### Constant fractional decay

**What the activity is for**

This is a set of three linked activities designed to show how radioactive decay can be modelled. In each case you can discuss the success and the limitations of the model. It is good for students to see a range of models so that they see how they differ, particularly in their limitations. The running theme throughout the models is to think about the idea of how the model shows that there is constant fractional decay.

In the first activity, you make popcorn the old-fashioned way using popping corn and a saucepan. This is a purely qualitative experiment but helps students to start thinking about physical models and the idea of constant fractional decay.

In the second activity, students use a large number of dice to generate data that can be plotted on a graph.

The third is an experiment, again with sweets, that shows the importance of using large numbers, of nuclei, popcorn or sweets, when plotting graphs.

Finally, we'd suggest a brief discussion of the significance of radioactive decay in the wider context.

### Popping corn

**What to prepare**

- popping corn
- saucepan with a lid that fits
- a portable gas hob or small gas camping stove (or electric, but gas is better)
- cooking oil
- a marker pen

**Safety note: **Make sure that you keep the lid on. The oil will be hot. You'll probably have rules about (not) eating in the laboratory.

**What happens during this activity**

Set up the gas hob with the pan on top. You can mark some of the kernels with a marker pen at this point. Later you can try to find them to illustrate that you cannot predict which of the kernels will decay

. Coat the bottom of the pan with oil. Then coat the bottom of the pan with popcorn and put the lid on. While the popcorn is cooking you can start discussing the model. Make sure that you keep shaking the pan as you talk or the popcorn will burn.

### Opening up a discussion

Teacher: So in this experiment we are modelling what happens in radioactive decay. What does one of these popcorn kernels represent?

Lydia: A radioactive atom?

Teacher: Not quite. Is it the atom that decays?

Lydia: It's the nucleus.

Teacher: Good. It is the unstable nuclei that decay. So in a minute (or this may have happened already) the popcorn will start popping. If that was a nucleus what would be happening?

Lydia: It is decaying?

Teacher: Yes, but what does that actually mean?

Lydia: It is giving out radiation.

Teacher: Good. The nucleus emits alpha, beta or gamma radiation when it decays. So in this model what represents an un-decayed nucleus?

Lydia: An un-popped popcorn.

Teacher: And a decayed nucleus?

Lydia: A popped popcorn.

Teacher: Good. So let's listen to the popcorn popping.

### Listening to corn popping

Let the popcorn pop for a while, continuing to shake the pan while holding it a few centimetres from the flame to prevent it from burning. When the popping is down to one or two every few seconds, turn the gas off and put the pan down.

Teacher: While we were listening to that there didn't seem to be any fixed pattern to the popping. Is there a word that we could use to describe that?

Lydia: Random?

Teacher: Excellent. But the word random has a very special meaning in physics. It isn't like how we think of things in everyday life. When do you use the word random in everyday life?

Lydia: Like if it's unpredictable? Or has no pattern?

Teacher: Exactly. We use it every day like that. But in physics it is a special word that has to do with probability. If a process is random then the probability that it will happen in a period of time, like a week or a second, is always the same. In radioactive decay there is a fixed probability that a nucleus will decay in one second. So let's say that the probability of a kernel decaying is one-tenth every second. In the first second one-tenth will pop. So there are now fewer. But in the next second one-tenth will still pop – a constant fraction every second. Now let's think about how good a model this is for radioactive decay. If I heat the oil up to a particular temperature will all the kernels have the same probability of popping? No. Why not?

Lydia: They aren't all exactly the same – some are smaller. And they might be different types.

Teacher: Good. That's right. And this isn't true of unstable atoms – they are all identical. So that is where this model falls down. Think about the number of pops per second – the rate of popping. The rate of popping depends on the number of un-popped kernels. How could we tell that from listening to the popping?

Lydia: It started with lots and then it got less.

Teacher: Good. So what can we say about the rate of decay of nuclei?

Lydia: The rate of decay depends on how many are left.

Teacher: Excellent. So in radioactive decay we say that the rate of decay is proportional to the number of un-decayed nuclei, and this model can help us to remember that. This is what we mean when we say exponential. That has a special meaning also – it means that the rate of decay of something, or growth, depends on how much you have at the time.

If you marked the popcorn with a marker pen before you put them in then pour out the popcorn into a tray and see if you can find the ones that have decayed

.

### Decay modelled with dice

**What to prepare**

- about 500 dice or wooden cubes with dot or a mark on one of the faces.
- support sheet
- computer projector and laptop with spreadsheet ready for results (optional)

**What happens during this activity**

Students throw dice where getting a 1 (or a dot) means that the dice has decayed

. A set of data for the number of dice remaining after each throw is collected.

Do the experiment before you have the discussion.

Teacher: What is the probability that a die will decay

on each throw?

Lydia: It's one-sixth.

Teacher: Good. Is that probability fixed?

Lydia: Yes, unless the dice are biased!

Teacher: That's a very good point. We are making assumptions here, assumptions that we couldn't make about the popcorn but that are true for atoms. What is that?

Lydia: That they are all the same.

Teacher: Excellent. So that is two reasons why this is a good model for decay. Let's plot a graph and see what happens.

### Understanding the shape of the decay curve

There are two ways of proceeding here.

You can get each student to plot the data on a piece of graph paper, which is a good idea if you want to develop graph-plotting skills. There is also a sense of ownership, and a chance to extend the discussion later about drawing lines to work out the half-life.

You can plot the data electronically and project the graph onto a whiteboard or screen.

Teacher: What can you tell me about how the number of un-decayed

dice changes with time?

Lydia: It's obvious, it goes down!

