Radioactive isotopes decay at a rate that never changes with temperature, pressure or chemistry. Each isotope has a characteristic half-life, the time for half of any sample to decay, and that single number predicts how much will remain after any length of time. This calculator applies the half-life law in all directions: how much is left after a given time, how much there was to begin with, how long a decay has been running (the basis of radiometric dating) and what the half-life is from two measurements.
How to use the half-life calculator
- Choose what to Solve for: amount remaining, initial amount, time elapsed or half-life.
- Pick an Isotope from the list (carbon-14, tritium, cobalt-60, strontium-90, cesium-137, americium-241, radium-226, plutonium-239, uranium-235 and -238, potassium-40, iodine-131, phosphorus-32, radon-222, technetium-99m or fluorine-18) or choose “Custom half-life” and enter it.
- Choose a Unit of the amounts: %, g, mg, µg, kg, mol, atoms, Bq, mCi, Ci or counts per minute.
- Enter the known amounts and the Time elapsed in seconds, minutes, hours, days or years.
- Read the result, the fraction remaining, the number of half-lives, the decay constant and mean lifetime, plus a decay curve and a table of the first ten half-lives.
Half-life formulas
N0 is the starting amount, N the amount after time t, T½ the half-life and λ = ln 2 ÷ T½ the decay constant. Activity, the number of decays per second, follows the same curve, because it is proportional to the number of nuclei present.
Worked example
Iodine-131 after a month
A hospital receives 100 mg of iodine-131 (half-life 8.025 days). How much is left after 30 days?
Half-lives elapsed: 30 ÷ 8.0252 = 3.738
N = 100 × 0.53.738 = 7.49 mg
About 92.5% has decayed. This rapid decay is why I-131 can be used in thyroid treatment yet is mostly gone within a couple of months.
Dating a bone. A bone sample retains 25% of the carbon-14 of living tissue. With T½ = 5,700 years, t = 5,700 × log₂(4) = 11,400 years.
Measuring a half-life. A detector shows a sample falling from 1,000 to 125 counts per minute in 3 hours. That is three halvings, so T½ = 3 h × ln 2 ÷ ln 8 = 60 minutes.
Half-lives of common isotopes
| Isotope | Half-life | Known for |
|---|---|---|
| Fluorine-18 | 109.7 minutes | PET scans |
| Technetium-99m | 6.007 hours | Medical imaging |
| Iodine-131 | 8.025 days | Thyroid treatment |
| Cobalt-60 | 5.271 years | Radiotherapy, sterilization |
| Cesium-137 | 30.08 years | Fission product |
| Carbon-14 | 5,700 years | Radiocarbon dating |
| Uranium-238 | 4.468 billion years | Dating rocks and Earth |
Half-lives are rounded from evaluated nuclear data published by the National Nuclear Data Center.
Radiometric dating in practice
Carbon-14 dating works for organic material up to about 50,000 years old, roughly nine half-lives, after which too little remains to measure. Laboratories report “conventional radiocarbon ages” using the older Libby half-life of 5,568 years and then calibrate them against tree rings and other records, so a published calendar age differs from this simple calculation. For rocks, geologists use long-lived pairs such as uranium-238 to lead-206 or potassium-40 to argon-40, measuring both parent and daughter to recover the starting amount.
The same exponential math describes many other processes, from drug elimination to cooling coffee. The logarithm calculator shows how the log steps work, the exponent calculator evaluates the powers of ½, and the compound interest calculator runs the same mathematics in the growth direction.
Frequently asked questions
What does half-life mean?
The time it takes for half of the radioactive nuclei in a sample to decay. After one half-life 50% remains, after two 25%, after three 12.5%, and so on. Each nucleus decays at random, but in a large sample the overall pattern is exactly predictable.
How is carbon-14 dating calculated?
Living things keep a steady proportion of carbon-14; after death it decays with a half-life of 5,700 years. If a sample has 25% of the original carbon-14, two half-lives have passed: t = 5,700 × log₂(100 ÷ 25) = 11,400 years.
Can a sample decay to exactly zero?
Mathematically, never: the amount halves forever. In practice, after about 50 half-lives fewer than one atom in 10¹⁵ is left, so for any real sample essentially none of the original nuclei remain.
What units should I use for the amount?
Any quantity proportional to the number of radioactive nuclei: grams, moles, atoms, percent, or activity in becquerels, curies or counts per minute. The unit only labels the result; the calculation uses ratios.
What is the decay constant?
λ = ln 2 ÷ T½, the probability per unit time that a given nucleus decays. Iodine-131, with an 8.025-day half-life, has λ ≈ 0.0864 per day. The mean lifetime, 1/λ, is 1.4427 times the half-life.