The half-life period of a radioactive element is 100 days. After 400 days, one gram of the element will reduced to:

Options

A. 0.5g
B. 0.25g
C. 0.0625g
D. 0.125g

Explanation

To solve the problem of how much of a radioactive element remains after a certain period, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Half-life period (t_half) = 100 days
- Total time (t) = 400 days
- Initial amount of the element (N_0) = 1 gram

2. **Calculate the Number of Half-Lives:**
- The number of half-lives (n) can be calculated using the formula:
\[ n = \frac{t}{t_{half}} \]
- Substitute the values:
\[ n = \frac{400 \text{ days}}{100 \text{ days}} = 4 \]

3. **Calculate the Remaining Amount:**
- The remaining amount of the radioactive element after n half-lives can be calculated using the formula:
\[ N = N_0 \times \left(\frac{1}{2}\right)^n \]
- Substitute the values:
\[ N = 1 \text{ gram} \times \left(\frac{1}{2}\right)^4 \]
- Calculate \(\left(\frac{1}{2}\right)^4\):
\[ \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]
- Thus,
\[ N = 1 \text{ gram} \times \frac{1}{16} = \frac{1}{16} \text{ grams} \]

4. **Convert to Decimal Form:**
- To express \(\frac{1}{16}\) in decimal form:
\[ \frac{1}{16} = 0.0625 \text{ grams} \]

5. **Final Answer:**
- After 400 days, the remaining amount of the radioactive element will be 0.0625 grams.

### Summary:
After 400 days, one gram of the radioactive element will reduce to 0.0625 grams.