Half-Life Calculator
Find the amount remaining, the starting amount, the half-life or the time elapsed for exponential decay, with decay constant, mean lifetime and isotope presets.
The interactive tool loads here when JavaScript runs.
About this toolWhat is the Half-Life Calculator?
This half-life calculator solves the exponential decay equation in whichever direction you need. Choose the unknown (remaining amount, initial amount, half-life or elapsed time), enter the other three in any time units from seconds to years, and the answer updates as you type together with the decay constant, the mean lifetime and the number of half-lives that have passed.
Presets load published half-lives for common isotopes such as carbon-14, iodine-131, technetium-99m, caesium-137 and uranium-238. A decay curve plots the amount against time with your point marked, and the steps panel shows the formula with your numbers in it.
The amount can be in any unit, including grams, percent, becquerels or counts per minute, because only the ratio between the start and the end enters the maths.
- Solve for remaining, initial, half-life or time
- Decay constant, mean lifetime and number of half-lives
- 16 isotope presets with evaluated half-lives
- Decay curve with your point marked
- Mixed time units; copy and share links
How to use the Half-Life Calculator
- Choose what to solve for: Remaining, Initial, Half-life or Time.
- Pick an isotope preset to load its half-life, or choose Custom and type your own.
- Enter the other values. Half-life and elapsed time each have their own unit, so a half-life in days and a time in hours is fine.
- Read the result, the fraction remaining, the number of half-lives, the decay constant and the mean lifetime.
- Use the decay curve to see where your point sits, then copy the summary or share a link to the same calculation.
Worked examples
Carbon-14 after 10,000 years
Input
Remaining; initial 100 g, half-life 5,700 years, time 10,000 years
Output
29.6399 g left (1.754 half-lives); λ = 0.0001216 per year; τ = 8,223.36 years
100 × 0.5^(10,000 / 5,700). The mean lifetime is longer than the half-life because a few nuclei survive much longer than average.
Dating from the fraction remaining
Input
Time; initial 100, remaining 25, carbon-14
Output
11,400 years
25% is exactly two half-lives, so the age is 2 × 5,700 years.
Iodine-131 a month after a treatment
Input
Remaining; initial 100%, half-life 8.0252 days, time 30 days
Output
7.4935% remains (3.74 half-lives)
This is physical decay only. In the body, biological excretion removes iodine as well, so the effective half-life is shorter.
Technetium-99m over a day
Input
Remaining; 100%, half-life 6.0067 hours, time 24 hours
Output
6.2694% remains
Almost exactly four half-lives in a day, which is why this imaging isotope is made on site from a molybdenum-99 generator.
Background and insight
Radioactive nuclei decay at random, but a large population follows N(t) = N₀ × (1/2)^(t / T½), where T½ is the half-life: the time for half of the remaining nuclei to decay. The same curve can be written N₀ × e^(−λt) with the decay constant λ = ln 2 / T½, and the mean lifetime τ = 1 / λ is about 1.4427 half-lives. After one half-life 50% remains, after two 25%, after five about 3.1% and after ten about 0.098%, which is why ten half-lives is a common rule of thumb for "effectively gone".
Solving for time uses logarithms: t = T½ × log₂(N₀ / N). This is the basis of radiometric dating. Carbon-14, made in the atmosphere and taken up by living things, decays with a half-life of about 5,700 years in current nuclear data evaluations; a sample with 25% of the modern carbon-14 level is about 11,400 years old. Conventional radiocarbon ages are still reported with the older Libby half-life of 5,568 years and then calibrated, so a raw half-life calculation is not a calibrated calendar date.
Preset half-lives come from the evaluated nuclear data in NNDC NuDat (ENSDF), rounded: iodine-131 8.0252 days, technetium-99m 6.0067 hours, cobalt-60 5.2714 years, caesium-137 30.08 years, uranium-238 4.468 billion years. The same equation also describes first-order chemical kinetics, drug elimination in pharmacokinetics and the discharge of a capacitor, where the half-life is ln 2 times the time constant.
Common use cases
Physics and chemistry homework
Solve half-life problems in any direction and check the logarithm steps.
Radiometric dating
Estimate an age from the fraction of a parent isotope that remains.
Nuclear medicine and lab sources
Work out how much activity a source or dose has left after shipping or storage.
Drug and first-order kinetics
Model any process that loses a fixed fraction per unit time, such as drug clearance with a known half-life.
Tips and limitations
- A "year" here is the Julian year of 365.25 days, the convention used in nuclear data.
- Activity (Bq or Ci) decays with the same half-life as the number of atoms, so you can enter activities directly.
- For a medical dose, the effective half-life combines physical and biological removal: 1/T_eff = 1/T_phys + 1/T_bio.
- Half-life is a statistical average. For a handful of atoms the result is an expected value, not a prediction for each atom.
- Raw radiocarbon results need calibration curves (such as IntCal) to become calendar dates.
Frequently asked questions
What is the formula for half-life?
N = N₀ × (1/2)^(t / T½), where N₀ is the starting amount, N the amount left after time t, and T½ the half-life. Rearranged for time: t = T½ × log₂(N₀ / N).
How do you find the half-life from two measurements?
Divide the elapsed time by the number of halvings: T½ = t / log₂(N₀ / N). If a sample falls from 80 to 10 in 24 hours, that is three halvings, so the half-life is 8 hours.
How much is left after 3 half-lives?
One eighth, or 12.5%. Each half-life halves what is left: 50%, 25%, 12.5%. After 10 half-lives about 0.1% remains.
What is the decay constant?
It is the fraction of nuclei decaying per unit time, λ = ln 2 / T½. For carbon-14, λ = 0.693 / 5,700 ≈ 1.216 × 10⁻⁴ per year.
What is the difference between half-life and mean lifetime?
Half-life is the time for half to decay. Mean lifetime is the average time a nucleus survives, τ = T½ / ln 2, about 1.44 times the half-life.
Why is the carbon-14 half-life given as 5,730 or 5,568 years?
5,568 years is the Libby value still used for conventional radiocarbon ages; 5,730 was a later measurement; current evaluations give about 5,700 years. Pick the value your course or lab uses.
Is there an in-depth guide to the Half-Life Calculator?
Yes. “Molar Mass, Molarity, Dilutions and Half-Life: Lab Maths” covers the background, how it works, common pitfalls and worked examples. Read the full guide.
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