Radioactive decay model using dice

Practical method

Aim

Model radioactive decay with dice to show that decay is random and exponential, and find the decay constant and half-life.

Variables

Independent variable: number of throws
Dependent variable: number of dice remaining
Control variables: the same probability of 'decay' (one face of six); starting number of dice.

Apparatus

• 600 dice (or a computer simulation)
• Large tray

Method

1. Throw all 600 dice. Remove every die showing a six: these have 'decayed'.
2. Count and record the dice remaining. Throw them again.
3. Repeat until fewer than 50 remain.
4. Plot N against throw number and ln N against throw number.
5. The gradient of ln N is −λ, the decay probability per throw; half-life = \(\dfrac{\ln 2}{\lambda }\).

Sample results

Expected values for a decay probability of \(\dfrac{1}{6}\) per throw.

Analysis

ln N falls linearly with gradient −0.182 per throw, so λ = 0.182 (close to \(\dfrac{1}{6}\) for small steps; exactly −\(\ln(\dfrac{5}{6})\)). The half-life is \(\dfrac{\ln 2}{0.182}\) = 3.8 throws. Real class data scatter around these values, more so as N falls.

Uncertainty

Like real decay, the number removed each throw fluctuates by about √N; with small N the scatter is large. Pooling class results reduces it.

Errors and improvements

Risk assessment

Common mistakes

• Expecting exactly 100 dice to decay each throw.
• Reading the half-life from one step.
• Confusing λ with the half-life.

Worked analysis

Gradient = \(\dfrac{5.303 - 6.397}{6}\) = −0.182. λ = 0.182 per throw. Half-life = \(\dfrac{\ln 2}{0.182}\) = 3.8 throws.

Common misconception

Decay is random for each die (or nucleus), yet a large number follows a predictable exponential pattern; you cannot predict which die will decay next.