Half-Life Calculator

The half-life calculator works out how much of a decaying quantity is left after a given amount of time. You enter three things — the initial quantity, the half-life period, and the elapsed time — and it returns the remaining amount, the amount that has decayed, the percentage still present, and how many half-lives have passed. Half-life is the time required for a quantity to fall to exactly half its starting value, and it is a cornerstone idea in nuclear physics (radioactive decay), pharmacology (how long a drug stays in the body), chemistry (first-order reaction kinetics), and archaeology (radiocarbon dating). It is useful for students checking physics or chemistry homework, medical and pharmacy learners reasoning about drug clearance, and anyone curious about how dating methods or nuclear medicine work.

The formula

N(t) = N₀ × (1/2)^(t / t½)

Here N₀ is the initial quantity, N(t) is the quantity remaining after time t, and t½ is the half-life. The exponent t / t½ is simply the number of half-lives that have elapsed — and that is exactly the figure this calculator reports alongside the remaining amount. The half-life and the elapsed time must be expressed in the same time unit; the quantity unit itself can be anything consistent — grams, moles, atoms, becquerels or millicuries.

The same physics can be written with the decay constant λ, which is the probability per unit time that any single atom decays. It is related to the half-life by λ = ln(2) / t½ (where ln 2 ≈ 0.693), and the decay law becomes the continuous form N(t) = N₀ × e^(−λt). The two expressions are mathematically identical — one is written base ½ over whole half-lives, the other base e over continuous time — and both describe the same smooth exponential curve. The decay constant is handy when you want an instantaneous rate, while the half-life form is more intuitive for "how much is left after so many periods".

Why decay is exponential

Radioactive decay is a memoryless, probabilistic process: every atom has the same fixed chance of decaying in the next instant, regardless of how long it has already survived or how many of its neighbours have gone. Because the number of decays per second is proportional to the number of atoms still present, the population shrinks by the same fraction in every equal time slice — and that constant-fraction behaviour is precisely what produces an exponential curve. Mathematically the quantity approaches zero asymptotically but never reaches it exactly.

Worked example

Suppose you start with 1,000 units of a substance whose half-life is 5,730 years (the half-life of carbon-14), and 11,460 years have passed. The number of half-lives is 11,460 / 5,730 = 2. Apply the formula: N(t) = 1000 × (1/2)² = 1000 × 0.25 = 250 units remaining. That means 750 units have decayed and 25% of the original remains. After one half-life you would have had 500 (50%); after three you would have 125 (12.5%).

Key milestones

After 1 half-life 50% remains · after 2, 25% · after 3, 12.5% · after 4, 6.25% · after 5, about 3.1% · after 10, roughly 0.1%. In pharmacology a drug is usually treated as effectively cleared after about 4–5 half-lives, when 94–97% has been eliminated.

Tips and common mistakes

Tips: keep the half-life and elapsed time in matching units — if the half-life is in days, enter the elapsed time in days too. You do not need a whole number of half-lives; the formula handles fractional values such as 2.5 half-lives perfectly well. To go the other way and find an unknown elapsed time or half-life, rearrange the formula using logarithms.

Common mistakes: mixing time units (for example a half-life in years against a time in days) is the single most frequent error and will badly distort the answer. Do not assume two half-lives leave nothing — they leave a quarter, not zero. And remember half-life describes the time to halve, not a fixed amount removed each period: each step halves whatever remains, so the absolute amount lost shrinks every cycle.