A radioactive isotope decays such that after 24 hours, only 25% of the original sample remains. What is the half-life of this isotope?

• A. 48 hours
• B. 6 hours
• C. 12 hours
• D. 24 hours

Answer: (C)

Verified step by step guidance

1. Identify the given information: after 24 hours, only 25% (or \( \frac{1}{4} \)) of the original radioactive sample remains.
2. Recall the relationship between the remaining fraction of a radioactive sample and the number of half-lives elapsed: \( \text{Remaining fraction} = \left( \frac{1}{2} \right)^n \), where \( n \) is the number of half-lives.
3. Set up the equation using the given data: \( \left( \frac{1}{2} \right)^n = \frac{1}{4} \). Recognize that \( \frac{1}{4} = \left( \frac{1}{2} \right)^2 \), so \( n = 2 \).
4. Use the definition of \( n \) as the ratio of total time elapsed to the half-life: \( n = \frac{t}{t_{1/2}} \), where \( t = 24 \) hours and \( t_{1/2} \) is the half-life.
5. Solve for the half-life \( t_{1/2} \) by rearranging the equation: \( t_{1/2} = \frac{t}{n} = \frac{24 \text{ hours}}{2} \).