The probaility that a certaun radioactive atom would get disintefrated in a time equal to the mean life fo the radioactive sample is

Options

A. `0.37`
B. `0.63`
C. `0.50`
D. `0.67`

Explanation

To solve the problem of finding the probability that a certain radioactive atom would get disintegrated in a time equal to the mean life of the radioactive sample, we can follow these steps:

### Step-by-Step Solution:

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

2. **Expression for Remaining Particles**:
The number of radioactive particles remaining at time \( t \) is given by:
\[ N(t) = N_0 e^{-\lambda t} \]
where \( N_0 \) is the initial number of particles.

3. **Finding the Number of Disintegrated Particles**:
The number of particles that have disintegrated by time \( t \) can be expressed as:
\[ N_{\text{disintegrated}} = N_0 - N(t) = N_0 - N_0 e^{-\lambda t} \]
Simplifying this gives:
\[ N_{\text{disintegrated}} = N_0 (1 - e^{-\lambda t}) \]

4. **Calculating Probability**:
The probability \( P \) that a particle disintegrates in time \( t \) is given by:
\[ P = \frac{N_{\text{disintegrated}}}{N_0} = 1 - e^{-\lambda t} \]

5. **Substituting Mean Life into the Probability Expression**:
Since we are interested in the probability at time \( t = \tau \):
\[ P = 1 - e^{-\lambda \tau} \]
Substituting \( \tau = \frac{1}{\lambda} \):
\[ P = 1 - e^{-\lambda \cdot \frac{1}{\lambda}} = 1 - e^{-1} \]

6. **Calculating \( e^{-1} \)**:
The value of \( e^{-1} \) is approximately \( 0.3679 \). Therefore:
\[ P = 1 - 0.3679 \approx 0.6321 \]

7. **Final Answer**:
Thus, the probability that a certain radioactive atom would get disintegrated in a time equal to the mean life of the radioactive sample is approximately \( 0.63 \).

### Conclusion:
The required probability is \( 0.63 \), which corresponds to option number 2.