To find the amount of iodine-131 remaining after a certain time, we can use the formula for radioactive decay based on the half-life:
\[ N = N_0 \left( \frac{1}{2} \right)^{t/T_{1/2}} \]
where:
- \(N\) is the remaining quantity of substance,
- \(N_0\) is the initial quantity of substance,
- \(t\) is the elapsed time,
- \(T_{1/2}\) is the half-life of the substance.
Given:
- \(N_0 = 53.0 , \text{mg}\)
- \(t = 40.11 , \text{days}\)
- \(T_{1/2} = 8.021 , \text{days}\)
First, we calculate the number of half-lives that have elapsed during the time \(t\):
\[ \text{Number of half-lives} = \frac{t}{T_{1/2}} = \frac{40.11 , \text{d}}{8.021 , \text{d}} \approx 5.000 \]
Next, we can plug this value into the decay formula:
\[ N = 53.0 , \text{mg} \left( \frac{1}{2} \right)^{5.000} \]
Now we calculate \(\left( \frac{1}{2} \right)^{5.000}\):
\[ \left( \frac{1}{2} \right)^{5.000} = \frac{1}{32} = 0.03125 \]
Now, we can calculate \(N\):
\[ N = 53.0 , \text{mg} \times 0.03125 = 1.65625 , \text{mg} \]
Finally, we need to report \(N\) with the correct number of significant figures. The initial amount (53.0 mg) has three significant figures. Thus, we round the final answer to three significant figures:
\[ N \approx 1.66 , \text{mg} \]
Therefore, the amount of iodine-131 remaining after 40.11 days is approximately 1.66 mg.