Radioactive Decay Series & Activity Calculator

Radioactive Decay Guide

What should I know about Decay Equations?

N = N₀ × e^(−λt) = N₀ × (½)^(t/t½). Where N = atoms remaining, N₀ = initial atoms, λ = decay constant = ln2/t½, t = elapsed time, t½ = half-life. Activity A = λN (decays per second, measured in becquerels, Bq). 1 Bq = 1 decay per second. 1 Curie (Ci) = 3.7×10¹⁰ Bq (defined as the activity of 1g of radium-226). After 1 half-life: 50% remains. After 2: 25%. After 10: 0.098% — essentially negligible for practical purposes. After 20 half-lives: less than 1 part per million.

What's the key thing to understand about Carbon-14 Dating?

Living organisms exchange carbon with the atmosphere (via food/photosynthesis), maintaining a constant ratio of ¹⁴C to ¹²C (approximately 1.2×10⁻¹² by mass). At death, ¹⁴C stops being replenished and decays (t½ = 5,730 years). Measuring the current ¹⁴C/¹²C ratio allows calculation of time since death. Range: up to approximately 50,000 years (beyond that, too little ¹⁴C remains). Calibration: atmospheric C-14 has varied over time — calibration curves built from dendrochronology (tree-ring dating) and other independent methods convert raw radiocarbon ages into more accurate calendar-year dates.

What do I need to know about Medical Isotopes?

Technetium-99m (6-hour half-life): most widely used medical isotope. Gamma emitter — photons detected by gamma camera for organ imaging. Short half-life minimises patient dose. Used for: bone scans, heart perfusion, kidney and thyroid imaging. Iodine-131 (8-day half-life): concentrates in thyroid tissue — used to treat hyperthyroidism and thyroid cancer. Beta and gamma emitter. Iodine-123 (13-hour half-life): gamma only — used for thyroid imaging (lower dose than I-131). Fluorine-18, with a 110-minute half-life, is the isotope most commonly used in PET scans, chosen specifically because its short half-life limits patient radiation exposure while still allowing enough time for imaging.

What should I know about Radiation Types and Penetration?

Alpha (α) particles: helium-4 nuclei (+2 charge). Range in air: ~5cm. Stopped by paper or skin. High ionising power — very damaging if ingested or inhaled (radon gas in homes). External alpha source: essentially harmless. Beta (β) particles: high-energy electrons or positrons. Range: up to ~3m in air. Stopped by a few mm of aluminium. Moderate ionising power. Gamma (γ) rays: electromagnetic radiation (photons). Very penetrating — requires cm of lead or metres of concrete to attenuate significantly, compared to alpha particles that are stopped by a sheet of paper.

What should I know about Decay Chains and Secular Equilibrium?

A decay series (or decay chain) is a sequence of radioactive decays where one unstable isotope decays into another unstable isotope, which decays again, continuing until a stable isotope is reached — uranium-238 passes through 14 steps before finally reaching stable lead-206. When a parent isotope's half-life is far longer than its daughter's, the chain reaches 'secular equilibrium': the daughter's activity (decays per second) matches the parent's activity, because the daughter is being produced and destroyed at the same rate. This is why uranium ore contains measurable radium and radon even though these intermediate isotopes have much shorter half-lives than uranium itself — they're continuously being replenished by the slow decay of uranium above them in the chain, maintaining a roughly constant concentration over geological timescales rather than decaying away to nothing.

What do I need to know about Branching Decay?

Not every isotope decays via a single path — some undergo branching decay, splitting into two or more different decay routes with fixed probabilities. Potassium-40, used in potassium-argon dating, decays about 89% of the time by beta decay into calcium-40, and about 11% of the time by electron capture into argon-40. The overall half-life (1.25 billion years) describes the total decay rate combining both branches; the branching ratio determines what fraction of decayed atoms end up as each daughter product. This matters for dating methods and nuclear physics calculations because you need to track not just 'how much parent isotope remains' but 'how much of each specific daughter product has accumulated' — potassium-argon dating specifically measures the accumulated argon-40, so it relies on knowing the 11% branching ratio to correctly convert an argon measurement back into elapsed time.

What should I know about the Uranium-238 and Thorium-232 Decay Series?

