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Atomic Physics / Nuclear Chemistry

Radioactive Decay Chain Simulator

Real-time animation of the decay chain parent A → daughter B → granddaughter C (Bateman equations). Watch the daughter build up and peak, while a long-lived parent drives the chain into secular equilibrium, where the daughter activity matches the parent activity.

Decay-Chain Presets

Parameters

Basic Equation of Radioactive Decay

$N(t) = N_0 e^{-\lambda t},\quad \lambda = \frac{\ln 2}{T_{1/2}}$
$A(t) = \lambda N(t)$ (Radioactivity [Bq])
$t_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}$
Live values
0.00
Time t [T½ᴬ]
100.0
Parent A fraction [%]
0.0
Daughter B fraction [%]
0.0
Granddaughter C fraction [%]
1.000
Parent activity A_A / A₀
0.000
Daughter activity A_B / A₀
Decay-chain real-time animation
Parent A Daughter B Granddaughter C (stable) Daughter activity (dashed)
Theory & Key Formulas (Bateman chain)

Sequential decay A → B → C (C stable), coupled equations:

$$\frac{dN_A}{dt}=-\lambda_A N_A,\quad \frac{dN_B}{dt}=\lambda_A N_A-\lambda_B N_B,\quad \frac{dN_C}{dt}=\lambda_B N_B$$

Bateman solution: $$N_A=N_0 e^{-\lambda_A t},\quad N_B=\frac{N_0\,\lambda_A}{\lambda_B-\lambda_A}\left(e^{-\lambda_A t}-e^{-\lambda_B t}\right)$$

Daughter peaks at $t_{\max}=\frac{\ln(\lambda_B/\lambda_A)}{\lambda_B-\lambda_A}$; activity $A=\lambda N$.

When the parent is far longer-lived than the daughter ($\lambda_A\ll\lambda_B$), the daughter activity matches the parent (secular equilibrium $A_B\to A_A$). For $\lambda_A\approx\lambda_B$ use the limit $N_B=N_0\lambda_A t\,e^{-\lambda_A t}$.

💬 Ask the Professor

🙋
Why does radioactive decay follow an exponential curve? I learned exponentials in math, but I do not see why they appear in physics.
🎓
It comes directly from the assumption that the number of decays per unit time is proportional to the number of atoms present. The differential equation is dN/dt = -λN: the more atoms you have now, the more will decay in the next moment. Solving it gives N(t)=N₀e^(-λt). Compound interest, bacterial growth, and Newton cooling all become exponential for the same reason: the rate of change is proportional to the current value.
🙋
Why do people say radiocarbon dating cannot measure samples older than about 50,000 years?
🎓
The half-life of ¹⁴C is 5,730 years. Fifty thousand years is about 8.7 half-lives, leaving only N₀×(1/2)^8.7 ≈ 0.24% of the original amount. Measuring such a tiny fraction accurately is technically difficult. On the other hand, uranium-238 has a half-life of about 4.5 billion years, so it changes too slowly for historical samples but works well for geologic ages. The nuclide must match the time scale you want to measure.
🙋
Cesium-137 became a concern after nuclear accidents. What does its 30-year half-life imply?
🎓
It means the activity falls to half after 30 years. After 10 half-lives, or about 300 years, it falls to roughly 1/1024, about 0.1%. That is the basis for long-term cleanup planning. Iodine-131, by contrast, has an 8-day half-life and mostly disappears within a few months. Understanding each nuclide's half-life is fundamental to exposure assessment.

Frequently Asked Questions

What is the difference between alpha, beta, and gamma decay?

Alpha decay emits a helium nucleus (⁴He) and has low penetration, stopped by paper. Beta decay emits an electron or positron and has moderate penetration, stopped by a few millimeters of metal. Gamma decay emits high-energy photons and has high penetration, requiring lead shielding.

What is a decay chain?

A decay chain is a sequence in which the daughter nuclide produced by one decay is also radioactive and decays again. Uranium-238 becomes stable lead-206 after 14 decay steps in the uranium series. Because each step has a different half-life, long-term activity prediction requires solving coupled differential equations.

How are radionuclides used in medicine?

PET imaging uses fluorine-18, with a half-life of about 110 minutes, in glucose-like tracers to image metabolically active regions such as tumors. Short half-lives help reduce exposure while preserving diagnostic usefulness. Iodine-131, with an 8-day half-life, is used in thyroid therapy because it concentrates selectively in thyroid tissue.

What is the difference between becquerel (Bq) and sievert (Sv)?

Becquerel (Bq) counts decays per second, while sievert (Sv) estimates biological dose effect in the human body. The same Bq value can have different biological impact depending on radiation type; alpha radiation has a much larger weighting factor than beta or gamma radiation.

List of Half-Lives for Major Radioactive Nuclides

NuclideHalf-lifeDecay TypeMain Uses / Issues
Hydrogen-3 (tritium)12.3 yrβ⁻Fusion research, water contamination tracing
Carbon-145,730 yrβ⁻Radiocarbon dating
Iodine-1318.02 daysβ⁻+γThyroid cancer therapy, nuclear accident monitoring
Cesium-13730.17 yrβ⁻+γNuclear accident contamination
Radium-2261,600 yrαHistorical radium therapy, luminous paint
Plutonium-23924,110 yrαNuclear weapons, reactor fuel
Uranium-235704 million yrαFission fuel, nuclear weapons
Uranium-2384.47 billion yrαUranium-lead dating in geology

💬 Conversation about Radioactive Decay

🙋

Half-life means the time for half of this nuclide to remain, right? But all the atoms do not decay at the same time, do they?

