Half-Life Calculator
Find how much is left after a given time, how much you started with, how long it took, or the half-life itself. Mix time units freely, and see the decay constant, mean lifetime and how many half-lives have passed.
Rounded values as printed in common textbooks, to try the sums. For real work use a current nuclear data table.
N = N₀ × (1/2)^(t ÷ T½)
The answer appears in the highlighted box; change its unit to see it another way. Big numbers like 4.468e9 or 4.468 billion work too.
How your half-life sum is worked out
Your numbers: count the half-lives, halve that many times, then scale the start.
- 1
Count half-lives
11,460 years ÷5,730 years= 2 half-lives
- 2
Halve that often
(1/2)^2= 0.25 left25% left
- 3
Scale the start
100 g × 0.25= 25 g
- 4
Rate and lifetime
λ = ln 2 ÷ T½= 0.000121/yearτ = 8,266.64 years
Read more: the formula, worked examples, isotopes and drug half-lives
Things that decay at random, like radioactive atoms, lose the same share of what is left in each equal stretch of time. The half-life is the time for that share to be one half. So the amount falls 100 → 50 → 25 → 12.5 and so on, one halving per half-life, and the curve never quite reaches zero.
The formula
amount left N = starting amount N₀ × (1/2)^(time t ÷ half-life T½)
starting amount N₀ = N × 2^(t ÷ T½)
time t = T½ × log₂(N₀ ÷ N)
half-life T½ = t ÷ log₂(N₀ ÷ N)
half-lives passed n = t ÷ T½
decay constant λ = ln 2 ÷ T½ (ln 2 ≈ 0.693147)
mean lifetime τ = 1 ÷ λ = T½ ÷ ln 2 ≈ 1.4427 × T½
the same curve: N = N₀ × e^(−λt)
Worked example
Amount left (the default): 100 g of carbon-14, half-life 5,730 years, after 11,460 years. Half-lives passed: 11,460 ÷ 5,730 = 2. Share left: (1/2)² = 0.25, so 25%. Amount left: 100 × 0.25 = 25 g, and 75 g has decayed. Decay constant: λ = 0.693147 ÷ 5,730 = 0.000120968 per year. Mean lifetime: τ = 5,730 ÷ 0.693147 = 8,266.64 years.
Starting amount: 10 mg is left after 24 hours, half-life 6 hours. 24 ÷ 6 = 4 half-lives, and 2⁴ = 16, so you started with 10 × 16 = 160 mg.
Time passed: iodine-131 (half-life about 8.02 days) falls from 80 MBq to 10 MBq. 10 ÷ 80 = 0.125 = (1/2)³, so 3 half-lives passed: 3 × 8.02 = 24.06 days.
Half-life: a sample falls from 1,000 counts to 125 counts in 15 hours. 1,000 ÷ 125 = 8 = 2³, so 3 half-lives passed, and the half-life is 15 ÷ 3 = 5 hours.
When it isn't a whole number of half-lives: how long until 100 g of carbon-14 falls to 30 g? log₂(100 ÷ 30) = log₂ 3.3333 = 1.736966, and 1.736966 × 5,730 = 9,952.81 years. On a calculator without log₂, use ln(100 ÷ 30) ÷ ln 2 = 1.203973 ÷ 0.693147.
Half-lives and what is left
| Half-lives | Share left | % left | % decayed |
|---|---|---|---|
| 1 | 1/2 | 50% | 50% |
| 2 | 1/4 | 25% | 75% |
| 3 | 1/8 | 12.5% | 87.5% |
| 4 | 1/16 | 6.25% | 93.75% |
| 5 | 1/32 | 3.125% | 96.875% |
| 7 | 1/128 | 0.78125% | 99.21875% |
| 10 | 1/1,024 | 0.09765625% | 99.90234375% |
Isotope half-lives (approximate textbook values)
These are the rounded figures you usually see in school and college books, given so you can try the sums. They have not been checked against an official data table here, and the best measured values can differ slightly in the last digit. For lab, medical or safety work, use a current nuclear data source.
