Half-Life Calculator

Some things do not shrink steadily; they halve, again and again. A radioactive sample, a drug leaving the body, and a chemical breaking down can all follow the same pattern: after one half-life, half remains. After two half-lives, a quarter remains. After three, an eighth. This calculator works with that pattern, finding the remaining amount, the elapsed time, or the half-life itself from the other values.

The rule is a power of one half. Remaining amount equals starting amount times one half raised to the number of half-lives that passed, where the number of half-lives is elapsed time divided by the half-life. That is the whole model, and it explains why decay slows down: each half-life removes half of what is left, so the amount removed gets smaller every time, even though the fraction removed stays the same.

Here is a worked example. Start with 100 grams of a substance with a half-life of 10 days. After 20 days, two half-lives have passed, so the remaining amount is 100 times one half squared, which is 100 times 0.25, or 25 grams. After 30 days, three half-lives give 12.5 grams. The tool also runs backwards: if 100 grams fell to 25 grams, the elapsed time is two half-lives, and if that took 20 days, the half-life is 10 days.

Half-life is a property of the substance and the process, not of the amount. A small sample and a large sample of the same isotope share the same half-life, and starting with twice as much does not change how fast each half disappears. That is why half-life is so useful: one number describes the decay of any amount. But it only holds while the decay is a simple first-order process, which covers many textbook cases and not every real one.

Medicine uses the same maths with a different name, biological half-life. A drug with a 6-hour half-life falls to half its level in 6 hours, a quarter in 12, and an eighth in 18, if the body clears it steadily. Doctors use this to set dose timing, but real bodies vary: liver and kidney function, age, and other drugs all shift the rate. A calculator answer is a study model, never a dosing guide, and no medicine decision should come from a web tool.

Two mistakes show up often. The first is using different time units for half-life and elapsed time: a half-life in days with an elapsed time in hours gives nonsense unless you convert one to match the other. The second is expecting the amount to hit zero: halving never reaches zero exactly, it only gets smaller and smaller. After ten half-lives, about a thousandth remains, which is tiny but not zero, and that matters for safety with radioactive materials.

Use this calculator as a practice aid for study and homework, not as professional guidance. It is an educational estimator for ideal decay. Radioactive materials, medicines, and chemicals follow safety rules, regulations, and professional advice that no web tool can replace. If a question touches real health or safety, stop and ask a qualified person. Check every result with a worked example, and treat the halving pattern as a model, not a promise.

Carbon dating is half-life maths applied to history. Living things keep replacing their carbon, so the fraction of radioactive carbon-14 stays steady while they live. When they die, the replacement stops and the carbon-14 halves every 5,730 years, while ordinary carbon-12 stays put. Measure the ratio that remains, and the halvings tell the age: a sample with half the modern ratio died about 5,730 years ago, and one with a quarter died about 11,460 years ago. The method fades past about ten half-lives, near 50,000 years, because too little carbon-14 is left to measure well, and contamination with modern carbon is the classic error that makes samples look too young.

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