Half-Life Calculator

The Decay Formula

Half-life is the time it takes for half of a quantity to decay or reduce, and it describes exponential decay — a process where the rate of decrease is proportional to the amount remaining. The core formula is N(t) = N₀ × (0.5)^(t/t½), where N₀ is the initial amount, t is the elapsed time, and t½ is the half-life. In words: divide the elapsed time by the half-life to get the number of half-lives that have passed, then multiply the starting amount by one-half raised to that power. An equivalent form used in physics and pharmacology is N(t) = N₀ × e^(−λt), where λ (the decay constant) equals ln(2)/t½ ≈ 0.693/t½ — the two forms give identical results, just expressed differently. A worked example: starting with 80 mg of a substance with a 6-hour half-life, after 6 hours you have 40 mg, after 12 hours 20 mg, after 18 hours 10 mg, and so on. The defining feature of exponential decay is that the same fraction is lost in each equal time interval — always half per half-life — regardless of how much you started with. This calculator lets you find the amount remaining after any time, or work out the half-life from measured decay.

After N Half-Lives

Tracking how much remains after a whole number of half-lives reveals how quickly exponential decay reduces a quantity. After 1 half-life, 50% remains; after 2, 25%; after 3, 12.5%; after 4, 6.25%; after 5, about 3.1%. By 7 half-lives only about 0.78% is left, and after 10 half-lives just 0.098% — less than a thousandth of the original. A useful rule of thumb in many fields (especially medicine) is that after about 5 half-lives, a substance is effectively gone (around 97% cleared), which is why drug dosing intervals and 'washout' periods are often set at roughly five half-lives. Mathematically, though, exponential decay never reaches exactly zero in finite time — it approaches zero asymptotically, halving forever. This is why radioactive waste with long half-lives remains measurably radioactive for enormous timescales, and why trace amounts of long-lived isotopes persist. The same pattern applies whether you're tracking radioactive atoms, drug concentration in the blood, the charge on a capacitor, or the cooling of a hot object — all follow the same exponential curve. Counting in half-lives is often easier than using the formula directly: just halve repeatedly for each half-life elapsed, and interpolate for partial half-lives.

Drug Half-Lives

Half-life is central to pharmacology because it determines how long a drug stays active and how often you need to take it. Caffeine is a familiar example, with a half-life of roughly 5–6 hours in a typical adult: a 200 mg dose at 2pm leaves about 100 mg by 8pm and 50 mg by 2am — which is exactly why an afternoon coffee can disrupt sleep even hours later, as a meaningful amount is still circulating at bedtime. Drugs vary enormously: aspirin has a short half-life of 2–3 hours so clears quickly, while some medications have half-lives of days, meaning they accumulate to a steady level over repeated doses and take a long time to clear after stopping. This is why dosing schedules differ — a drug with a 4-hour half-life might need taking several times a day to maintain effect, while one with a 24-hour half-life is taken once daily. Half-life also explains 'steady state': taking a drug regularly, levels build up over about 5 half-lives until intake balances clearance. Individual factors change half-life — liver and kidney function, age, genetics, and other medications can all lengthen or shorten it, which is why the same dose affects people differently. (This is general educational information, not medical advice — always follow the guidance of a doctor or pharmacist for any medication.)

Units and Significant Figures

Half-life calculations require consistent time units throughout — the elapsed time and the half-life must be in the same units (both hours, or both years, etc.) before you divide them, since mixing units is the most common error. If a half-life is given in days but you want the amount after a number of hours, convert one to match the other first. For the decay-constant form (using e and λ), ensure λ and t use consistent time units too, since λ = 0.693/t½ carries the units of the half-life. Significant figures should reflect the precision of your inputs: if the half-life is known to 2 significant figures, don't quote the remaining amount to 5. For radioactive decay and other scientific contexts, results often span many orders of magnitude, so scientific notation (e.g. 6.2 × 10⁻³ for the fraction remaining after 7 half-lives) is clearer and avoids ambiguity about significant figures. When working with very long or very short half-lives, watch the time units carefully — geological dating uses half-lives of billions of years, while some particle physics deals in fractions of a second. Always sanity-check: the amount remaining should always be less than the starting amount and should halve over each half-life period, which is a quick way to catch arithmetic slips.