What is the probability of a radioactive nucleus to survive one mean life?

Options

A. (a) `1/e`
B. (b) `(1)/(e+1)`
C. (c) `1-1/e`
D. (d) `1/e-1`

Explanation

To find the probability of a radioactive nucleus surviving one mean life, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Mean Life**:
- The mean life (or average lifetime) of a radioactive nucleus is the average time that a nucleus exists before it decays. It is denoted by the symbol \( \tau \) and is related to the decay constant \( \lambda \) by the equation:
\[ \tau = \frac{1}{\lambda} \]

2. **Initial Number of Nuclei**:
- Let the initial number of radioactive nuclei be \( N_0 \).

3. **Number of Nuclei After One Mean Life**:
- After one mean life, the number of remaining nuclei \( N \) can be expressed using the exponential decay formula:
\[ N = N_0 e^{-\lambda t} \]
- Here, \( t \) is the time elapsed. For one mean life, \( t = \tau = \frac{1}{\lambda} \).

4. **Substituting for One Mean Life**:
- Substitute \( t = \tau \) into the decay formula:
\[ N = N_0 e^{-\lambda \cdot \tau} = N_0 e^{-\lambda \cdot \frac{1}{\lambda}} = N_0 e^{-1} \]

5. **Calculating the Probability**:
- The probability \( P \) of a nucleus surviving after one mean life is given by the ratio of the number of remaining nuclei to the initial number:
\[ P = \frac{N}{N_0} = \frac{N_0 e^{-1}}{N_0} = e^{-1} \]

6. **Final Result**:
- Therefore, the probability of a radioactive nucleus surviving one mean life is:
\[ P = \frac{1}{e} \approx 0.3679 \]

### Summary:
The probability of a radioactive nucleus surviving one mean life is \( \frac{1}{e} \).