How to use the Half-Life Calculator
- Enter the initial amount — grams, becquerels, milligrams of drug, any unit you like.
- Enter the half-life of the substance.
- Enter the elapsed time using the same time unit as the half-life.
- Read the amount remaining and the percentage of the original still present.
- Use the decay constant and mean lifetime when a formula calls for λ or τ rather than t½.
How the calculation works
Exponential decay describes any process where the rate of loss is proportional to the amount present. Radioactive nuclei are the canonical case — each nucleus has a fixed probability of decaying per unit time, independent of age or of what its neighbours are doing — but the same maths governs first-order drug elimination, capacitor discharge and the cooling of a body toward ambient.
The half-life is simply a convenient restatement of the decay constant: t½ = ln2 ÷ λ. Half-lives are easier to reason about because the arithmetic is doubling in reverse. After five half-lives about 3% remains; after seven, under 1%. This is why clinical pharmacology treats five half-lives as effectively complete elimination and as the time to reach steady state on repeated dosing.
The mean lifetime τ = 1/λ is longer than the half-life by a factor of about 1.44, because the distribution has a long tail. Both appear in the literature and they are not interchangeable; substituting one for the other misstates the timescale by 44%. Carbon dating uses the 5,730-year half-life of carbon-14 in this framework, with calibration curves correcting for historical variation in atmospheric C-14.
N(t) = N₀ × (½)^(t ÷ t½) = N₀e^(−λt), with λ = ln2 ÷ t½ and mean lifetime τ = 1 ÷ λSource: IAEA Live Chart of Nuclides; Rutherford & Soddy, 'The Cause and Nature of Radioactivity' (1902); Goodman & Gilman's The Pharmacological Basis of Therapeutics on first-order elimination.
Worked example
A 400 mg dose of a drug with a 6-hour elimination half-life. How much remains after 24 hours?
- Half-lives elapsed: 24 ÷ 6 = 4.
- Remaining fraction: (½)⁴ = 0.0625.
- Amount: 400 × 0.0625 = 25 mg.
- Decay constant: ln2 ÷ 6 = 0.1155 per hour; mean lifetime 8.66 hours.
25 mg remains after 24 hours — 6.25% of the dose — and it would take about 30 hours (five half-lives) to fall below 3%.
Frequently asked questions
Does anything ever fully decay?+
Mathematically no — the curve approaches zero asymptotically. Physically, once the count reaches individual atoms the process is discrete and does finish.
What is the difference between half-life and mean lifetime?+
Mean lifetime is 1/λ and is about 1.44 times the half-life. Half-life is the median; mean lifetime is the average.
Can I use this for drug dosing?+
It models first-order elimination correctly, which covers most drugs at therapeutic doses. Never use it to make clinical decisions without professional advice.
Why is five half-lives the rule of thumb?+
After five, 3.1% remains; after seven, 0.8%. Five is the usual point where the residue stops being clinically or practically relevant.
Last reviewed September 1, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.