The half-life of \(^{131}I\) is approximately 15.87 days.
The half-life (\(t_{1/2}\)) of a radioactive isotope is the time it takes for half of a sample of that isotope to decay. The relationship between the remaining fraction (\(R\)) and the number of half-lives (\(n\)) can be expressed using the formula:
\[ R = \left( \frac{1}{2} \right)^n \]
In this case, after 20.8 days, 41.1% of the sample remains. The remaining fraction (\(R\)) is 0.411.
\[ 0.411 = \left( \frac{1}{2} \right)^n \]
Now, solve for \(n\):
\[ n = \frac{\log(0.411)}{\log(1/2)} \]
\[ n \approx 1.31 \]
Since \(n\) is the number of half-lives, and we know that after 20.8 days, 1.31 half-lives have passed, we can find the half-life (\(t_{1/2}\)):
\[ t_{1/2} = \frac{20.8}{1.31} \]
\[ t_{1/2} \approx 15.87 \, \text{days} \]
So, the half-life of \(^{131}I\) is approximately 15.87 days.