About the Half-Life Calculator
This half-life calculator models exponential decay with N(t) = N₀ × (½)^(t / T½). Pick what you want to find — the amount remaining after a given time, the time it takes to decay to a given amount, or the half-life itself from two measurements — and the calculator solves the equation and draws the decay curve.
It works for anything that decays at a rate proportional to how much is left: radioactive isotopes such as carbon-14 or iodine-131, the elimination of many medicines from the bloodstream, and first-order chemical reactions. Students use it for nuclear physics and chemistry problems, and it is useful for understanding how long a drug stays in your system.
Quantities can be in any unit (grams, becquerels, mg/L, number of atoms) as long as the initial and remaining amounts use the same one, and the time and half-life share the time unit you select. The default uses the conventional 5,730-year half-life of carbon-14 found in most textbooks; the latest evaluated value from the Laboratoire National Henri Becquerel (DDEP) is 5,700 ± 30 years, and iodine-131 is 8.0233 days. Drug half-lives vary between people and with age, liver and kidney function, so the calculator is an educational model, not dosing advice.
How to use the half-life calculator
- 1Choose whether to find the remaining amount, the elapsed time or the half-life.
- 2Enter the initial amount N₀ in any unit.
- 3Enter the two other known values (remaining amount, time or half-life).
- 4Select the time unit used for both time and half-life.
- 5Read the answer, decay constant and the decay curve.
Formula and method
In exponential (first-order) decay a fixed fraction of the remaining material disappears in each equal time interval. After one half-life half remains, after two a quarter, after three an eighth, and so on — so the remaining amount is N₀ multiplied by ½ raised to the number of half-lives elapsed.
Taking logarithms solves for time or half-life: t = T½ × log₂(N₀/N) and T½ = t ÷ log₂(N₀/N). The same process is often written N = N₀e^(−λt), where the decay constant λ = ln 2 ÷ T½ and the mean lifetime τ = 1 ÷ λ ≈ 1.4427 × T½.
- N₀
- Initial quantity
- N
- Quantity remaining after time t
- t
- Time elapsed
- T½
- Half-life
- λ
- Decay constant (ln 2 ÷ T½)
- τ
- Mean lifetime (1 ÷ λ)
Worked examples
Carbon-14 left after 10,000 years
With a 5,730-year half-life, 10,000 years is 1.745 half-lives. 100 × 0.5^1.745 ≈ 29.8, so about 29.8% of the original carbon-14 remains.
How long until iodine-131 falls to 10%
Falling to 10% takes log₂(10) = 3.32 half-lives. At 8.02 days each, that is about 26.6 days.
Half-life from two measurements
Dropping from 80 to 20 is a factor of 4 — exactly two half-lives. Two half-lives in 12 hours means T½ = 6 hours, and the mean lifetime is 6 ÷ ln 2 ≈ 8.66 hours.
Frequently asked questions
What is half-life?+
Half-life is the time it takes for half of a quantity to decay or be eliminated. It is constant for a first-order process, so a sample loses half its remaining amount in every half-life regardless of how much you start with.
What is the half-life formula?+
N(t) = N₀ × (1/2)^(t/T½), where N₀ is the starting amount, t the elapsed time and T½ the half-life. Equivalently N = N₀e^(−λt) with λ = ln 2 ÷ T½.
How many half-lives until something is gone?+
Exponential decay never reaches exactly zero, but after 5 half-lives about 3.1% remains and after 10 half-lives less than 0.1% remains. Pharmacology often treats a drug as effectively cleared after 4–5 half-lives.
How is carbon dating done with half-life?+
Living things keep a steady ratio of carbon-14. After death it decays with a half-life of about 5,730 years (the current evaluated value is 5,700 ± 30 years), so measuring the fraction left and using t = T½ × log₂(N₀/N) estimates the age. Published radiocarbon ages use the older Libby half-life of 5,568 years by convention and are then calibrated.
What is the decay constant?+
The decay constant λ is the fraction of the substance that decays per unit time in the continuous model. It equals ln 2 divided by the half-life, so a shorter half-life means a larger λ.
This tool gives general estimates and is not medical advice. Talk to a doctor or qualified professional about your health, diet or training.