Interactive practical

Half-life of a model radioactive source

Throw 120 dice fifteen times, removing the sixes as decays, then plot the number remaining against throw number and read two half-lives from the curve.

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GCSE Physics

Modelling radioactive half-life

WJEC 3430QD — matched to your course.

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Practical focuses1

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Practical focus

Use dice or cubes to model random decay and determine a model half-life from a decay curve.

Practical method

Aim

Model radioactive decay using dice and find the half-life of the model.

Variables

Independent variable: number of throws (representing time)
Dependent variable: number of dice remaining (undecayed nuclei)
Control variables: the rule for 'decay' (a six); the same type of six-sided dice; throwing all remaining dice each time.

Apparatus

100 six-sided dice
Tray
Results table

Method

Throw 100 dice into a tray.
Remove every die showing a six; these represent nuclei that have decayed.
Count and record the dice remaining.
Throw the remaining dice again and repeat until fewer than 20 remain.
Repeat the whole experiment (or combine class results) to reduce random variation.
Plot the number remaining against the number of throws and draw a smooth curve. Read off the half-life: the number of throws for the number remaining to halve.

Sample results

100 dice; a six counts as a decay.

Throw	Dice remaining
0	100
1	84
2	69
3	58
4	49
5	41
6	33
7	28
8	23

Analysis

The number of dice remaining falls by about one sixth each throw, giving an exponential decay curve. Reading from the graph, 100 falls to 50 after about 3.8 throws, and 50 falls to 25 after about another 3.8 throws, so the half-life is about 3.8 throws. Each die has a fixed probability \(\dfrac{1}{6}\) of decaying per throw, just as each nucleus has a fixed probability of decaying per second.

Errors and improvements

Problem	Effect	Improvement
Random variation with a small number of dice	Curve is uneven	Use more dice or combine class results
Dice miscounted	Wrong values	Count twice or in groups of ten
Drawing straight lines between points	Half-life read inaccurately	Draw a smooth curve of best fit

Risk assessment

Hazard	Risk	Control measure
Dice on the floor	Slipping	Throw into a tray and pick up any dropped dice
Low-risk activity	-	No radioactive sources are used

Common mistakes

Thinking all the dice will decay after two half-lives.
Reading the half-life from a single throw rather than the curve.
Joining the points with straight lines.

Worked analysis

100 dice with probability \(\dfrac{1}{6}\) of decaying each throw. Expected after one throw: \(\dfrac{100 \times 5}{6}\) ≈ 83. After two throws: 100 × \(\left(\dfrac{5}{6}\right)^{2}\) ≈ 69. The half-life satisfies \(\dfrac{5}{6}\)ⁿ = 0.5, giving n ≈ 3.8 throws.

Common misconception

After two half-lives a quarter of the nuclei remain, not none. Each half-life halves whatever is left.

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