Half-Life & Radioactive Decay
How long does it take for a radioactive isotope to decay? The rate of decay of isotopes depends on how stable (or unstable) a radioisotope (radioactive isotope) is. The most unstable, the more quickly it will decay. But not even all atoms of the same isotope decay at the same time, so instead we describe it as a half-life—the amount it takes for about half of the atoms of an unstable isotope to go through radioactive decay. This is useful because it can tell us what amount of a sample will be left after a certain amount of time, or it can tell us how long has passed if we know how much we started with. Half-lives can be short (less than a second) or very long (billions of years).
The fraction of remaining sample of an isotope can be calculated with the formula , where n is the number of half-lives that have passed. For example, after 1 half-life, ½ of the sample would be left. After 2 half-lives, of the sample would be left. We can then apply this to figure out the amount of sample remaining in grams, using the formula:
where n = time/length of half-life.
This can also be converted to a percent—just take the fraction as a decimal and multiply by 100. Try applying this to the following problems:
1)Chromium-48 decays. After 6 half-lives, what fraction of the original nuclei would remain?
2)Fluorine-21 has a half-life of approximately 5 seconds. What fraction of the original nuclei would remain after 1 minute?
3)Iodine-131 has a half-life of 8 days. What percent of the original sample would remain at the end of 32 days?
4)The half-life of chromium-51 is 28 days. If the sample contained 510 grams, how much chromium would remain after 56 days?
5)Titanium-51 decays with a half-life of 6 minutes. If you started with 75 g, how much would remain after 1 hour?
You can also work backwards to see how much of something you started with, the half-life, or how old something is. Remember that if you are going backward, each half-life would double the amount you have.
6)A medical institution requests 1 g of bismuth-214, which has a half-life of 20 min. How many grams of bismuth-214 must be prepared if the shipping time is 2 h?
7)You have an 80 g sample of an isotope. After 60 minutes, only 20 grams remaining. What is the half-life of the isotope?