8.5 - Half-Life (Model Radioactive Source)
The experiment
Aim
To model radioactive decay using a large number of dice, plot a decay curve from the results, and use the curve to find the half-life of the model source.
Background
- Radioactive decay is random. You can't say when a particular unstable nucleus will decay, only that every nucleus has the same fixed chance of decaying in a given time.
- Across a very large number of nuclei, the number that are undecayed falls by the same fraction in each equal period of time.
- The half-life is the time taken for the number of undecayed nuclei, and so the activity, to halve.
- You can't see individual nuclei decay, but dice follow the same rule. Every dice has a one in six chance of landing on a 6 on each throw, and you can't predict which ones will.
- So in this practical you roll a large number of dice, remove the 6s as decayed nuclei, and plot the number left after each roll to find the half-life.
Variables
- Independent variable - The number of rolls, which stands for time.
- Dependent variable - The number of dice left, which stands for the number of undecayed nuclei.
- Control variables - The rule for decaying (only a 6 counts), rolling every remaining dice on every turn, and the same set of dice throughout.
Method
- Count the dice - Count out at least 50 dice and record the total.
- Roll them all - Roll every dice at once into a tray, so none are lost.
- Remove the 6s - Take out every dice that has landed on a 6. These stand for the nuclei that have decayed.
- Record the number left - Count the dice left and record the number against the roll number.
- Keep rolling - Roll the remaining dice again, remove the 6s and record the number left, until every dice has decayed.
- Pool the class results - Add every group's numbers together for each roll.
- Plot a decay curve - Number of dice left on the y-axis against number of rolls on the x-axis, with a smooth curve of best fit rather than straight lines between points.
Worked example - Half-life from a decay curve
A class pooled a set of 600 dice. They rolled all the dice, removed every dice showing a 6 after each roll, and recorded how many were left.
| Number of rolls | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Dice remaining | 600 | 497 | 419 | 342 | 291 | 238 | 205 | 168 | 137 | 118 | 95 |
Find the half-life, checking it from two different starting points, and explain why the points don't sit perfectly on a smooth curve.
Step 1 - Plot the points and draw a smooth curve
Plot the number of rolls on the x-axis and the dice remaining on the y-axis. Draw a single smooth curve of best fit through the points.
Step 2 - Read the half-life from the start
The starting number is 600, and half of it is 600 ÷ 2 = 300. The curve crosses 300 at about 3.8 rolls.
Half-life ≈ 3.8 rolls
Step 3 - Read it again from a different starting point
Start from 400 dice instead. The curve passes 400 at about 2.2 rolls, and passes 200, half of 400, at about 6.1 rolls.
6.1 - 2.2 = 3.9 rolls
This closely matches the value from step 2.
Step 4 - Say what the two readings show
The half-life comes out the same whichever starting point is chosen. Half-life is a fixed property of the decay, and it doesn't change as the sample shrinks.
Step 5 - Explain the scatter around the curve
- Each roll is random, so the number of 6s on each turn varies around the expected one in six.
- With fewer dice left towards the end, these random variations have a larger effect.
- A real sample has so many nuclei that the random variations average out, giving a smooth decay curve.
Common exam questions
Question 1
Why is a model used rather than a real radioactive source?
Real sources give out ionising radiation, which is a hazard. Their half-lives are also usually far too long or far too short to plot a decay curve in a lesson.
Question 2
Why is every dice showing a 6 removed, rather than a chosen number of dice?
The decay has to stay random. Removing a set number each time would give a straight-line fall, not a decay curve, so the model would no longer behave like a radioactive source.
Question 3
A student flips 800 coins, removes every coin that lands on heads, and repeats with the coins that are left. What is the half-life of this model, in flips?
One flip. Each coin has a one in two chance of landing on heads, so about half of the remaining coins are removed each time.
Question 4
Using the coin model above, how many coins would be left after 4 flips?
Four flips is four half-lives, so the fraction left is (1/2)4 = 1/16. That gives 800 ÷ 16 = 50 coins.
Question 5
A model source starts with 480 dice and 120 are left after 8 rolls. Find the half-life.
120 is a quarter of 480, which is two half-lives. So one half-life is 8 ÷ 2 = 4 rolls.