Half-Life: Working Out What Is Left After Any Amount of Time

The same formula covers a drug clearing your bloodstream, a radioactive isotope decaying, and a carbon date on a bone. Only the value of T changes.

One formula, three quite different problems

Write it as N = N0 x (0.5)^(t/T). N0 is what you started with, t is elapsed time, T is the half-life, and N is what is left. The exponent t/T is just the number of half-lives that have passed, and it does not need to be a whole number.

Working forwards

Caffeine has a half-life of around five hours in a typical adult, though it varies a lot between people and drops sharply for smokers. Drink a 200 mg coffee at 8am and the arithmetic is easy because the times land on whole half-lives: 100 mg at 1pm, 50 mg at 6pm, 25 mg at 11pm. That last figure is an eighth of the dose still circulating at bedtime. Push the cup back to 4pm and 11pm is only 1.4 half-lives later, so about 38% of it is still in you.

Fractional half-lives work the same way. At 3.5 hours, t/T is 0.7, and 0.5^0.7 is 0.616, so about 123 mg of the original 200 mg remains.

Working backwards to find elapsed time

Rearranging for t gives t = T x log(N/N0) / log(0.5). This is the version used for dating things.

Carbon-14 has a half-life of 5,730 years. A sample holding 25% of its original carbon-14 has been through exactly two half-lives, so it is about 11,460 years old, and you can do that one in your head. For 30% remaining, log(0.30) / log(0.5) is 1.737 half-lives, and 1.737 x 5,730 gives roughly 9,950 years. The awkward percentages are why the log form exists.

Why it never quite reaches zero

Each half-life removes half of whatever is left, not a fixed amount, so the curve approaches zero without arriving. After 1 half-life 50% remains, after 2 it is 25%, after 3 it is 12.5%, after 4 it is 6.25%, and after 5 it is 3.125%.

That 5 half-life mark is the practical rule of thumb in pharmacology: a drug is treated as effectively cleared once about 97% has gone. It is also why exponential decay and compound interest are the same shape drawn in opposite directions. One multiplies by a number above 1 each period, the other by a number below it, and both curve rather than run straight.

Using the half-life calculator

The tool solves for whichever box you leave empty, so decide first what you are missing. Enter the starting quantity, the half-life and the elapsed time to get the remaining amount. Enter the starting and remaining quantities plus the half-life to get elapsed time. Enter starting, remaining and elapsed to work out the half-life itself, which is what you do with experimental data.

Keep the time units consistent. A half-life in hours and an elapsed time in days is the most common mistake, and the answer will look wrong by orders of magnitude rather than subtly wrong, which at least makes it easy to catch. Some tools also accept the decay constant, related by lambda = 0.693 / T. If you would rather see the result as a percentage remaining than a quantity, a percentage calculator converts between the two.

Common questions

How long before a medicine is out of my system? Four to five half-lives for the drug itself, which leaves 6.25% and 3.125% respectively. Active metabolites can hang around much longer than the parent drug, so the label matters more than the arithmetic.

Does taking a bigger dose make the half-life longer? Not for most substances. First order kinetics means a constant fraction is cleared per unit time, so the half-life is the same whether you started with 200 mg or 400 mg. A few drugs, and alcohol, behave differently and clear at a fixed rate instead.

What is mean lifetime, and how does it differ? Mean lifetime is the average survival time of a single particle, equal to T divided by 0.693, so about 1.44 times the half-life. Physics papers often quote it instead of half-life, which makes numbers look inconsistent when they are not.

Why does carbon dating stop working on very old samples? After about 50,000 years, roughly nine half-lives, only about 0.2% of the original carbon-14 is left. There is too little signal to measure reliably against background, so older samples need isotopes with longer half-lives.