If 87.5% of a sample of pure 131I decays in 24 days, what is the half-life of 131I?

• A. 24 days
• B. 6 days
• C. 12 days
• D. 8 days

Answer: (D)

Verified step by step guidance

1. Identify the given information: 87.5% of the sample decays in 24 days, which means 12.5% of the sample remains after 24 days.
2. Express the remaining fraction as a decimal: remaining fraction = 0.125 (since 12.5% = 0.125).
3. Use the radioactive decay formula relating remaining amount to initial amount and half-life: \(N = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}\), where \(N\) is the remaining amount, \(N_0\) is the initial amount, \(t\) is the elapsed time, and \(t_{1/2}\) is the half-life.
4. Substitute the known values into the equation: \$0.125 = \left(\frac{1}{2}\right)^{\frac{24}{t_{1/2}}}$.
5. Solve for the half-life \(t_{1/2}\) by taking the logarithm of both sides and isolating \(t_{1/2}\): \(\log(0.125) = \frac{24}{t_{1/2}} \times \log\left(\frac{1}{2}\right)\), then rearrange to find \(t_{1/2} = \frac{24 \times \log\left(\frac{1}{2}\right)}{\log(0.125)}\).