Enter an initial amount, its half-life, and an elapsed time (in the same units) to calculate how much remains after exponential decay.
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Half-life describes the time it takes for a quantity to reduce to half its value through exponential decay. This calculator applies the formula: remaining amount = initial amount × 0.5^(elapsed time ÷ half-life), directly modeling how the substance halves repeatedly over each full half-life period.
Half-life is most commonly associated with radioactive decay (like carbon-14 dating, half-life about 5,730 years), but the same math applies to drug elimination from the body, and any other process that decays at a rate proportional to its current amount.
For an initial amount of 100 (grams, say), a half-life of 5,730 years (carbon-14), and 1,000 years elapsed: the number of half-lives elapsed is 1000 ÷ 5730 ≈ 0.1745. Remaining amount = 100 × 0.5^0.1745 ≈ 88.61, meaning about 88.61% of the original amount remains after 1,000 years, since not even one full half-life has passed yet.
Entering half-life and elapsed time in different units (like half-life in years but elapsed time in days), both must use the exact same time unit for the formula to give a correct result.
Assuming the substance is completely gone after exactly one half-life, it isn't, half of it remains after one half-life, a quarter after two, an eighth after three, the amount keeps approaching zero without ever fully reaching it mathematically.
Expecting a linear decay rate, half-life decay is exponential, not linear, so the amount lost in the first half-life period is much larger in absolute terms than the amount lost in a later equal-length period, even though each period always removes half of what remained.
Yes, both values must use the same time unit (both in years, both in hours, and so on) for the calculation to be correct.
The formula works for any elapsed time, the remaining amount just keeps approaching (but never quite reaching) zero.
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