Free half-life calculator
Calculate radioactive decay and half-life — enter initial quantity, half-life period, and elapsed time to see how much remains, updated live, as you type.
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N(t) = N₀ × (1/2)^(t/T½). Used for radioactive decay, pharmacokinetics, etc.
Results are estimates. Consult a professional.
How the half-life calculator works
Radioactive decay is a first-order process: at any moment, a fixed fraction of the remaining atoms decays per unit time. The half-life T½ is the time for exactly half the atoms to decay. Given an initial quantity N₀, the amount remaining after time t is determined by how many half-lives have elapsed. The same formula applies to any exponentially decaying quantity — drug concentrations, capacitor discharge, or carbon-14 dating.
Worked example: Carbon-14 after 11,460 years
Carbon-14 has a half-life of 5,730 years. An ancient wood sample originally contained 100 g of ¹⁴C. How much remains after 11,460 years?
Half-lives of common isotopes
Half-lives span an enormous range — from nanoseconds for unstable isotopes to billions of years for nearly stable ones. This table covers isotopes commonly encountered in medicine, archaeology, and nuclear engineering.
| Isotope | Half-life | Common use |
|---|---|---|
| Carbon-14 (¹⁴C) | 5,730 years | Radiocarbon dating of organic material up to ~50,000 years old |
| Uranium-238 (²³⁸U) | 4.47 billion years | Uranium-lead dating of rocks and Earth's age |
| Potassium-40 (⁴⁰K) | 1.25 billion years | Potassium-argon dating of volcanic rocks |
| Iodine-131 (¹³¹I) | 8.02 days | Medical thyroid treatment and diagnostic imaging |
| Technetium-99m (⁹⁹ᵐTc) | 6.01 hours | Most widely used medical radioisotope for imaging |
| Cobalt-60 (⁶⁰Co) | 5.27 years | Cancer radiotherapy and food irradiation |
| Radon-222 (²²²Rn) | 3.82 days | Naturally occurring indoor air-quality hazard |
| Tritium (³H) | 12.32 years | Luminescent watch dials and nuclear fusion research |
Source: NNDC Chart of Nuclides (Brookhaven National Laboratory, 2024). Half-lives rounded to 3 significant figures.
Tips for half-life calculations
Half-life calculations come up in nuclear physics, pharmacokinetics, carbon dating, and environmental science. These tips prevent the most common errors.
- Match units — If the half-life is in days, the elapsed time t must also be in days. Unit mismatch is the single most frequent calculation error.
- Count half-lives for round numbers — When t is an exact multiple of T½, you can simply halve N₀ that many times: no exponents needed. Two half-lives → 25%; three → 12.5%.
- Percent remaining vs. percent decayed — The formula gives the fraction remaining. Percent decayed = 100% − percent remaining. After one half-life: 50% remains, 50% has decayed.
- Drug elimination half-life — The same formula applies to how the body clears drugs. After 4–5 half-lives, a drug is considered effectively eliminated (< 5% remains).
- Inverse problem: finding age — Rearrange to t = T½ × log₂(N₀/N). In carbon dating, measure the ratio of ¹⁴C to ¹²C in a sample and compare to the known atmospheric ratio.
Accuracy and limitations
The half-life formula is mathematically exact within the continuous exponential decay model. In reality, radioactive decay is a quantum statistical process — for small numbers of atoms, individual decay events are random and the formula gives only the expected (mean) value. For large populations of atoms (as in practical samples), the statistical fluctuations are negligible and the formula is highly accurate. Half-life values themselves carry measurement uncertainties; the NNDC values are the best available experimental determinations.
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