Free Half-Life Calculator – Solve Any of the Four Decay Variables

What Is Half-Life and How This Calculator Solves It

A half-life is the time it takes for half of a decaying quantity to disappear, whether that quantity is a radioactive isotope, a dose of medicine in the bloodstream, or any process that loses a fixed fraction of itself in a fixed interval. Physicists, chemistry students, nuclear-medicine technicians, geologists dating rock samples, and pharmacology students all reach for the same underlying law even though the substance and the timescale are wildly different — five hours for a diagnostic isotope, five thousand years for radiocarbon, or four and a half billion years for uranium-238. This calculator handles the same law from any of four starting points, so you can plug in whichever three numbers you already know and solve for the fourth, rather than being locked into one direction like most calculators online. The mechanism is the exponential decay law N = N₀ × (1/2)^(t/t½), where N₀ is the amount you start with, N is what remains after time t, and t½ is the half-life. Two related quantities fall out of the same law: the decay constant λ = ln2 / t½, which is the instantaneous fractional decay rate, and the mean lifetime τ = 1/λ = t½/ln2, which is the average time a single atom or molecule survives before decaying — about 44% longer than the half-life itself, since τ/t½ = 1/ln2 ≈ 1.4427. Multiplying λ by the remaining amount N gives the activity A = λN, the number of decays per second, measured in becquerels (Bq) when time is in seconds. This tool computes N using Math.pow(0.5, t/t½) rather than the mathematically equivalent Math.exp(−λt), because the power form lands on an exact result at whole numbers of half-lives — three half-lives gives exactly 0.125, not 0.12499999999999998 — which matters when you are trying to sanity-check a textbook answer. Here is a concrete walk-through using carbon-14, whose half-life is 5,730 years by the value NOAA's education program and NIST's dating literature both use. If a sample has decayed to 25% of its original carbon-14, that is two half-lives (0.5² = 0.25), so the elapsed time is 2 × 5,730 = 11,460 years — matching NOAA's own published statement that a quarter of the original carbon-14 remains at 11,460 years. Run the same logic on a simpler case: 100 grams with a 10-year half-life, after 30 years (three half-lives), leaves exactly 12.5 grams, because 100 × 0.5³ = 12.5. The calculator shows this same substituted arithmetic underneath your own numbers, in the same “t½ in seconds, ratio, then N” order, so you can check every step rather than trust a black box. Outside a physics classroom, this law shows up constantly. A hospital pharmacist times a technetium-99m dose (6.01-hour half-life) so the scan happens while enough activity remains to image clearly but the patient's radiation exposure drops off quickly afterward. A geologist estimating the age of a uranium-bearing rock uses U-238's 4.468-billion-year half-life to read a decay ratio as an age. An archaeologist dating a bone fragment uses the carbon-14 fraction remaining to place death within a few centuries. A pharmacology student modeling how a drug clears the bloodstream uses the identical N = N₀(1/2)^(t/t½) form, because first-order elimination kinetics is mathematically the same exponential law, just with a biological half-life instead of a nuclear one. One genuine limitation worth knowing: this law describes a single decay step with one constant decay constant, and does not model a decay chain — uranium-238 does not stop at one transformation, it runs through more than a dozen daughter isotopes before reaching stable lead-206, and modeling that requires secular-equilibrium equations this tool does not attempt. It also does not compute a drug's effective half-life, which nuclear-medicine dosing combines from both physical decay and the body's own biological clearance using 1/t_eff = 1/t_physical + 1/t_biological. A common trap when solving for elapsed time or half-life is entering a remaining amount N that is equal to or larger than the initial amount N₀ — since decay only removes material, that combination has no valid solution, and the calculator names the problem rather than returning a confusing negative or infinite result. Every calculation here runs locally in your browser; nothing you type is uploaded, logged, or stored anywhere.

Half-Life Calculator Formula & Method

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