OneTaskBy Tim King · Updated
Half-Life Calculator
Radioactive Decay Time, Amount Left and Carbon Dating With Common Isotopes
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How the Half-Life Calculator works
This half-life calculator works out how much of a radioactive substance is left after a given time, how long it takes to decay to an amount, the half-life itself, or how much you started with. Pick an isotope such as carbon-14 or iodine-131 to load its half-life, or choose Custom and enter your own for drugs, caffeine or any process with exponential decay.
A half-life is the time it takes for half of a sample to decay. After one half-life 50% remains, after two 25%, after three 12.5%, and so on. The pattern is the same whether the half-life is microseconds or billions of years, which is why the same maths powers carbon dating, medical imaging and drug dosing.
How the half-life calculator works
The formula is N = N₀ × (½)^(t ÷ T½), where N₀ is the starting amount, t the time elapsed and T½ the half-life. Rearranged, the time to fall to N is t = T½ × log₂(N₀ ÷ N), and the half-life from two measurements is T½ = t ÷ log₂(N₀ ÷ N). Amounts can be grams, percent, becquerels or counts, as long as both use the same unit.
First example. 100 units of carbon-14, half-life 5,730 years, left for 10,000 years: 10,000 ÷ 5,730 = 1.745 half-lives, and 100 × 0.5^1.745 = 29.83 units remain.
Second example. Iodine-131 has a half-life of 8.0252 days. To fall from 100 to 10 takes 8.0252 × log₂(10) = 26.66 days, about 3.32 half-lives.
How many half-lives until less than 1% is left? Seven: 0.5^7 = 0.78%, while six still leaves 1.56%.
Amount remaining after each number of half-lives
| Half-lives elapsed | Fraction left | Percent left | Carbon-14 age (years) |
|---|---|---|---|
| 0 | 1/1 | 100% | 0 |
| 0.5 | 0.7071 | 70.71% | 2,865 |
| 1 | 1/2 | 50% | 5,730 |
| 1.5 | 0.3536 | 35.36% | 8,595 |
| 2 | 1/4 | 25% | 11,460 |
| 2.5 | 0.1768 | 17.68% | 14,325 |
| 3 | 1/8 | 12.5% | 17,190 |
| 3.5 | 0.08839 | 8.839% | 20,055 |
| 4 | 1/16 | 6.25% | 22,920 |
| 4.5 | 0.04419 | 4.419% | 25,785 |
| 5 | 1/32 | 3.125% | 28,650 |
| 5.5 | 0.0221 | 2.21% | 31,515 |
| 6 | 1/64 | 1.563% | 34,380 |
| 6.5 | 0.01105 | 1.105% | 37,245 |
| 7 | 1/128 | 0.7813% | 40,110 |
| 7.5 | 0.005524 | 0.5524% | 42,975 |
| 8 | 1/256 | 0.3906% | 45,840 |
| 8.5 | 0.002762 | 0.2762% | 48,705 |
| 9 | 1/512 | 0.1953% | 51,570 |
| 9.5 | 0.001381 | 0.1381% | 54,435 |
| 10 | 1/1024 | 0.09766% | 57,300 |
| 10.5 | 0.000691 | 0.06905% | 60,165 |
| 11 | 1/2048 | 0.04883% | 63,030 |
| 11.5 | 0.000345 | 0.03453% | 65,895 |
| 12 | 1/4096 | 0.02441% | 68,760 |
Tips and pitfalls
- Match the time units. A half-life in years and a time in days gives nonsense; pick one time unit for both, and the built-in isotopes are converted for you.
- Decay never reaches zero. Mathematically something always remains, but after about ten half-lives less than 0.1% is left, which is often treated as gone.
- Carbon dating limits. Beyond about 50,000 years too little carbon-14 remains to measure.
- Activity decays too. Becquerels or counts per minute fall at the same rate as the amount.
FAQ
How do you calculate half-life?
Measure the amount at the start and after some time t, then use T½ = t ÷ log₂(N₀ ÷ N). If a sample drops from 80 to 20 counts in 12 hours, that is two halvings, so the half-life is 6 hours. The calculator does this in the Half-life mode.
How much is left after three half-lives?
One eighth, or 12.5%, of the original amount. Each half-life halves what is left: 100% to 50%, then 25%, then 12.5%. After four half-lives 6.25% remains and after ten about 0.1%, roughly a thousandth of the starting amount.
How does carbon dating use half-life?
Living things keep a steady level of carbon-14. When they die it decays with a 5,730 year half-life. Measuring the fraction left gives the age: a bone with 25% of the living level is two half-lives old, about 11,460 years.
Can I use this for drugs or caffeine?
Yes. Many drugs leave the body with roughly exponential decay. Choose Custom, enter the half-life, such as about five hours for caffeine in a healthy adult, and the calculator shows how much remains after any time. Real clearance varies from person to person.
What is the difference between half-life and decay constant?
The decay constant λ is the fraction decaying per unit time, and the two are linked by T½ = ln 2 ÷ λ, about 0.693 ÷ λ. The calculator shows the decay constant and the mean lifetime alongside each answer.
See also: all Chemistry tools, Physics, Electronics.
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