Half-Life Calculator
Radioactive decay describes the process by which an unstable atomic nucleus loses energy by emitting radiation and transforming into a different nucleus or a lower energy state. The rate of decay is characterized by the half-life, the time it takes for exactly half of the nuclei in a sample to decay. The fundamental decay formula is N(t) = N0 times (0.5)^(t / t_half), where N0 is the initial quantity, t is the elapsed time, and t_half is the half-life. This formula applies universally to all radioactive isotopes and is also expressed as N(t) = N0 times e^(negative lambda times t), where the decay constant lambda equals ln(2) divided by the half-life. This calculator accepts the initial amount (in any consistent unit: atoms, grams, moles, or percent), the half-life with your choice of time unit, and the elapsed time. It then returns the remaining quantity, the fraction remaining as a percentage, and how many half-lives have elapsed. Applications include nuclear medicine, carbon dating, nuclear power safety, and environmental monitoring. Figures are computed deterministically from the formula above.
Remaining: -- | Fraction remaining: --%
Half-life formula explained
Radioactive decay follows first-order kinetics, meaning the rate of decay is proportional to the number of undecayed nuclei present at any moment. This leads to the exponential decay law.
N(t) = N0 * (0.5)^(t / t_half)
Equivalently: N(t) = N0 * e^(-lambda * t)
Decay constant: lambda = ln(2) / t_half
Worked example (Carbon-14 dating)
- Initial C-14 amount: 1,000 atoms
- Half-life of C-14: 5,730 years
- Elapsed time: 11,460 years (= 2 half-lives)
- N(t) = 1,000 * (0.5)^(11,460 / 5,730) = 1,000 * (0.5)^2 = 250 atoms remaining
- Fraction remaining = 250 / 1,000 = 25%
Half-lives of common isotopes
| Isotope | Half-life | Common use |
|---|---|---|
| Carbon-14 | 5,730 years | Archaeological dating |
| Iodine-131 | 8.02 days | Medical imaging and therapy |
| Uranium-238 | 4.47 billion years | Geological dating |
| Technetium-99m | 6.01 hours | Nuclear medicine scans |
| Radon-222 | 3.82 days | Environmental monitoring |
| Cobalt-60 | 5.27 years | Radiation therapy |
Frequently asked questions
What is a half-life?
A half-life is the time required for half of a given quantity of a radioactive substance to decay into a different element or isotope. After one half-life, 50% of the original material remains. After two half-lives, 25% remains. The half-life is a fixed characteristic of each radioactive isotope and does not depend on the initial amount or external conditions such as temperature or pressure.
What is the half-life formula?
The radioactive decay formula is N(t) = N0 * (0.5)^(t/t_half), where N(t) is the remaining quantity at time t, N0 is the initial quantity, t is elapsed time, and t_half is the half-life. Equivalently, N(t) = N0 * e^(-lambda*t) where the decay constant lambda = ln(2) / t_half.
What are common half-lives for well-known isotopes?
Half-lives span an enormous range. Carbon-14 has a half-life of about 5,730 years, making it useful for archaeological dating. Uranium-238 has a half-life of about 4.47 billion years. Iodine-131 (used in medical imaging) has a half-life of about 8 days. Polonium-214 decays in microseconds. The half-life determines the practical use of each isotope.
How is radioactive decay used in carbon dating?
Radiocarbon dating uses the known half-life of carbon-14 (5,730 years) to estimate the age of organic materials. Living organisms continuously exchange carbon with the environment, maintaining a known ratio of carbon-14 to carbon-12. When an organism dies, carbon-14 decays without replenishment. By measuring the remaining ratio and applying the decay formula, scientists can estimate when the organism died, typically up to about 50,000 years ago.
Does the half-life change with temperature or pressure?
No. Radioactive decay is a quantum mechanical process governed by the strong and weak nuclear forces. It is not affected by temperature, pressure, chemical state, or other environmental conditions that influence ordinary chemical reactions. This stability is why radioactive decay is reliable for dating and for medical dosing calculations. The only exception is electron-capture decay, which can be very slightly influenced by electron density around the nucleus.
Sources
- US Nuclear Regulatory Commission: Half-Life Glossary.
- NIST: Radionuclide Half-Life Measurements.
- EPA: Radioactive Decay.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology. General information only.