In one average-life,

Options

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

Explanation

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

### Step-by-Step Solution:

1. **Understanding Average Life**:
The average life (mean life) of a radioactive substance is denoted by \( t_{mean} \) and is related to the decay constant \( \lambda \) by the formula:
\[ t_{mean} = \frac{1}{\lambda} \]

2. **Finding Remaining Nuclei**:
The number of nuclei remaining after a time \( t \) can be expressed using the formula:
\[ N_t = N_0 e^{-\lambda t} \]
where \( N_0 \) is the initial number of nuclei.

3. **Substituting Average Life**:
To find the number of nuclei remaining after one average life (\( t = t_{mean} \)):
\[ N_{t_{mean}} = N_0 e^{-\lambda t_{mean}} = N_0 e^{-\lambda \cdot \frac{1}{\lambda}} = N_0 e^{-1} \]

4. **Calculating Remaining Nuclei**:
We know that \( e^{-1} \) is approximately \( \frac{1}{e} \). Thus:
\[ N_{t_{mean}} = \frac{N_0}{e} \]

5. **Finding Decayed Nuclei**:
The number of decayed nuclei after one average life can be calculated as:
\[ N_{decayed} = N_0 - N_{t_{mean}} = N_0 - \frac{N_0}{e} = N_0 \left(1 - \frac{1}{e}\right) \]

6. **Calculating the Fraction of Decayed Nuclei**:
The fraction of decayed nuclei is:
\[ \frac{N_{decayed}}{N_0} = 1 - \frac{1}{e} \]
The value of \( e \) is approximately 2.7, so:
\[ 1 - \frac{1}{e} \approx 1 - \frac{1}{2.7} \approx 1 - 0.37 \approx 0.63 \]

7. **Conclusion**:
This indicates that approximately 63% of the nuclei have decayed after one average life, which is more than half of the original nuclei.

### Final Answer:
Thus, the correct option is **more than half the active nuclei decay**.

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