Half-Life Calculator
How much of a decaying quantity remains after a given time, or the half-life implied by a measured drop.
Last updated
Your details
Use the same time unit throughout — years, days, seconds.
Result
- Amount remaining
- 25
- Amount decayed
- 75
- Half-lives elapsed
- 2
- Decay constant λ
- 0.00012097
25% of the original
Mean lifetime 8,266.6426
100 after 11460…
N = N₀ × (½)^(t ÷ t½)
t ÷ t½ = 11460 ÷ 5730 = 2
(½)^2 = 0.25
100 × 0.25
N = 25
Formula — written step by step
Carbon-14 after two half-lives…
N = N₀ × (½)^(t ÷ t½)
t ÷ t½ = 11,460 ÷ 5,730 = 2
(½)² = 0.25
100 × 0.25
N = 25
N₀ is the starting amount, t the time elapsed and t½ the half-life. Each half-life removes half of whatever remains, so two leave a quarter and three an eighth — never zero, however long you wait.
A worked example
Dating an organic sample
- Carbon-14 remaining
- 35% of the living level
- Half-life of carbon-14
- 5,730 years
- N ÷ N₀ = 0.35
- Half-lives = ln(0.35) ÷ ln(0.5)
- = −1.049822 ÷ −0.693147
- = 1.5146 half-lives
About 8,680 years old
Radiocarbon dating works because living things exchange carbon with the atmosphere and stop at death, so the clock starts then. Beyond about ten half-lives — roughly 60,000 years — too little carbon-14 remains to measure reliably, which sets the practical limit of the method.
How to use this calculator
- Choose whether you know the half-life and want the amount left, or know the drop and want the half-life.
- Use the same time unit for the half-life and the elapsed time — years, days or seconds, but consistently.
- The starting amount can be in any unit: grams, becquerels, a percentage, or a count. Only the ratio matters.
- The table shows the decay over successive half-lives, which makes the shape of the curve easier to read than a single figure.
What a half-life is
A half-life is the time for half of a decaying quantity to disappear. It is a property of the substance, not of how much you have: a gram of carbon-14 and a tonne of it both halve in 5,730 years.
The decay is exponential, so the quantity is always halving and never quite reaches zero. After ten half-lives less than a thousandth remains, which is where most practical work treats the process as finished.
Half-life and the decay constant
Decay is often written as N = N₀e^(−λt), where λ is the decay constant — the probability per unit time that any given atom decays. The two descriptions are equivalent: λ = ln 2 ÷ t½.
The mean lifetime, 1 ÷ λ, is the average time an individual atom survives, and is about 1.44 half-lives. It is longer than the half-life because the distribution has a long tail — most atoms decay relatively early, a few last a very long time.
Beyond radioactivity
The same mathematics governs any process where the rate of loss is proportional to the amount present. Drug clearance from the bloodstream is described by a half-life, which is what sets dosing intervals; caffeine’s is around five hours, so a late-afternoon coffee still has a quarter of its dose present at midnight.
Capacitor discharge, atmospheric CO₂ absorption and the cooling of a hot object all follow the same exponential form. Only the constant changes, which means the intuitions built here transfer directly.
Formula & methodology
What this calculator assumes
- Assumes a first-order process, where the rate of loss is proportional to the amount present. Radioactive decay, drug clearance and capacitor discharge all behave this way; saturable processes do not.
- Treats the quantity as continuous. For a small number of atoms, decay is stochastic and the smooth curve is only an expectation.
- The half-life must be constant. It is for radioisotopes; it is not for drug clearance in someone whose metabolism changes.
Sources
- Krane, Introductory Nuclear Physics, Wiley, 1987 — radioactive decay law.
What each half-life leaves
| Half-lives | Fraction left | Percentage |
|---|---|---|
| 0 | 1 | 100% |
| 1 | 1/2 | 50% |
| 2 | 1/4 | 25% |
| 3 | 1/8 | 12.5% |
| 5 | 1/32 | 3.125% |
| 10 | 1/1024 | 0.098% |
| 20 | 1/1,048,576 | 0.0001% |
Ten half-lives leave under a thousandth of the original, which is the rule of thumb usually used for when a radioactive source can be treated as spent.
Frequently asked questions
What time unit should I use?
Any, provided the half-life and the elapsed time use the same one. Enter both in years, or both in seconds — the result comes back in that unit.
Does the starting amount matter?
Only for the absolute figure returned. The fraction remaining depends solely on how many half-lives have passed, which is why the method identifies age regardless of sample size.
What is the decay constant?
The probability per unit time that a given atom decays, equal to ln 2 divided by the half-life. It appears in the exponential form of the decay law.
Why does the amount never reach zero?
Because each half-life removes half of what is left, not a fixed quantity. Mathematically the curve approaches zero without arriving; physically you eventually run out of atoms.
Decay over successive half-lives
| Half-lives | Time | Remaining | Share left |
|---|---|---|---|
| 0 | 0 | 100 | 100% |
| 1 | 5,730 | 50 | 50% |
| 2 | 11,460 | 25 | 25% |
| 3 | 17,190 | 12.5 | 12.5% |
| 4 | 22,920 | 6.25 | 6.25% |
| 5 | 28,650 | 3.125 | 3.125% |