🎓

Exactly. Individual atoms decay randomly, and half-life is a statistical average. With about $10^{23}$ atoms, the law of large numbers makes the sample follow $N(t)=N_0 e^{-\lambda t}$ very closely. For a single atom, it means there is a 50% chance it has decayed after one half-life.

🙋

How does carbon-14 dating tell us the age of something from the past?

🎓

Living organisms continuously exchange carbon with the atmosphere through respiration and food, so their ${}^{14}C/{}^{12}C$ ratio stays close to the atmospheric value. After death, intake stops and ${}^{14}C$ decays with a 5,730-year half-life. Measuring the current ratio gives the age from $t = \frac{t_{1/2}}{\ln 2} \ln\!\left(\frac{N_0}{N}\right)$.

🙋

Cesium-137 became a problem after nuclear accidents because its half-life is about 30 years, right?

🎓

Right. Iodine-131 has an 8-day half-life, so it mostly disappears within a few months. Cesium-137 lasts about 30 years, roughly a human generation. It can bind to soil, enter food chains, and keep internal-exposure risk relevant for a long time. Tritium and plutonium-239 pose different problems because their half-lives are very different.

🙋

What is a decay chain? Uranium-238 does not become stable lead in one step, does it?

🎓

Correct. Uranium-238 reaches lead-206 through 14 decay steps. Intermediate nuclides include radon-222, a gas with a 3.8-day half-life, and polonium-210. Radon can enter buildings from soil and is a major natural radiation source linked to lung cancer risk. In a decay chain, the number of atoms of each intermediate nuclide is found by solving coupled differential equations.

Decay Types and Characteristics

Decay Type Emitted Particle Penetration Atomic Number Change Example
Alpha decay Helium nucleus (He-4) Stopped by paper Z: -2, A: -4 U-238 → Th-234
Beta-minus decay Electron + antineutrino A few mm of aluminum Z: +1, A: unchanged Cs-137 → Ba-137
Beta-plus decay Positron + neutrino Low, followed by annihilation radiation Z: -1, A: unchanged F-18 → O-18 (PET)
Gamma decay High-energy photon Requires lead or concrete No change; excited state relaxes Co-60 → Ni-60 + γ
Electron capture Characteristic X-ray + neutrino Low Z: -1, A: unchanged I-125 → Te-125

What is Radioactive Decay Chain?

Radioactive Decay Chain is a fundamental topic in engineering and applied physics. This interactive simulator lets you explore the key behaviors and relationships by directly manipulating parameters and observing real-time results.

By combining numerical computation with visual feedback, the simulator bridges the gap between abstract theory and physical intuition — making it an effective learning tool for students and a rapid-verification tool for practicing engineers.

Physical Model & Key Equations

The simulator is based on the governing equations behind Radioactive Decay Chain Simulator. Understanding these equations is key to interpreting the results correctly.

Each parameter in the equations corresponds to a slider in the control panel. Moving a slider changes the equation's solution in real time, helping you build a direct connection between mathematical expressions and physical behavior.

Real-World Applications

Engineering Design: The concepts behind Radioactive Decay Chain Simulator are applied across mechanical, structural, electrical, and fluid engineering disciplines. This tool provides a quick way to estimate design parameters and sensitivity before committing to full CAE analysis.

Education & Research: Widely used in engineering curricula to connect theory with numerical computation. Also serves as a first-pass validation tool in research settings.

CAE Workflow Integration: Before running finite element (FEM) or computational fluid dynamics (CFD) simulations, engineers use simplified models like this to establish physical scale, identify dominant parameters, and define realistic boundary conditions.

Common Misconceptions and Points of Caution

Model assumptions: The mathematical model used here relies on simplifying assumptions such as linearity, homogeneity, and isotropy. Always verify that your real system satisfies these assumptions before applying results directly to design decisions.

Units and scale: Many calculation errors arise from unit conversion mistakes or order-of-magnitude errors. Pay close attention to the units shown next to each parameter input.

Validating results: Always sanity-check simulator output against physical intuition or hand calculations. If a result seems unexpected, review your input parameters or verify with an independent method.

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How to Use

  1. Set the half-life (t12Val) in years using the slider or numeric input. For U-238, enter 4.468e9 years; for C-14, use 5730 years.
  2. Configure initial nuclei count (n0Slider) from 1e10 to 1e24 atoms, representing your sample mass via Avogadro's number conversion.
  3. Define maximum simulation time (tmaxVal) in appropriate units matching your half-life scale—milliseconds for Tc-99m decay, millions of years for uranium series.
  4. Execute the simulation to plot activity (Bq or Ci), remaining nuclei count, and decay rate versus time using Bateman equations.

Worked Example

Simulate U-238 decay in a 1 kg uranium ore sample. Set t12Val=4.468e9 years (U-238 half-life), n0Slider=2.56e21 atoms (approximately 1 kg U-238 mass), tmaxVal=1e6 years. The simulator computes initial activity: λN₀ = (ln2/4.468e9 yr) × 2.56e21 ≈ 1.24e8 Bq (3.4 Ci). After 500,000 years, remaining nuclei stays at essentially 2.56e21 atoms (about 99.99% of the original), demonstrating negligible decay over human timescales.

Practical Notes