Radioactive Decay Calculator
Compute how much of a radioactive isotope remains after a given elapsed time using its half-life. Useful for carbon dating, medical isotope planning, nuclear waste decay estimates, and radiometric geology.
Last updated: May 2026
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About this calculator
Radioactive decay is a first-order stochastic process: each unstable nucleus has a fixed per-second probability of decaying, independent of temperature, pressure, chemistry, or how many other nuclei have already decayed. The formula used here is N(t) = N₀ × (1/2)^(t / t½), where N₀ is the starting quantity, t is the elapsed time, and t½ is the half-life — the time required for exactly half the original atoms to decay. The same formula works whether N is measured in number of atoms, mass (mg, g, kg), or activity (Bq, Ci), because activity is proportional to atom count via A = λN with λ = ln(2)/t½. Variables and unit constraints: t and t½ must use the same time unit (both seconds, both years, etc.); mixing minutes and years silently produces grossly wrong answers. After 1 half-life 50% remains, after 3 half-lives ~12.5%, after 10 half-lives ~0.1%, after 20 half-lives ~10⁻⁶ — material is effectively (but not exactly) gone. Edge cases: the equation is an expectation value over many atoms, so for very small samples (< ~100 atoms) Poisson fluctuations dominate and a deterministic answer is misleading. The formula assumes a single decay channel; for isotopes with branching decay or decay chains (e.g., U-238 → Th-234 → … → Pb-206), this calculator gives only the parent population, not daughter buildup. Secular equilibrium and Bateman-equation effects are out of scope here.
How to use
Example 1 — carbon-14 archaeological sample. Start with N₀ = 100 mg of ¹⁴C; half-life t½ = 5,730 years; time elapsed t = 11,460 years (two half-lives). Step 1: t / t½ = 11,460 / 5,730 = 2. Step 2: (1/2)² = 0.25. Step 3: N(t) = 100 × 0.25 = 25 mg remaining. Verify: 2 half-lives means half then half again — 100 → 50 → 25, matches. Example 2 — iodine-131 medical isotope decay. A thyroid clinic receives 5,000 MBq of ¹³¹I (t½ = 8.02 days) on Monday and needs to know activity remaining on Friday (t = 4 days). Step 1: t / t½ = 4 / 8.02 = 0.4988. Step 2: (1/2)^0.4988. Using natural log: ln(0.5) × 0.4988 = −0.6931 × 0.4988 = −0.3458; exp(−0.3458) = 0.7077. Step 3: N(t) = 5,000 × 0.7077 ≈ 3,539 MBq. Verify against the equivalent A = A₀ × exp(−λt) form: λ = ln(2)/8.02 = 0.0864 day⁻¹; exp(−0.0864 × 4) = exp(−0.3456) = 0.7078 — matches within rounding.
Frequently asked questions
What is the half-life of common radioactive isotopes used in science and medicine?
Half-lives span 20 orders of magnitude across isotopes used in different fields. In nuclear medicine: technetium-99m has a half-life of 6 hours (ideal for short SPECT scans), fluorine-18 about 110 minutes (for PET imaging), iodine-131 about 8.02 days (thyroid therapy), and iridium-192 about 74 days (brachytherapy). In radiometric dating: carbon-14 is 5,730 years (organic material up to ~50,000 years old), potassium-40 is 1.25 billion years (volcanic rock and pottery), and uranium-238 is 4.47 billion years (oldest rocks on Earth). In nuclear power and waste: cesium-137 is 30 years (dominates first-century waste hazard), strontium-90 is 28.8 years, plutonium-239 is 24,100 years, and iodine-129 is 15.7 million years. The IAEA Nuclear Data Section publishes authoritative half-life tables maintained to ±0.001% accuracy for medical and metrology isotopes.
How is half-life related to the decay constant λ used in nuclear physics equations?
The decay constant λ is the probability per unit time that any single nucleus decays, and it relates to half-life by λ = ln(2) / t½ ≈ 0.6931 / t½. The continuous form of the decay law is N(t) = N₀ × e^(−λt), which is mathematically equivalent to the half-life form N(t) = N₀ × (1/2)^(t / t½). Use λ when integrating activity over time (cumulative dose calculations), when chaining decays in Bateman equations, or when working with mean lifetime τ = 1/λ = t½/ln(2). The mean lifetime is about 1.443 × t½ — that is the average lifetime of an atom, but the median lifetime equals exactly t½. For very short-lived states (μs and below), nuclear physicists often quote the natural linewidth Γ = ℏ/τ instead of t½.
Can the rate of radioactive decay be sped up, slowed down, or altered by any means?
For practical purposes, no — radioactive decay rates are fixed by nuclear structure and effectively immune to temperature, pressure, chemical environment, magnetic fields, and electric fields under any conditions achievable on Earth. This independence from chemistry is what makes radiometric dating reliable: the decay rate observed today equals the rate millions of years ago. The tiny exceptions involve electron-capture isotopes (like beryllium-7) whose rates change by less than 1% under extreme chemical bonding or pressures of millions of atmospheres because the electron orbital overlap with the nucleus changes. A more dramatic case is bound-state beta decay in fully ionized atoms in heavy-ion storage rings: rhenium-187 half-life drops from 42 billion years to 33 years because the ejected electron has nowhere to go in a neutral atom but has free space in a stripped ion. These laboratory curiosities do not apply to nuclear power, waste storage, medical isotopes, or radiometric dating. The half-life you read in a nuclear data table is the half-life you will measure in any normal terrestrial setting.
What are common mistakes when applying the radioactive decay formula?
The most frequent mistake is mixing time units — entering elapsed time in seconds and half-life in years, or vice versa, which can produce errors of factors of 10⁷ or more. Always check that both inputs use the same unit before computing. Another error is treating mass equivalently to activity in decay chains: the formula tracks the parent isotope only, so for U-238 the "remaining U-238" decreases but the total radioactive mass in the sample stays roughly constant for billions of years because daughters (Th-234, Pa-234m, U-234, etc.) build up. People also confuse the half-life formula with the exponential formula and use them inconsistently — e.g., applying λ = ln(2)/t½ but then using base-2 exponent of t. Finally, applying the formula to very small samples (a handful of atoms) ignores Poisson statistics; the expected value is correct, but actual measurements will fluctuate by ±√N around it.
When should I NOT use this calculator?
Skip this calculator for decay chains with significant daughter buildup — e.g., U-238 → Th-234 → Pa-234m, or radon-222 in a sealed sample where polonium and lead daughters build up over hours. For those cases, use the Bateman equations or a nuclear-decay simulator like ICRP-107 software or Nucleonica. Avoid it for radiation dose calculations: this formula tells you how much parent remains, not how much dose someone received — for that, integrate activity × dose-coefficient over time. Do not use it for branching-decay isotopes where decay can go down multiple paths with different products (e.g., copper-64 decays by β+, β−, and electron capture); the bulk parent number is correct, but specific daughter populations require branching ratios. The formula is also inappropriate for spontaneously fissioning isotopes where multiple fragments are produced — track those with full fission-product yield tables. Finally, for very short-lived nuclear states (lifetime < 1 ps), quantum-mechanical considerations dominate and the classical exponential law breaks down.