Half-Life Calculator

The one formula behind every half-life question

Half-life is the time it takes for half of something to disappear. The something can be radioactive atoms, caffeine in your bloodstream, a dose of ibuprofen, or the concentration of a pollutant in a lake. As long as a fixed fraction vanishes per unit of time — what chemists call first-order decay — one equation covers all of them: N = N₀ x 0.5^(t/T), where N₀ is what you started with, T is the half-life, t is how long you have waited, and N is what is left.

Rearranged, the same relationship answers the two reverse questions. Time elapsed is t = T x log₂(N₀/N), and the half-life itself is T = t / log₂(N₀/N). The calculator above simply switches between the three forms, which is why it asks what you want to find before it asks for numbers.

Worked example: 200 mg of caffeine, 24 hours later

A large coffee delivers roughly 200 mg of caffeine, and caffeine's half-life in a healthy adult averages about 6 hours. Twenty-four hours is 24 / 6 = 4 half-lives. Halving four times gives 200 → 100 → 50 → 25 → 12.5 mg, so 12.5 mg is still circulating a full day later — 6.25% of the original dose. That residue is small, but it is why a 4 p.m. coffee can still shave measurable minutes off deep sleep, and it stacks with whatever you drink the next morning.

Run the same numbers backwards to see the other two modes. Enter 200 as the starting amount, 25 as the remaining amount and 6 hours as the half-life: the tool returns 18 hours, because 200 → 25 is exactly 3 half-lives. Swap the inputs — 200 down to 25 in 18 hours — and it returns a 6-hour half-life. Any two of the three unknowns pin down the third.

Why "five half-lives" is the number to remember

Decay never mathematically reaches zero, so pharmacology uses a practical threshold instead. After each half-life you have 50%, 25%, 12.5%, 6.25%, then 3.125% remaining. Five half-lives clear about 97% of a dose, which is close enough that most drugs stop producing measurable effects, and it is the standard rule for how long a steady-state drug level takes to build up when you start a new prescription as well as how long it takes to wash out when you stop. Seven half-lives take you past 99%, the figure used when a drug must be essentially undetectable before surgery or a switch to an interacting medicine.

The same rule maps onto radiation safety. Iodine-131, used to treat thyroid conditions, has an 8-day half-life, so a patient's activity is down by about 97% after 40 days. Technetium-99m, the workhorse of nuclear imaging, has a 6-hour half-life, which is exactly why it can be injected in the morning and be almost gone by bedtime.

Where else this shows up

Outside chemistry, the same maths describes capacitor discharge, the cooling of a hot object toward room temperature, and the depreciation of an asset losing a fixed percentage of value each year. If a process removes a constant proportion rather than a constant quantity, it has a half-life.

The limits worth knowing

First, this model only fits first-order processes. Alcohol is the famous exception: the liver clears it at a roughly constant rate — around one standard drink per hour, regardless of how much you drank — which is zero-order kinetics and does not have a half-life at all. High doses of aspirin and phenytoin behave the same way once the clearing enzymes saturate. Feeding those into a half-life calculator gives an answer that is confidently wrong.

Second, published drug half-lives are population averages. Liver and kidney function, age, genetics, pregnancy, smoking and interacting medicines routinely move a drug's half-life by a factor of two in either direction, so treat the output as a ballpark and never as dosing advice — that conversation belongs with a pharmacist or doctor. Radioactive half-lives, by contrast, are fixed constants of nature, unmoved by heat, pressure or chemistry, which is precisely what makes them usable as clocks.

Third, keep your units consistent. The calculator uses one time unit for both the half-life and the elapsed time, because mixing hours and days is by far the most common way to get a plausible-looking but meaningless answer.