Teach radioisotopes and decay interdisciplinarily at a low cost Teach article

Author(s): Patrick-Joshua Biro

How to teach radioactive decay and radioisotopes to students who feel that equations are boring? Here are two inexpensive and captivating activities to apply in your classroom!

*Images: Fossil: David Clode/Unsplash, CC0; Radium Girls: Unknown/Wikimedia Commons, Public Domain; edited by Chiara Obermüller*

High-school students might have heard of radiocarbon dating for organic materials, but they might be intimidated by the decay equation as it can feel abstract and dry. While promoting active-participative learning in a 12^th grade mathematics-informatics class, I aimed to help students understand the connections between the solutions of the mathematical equations and the behaviours that are observed in the real world. As such, I created a board game with cards to make the connection for given radioisotopes even more obvious. I focused on the probabilistic nature of radioactive decay and on the proportionality between activity and the size of the undecayed population. The students were getting excited not only for working in teams but also because the activity creates just enough of a challenge to spark questions, before imposing answers.

This activity takes between 20 and 35 minutes.

Make sure your students are familiar, at a basic level, with the following concepts, operations, and equations:

Radioactivity
Logarithm
Radioactive decay equation (e.g., mathematical expression, undecayed nuclide population, decay constant, half-life)

If that is not the case, your students may use the theoretical key points handout.

Activity 1: Decay dash – a simple-to-use classroom activity for teaching radioactive decay

This activity simulates radioactive decay using a competitive, teamwork-based classroom game. We have chosen a number of five teams for a variety of radionuclides, but you may adapt the number of teams depending on your classroom’s configuration. This is a competition between the teams, so you are able to promote fair play and perseverance among students with the help of science. Obviously, while each team benefits from learning about radioactive decay, the team that first and correctly arranges the snapshot cards from the earliest timepoint to the latest wins!

Materials

Printed sheets of radionuclide cards
Note: you may choose any radionuclides you like (e.g., Technetium-99m (^99mTc), Cobalt-60 (⁶⁰C), Carbon-14 (¹⁴C), Caesium-137 (¹³⁷Cs), Iodine-131 (¹³¹I), and Radium-226 (²²⁶Ra)). Keep in mind that each team has its own radionuclide; you need to choose five different radionuclides for this activity. You can prepare your own cards or use the template provided in the supporting material.

Example of radionuclide cards
*Image courtesy of the author*

Each card should contain the following information:
- The symbol of the radionuclide, including the mass number and metastability (if applicable)
- The value N₀ (i.e., the number of undecayed nuclei at the timepoint t = 0)
- The value for the decay constant, λ, or the value for the half-life, t_1/2 (you may use IAEA’s Live Chart of Nuclides to identify the half-life of your preferred radionuclides)

When you have limited time for the activity or for lower level-classrooms:

Choose a more ‘manageable’ N₀ value for the calculations, such as N₀ = 1 000 nuclei. Any value between 1 000 and 10 0000 nuclei is acceptable for the activity’s objective.
Explicitly give the value for the decay constant. Otherwise, you may give them only the value for t_1/2. We will elaborate upon this in the ‘Procedure’ section.

Seven ‘snapshot’ cards for each team, each containing a number of undecayed nuclei at a particular (unknown) timepoint

Scientific calculator

Example of snapshot cards
*Image courtesy of the author*

Procedure

Divide the class into five teams, each with its own radionuclide. To make things more exciting, you may let a representative of each team choose a radionuclide randomly.
For each team, hand out seven ‘snapshot’ cards.
If you chose to give only the value for t_1/2, instruct them to find and write down the value for λ using the radioactive decay equation.
Instruct your students to calculate the timepoints that correspond to each given number of undecayed radioactive nuclei.
Encourage your students to ‘sort the snapshots out’, i.e., to arrange the ‘snapshot’ cards from the earliest timepoint to the latest. The cards do not need to be spaced according to the successive time intervals!
Ask your students to consider the relationship between the initial population of radionuclides and the number of undecayed nuclei when the values for the time that has passed increase.

Hints for students who get stuck

If they don’t understand how to find the expression for the decay constant (λ), ask them where it appears in the decay equation. Is it at an exponent or a base? Then, instruct them to consider the inverse of the exponential function. They should understand that taking the logarithm allows them to derive the quantity at the exponent.
This is also the essence of finding the values for specific timepoints t. If needed, insist upon the importance of the negative sign at the exponential and make sure the students include it in their calculations.

Results

Here is a worked example for the following case:

Example of cards
Image courtesy of the author

Since we are given the half-life of ¹⁵O, we can find the value of the decay constant:

We will write this value on the card with the radionuclide, which will be useful for the next calculations.

According to the law of radioactive decay, t is expressed as a function of ln(N/N₀) in the numerator and λ in the denominator (See the theoretical key points handout for a step-by-step guide to calculate t ). Therefore, we can calculate the necessary values for t.

For example, when N (t) = 5 000 nuclei:

122.26 s corresponds to the half-life (t_1/2) of ¹⁵O.

The same way we find the other values for t :

By arranging the ‘snapshot’ cards in order of increasing time values, we can see that the population of undecayed nuclei decreases exponentially over time:

Discussion

In my experience, students have no significant difficulty in realising the pattern: the numbers don’t decrease linearly, but exponentially. They begin calculating, estimating, arguing. One of the students suggested plotting the data to verify the curve, which could be done using Microsoft Excel or graph paper. Another student even noticed that the activity mirrored the unpredictability of decay in real systems. Most importantly, they internalized the concept without requiring me to provide incremental values for timepoints or to derive the radioactive decay equation.

Extension activity: Speaking mathematically about radioactive decay

These questions are meant to evaluate how well the students understand the quantitative aspects of radioactive decay.

If we double the time, does this halve the population of undecayed nuclei? What if we multiply the time by an integer n? Does this make the population n times smaller?
Which part of the decay equation ‘shows’ us the exponential decrease? Can you give other examples of similar equations in physics?
Would it make sense for the undecayed population to (also) be expressed as a decimal for certain values of timepoints? Why or why not? (Hint: this helps explain the probabilistic nature of radioactive decay.)
Does the population of the undecayed nuclide ever become exactly zero? If yes, when? If not, why?

Activity 2: Investigating how radionuclides shape science and society

This activity takes the form of an inquiry-based group project. Students are invited to work as scientific consultants, using data and background knowledge to analyse how certain properties of radionuclides (half-life, decay constant, energy levels) are applied in real-life scenarios. While this activity typically takes longer than Activity 1 (35–50 min), bear in mind that it may be useful even in classes where mathematics is not taught intensively. As such, it strengthens interdisciplinary understanding by solving realistic problems that require both scientific reasoning and communication.