The half life ofa particular isotope of iodine is 8.0 days. How much of a 10.0g sample of this isotope will remian after 30 days?

Options

A. 0.37 g
B. 0.60 g
C. 0.45 g
D. 0.74 g

Answer

To find out how much of a 10.0 g sample of iodine will remain after 30 days, we will use the concept of half-life.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Half-life (T_half) of iodine = 8.0 days
- Initial amount (N0) = 10.0 g
- Total time (t) = 30.0 days

2. **Calculate the Number of Half-Lives:**
- The number of half-lives (n) can be calculated using the formula:
\[ n = \frac{t}{T_{half}} \]
- Substituting the values:
\[ n = \frac{30.0 \text{ days}}{8.0 \text{ days}} = 3.75 \]

3. **Use the Half-Life Formula:**
- The remaining amount of the isotope (N) after n half-lives can be calculated using the formula:
\[ N = N_0 \left(\frac{1}{2}\right)^n \]
- Substituting the values:
\[ N = 10.0 \text{ g} \left(\frac{1}{2}\right)^{3.75} \]

4. **Calculate \(\left(\frac{1}{2}\right)^{3.75}\):**
- This can be calculated using a scientific calculator or logarithmic tables:
\[ \left(\frac{1}{2}\right)^{3.75} \approx 0.0743 \]

5. **Final Calculation of Remaining Amount:**
- Now, substitute this value back into the equation for N:
\[ N = 10.0 \text{ g} \times 0.0743 \approx 0.743 \text{ g} \]

6. **Conclusion:**
- After 30 days, approximately **0.743 grams** of the iodine isotope will remain.