Uranium-238, with a half-life of about 4.5 billion years, decays through a chain of 14 successive alpha and beta decays before finally reaching stable lead-206 — a journey that takes it through well-known intermediate isotopes including radium-226 and radon-222. Thorium-232 similarly decays through its own multi-step series to stable lead-208. These two natural decay series, alongside the shorter uranium-235 (actinium) series, are the ultimate source of most natural background radiation from rocks and soil, and the radioactive gas radon-222 produced partway through the uranium series is a well-documented indoor air-quality hazard.

What should I know about Radon Gas in the Uranium Decay Series?

Radon-222 is a radioactive noble gas that appears partway through the uranium-238 decay series, formed when radium-226 undergoes alpha decay. Unlike the solid daughter products that make up most of the series, radon is a gas at room temperature, so it can escape from the rock, soil, and building materials in which uranium-238 and its decay products occur naturally, seeping upward and accumulating in enclosed spaces such as basements and poorly ventilated buildings. Radon-222 itself has a half-life of only about 3.8 days and decays by alpha emission into polonium-218, continuing the decay chain toward further short-lived, solid alpha- and beta-emitting isotopes before eventually reaching stable lead-206. While radon gas itself is chemically inert and largely exhaled before it decays, its solid decay products are not: they readily attach to dust and lung tissue, where their continued alpha decay can damage DNA in lung cells. This makes radon exposure, particularly in enclosed indoor spaces with poor ventilation, the second-leading cause of lung cancer after smoking in several countries, and is why building codes in uranium-rich regions often require radon testing and mitigation systems such as sub-slab ventilation.

Decay Chain Steps: Uranium-238 to Stable Lead-206

The U-238 decay series passes through 14 intermediate radioactive isotopes before finally reaching the stable end product, lead-206. A few of the key steps:

The chain alternates between alpha decays (which drop mass number by 4 and atomic number by 2) and beta decays (which keep mass number the same but raise atomic number by 1), ultimately converting U-238 (92 protons, mass 238) down to Pb-206 (82 protons, mass 206) — a net loss of 8 alpha particles and 6 beta particles' worth of nuclear transformation.

Worked Example: Secular Equilibrium Activity in a Decay Chain

When a parent isotope's half-life is far longer than its daughter's, the chain reaches "secular equilibrium," where the daughter's activity equals the parent's activity (A_parent = A_daughter), even though their quantities (N) differ hugely. At equilibrium: N_daughter / N_parent = t½,daughter / t½,parent. Take Radium-226 (t½ = 1,600 years) decaying into Radon-222 (t½ = 3.82 days = 0.010466 years). At secular equilibrium: N_Rn / N_Ra = 0.010466 / 1,600 = 6.54 × 10⁻⁶. For 1 mole of Radium-226 (6.022 × 10²³ atoms), the equilibrium quantity of Radon-222 present at any instant is N_Rn = 6.022 × 10²³ × 6.54 × 10⁻⁶ ≈ 3.94 × 10¹⁸ atoms — a tiny fraction of the parent's quantity, yet its activity (decays per second) exactly matches the parent's, because what Radon-222 lacks in quantity it makes up for with a decay rate roughly 153,000 times faster (1,600 years ÷ 3.82 days ≈ 153,000, the ratio of their decay constants). This is why radon gas, despite being generated from a tiny equilibrium population, still poses a measurable indoor air hazard in homes built on uranium- or radium-rich soil.

Alpha, Beta, and Gamma Decay: Mass and Charge Changes

Each decay step in a chain changes the nucleus's mass number (A) and atomic number (Z) differently depending on the type of radiation emitted:

Gamma decay changes neither mass nor atomic number because it simply releases excess energy from an already-formed nucleus (often immediately following an alpha or beta decay) without ejecting any particle — which is why gamma rays frequently accompany alpha and beta decay steps in a chain rather than occurring as standalone transformations. Across the full U-238 to Pb-206 chain, the cumulative changes (8 alpha decays, 6 beta decays) account precisely for the net mass drop of 32 (238 − 206) and net charge change of −10 (92 − 82): 8 alpha decays contribute −32 to mass number and −16 to atomic number, while 6 beta decays contribute 0 to mass and +6 to atomic number, giving a net atomic number change of −16+6 = −10, matching 92 − 82 = 10 exactly.