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Half-Life Calculator

Fill in any three of what remains, what you started with, time passed and half-life - the fourth solves itself, with steps. Includes a half-life ⇄ mean lifetime ⇄ decay constant converter.

Find any missing value

Provide any three of the following to calculate the fourth. Answers update as you type; the solved field is highlighted.

- fill any three
- percent remaining
- half-lives elapsed

Half-life, mean lifetime & decay constant converter

Provide any one of the following to get the other two.

Legend

Nt:
How much of the substance is left after time t, in any unit (grams, atoms, becquerels).
N0:
How much you began with, in the same unit as Nₜ.
t:
How long the substance has been decaying, in any time unit.
t1/2:
How long it takes for half of the substance to decay, in the same time unit as t.
τ:
The average time a single atom survives before decaying; always a bit longer than the half-life (τ = t½ ÷ ln 2).
λ:
The fraction of atoms decaying per unit time; larger λ means faster decay (λ = ln 2 ÷ t½).

Runs entirely on your device - nothing is ever uploaded.

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How to use the half-life calculator

  1. Enter any three of the four values: what remains (Nₜ), what you started with (N₀), time passed (t) and half-life (t½).
  2. Leave the value you want blank. It solves itself and is highlighted; amounts can use any unit, and both times must share a unit.
  3. Read the answer plus what percent remains and how many half-lives elapsed. Fill all four to check an existing set instead.
  4. For the converter, enter any one of half-life, mean lifetime or decay constant - the other two fill in.

The half-life formula

After each half-life, half of what was there decays, so the remainder follows Nₜ = N₀ × (1/2)^(t ÷ t½). The same decay can be written with the mean lifetime τ or the decay constant λ: Nₜ = N₀ × e^(−t ÷ τ) = N₀ × e^(−λt), where τ = t½ ÷ ln 2 and λ = ln 2 ÷ t½.

Worked examples

100 g with a 5-day half-life after 12 days: 12 ÷ 5 = 2.4 half-lives, so 100 × (1/2)^2.4 ≈ 18.946 g remains, about 18.946% of the start.

Going the other way: 10 g left from 100 g after 50 days. The half-life is t½ = 50 × ln 2 ÷ ln(100 ÷ 10) ≈ 15.051 days.

Carbon-14 dating

Carbon-14, with a half-life of about 5,730 years, is the best-known use of half-life math: living things keep a steady level of it, and it decays once they die, so the fraction left dates the sample. A fossil holding 25% of the carbon-14 of a living sample has been through two half-lives - about 11,460 years old.

Half-life questions, answered

What exactly is a half-life? The time it takes for half of a quantity to decay. After one half-life 50% remains, after two 25%, after three 12.5%, and so on.

How do I find a half-life from two measurements? Divide the earlier amount by the later one, take the natural log, and divide the elapsed time × ln 2 by it. Enter your two amounts and the time above with the half-life blank to get it.

What is the difference between half-life and mean lifetime? Both describe the same decay speed in different units: the mean lifetime τ is the average survival time of one atom and is always longer (τ = t½ ÷ 0.693). The converter above switches between them.

Does half-life apply outside radioactivity? Yes - anywhere a quantity shrinks by a fixed fraction per unit time: drug levels in the body, caffeine, capacitor discharge and depreciation all follow the same curve.

What it is for

Use it for radioactive decay and carbon dating, medicine and pharmacology half-lives, lab isotopes, and any exponential decay - anywhere you know three of amount, time and rate and need the fourth. For mixing stock solutions instead, use the Dilution Calculator.