Half Life Example Problem

This example problem demonstrates how to use an isotope's half life to determine the amount of the isotope present after a period of time.

Problem:

²²⁸Ac has a half life of 6.13 hours. How much of a 5.0 mg sample would remain after one day?

Solution:

The first step is to determine the number of half lives that have elapsed.

number of half lives = 1 half life/6.13 hours x 1 day x 24 hours/day
number of half lives = 3.9 half lives

For each half life, the total amount of the isotope is reduced by half.

Amount remaining = Original amount x 1/2^{(number of half lives)}

Amount remaining = 5.0 mg x 2^-(3.9)
Amount remaining = 5.0 mg x (.067)
Amount remaining = 0.33 mg

Answer:

After 1 day, 0.33 mg of a 5.0 mg sample of ²²⁸Ac will remain.