Teacher: That is very true. Let me draw another graph for which that is true.

On the board sketch a graph with a straight line and a negative gradient.

Teacher: It's true for this graph too. What's different?

Lydia: That one doesn't change, the other one does.

Teacher: Can you be more precise? How does the dice graph change?

Lydia: It is steep to start with, then it gets less steep.

Teacher: Excellent! So think back to the popcorn. The rate of popping depends on the number of unpopped kernels. So the steepness of the graph tells us about how many dice decay

per second – loads at the start and then fewer and fewer. Let's think about this another way. What did we say the probability of {decay} per throw was?

Lydia: One-sixth.

Teacher: That's right, and assuming that all the dice are the same and that nothing changes it is always one-sixth. So another way of thinking about it is to say the same fraction of the dice decay per throw. How could we explain the shape of the graph with this idea?

Lydia: One-sixth decay

each time, so the graph goes down.

Teacher: That's a good start. Why is the one-sixth that decay at the start bigger than the one-sixth that decay at the end?

Lydia: Because there are more at the start. So one-sixth of a big number is bigger.

### Linking the shape of the decay curve to the half life

Teacher: Excellent. This model shows that a fixed probability of decay per second for radioactive nuclei means that a constant fraction decays each second. That gives a graph this shape for radioactive decay. Let's go back to the graph. What happens towards the end of the experiment? Why had some people not got any dice, while other people had?

Lydia: Because you can't tell which dice are going to decay.

Teacher: Exactly, just like we didn't know which popcorn were going to decay, and we don't know which atoms are going to decay. So that is also a good feature of this model. So far there seem to be only good things to say about this model. Now let's work out the half life.

Either use the graph on the screen to work out the half-life, or get them to work it out from the graph. For constant fractional decay you'd expect the number of dice to go down to half the original number in about 4 throws.

Teacher: So we can think about the half-life in terms of the probability of decay. If we expect one-sixth to decay, what fraction should still be there?

Lydia: Five-sixths.

Teacher: Great. Then the next time you throw them you would still have five-sixths left, just five-sixths of a smaller number and so on. With your calculator find five-sixths of 500, then five-sixths of that and so on. How many times do you have to do it to get about a half?

Lydia: Three is a bit over and four is a bit under.

Teacher: OK, and that matches what we have found on our graph.

### Decay with sweets

**What to prepare**

- enough M&M sweets for each pair to have 10 each
- a small jar or tub to put the M&M sweets in for shaking

You need M&Ms that have M&M printed on one side. If not, use pennies and say that tails → decay.

**What happens during this activity**

Draw a simple table on the board with Throw

and Number of M&Ms left

as the column headings for the students to fill in. Fill in 0

and 10

in the first row.

Tell them to put the M&Ms in their tub, throw them, and remove any that have M&M showing. Record how many M&Ms remain for throw 1, put the un-decayed

M&Ms back in the tub and repeat. Depending on where you are you can let them eat the decayed

M&Ms. Depending on time you could get them to plot the data. The graphs should vary quite a bit, and this is the talking point.

### Graphs – variations and patterns

Teacher: Compare your graph with those of other groups around you. What do you notice?

Lydia: My graph is different. They must have done it wrong, or I did.

Teacher: Really? But you all did the same thing. Could there be another reason why they differ so much?

Lydia: The M&Ms are different in different bags?

Teacher: Not enough to make any difference. The people making them would make sure of that. Let's think about it in a different way. What is the average number of throws to halve the number of M&Ms? Obvious? It's one! But is it possible to have one M&M that just doesn't decay?

Lydia: No, eventually it would come up with the M&M side up.

Teacher: Really? Are you sure?

Lydia: Well, I suppose it could but it wouldn't be very likely.

Teacher: Exactly. So when we get down to very small numbers of M&Ms, or popcorn or atoms, then you could get strange things happening. That's why your graphs are so different. The unlikely events show up more for smaller numbers, so this is why using very large numbers is a much better idea. In our dice experiment we did not encounter this problem as we had quite a large number of dice. Finding the half life from the results at the start is much more reliable than from results at the end for this reason.

### Exponential decays – a closing discussion

Students may ask what the point is of studying this idea of exponential decay, or they may not. Either way, there is a discussion to be had about the fact that radioisotopes have different half-lives and the implications of this fact. If they have not studied uses of radioisotopes at this point, you could leave the discussion until later.

Teacher: So this is all a bit theoretical! Where would we meet situations that matter in real life that involve the ideas we have talked about – exponential decay, constant fractional decay and half life?

Lydia: Anything radioactive. Nuclear power?

Teacher: Exactly. Knowing the rate at which the isotopes in nuclear waste will decay is extremely important if we are going to dispose of it safely. Some isotopes have half-lives of many thousands of years, so we need to be able to store them safely for a very long time. Others we can bury in shallow sites. What about places where we use radioisotopes to help people?

Lydia: Like hospitals?

Teacher: Yes, exactly. How do doctors make use of radioactivity in hospitals?

Lydia: To see inside, like your kidneys and bladder.

Teacher: Good. So if you are giving somebody something to eat that is radioactive what would you like the half life to be?

Lydia: Really short.

Teacher: OK, but you need a balance. If is is too short then it won't last long enough for the doctor to 'see' what is going on with their gamma camera. Several of these radioisotopes have half lives in days. What advice do you think patients get before they leave hospital?

Lydia: Does that mean that they are emitting gamma rays at home?

Teacher: Yes, they are advised to not spend time close to people, less than a metre away, for several days afterwards. However, the benefits of the treatment would usually outweigh the risks involved in the treatment. That's why knowing about the rate of decay is so important.