Radioactive Decay & Half-Life Calculator

Radioactive Decay Guide

What's the key thing to understand about The Decay Equations?

N(t) = N₀ × e^(−λt). Where N₀ = initial number of nuclei, λ = decay constant (s⁻¹), t = time (s). Relationship between half-life and λ: T½ = ln2 / λ = 0.693 / λ. Activity: A = λN (in Becquerels, Bq). 1 Bq = 1 decay per second. A(t) = A₀ × e^(−λt) — activity falls at the same rate as the number of nuclei. After n half-lives: N = N₀ × (1/2)ⁿ. Example: C-14 (T½ = 5730 years). After 11,460 years (2 half-lives): N = N₀ × (1/2)² = N₀/4 = 25% remaining.

What should I know about Carbon Dating?

Radiocarbon dating uses C-14 to date organic material up to approximately 50,000 years old. C-14 is produced in the upper atmosphere by cosmic ray bombardment of N-14. Living organisms continually exchange carbon with the atmosphere — maintaining a constant C-14/C-12 ratio (approximately 1 in 10¹²). After death: no more C-14 intake. C-14 decays with T½ = 5,730 years. By measuring the C-14 activity relative to living organisms, the time since death can be calculated. Calibration curve (IntCal20): the current internationally agreed standard used to correct raw radiocarbon ages for historical fluctuations in atmospheric C-14, giving a more accurate calendar-year date.

What should I know about Medical Applications?

Technetium-99m (T½ = 6.01 hours): the most widely used medical radioisotope. Used in over 30 million diagnostic scans per year worldwide. Very short half-life means it decays rapidly, minimising patient radiation dose. Iodine-131 (T½ = 8.02 days): thyroid cancer treatment and hyperthyroidism diagnosis. The thyroid specifically absorbs iodine — so I-131 delivers targeted radiation to thyroid tissue. PET scanning: uses short-lived positron emitters (F-18, T½ = 110 minutes). Tracer in a glucose analogue that concentrates in metabolically active tissue, such as tumours, making it detectable on the scan.

What's the key thing to understand about Nuclear Waste and Long-Term Storage?

Half-life determines how long radioactive waste must be stored. Short-lived waste (T½ < 30 years): Cs-137 (30.17 yr), Sr-90 (28.8 yr). After 300 years (10 half-lives): activity reduced to less than 0.1% of original. Intermediate storage for decades. Long-lived waste: I-129 (T½ = 15.7 million years), Tc-99 (T½ = 211,000 years). Must be stored for geological timescales (>100,000 years). Deep geological disposal (DGD) is the internationally agreed solution — stable rock formations at 300-1000m depth are designed to isolate waste safely for the tens of thousands of years it remains hazardous.

What should I know about the Decay Equation and Half-Life Relationship?

The decay equation N = N₀×e^(−λt) relates remaining quantity N to initial quantity N₀, decay constant λ, and elapsed time t. The decay constant relates to half-life via λ = ln(2)/t½ ≈ 0.693/t½. For carbon-14 (half-life 5730 years): λ = 0.693/5730 ≈ 0.000121 per year. To find remaining fraction after 10,000 years: N/N₀ = e^(−0.000121×10000) = e^(−1.21) ≈ 0.298, meaning about 29.8% of the original carbon-14 remains — consistent with the half-life rule of thumb, since 10,000 years is just under two half-lives (11,460 years would give exactly 25%). This exponential formula and the simpler 'halving every half-life' rule are mathematically identical; the exponential form is just needed for times that aren't exact multiples of the half-life.

What do I need to know about Activity vs Remaining Quantity?

Activity (measured in becquerels, decays per second) is proportional to the remaining quantity of radioactive material: A = λN, so activity decays with exactly the same half-life as the quantity itself — if you halve the number of atoms, you halve the decay rate too, since there are half as many atoms available to decay at any instant. This is why a Geiger counter reading (which measures activity, not mass) can be used to estimate remaining radioactive material and why it drops by half every half-life, just like the quantity itself. It's a common point of confusion: people sometimes assume activity and remaining mass behave differently, but because activity is directly proportional to N, they always track together, both following the same exponential decay curve and the same half-life.

What's the difference between the Becquerel and the Curie?

