chemistry

Radioactive Decay Calculator

Radioactive decay is the spontaneous transformation of unstable atomic nuclei into more stable forms, releasing energy as particles or radiation in the process. Physicists, nuclear engineers, and radiation safety professionals use decay calculations to predict isotope behaviour, manage hazardous materials, and assess contamination risks. This calculator determines activity from sample mass and half-life, or reverses the calculation to find unknown parameters—essential for working with everything from medical radioisotopes to nuclear waste assessment.

Last updated: April 24, 2026

Creators Hanna Pamuła, PhD

Reviewers Davide Borchia and Dominik Czernia

5,963 people find this calculator helpful

Understanding Radioactive Decay

An atomic nucleus becomes unstable when it contains too many protons, too many neutrons, or an unfavourable ratio of the two. To reach a lower energy state, these nuclei spontaneously emit radiation through several well-defined mechanisms.

Alpha decay releases an alpha particle—two protons and two neutrons bound together, equivalent to a helium-4 nucleus. Polonium-210 decaying to lead-206 is a classic example. Beta-minus decay converts a neutron into a proton, ejecting an electron and an antineutrino; carbon-14 transforming to nitrogen-14 demonstrates this process. Beta-plus decay does the reverse, and gamma decay emits high-energy photons without changing the nucleus's proton or neutron count.

Half-life—the time for a sample to decay to half its original quantity—characterises each isotope. Some decay in fractions of a second; others persist for billions of years. This property is intrinsic to the nucleus and independent of sample size, temperature, or chemical state.

Calculating Activity from Mass and Half-Life

Activity quantifies the decay rate: the number of disintegrations per second. To find it, multiply the number of atoms in your sample by the decay constant, which depends on half-life.

Activity = (m / M) × Nₐ × (ln(2) / t₁/₂)

Specific Activity = (Nₐ / M) × (ln(2) / t₁/₂)

m — Mass of the radioactive sample in grams
M — Molar mass of the isotope in g/mol
Nₐ — Avogadro's number (6.022 × 10²³ atoms/mol)
ln(2) — Natural logarithm of 2, approximately 0.693
t₁/₂ — Half-life of the isotope in seconds (or your chosen time unit)

Measurement Units and Standards

Activity is measured in Becquerels (Bq), where 1 Bq equals one disintegration per second. For large samples, activity is often expressed in kilobecquerels (kBq), megabecquerels (MBq), or gigabecquerels (GBq).

The Curie (Ci), an older unit still encountered in legacy data, was originally defined as the activity of one gram of radium-226. The conversion is straightforward: 1 Ci = 37 GBq (approximately 3.7 × 10¹⁰ Bq). While the Becquerel is now the SI standard, Curies remain in use in medical and industrial contexts, particularly in the United States.

Specific activity normalises activity by mass, expressed in Bq/g. It reveals how intrinsically radioactive a substance is, independent of sample size. Two samples of the same isotope always have identical specific activity, even if their total activity differs dramatically.

Practical Examples and Real-World Context

A ripe banana contains roughly 15 Bq of activity, mostly from potassium-40. This natural radioactivity is harmless—the potassium content needed for that activity would be diluted across the entire fruit.

Radon-222, the only naturally occurring radioactive gas, is a major indoor air hazard in regions with granite or uranium-rich soil. Basement concentrations can exceed 1,000 Bq/m³; extreme cases have reached 100,000 Bq/m³. Plutonium-239, with a half-life of 24,110 years, has extremely high specific activity: just 16 grams can sustain a critical mass in a nuclear weapon.

Medical applications rely on short-lived isotopes like technetium-99m (6-hour half-life) for diagnostic imaging or iodine-131 (8-day half-life) for thyroid therapy. The rapid decay minimises long-term radiation burden on patients.

Common Pitfalls and Important Caveats

Avoid these mistakes when working with radioactive decay calculations.

Consistency in time units — Half-life, decay time, and activity measurement periods must all use the same time unit. If half-life is in years but you're calculating seconds, the formula breaks down. Always convert beforehand—this is the source of most numerical errors.
Confusing activity with mass — A tiny sample can have enormous activity if its half-life is very short. Conversely, a large sample of a long-lived isotope may show minimal activity. Activity and mass are independent properties; never assume high mass means high activity.
Forgetting to account for decay over time — The formulas above calculate activity at a single moment. If you need activity after a sample has been sitting for weeks or months, you must apply the exponential decay law: A(t) = A₀ × (1/2)^(t/t₁/₂). Initial activity halves every half-life period.
Mislabeling molar mass — Molar mass is always in grams per mole, not atomic mass units. For example, plutonium-239 has a molar mass of 239 g/mol, not 239 u. Using the wrong value by orders of magnitude is common and ruins results.

Frequently Asked Questions

What causes a nucleus to become radioactive?

Nuclei with too many protons, too many neutrons, or an unstable neutron-to-proton ratio occupy high-energy states. They spontaneously undergo decay—emitting particles or radiation—to reach lower, more stable configurations. This transition releases energy because the products are more tightly bound than the original nucleus. Heavier nuclei (above iron-56) are especially prone to instability because the strong nuclear force struggles to hold many protons together against electromagnetic repulsion.

How is activity different from specific activity?

Activity measures total disintegrations per second in a sample, expressed in Becquerels. Specific activity normalises this by mass, giving disintegrations per second per gram (Bq/g). Two samples of the same isotope always have identical specific activity, regardless of their size. Activity scales linearly with mass—double the mass, double the activity. Specific activity is an intrinsic property of the isotope itself and depends only on half-life and molar mass.

Why is the Becquerel preferred over the Curie?

The Becquerel is a straightforward SI unit: one disintegration per second. It scales naturally across orders of magnitude (kBq, MBq, GBq). The Curie, defined as the activity of 1 gram of radium-226, is an arbitrary reference tied to a specific isotope and historical convention. Its use complicates international standardisation. The Becquerel was adopted by the SI system in 1975 and is now standard in physics, medicine, and regulation, though Curies persist in older literature and some American medical facilities.

Can I use this calculator to find unknown half-lives?

Yes. Rearranging the activity formula gives: half-life = (Nₐ × m × ln(2)) / (M × activity). If you measure activity experimentally (with a radiation detector) and know the sample's mass and molar mass, you can solve for half-life. This reverse calculation is common in analytical chemistry and archaeology. Radiocarbon dating relies on this principle, measuring the activity of carbon-14 in organic samples to determine age.

What happens to activity if I wait several half-lives?

Activity decays exponentially, halving with each half-life period. After one half-life, 50% remains; after two, 25%; after three, 12.5%, and so on. Mathematically, remaining activity = initial activity × (0.5)^n, where n is the number of half-life periods elapsed. For example, iodine-131 (8-day half-life) drops to 1.6% of its initial activity after 40 days (five half-lives). This exponential behaviour is why short-lived isotopes vanish quickly from the environment, while long-lived ones persist as chronic hazards.

How does sample composition affect radioactive decay rate?

The decay rate (fraction of nuclei decaying per unit time) is independent of sample composition, temperature, pressure, or chemical form. Only the isotope's intrinsic half-life matters. However, the total activity of a sample depends on how many radioactive nuclei it contains. Pure plutonium-239 metal and plutonium-239 oxide have identical decay rates per nucleus, but the oxide's lower density (fewer nuclei per gram) means lower total activity for the same mass. The decay constant itself never changes.

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