In one average-life,

Options

A. A half the active decay
B. less than half the active nuclei decay
C. more than half the active nuclei decay
D. all the nuclei decay

Answer

To solve the question regarding how many nuclei decay in one average life, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Average Life**:
- The average life (or mean life) of a radioactive substance is defined as the average time a nucleus exists before it decays. It is denoted by the symbol \( \tau \).

2. **Decay Constant**:
- The decay constant \( \lambda \) is related to the average life by the formula:
\[ \tau = \frac{1}{\lambda} \]

3. **Exponential Decay**:
- The number of undecayed nuclei \( N(t) \) at time \( t \) is given by:
\[ N(t) = N_0 e^{-\lambda t} \]
- Here, \( N_0 \) is the initial number of nuclei.

4. **Calculating Decayed Nuclei**:
- In one average life \( \tau \), we can substitute \( t = \tau \):
\[ N(\tau) = N_0 e^{-\lambda \tau} \]
- Since \( \tau = \frac{1}{\lambda} \), we have:
\[ N(\tau) = N_0 e^{-1} \approx N_0 \times 0.3679 \]

5. **Finding the Decayed Nuclei**:
- The number of decayed nuclei in one average life is:
\[ N_{\text{decayed}} = N_0 - N(\tau) = N_0 - N_0 e^{-1} \]
- This simplifies to:
\[ N_{\text{decayed}} = N_0 (1 - e^{-1}) \approx N_0 (1 - 0.3679) \approx N_0 \times 0.6321 \]

6. **Percentage of Decayed Nuclei**:
- The percentage of nuclei that decay in one average life is approximately:
\[ \frac{N_{\text{decayed}}}{N_0} \times 100 \approx 63.21\% \]
- This indicates that more than 50% of the original sample decays in one average life.

7. **Conclusion**:
- Therefore, in one average life, approximately 63.2% of the original sample decays, which is indeed greater than 50%. Hence, the correct answer is option C.