Final answer:
The half-life of a radioactive isotope, given that a 500.0 g sample decays to 62.5 g in 24.3 hours, is 8.1 hours. This is calculated by dividing the total time by the number of half-lives that have occurred.
Step-by-step explanation:
The half-life of a radioactive isotope is the time required for half of the isotope to decay. To find the half-life when given the initial amount of substance and the amount remaining after a certain time, we observe the number of half-lives that have passed. In the example given, a 500.0 g sample decays to 62.5 g in 24.3 hours. By noticing that the sample's mass has halved multiple times (500 g to 250 g, 250 g to 125 g, 125 g to 62.5 g), we can see that three half-lives have passed. The time for these three half-lives is the given 24.3 hours.
To calculate one half-life, we simply divide the time elapsed by the number of half-lives:
Half-life = 24.3 hours / 3 = 8.1 hours
So, the half-life of the isotope in question is 8.1 hours.