READ: Half-Life Calculations
READ: Half-Life Calculations
Half-lives have a very wide range, from billions of years to fractions of a second. Listed in the table below are the half-lives of some common and important radioisotopes.
Before you start making half-life calculations, there are some terms you should be familiar with. In every problem, you will be using or looking for the following variables:
- Original amount: This is the amount of the isotope you start with. Amount is usually given in grams or milligrams of the substance.
- Remaining amount: This is the amount of radioactive substance remaining after a certain amount of time. Amount is usually given in grams or milligrams of the substance.
*TIP: Remember that we are talking about a radioactive isotope breaking down. Therefore, the original amount will always be greater than the remaining amount. This is a good check to make sure that you have calculated the problem correctly.
- Total Time: This is the amount of time elapsed or gone by. Time can be given in seconds, minutes, days, years, etc.
- Time of half-life: This is the given amount of time for a specific isotope in which half of the original substances decays. In other words, this is the time of one half-life for a particular isotope.
*TIP: Total time is always going to be greater than the time of one half-life. This is a good check to make sure that you have calculated the problem correctly.
- Number of half-lives: You will always want to determine how many half-lives have occurred. This can be found by 2 methods depending on what is given in the problem.
- Calculated when given both original amount and remaining amount.
- For example: If you are given that one half-life for an isotope is 11 days and the total time that has elapsed is 44 days, how many half-lives have occurred?
- Calculated when given both original amount and remaining amount.
= 4 half-lives
By knowing 4 half-lives have gone by, you can know that you have
of the original amount remaining (look at the chart below).
| Number of Half-lives Passed | Percentage of Radioactive Isotope Remaining | Fraction of Radioactive Isotope Remaining |
|---|---|---|
1 | 50% |
|
2 | 25% |
|
3 | 12.5% |
|
4 | 6.25% |
|
5 | 3.125% |
|
- Calculated when given total time and time of half-life.
- For example: If you are given that you start with an amount of 80 grams of a substance and the remaining amount after a certain period of time unknown to you is 2.5 grams, how many half-lives have occurred?
= 0.03125, changing to a percent you have 3.125% remaining
By knowing 3.125% (or 1/32), you can know that 5 half-lives have occurred (look at the chart above).
Knowing the vocabulary of what will be needed to calculate half-lives, look at the example problem below. You will see how to use the half-life of a sample to determine the amount of radioisotope that remains after a certain period of time has passed without using a formula.
Problem: Strontium-90 has a half-life of 28.1 days. If you start with a 5.00 mg sample of the isotope, how much remains after 140.5 days have passed?
Step 1: List the known values and plan the problem.
Known Variables
- Original amount = 5.00 mg
- t ½ = 28.1 days
- Total time = 140.5 days
Unknown Variable
- Remaining amount of Sr-90 = ?
Step 2: Determine how many half-lives have passed.
One half-life for Sr-90 is 28.1 days and the total time that has elapsed is 140.5 days, how many half-lives have occurred?
= 5 half-lives
By knowing 5 half-lives have gone by, you can know that you have
of the original amount remaining (look at the chart above).
Step 3: Solve for how much of the substance is remaining.
Remaining amount of Sr-90 = 5.00 mg × 
= 0.156 mg
Step 4: Think about your result.
The remaining amount is less than the starting amount. Is this correct? Yes, since the nucleus is breaking down you should expect less.
Practice Problems
Georgia Virtual, Nuclear Chemistry, CC BY-NC-SA 3.0
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