Half-life is the single number that captures how fast something decays — whether it is a radioactive isotope, a drug in your bloodstream, or any process that loses a fixed fraction of itself in a fixed time. This calculator solves the exponential decay law for any unknown and translates the answer into the decay constant, mean lifetime, and percent remaining. This guide explains how those quantities relate, how radiocarbon dating works, and what 'five half-lives' means for medicines.
How exponential decay works
Radioactive decay and first-order drug elimination both follow the same rule: in each fixed interval, a constant fraction of what remains disappears. After one half-life, 50% is left; after two, 25%; after three, 12.5%; and so on — never quite reaching zero. The governing equation is N = N₀ · (½)^(t/t½). Crucially, the half-life does not depend on the starting amount: a gram of cobalt-60 and a kilogram of it both lose half their atoms in 5.27 years. Because the same fraction is lost each interval, plotting the amount against time gives the characteristic exponential curve you see on the Decay Curve tab, and plotting it on a logarithmic axis gives a straight line whose slope is the decay constant.
Half-life, decay constant, and mean lifetime
Three numbers describe the same decay. The half-life t½ is the most intuitive — the time for half the sample to go. The decay constant λ = ln(2)/t½ is the probability per unit time that any single atom decays, and it is what appears in the fundamental rate law dN/dt = −λN. The mean lifetime τ = 1/λ = t½/ln(2) is the average time an individual atom actually survives; it is always about 1.443 times the half-life, and after one mean lifetime exactly 1/e ≈ 36.8% of the sample remains. The activity of a source — its decays per second — is A = λN, so a short half-life means high activity for the same number of atoms. This calculator reports all three so you can move between the physicist's λ, the clinician's t½, and the statistician's τ without re-deriving them.
Radiocarbon dating and geological clocks
Solving the decay law for time turns half-life into a clock. Living things absorb carbon-14 from the atmosphere; when they die, the C-14 (half-life 5,730 years) decays without replacement, so the fraction remaining reveals the age. A measured 25% means two half-lives, or ~11,460 years. Longer-lived isotopes date deeper time: potassium-40 (1.25 billion years) and uranium-238 (4.468 billion years) are used to date rocks and meteorites, and U-238 dating set the age of the Earth at about 4.54 billion years. Each method has a useful range — roughly one-tenth to ten half-lives — beyond which too little or too much remains to measure precisely, which is why no single isotope can date everything.
Drug half-life and the five-half-life rule
Most drugs leave the body by first-order kinetics, so the same math applies. Two practical rules of thumb fall out of it. First, a single dose is about 97% gone after five half-lives (3.125% remaining) and effectively gone — under 1% — after about seven. Second, on a regular dosing schedule, blood levels build up and plateau at steady state after roughly five half-lives, because that is how long it takes the eliminated amount to catch up with the dose being added. A drug with a 5-hour half-life like caffeine reaches steady state in about a day; one with a 4-day half-life like fluoxetine takes weeks. The Drug Half-Life tab estimates both clearance and steady-state timing, but real pharmacokinetics vary with age, organ function, and genetics, and some substances (notably alcohol and phenytoin) do not follow first-order decay at all — so treat these figures as educational, not medical, guidance.
Edge cases and limits
A few situations break the simple model. The remaining amount can never exceed the initial amount, so if you ask the calculator to solve for time or half-life with N greater than N₀, it flags the inputs as invalid — decay only removes material. A half-life of zero or below is physically meaningless and is rejected. The model also assumes a single, pure decay process: it does not handle decay chains where a daughter isotope is itself radioactive (secular equilibrium), mixtures of isotopes with different half-lives, or biological half-lives that combine metabolism and excretion. For those, the simple law gives a useful first approximation but a specialist model is needed for precise work.