Radioactive Decay Calculator

Calculate the remaining amount of a radioactive substance after a given time, or find how long until a substance decays to a target fraction.

Nuclear

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How to Use

1
Enter the initial amount and half-life
Provide the starting quantity (in any unit of mass, activity, or moles) and the half-life of the isotope along with its time unit (seconds, days, years, etc.).
2
Specify elapsed time
Enter the time that has elapsed since the measurement started. Use the same time unit as the half-life for direct comparison, or let the calculator convert units.
3
Read remaining quantity and decay data
The result shows remaining amount, fraction decayed, number of half-lives elapsed, and the decay constant λ = ln(2)/t½.

About

Radioactive decay is the process by which unstable atomic nuclei transform spontaneously toward stability, emitting ionizing radiation in the process. Unlike chemical reactions, nuclear decay is governed purely by quantum mechanics and occurs at a characteristic rate described by a simple exponential law: N(t) = N₀ e^(-λt), where the decay constant λ is unique to each isotope and independent of all external conditions.

Understanding radioactive decay is essential across a wide range of scientific and technological applications. In medicine, radiopharmaceuticals exploit the predictable decay of short-lived isotopes for diagnostic imaging (PET, SPECT) and targeted radiotherapy. In geology and archaeology, decay chains of long-lived isotopes serve as precision clocks for dating rocks and artifacts. In nuclear energy, decay heat from fission products drives the requirement for long-term cooling of reactor cores. In radiation safety, half-life determines how long a contaminated area remains hazardous.

This calculator implements the standard exponential decay law and handles unit conversions between seconds, minutes, hours, days, and years, making it applicable to isotopes across the full range of half-lives — from radon-222 (3.8 days) to potassium-40 (1.25×10⁹ years). It also computes the decay constant λ and the fraction remaining, giving a complete quantitative picture of a sample's radioactive evolution.

FAQ

What is radioactive decay and what causes it?

Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into a more stable configuration, releasing energy as alpha particles, beta particles, gamma rays, or other radiation. Instability arises from an imbalance between protons and neutrons, or excess nuclear energy. Decay is a quantum mechanical process governed by probability: each nucleus has a fixed probability per unit time of decaying, regardless of its age or environment, making the process memoryless and describable by an exponential function.

What is a half-life and what determines it?

The half-life (t½) is the time required for exactly half of a radioactive sample to decay. It ranges from fractions of a microsecond (some excited nuclear states) to billions of years (uranium-238 t½ = 4.47×10⁹ yr). Half-life is an intrinsic property of each isotope determined by nuclear structure — specifically by the overlap between initial and final nuclear wave functions. It cannot be altered by temperature, pressure, chemical bonding, or other external conditions (with very rare exceptions for specific electron-capture isotopes).

What is the decay constant and how does it relate to half-life?

The decay constant λ (lambda) is the probability per unit time that a given nucleus will decay. It is related to half-life by λ = ln(2)/t½ ≈ 0.693/t½. The number of nuclei remaining after time t is N(t) = N₀ e^(-λt), where N₀ is the initial count. Activity (decays per second, measured in Becquerels) equals λN, meaning activity decreases at the same exponential rate as the number of undecayed nuclei.

What is the difference between activity and quantity in radioactive decay?

Quantity refers to the number of radioactive atoms present (or their mass), while activity is the rate of decay — typically measured in Becquerels (1 Bq = 1 decay/s) or Curies (1 Ci = 3.7×10¹⁰ Bq). Activity is proportional to both quantity and decay constant: A = λN. Because short-lived isotopes have large λ, a small amount of a short-lived isotope can have very high activity. For example, 1 gram of Ra-226 has an activity of exactly 1 Curie by definition.

How is radioactive decay used in carbon dating?

Radiocarbon dating exploits the steady-state production of carbon-14 in the atmosphere by cosmic ray neutrons (n + N-14 → C-14 + H-1) and its subsequent decay (t½ = 5,730 years). Living organisms continuously exchange carbon with the atmosphere, maintaining a constant C-14/C-12 ratio. After death, the ratio decreases by radioactive decay. Measuring the current ratio and comparing it to the modern standard allows age determination up to about 50,000 years. The technique was developed by Willard Libby, who received the 1960 Nobel Prize in Chemistry.

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Half-lives elapsed
Decay constant λ