| Isotope | Half-life (approx.) | Known for |
|---|---|---|
| Carbon-14 | 5,730 years | carbon dating |
| Tritium (hydrogen-3) | 12.3 years | watch dials, tracers |
| Cobalt-60 | 5.27 years | radiotherapy, sterilising |
| Strontium-90 | 28.8 years | fallout |
| Caesium-137 | 30.2 years | fallout, gauges |
| Radium-226 | 1,600 years | old luminous paint |
| Plutonium-239 | 24,100 years | reactor fuel |
| Uranium-235 | 0.704 billion years | reactor fuel |
| Uranium-238 | 4.468 billion years | dating rocks |
| Potassium-40 | 1.25 billion years | dating rocks |
| Iodine-131 | 8.02 days | thyroid treatment |
| Phosphorus-32 | 14.3 days | tracers |
| Polonium-210 | 138.4 days | static removers |
| Radon-222 | 3.82 days | radon gas in homes |
| Technetium-99m | 6.01 hours | medical scans |
| Fluorine-18 | 109.8 minutes | PET scans |
Medicines and other first-order decay
The same maths works for anything that falls by a fixed share per unit of time (first-order decay), including how the body clears many drugs, caffeine and alcohol by-products, the charge on a capacitor, and some chemical reactions. A common rule of thumb is that a drug is mostly gone after about 4 to 5 half-lives (6.25% to 3.1% left). Real drug levels depend on the dose, repeat doses, kidneys and liver, age and other medicines, and some drugs do not follow this curve at all. This calculator is a maths tool, not medical advice. Ask a doctor or pharmacist about any medicine.
Notes
- A year here is 365.25 days (the Julian year used in science), so 1 year = 31,557,600 seconds.
- Activity (Bq or Ci) falls with exactly the same half-life as the number of atoms, so you can use either. 1 Ci = 37 GBq exactly.
- The two amounts can use different units of the same kind, for example g and mg, or MBq and mCi. Amounts in %, mg/L or units are taken as they are.
- Converting between mass and activity needs the isotope's atomic mass and is not done here.
Common questions
What is a half-life?
A half-life is the time it takes for half of something to decay or be removed. After one half-life 50% is left, after two 25%, after three 12.5%, and so on. It is the same length of time whatever amount you start with, which is why it is a handy way to describe radioactive decay and how fast the body clears many medicines.
How do I calculate how much is left after a certain time?
Divide the time by the half-life to get the number of half-lives, then multiply the starting amount by 1/2 to that power. For 100 g of carbon-14 (half-life about 5,730 years) after 11,460 years: 11,460 ÷ 5,730 = 2 half-lives, (1/2)² = 0.25, and 100 × 0.25 = 25 g left.
How do I find the half-life from two measurements?
Work out how many times the amount has halved, which is log₂(start ÷ amount left), then divide the time by that. If 1,000 counts fall to 125 in 15 hours, 1,000 ÷ 125 = 8 = 2³, so 3 half-lives passed and the half-life is 15 ÷ 3 = 5 hours.
What is the difference between half-life, decay constant and mean lifetime?
They are three ways to say the same thing. The decay constant λ = ln 2 ÷ half-life is the share that decays per unit of time (in the limit of a very short time). The mean lifetime τ = 1 ÷ λ = half-life ÷ ln 2 is the average time a single atom lasts, about 1.4427 times the half-life. For carbon-14, λ ≈ 0.000121 per year and τ ≈ 8,267 years.
Does anything ever reach zero?
In the formula, no: each half-life only halves what is left, so the amount gets smaller and smaller without reaching zero. In real life you count atoms, so eventually none are left. After 10 half-lives about 0.098% remains (1/1,024), which is often treated as practically gone.
Does this work for medicines too?
Yes, for drugs that the body clears at a steady rate in proportion to how much is there (first-order elimination), the same sum applies: after about 5 half-lives roughly 3% is left. Many drugs do not follow this exactly, and a doctor or pharmacist uses much more than one number, so treat this as a maths tool only, not medical advice.
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