Radioactivity (the rate of decay events) is measured in two units. The becquerel (Bq), the SI unit, equals exactly one decay per second — a very small, inconveniently sized unit for anything but lab-scale sources. The curie (Ci), an older unit named for Marie and Pierre Curie, equals 3.7 × 10¹⁰ Bq, originally defined as the activity of exactly 1 gram of radium-226. Medical and historical sources are often still quoted in curies or millicuries, while modern scientific literature and regulatory limits use becquerels almost exclusively, so converting between the two is a common practical step.

What should I know about Radioactive Decay in Smoke Detectors?

Many household ionisation smoke detectors contain a tiny amount of americium-241, an alpha-emitting radioactive isotope with a half-life of about 432 years, used precisely because of its predictable, steady decay. The americium source continuously emits alpha particles into a small ionisation chamber, knocking electrons off air molecules and creating a weak, steady electric current between two charged plates. Smoke particles entering the chamber attach to these ionised air molecules, disrupting the current; the detector's circuit senses this drop in current and triggers the alarm. Because alpha particles cannot travel more than a few centimetres through air and are blocked entirely by a sheet of paper or the detector's plastic casing, the radiation never reaches outside the sealed chamber in normal use, and the quantity involved (typically under one microgram) is far too small to pose a meaningful health risk even if the detector were opened. The choice of americium-241 over other isotopes reflects a careful balance: its half-life is long enough that the detector's sensitivity barely changes over its working life of a decade or more, yet its decay rate is high enough to sustain a usable ionisation current from a minuscule, safe quantity of material.

Half-Life Reference Table for Common Isotopes

Half-life (t½) and the decay constant (λ = ln 2 / t½) are two sides of the same coin — a shorter half-life means a larger decay constant and faster decay:

The enormous range — six hours to 4.47 billion years — is why no single detector or dating method works for every isotope: short-half-life isotopes like Tc-99m decay away within a day (ideal for medical use, since the radioactivity doesn't linger in the patient), while Uranium-238's half-life is comparable to the age of the Earth itself, making it useful for dating rocks billions of years old.

Worked Example: Carbon-14 Dating a Bone Sample

A fossilized bone is found to contain only 62.5% of the Carbon-14 it would have had when the organism was alive. How old is the bone? The decay constant for C-14 is λ = ln(2)/t½ = 0.693147/5,730 years = 0.00012097 per year. Using N/N₀ = e^(-λt), solve for t: t = -ln(N/N₀)/λ = ln(1/0.625)/λ = ln(1.6)/0.00012097. ln(1.6) = 0.470004, so t = 0.470004 / 0.00012097 = 3,886 years, which rounds to roughly 3,890 years old. As a sanity check: 62.5% = 5/8 = (1/2)^1.5, meaning the bone is 1.5 half-lives old. 1.5 × 5,730 = 8,595 years... that doesn't match, so let's verify directly instead: after 1 half-life (5,730 yr), 50% remains; after another half-life you'd be at 25%, not 62.5%. Since 62.5% sits between 50% and 100%, the correct age is under one full half-life — exactly the ≈3,886 years computed above, which is about 0.68 half-lives (3,886/5,730 = 0.678), consistent with 2^(-0.678) = 0.625. This single-equation calculation is the same one radiocarbon labs use, refined further with calibration curves to correct for historical fluctuations in atmospheric C-14 concentration.

Half-Life vs. Mean Lifetime (τ): Two Related but Different Numbers

Half-life (t½) is the time for half a sample to decay, while mean lifetime (τ, the average time a single atom survives before decaying) is a related but distinct quantity: τ = t½ / ln(2) = 1.4427 × t½. For Carbon-14 (t½ = 5,730 years): τ = 5,730 × 1.4427 ≈ 8,267 years — meaning the average individual atom survives considerably longer than the "half the sample is gone" milestone suggests, because a small fraction of atoms persist for very long times, pulling the average lifetime above the half-life. Mean lifetime is also the reciprocal of the decay constant: τ = 1/λ. For C-14, λ = 0.00012097/year, so τ = 1/0.00012097 = 8,267 years — matching the conversion above exactly. Mean lifetime is the more natural quantity in some physics contexts (like exponential decay in radioactive source design or electronics capacitor discharge, which follows the same mathematical form), while half-life remains the more intuitive and widely reported figure in everyday and medical contexts.