In a mean life of a radioactive sample

Options

A. About `1//3` of substance disintegrates
B. About `2//3` of the substance disintegrates
C. About `90 %` of the substance disintegrates
D. Almost all the substance disintegrates

Explanation

To solve the question regarding the mean life of a radioactive sample, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Mean Life**:
The mean life (or average life) of a radioactive sample is the average time that a nucleus of the sample will exist before it decays. It is denoted by \( T \) and is related to the decay constant \( \lambda \).

2. **Know the Relationship**:
The mean life \( T \) is given by the formula:
\[ T = \frac{1}{\lambda} \]
where \( \lambda \) is the decay constant.

3. **Use the Decay Formula**:
The number of undecayed nuclei \( n \) at time \( t \) is given by the formula:
\[ n = n_0 e^{-\lambda t} \]
where \( n_0 \) is the initial number of nuclei.

4. **Substituting Mean Life into the Decay Formula**:
To find the number of undecayed nuclei after one mean life, substitute \( t = T = \frac{1}{\lambda} \):
\[ n = n_0 e^{-\lambda \cdot \frac{1}{\lambda}} = n_0 e^{-1} \]

5. **Calculate the Value of \( n \)**:
Since \( e^{-1} \) is approximately equal to \( \frac{1}{e} \) (which is about 0.3679), we can express this as:
\[ n \approx \frac{n_0}{e} \approx 0.37 n_0 \]

6. **Determine the Disintegrated Nuclei**:
The number of disintegrated nuclei after one mean life can be calculated as:
\[ \text{Disintegrated} = n_0 - n = n_0 - 0.37 n_0 = 0.63 n_0 \]
This means that approximately 63% of the original sample has disintegrated.

7. **Conclusion**:
Therefore, in one mean life, approximately \( \frac{2}{3} \) of the original sample disintegrates. This corresponds to the option that states that \( \frac{2}{3} \) of the sample disintegrates.

### Final Answer:
The correct option is **B**, which states that approximately \( \frac{2}{3} \) of the original sample disintegrates in one mean life.