Half-Life Calculator
Exponential decay solved for remaining quantity, half-life or elapsed time, with decay constant, mean lifetime, a decay curve and a table of isotope half-lives.
Half-Life Calculator: with the default inputs, quantity remaining is 50.
Any unit — grams, atoms, becquerels, or 100 for percent.
Not used when solving for half-life.
Not used when solving for elapsed time.
Same unit as the initial quantity. Used when solving for half-life or elapsed time.
- Fraction remaining
- 50%
- Quantity decayed
- 50
- Half-lives elapsed
- 1
- Half-life
- 5,730 year
- Elapsed time
- 5,730 year
- Decay constant λ
- 3.833 × 10⁻¹² s⁻¹ = 3.312 × 10⁻⁷ d⁻¹ = 1.210 × 10⁻⁴ y⁻¹λ = ln 2 ÷ half-life; the fraction decaying per unit time when that fraction is small.
- Mean lifetime τ
- 8,266.6426 yearτ = 1 ÷ λ = half-life ÷ ln 2 — the average time a nucleus survives.
Assumptions
- Single-step first-order decay; daughter products are not tracked.
- The half-life is constant over the interval (true for nuclear decay).
- A 'year' is 365.2425 days.
| Isotope | Half-life | Where you meet it |
|---|---|---|
| Technetium-99m | 6.01 h | Medical imaging |
| Radon-222 | 3.82 d | Indoor air hazard |
| Iodine-131 | 8.02 d | Thyroid therapy |
| Cobalt-60 | 5.27 y | Radiotherapy, sterilization |
| Tritium (H-3) | 12.32 y | Luminous signs, fusion fuel |
| Strontium-90 | 28.8 y | Fission product |
| Cesium-137 | 30.08 y | Fission product, calibration |
| Carbon-14 | 5,730 y | Radiocarbon dating |
| Plutonium-239 | 24,110 y | Nuclear fuel |
| Uranium-235 | 7.04 × 10⁸ y | Fissile fuel |
| Potassium-40 | 1.248 × 10⁹ y | K-Ar dating |
| Uranium-238 | 4.468 × 10⁹ y | U-Pb dating |
Values from the IAEA Live Chart of Nuclides / NNDC NuDat. Carbon-14 is listed at the 5,730 y Cambridge half-life used in radiocarbon dating; the current nuclear-data evaluation is 5,700 ± 30 y.
How this is worked out
The formula
N(t) = N₀ × (½)^(t / T½) = N₀ × e^(−λt) λ = ln 2 ÷ T½ (decay constant) τ = 1 ÷ λ = T½ ÷ ln 2 (mean lifetime) Rearranged: T½ = t × ln 2 ÷ ln(N₀ ÷ N) t = T½ × log₂(N₀ ÷ N) N₀ = initial quantity, N = remaining quantity, t = elapsed time
Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- Solve for
- Choose one of 3 options.Quantity remaining — after an elapsed time · Half-life — from initial & remaining amounts · Elapsed time — to reach a remaining amount
- Initial quantity
- Any unit — grams, atoms, becquerels, or 100 for percent.0 or more · defaults to 100
- Half-life
- Not used when solving for half-life.in year · 0 or more · defaults to 5730
- Elapsed time
- Not used when solving for elapsed time.in year · 0 or more · defaults to 5730
- Remaining quantity
- Same unit as the initial quantity. Used when solving for half-life or elapsed time.0 or more · defaults to 25
What you get back
- Quantity remainingmain answer
- Fraction remaining
- Quantity decayed
- Half-lives elapsed
- Half-life
- Elapsed time
- Decay constant λ
- λ = ln 2 ÷ half-life; the fraction decaying per unit time when that fraction is small.
- Mean lifetime τ
- τ = 1 ÷ λ = half-life ÷ ln 2 — the average time a nucleus survives.
What this assumes
- Single-step first-order decay; daughter products are not tracked.
- The half-life is constant over the interval (true for nuclear decay).
- A 'year' is 365.2425 days.
About this calculator
Radioactive decay — and a lot of other things, from drug elimination to charge leaking off a capacitor — follows the same rule: a fixed fraction disappears in each equal interval of time. The half-life T½ is the interval in which half goes. After two half-lives a quarter remains, after three an eighth, after ten about a thousandth. This calculator works the relationship in whichever direction you need: how much is left after a given time, how long until only a given amount is left, or what the half-life must be given two measurements.
How to use it
Quantities can be in any unit as long as both use the same one — grams, atoms, counts per minute, or 100 for "percent". Time units are independent for the half-life and the elapsed time, so you can enter a half-life in days and an elapsed time in hours. The reference table lists common isotopes; the Solve for tool can also find, say, the initial quantity that leaves a target amount.
Reading the results
- Half-lives elapsed is the number to keep in your head: fraction remaining = (½) to that power.
- Decay constant λ is the probability per unit time that any one nucleus decays. Activity (decays per second) is λN, so a short half-life means a hot source that burns out fast.
- Mean lifetime τ = T½ ÷ ln 2 ≈ 1.44 T½ is the average survival time — longer than the half-life because a few nuclei last a very long time.
Common mistakes
- Assuming decay is linear. Two half-lives leave 25%, not 0%.
- Mixing time units between half-life and elapsed time when working by hand.
- Expecting the sample to reach zero. Mathematically it never does; practically, after 10 half-lives it is below 0.1% and after 20 below a millionth.
- Treating the "remaining amount" as the parent isotope's mass when the daughter is also radioactive — decay chains need a more elaborate model (Bateman equations).
Where the model breaks
Nuclear half-lives are astonishingly constant — temperature, pressure and chemistry do not change them (with exotic exceptions such as electron-capture isotopes in fully ionized atoms). Radiocarbon dating, however, also depends on the atmospheric ¹⁴C level at the time of death, which has varied; real dates are calibrated against tree rings, so a raw "radiocarbon age" from this formula is only the first step.
Frequently asked questions
▸What is the half-life formula?
N = N₀ × (½)^(t/T½): the remaining quantity equals the starting quantity times one-half raised to the number of half-lives elapsed. Equivalently N = N₀e^(−λt) with λ = ln 2 ÷ T½.
▸How do I calculate how much is left after a certain time?
Divide the elapsed time by the half-life to get the number of half-lives, then multiply the starting amount by ½ that many times. 100 g of I-131 (8.02 d) after 24 d is three half-lives: 100 × ½ × ½ × ½ = 12.5 g.
▸What is the difference between half-life and mean lifetime?
The half-life is when half the sample has decayed; the mean lifetime (τ = T½ ÷ 0.693) is the average time a single nucleus survives. τ is always about 44% longer than T½.
▸How does carbon dating use half-life?
Living things hold a steady ratio of ¹⁴C to ¹²C; after death the ¹⁴C decays with a 5,730-year half-life. Measuring the remaining fraction gives the elapsed time — the 'time' mode here — which is then calibrated against known atmospheric variations.
▸Does half-life apply to things other than radioactivity?
Yes — any first-order process: drug concentration in the blood (caffeine ≈ 5 h), foam settling on a beer, a capacitor discharging through a resistor. Same math, different constants.
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