0} = \frac{N_0 \left(1 - \frac{1}{e}\right)}{N_0} \]
This simplifies to:
\[ \text{Fraction} = 1 - \frac{1}{e} \]

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

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