Background:

Half-life is the period of time it takes one half of a sample of radioactive isotope to decay into another isotope, assuming decay is a random event.

\[ \newcommand\T{\Rule{0pt}{1em}{.3em}} \begin{array}{|l|l|l|l|l|l|l|l|} \hline \mbox{Half-lives elapsed} \T & 1 \T & 2 \T & 3 \T & 4 \T & 5 \T & \cdots \T & N \\ \\\hline \mbox{Mass of sample remaining} \T & \frac{1}{2} \T & \frac{1}{4} \T & \frac{1}{8} \T & \frac{1}{16} \T & \frac{1}{32} \T & \cdots \T & \frac{1}{2^N} \\\hline \end{array} \]

Key Concept:

As the number of half-lives increase, the mass and activity level of a radioactive substance decrease.



Practice:

How Many Half Lives Have Elapsed?

Problem 1:

How many half-lives occured if the mass of a certain isotope initially weighs g and is found to weigh g some time later?



The chain of reasoning is: Using the table above we find that we have had half-lives.


Problem 2:

How many half-lives occured if the mass of a certain isotope initially weighs g and is found to weigh g some time later?



The chain of reasoning is: Using the table above we find that we have had half-lives.



How much Mass Remains?

Problem 3:

How much mass was left after half-lives if the mass before decay is g?



The chain of reasoning is: We find that we have g remaining.


Problem 4:

What was the mass before decay started if half-lives occured and the mass after was g?



The chain of reasoning is: We find that we had started with g.


Problem 5:

Starting with kg of Plutonium-239 which has a half-life of approximately 24,000 years, how many grams will remain after years?



The chain of reasoning is: We would have g remaining.



How much Time Will Elapse?

Problem 6:

A sample of radioactive waste has a half-life of years and an activity level of curies. After how many years will the activity level of this sample be ?



The chain of reasoning is: It would take years.


Problem 7:

U-235 has a half-life of 710 million years. If it is determined that a certain amount of stored U-235 will be considered safe only when its radioactivity has dropped to less than %, how much time in half-life iterations must the U-235 be stored securely to be safe?



The chain of reasoning is: It would take million years which is half-life iterations. Notice that we don't need to achieve the exact percentage, but that we need to land below it in terms of half lives.