28 Project: Radioactive decay, half-lives, and C-14 dating
In this project we study exponential models for radioactive decay. The goal is to explore, understand, and use the concept of a half-life for radioactive isotopes.
28.1 Part 1: Iodine-131
Iodine-131 (called I-131 for short) is a radioactive isotope used in medical applications, especially for treating thyroid conditions. Any given quantity of I-131 will decay, losing roughly 8.3% per day.
Start your exploration with the following three tasks:
Suppose we start with 50 grams of I-131. Make a table of values showing how much remains each day for the first few days. Then construct a formula for the amount of I-131 remaining in terms of the number of days that has passed. After how many days has half of the Iodine-131 decayed away?
Suppose instead we start with 500 grams of I-131. Make a table of values showing how much remains each day for the first few days. Then construct a formula for the amount of I-131 remaining in terms of the number of days that has passed. After how many days has half of the Iodine-131 decayed away?
Suppose that we start with 1000 grams of I-131. Make a table of values showing how much remains each day for the first few days. Then construct a formula for the amount of I-131 remaining in terms of the number of days that has passed. After how many days has half of the Iodine-131 decayed away?
Based on the three tasks above, answer the following questions.
Compare the three formulas you constructed. How are they similar? How are they different?
In each of the three scenarios (starting with 50 grams, 500 grams, 1000 grams), compare how long it took for half of the Iodine-131 to decay away.
It should be the case that in each scenario, it took the same amount of time for half of the iodine to decay away. (If not, please go revisit your calculations before proceeding further!) This amount of time is called the half-life of Iodine-131.
Each radioactive isotope has a different value for its half-life, depending on the rate of decay. A list of common radioactive isotopes, and their half-lives, is available from the University of Alabama at Birmingham.
We can use the half-life of Iodine-131 as a unit of time. Suppose we start with 50 grams of I-131. Complete the following table.
| Half-lives | Days | I-131 (grams) | Percent of I-131 remaining |
|---|---|---|---|
| 0 | 0 | 50 g | 100% |
| 1 | 25 g | 50% | |
| 2 | 12.5 g | ||
| 3 | |||
| 4 |
How many half-lives must pass until there is less than 1% of the original I-131 remaining? How many days is this? Given that I-131 is used in medical applications, what is the consequence of this time scale?
28.2 Part 2: Cesium-137
Cesium-137 (Cs-137 for short) is a radioactive isotope formed in nuclear reactors and other nuclear fission events that involve uranium. The half-life of Cs-137 is approximately 30 years.
Suppose we start with 24 grams of Cs-137. Complete the following table.
| Half-lives | Years | Cs-137 (grams) | Percent of Cs-137 remaining |
|---|---|---|---|
| 0 | 0 | 24 g | 100% |
| 1 | 30 | 12 g | 50% |
| 2 | |||
| 3 | |||
| 4 |
How many half-lives must pass until there is less than 1% of the original Cs-137 remaining? How many years is this? Given that Cs-137 appear in radioactive waste, what are the practical consequences of this time scale?
Take a moment to compare the table you made for Iodine-131 and the table you made for Cesium-137. What parts of the tables are the same? What parts are different?
28.3 Part 3: Carbon-14
Carbon-14 (C-14) is an isotope of carbon created in earth’s atmosphere by cosmic rays. These isotopes bind with oxygen to form a special type of carbon dioxide, which is in turn taken in by plants and animals. The result is that all living creatures have a certain (and known) amount of C-14 in them.
When a plant or animal dies, it stops taking in new C-14. Meanwhile, the Carbon-14 that exists in the plant or animal slowly decays with a half-life of roughly 5730 years.
Consider a tree that died many, many years ago. Complete the following table.
| Half-lives | Years | Percent of C-14 remaining |
|---|---|---|
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| \(\phantom{\dfrac{2}{3}}\) | ||
| \(\phantom{\dfrac{2}{3}}\) | ||
| \(\phantom{\dfrac{2}{3}}\) |
Use your table (or an extension of your table) to address the following questions:
Suppose we find the remains of a tree that has 25% of the original C-14 in it. How long ago did the tree die?
Suppose we find the remains of a tree that has 5% of the original C-14 in it. Estimate how long ago the tree might have died.
Suppose less than 1% of the original C-14 remains in a tree sample. What can we say about how long ago the tree died?
28.4 Part 4: Exploration
Select one of the radioactive isotopes from the University of Alabama list above. Your goal is to construct a poster that contains the following features:
An introduction to this isotope. Using Wikipedia, or some other source, write a few sentences about the uses and/or origins of the isotope. Be sure to write down the link to your source!
A computational example. Suppose we have 5 grams of the isotope. Make a table showing half-lives, time, amount, and percent remaining. Be sure to include the time units.
A qualitative discussion of the isotope. Is the isotope long-lived or short-lived? Explain. What are the consequences of the lifespan of the isotope?
Any other fun or interesting elements you can add to your poster is appreciated!