The half - life of` `I ^ (131)` is 8 days. Given a sample of `I^(131)` at time `t = 0` , we can assert that

Options

A. no nucleus will decay before `t=4days`
B. no nucleus will decay before `t=8days`
C. all nucleus will decay before `t=16days`
D. a given nucleus may decay at any tiem after `t=0` .

Explanation

To solve the question regarding the half-life of I-131, we will analyze the options provided and apply our understanding of radioactive decay.

### Step-by-Step Solution:

1. **Understanding Half-Life**:
The half-life of a radioactive isotope is the time required for half of the radioactive nuclei in a sample to decay. For I-131, the half-life is given as 8 days.

2. **Initial Condition**:
At time \( t = 0 \), we have a sample of I-131. This means that all nuclei are intact and none have decayed yet.

3. **Decay Process**:
Radioactive decay is a random process. This means that while we can predict the average behavior of a large number of nuclei, we cannot predict when a specific nucleus will decay.

4. **Evaluating the Options**:
- **Option A**: "No nucleus will decay before \( t = 4 \) days."
- This is incorrect because, although the average decay time is 8 days, some nuclei may decay before 4 days.
- **Option B**: "No nucleus will decay before \( t = 8 \) days."
- This is also incorrect for the same reason; some nuclei can decay before the 8-day mark.
- **Option C**: "No nucleus will decay before \( t = 16 \) days."
- This is incorrect as well; it is highly unlikely that all nuclei will wait until 16 days to decay.
- **Option D**: "A given nucleus may decay at any time after \( t = 0 \)."
- This is correct because the decay of individual nuclei is random, and any nucleus can decay at any moment after \( t = 0 \).

5. **Conclusion**:
The correct assertion is that a given nucleus may decay at any time after \( t = 0 \). Therefore, the correct answer is **Option D**.