Half Life and Activity

Questions from email…

How would you solve a problem like:

A certain radioactive isotope has a half life of 6.25 hours. What percent of the original isotope is still present after 22.9 hours have passed?

And 

A certain radioactive isotope is found to have an activity of 0.567Ci per gram. What is the activity of 0.807g of the isotope? 

For the first one...

Almost all half-life problems use some form of the relationship:

(amount remaining) = (original amount)(1/2)ⁿ

where n = the number of half lives.

If one half-life is 6.25 hours, then 22.9 hours represents (22.9/6.25 = 3.664) half-lives. There are a LOT of different ways we can think about "amount" in these problems... in this context, let's say the "original amount" is "100%". That means:

(amount remaining) = (100%)(1/2)^3.664

I'll let you plug those numbers in... This is a problem where you can probably estimate a reasonable answer. If 3 half-lives have passed, there should be 1/8 of the original amount (0.125, or 12.5%); if 4 half-lives have passed, there should be 1/16 of the original amount (0.0625, or 6.25%). When you solve the equation above, your answer should be something less than 12.5% but more than 6.25%.

For the second problem... My guess is that you're trying to make it a super-complex chemistry problem. This is a good example of being attentive to the units on numbers in a problem and using them to get an answer, even if you might not "know" how the problem really works. You're given "Ci per gram", so if you have 1 gram of sample, the activity would be 0.567Ci. But you don't have 1 gram, you have 0.807g. If I told you that a blueberry cake had 28.0 blueberries per pound of cake (mmmm... pound cake...), how many blueberries would you expect to find in a 0.395 pound piece of cake?