Radioactive Decay and Radiocarbon Dating — 3D Lab Simulation
400 random C-14 atoms decay: compare with e^(−λt) and infer an age from the remaining fraction
Theoretical Foundation & Historical Context
Plants and animals take up carbon-14 from the atmosphere while alive; after death the 14C decays with a half-life of 5,730 years. Libby measured the remaining fraction to infer the time of death and won the 1960 Nobel Prize in Chemistry. Each atom decays at random and has no memory, yet large numbers follow a smooth exponential curve; the simulation shows both the fluctuations of a small sample and the average law.
Law: Exponential radioactive decay law N = N₀·e^(−λt) | Scientist: Willard Libby | Year: 1949
Key Learning Objectives & Competencies
- ✓ Compute the remaining fraction after any number of half-lives and invert it to an age.
- ✓ Explain why individual decays are random but a large sample follows a smooth law.
- ✓ Recognise the practical limit of radiocarbon dating near 50,000 years.
Mathematical Formulations & Differential Equations
N(t) = N₀ · exp(−λt) = N₀ · 2^(−t / t½)
λ = ln 2 / t½ = 1.21·10⁻⁴ yr⁻¹ for C-14
t = −(t½ / ln 2) · ln(N / N₀)
50 % remains after 5,730 yr, 25 % after 11,460 yr
P(sống sót đến t) = exp(−λt), τ = 1/λ = t½ / ln 2
Each atom in the sample draws an exponentially distributed decay time
Physical Parameters & Units of Measurement
| Symbol | Physical Quantity | SI Unit | Default |
|---|---|---|---|
| N / N₀ | Fraction of C-14 nuclei remaining | — | — |
| t½ | Half-life of C-14 = 5,730 years | — | — |
| λ | Decay constant (1/year) | — | — |
| t | Age of the sample (years) | — | — |
| F | Measured 14C fraction relative to a modern sample | — | — |
Real-World & Industrial Engineering Applications
Dating archaeological finds such as wood, bone and charcoal.
Calibrating climate records from tree rings, corals and lake sediments.
Other decay clocks (uranium-lead, potassium-argon) use the same mathematics on longer timescales.
3-Step Controlled Experimental Protocol
- Step 1: Baseline Setup — Initialize the simulation in standard conditions and observe undisturbed system dynamics.
- Step 2: Parameter Variation — Gradually modulate physical parameters using the control panel sliders.
- Step 3: Verification & Phase Space Analysis — Compare phase space trajectories and conservation metrics against analytical ODE solutions.
Frequently Asked Questions (FAQ)
Why can radiocarbon only date up to about 50,000 years?
After roughly nine half-lives less than 0.2 % of the 14C remains, comparable to background noise and contamination, so it cannot be measured reliably.
When will one particular atom decay?
It cannot be known; the decay probability per second is constant and independent of the atom's age, so only averages over many atoms are predictable.
Is the age calibrated against tree rings?
No. This is pure radiometric age from t½ = 5,730 yr; true calendar ages need calibration (for example the IntCal curve) because atmospheric 14C has varied over time.
Related Simulations & Experiments
Interactive Simulation: Interactive real-time 60 FPS WebGL simulation is active in your browser. Rotate, zoom, and inspect physical apparatus components with full camera freedom.