Dice as Radioactive Nuclei: Scaling a Model to Carbon-14
Students model radioactive decay by rolling 100 dice and removing every die showing a six after each throw. Their counts are: throw 0, 100; 1, 84; 2, 70; 3, 58; 4, 48; 5, 40; 6, 33. A student notes that the count halves after about four throws and says the half-life is '4 throws, which is not a real time, so the model cannot tell us anything about carbon-14 with a half-life of 5730 years.' Another student claims the model is wrong because real nuclei decay randomly, while dice rolls are deterministic. The teacher plans a short assessment task asking students to explain the model's assumptions and limitations. The class has already met exponential decay and the decay constant. The teacher has a spreadsheet and ten sets of dice for a whole-class simulation.
1. Explain how the dice model maps onto radioactive decay. Include what one throw represents, what the half-life is in the model, and why the count is not exactly halved every four throws.
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Each die represents a nucleus. A six is a decay; the probability of decay per throw is p = 1/6, constant and independent for each die. After n throws the expected number is N = N0(5/6)^n. Half-life is the number of throws for N to halve: (5/6)^n = 0.5, so n = ln 0.5 / ln(5/6) ≈ 3.8 throws. One throw represents a fixed time interval Δt, not a universal time. The count is not exactly halved every four throws because each throw is a random trial; with only 100 dice, fluctuations of several counts are expected. The model therefore shows exponential decay without assigning a real time to a throw.
2. Critique the two students' statements. Which parts are valid and which are misconceptions?
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The first student is right that '4 throws' is not a real time; the throw is a scaled time interval. But the model can be mapped onto carbon-14 by setting 3.8Δt = 5730 years, so Δt ≈ 1508 years. It then predicts the shape of decay, not the absolute half-life. The second student's claim that dice are deterministic is a misconception in this context: a die roll is used as a random event, and the model treats each die independently. Real nuclear decay is random at the individual level, so the model captures constant probability and independence, but not daughter products, background radiation, or the quantum nature of real decay. The model is an analogy for the mathematics, not a literal simulation of nuclei.
3. Design a question the teacher could use to assess whether students can transfer the model to carbon-14 dating, and give the expected answer.
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Question: 'A bone sample contains 25% of its original carbon-14. Using the dice model, how many throws and how many years does this represent?' Expected answer: 25% is two half-lives. Two half-lives = 2 × 3.8 = 7.6 throws. Setting 3.8 throws = 5730 years gives 7.6 throws = 11,460 years. Award marks for converting the percentage to half-lives, using the model's half-life, and scaling to years. A follow-up asks why the model cannot give an exact age for a single nucleus, and students should refer to random decay and the statistical nature of half-life.