How Half-Life Works: What Is Left After 1, 3 and 10 Half-Lives

One formula written two ways

The base-two form is the one to reason with: remaining = initial × (1/2)^(t / t½). The exponent t / t½ is a count of half-lives, and because it is a ratio of two times it has no unit — which is also why both times have to be expressed in the same unit before you divide them. Feed it 11,460 years against a half-life of 5,730 years and the exponent is 2, so a quarter is left. Feed it a time in hours against a half-life in minutes and you will be out by a factor of sixty in the exponent, which is a factor of about 10^18 in the answer.

Physics and pharmacology usually write the same curve as N(t) = N0 × e^(−λt), with λ = ln 2 / t½. There is nothing deeper in the second form; ln 2 is 0.6931, so a half-life of 10 units means λ = 0.06931 per unit, and both expressions return the identical number. What the exponential form does expose is the mean lifetime, τ = 1/λ = t½ / ln 2 = 1.4427 × t½. That is longer than the half-life, which regularly surprises people. The reason is that the surviving tail is unbounded: a small number of entities persist for many half-lives and drag the average up, while the median — the point where exactly half have gone — is by definition the half-life itself.

Nothing ever reaches zero, so practice uses a convention instead

Halving is a multiplication, not a subtraction, and multiplying a positive number by one half can be repeated forever without producing zero. The curve is asymptotic: it approaches the axis and never touches it. So the common question — how long until it is completely gone — is not a hard question, it is a question with no mathematical answer, and any source that gives you a number has silently chosen a threshold. The honest version is to name the threshold you want. Ten percent left is reached at 3.32 half-lives, one percent at 6.64, and a tenth of a percent at 9.97. Those figures come from log base two of 10, 100 and 1000, which is where the familiar rule of ten half-lives comes from.

There is a second, physical reason the smooth curve eventually stops describing reality: real samples contain a finite, whole number of entities, and decay is a random event per entity rather than a steady drain. The exponential is the expected value of that random process. Start with a mole of a radioactive isotope, about 6.022 × 10^23 atoms, and after ten half-lives roughly 5.9 × 10^20 atoms are still there — a fraction of one part in a thousand, and yet an enormous absolute number. Start instead with a thousand atoms and after ten half-lives you expect about one, at which point talking about a fraction stops being useful and you are counting individual events. The ten-half-life rule is a statement about proportions, never about absolute quantities, and a sample large enough can stay measurable far past it.

A half-life is only constant when the process is first order

Everything above assumes the rate of disappearance is proportional to how much is present. That is what first order means, and radioactive decay is the textbook case: each nucleus decays with a fixed probability per unit time, regardless of its neighbours. Under that assumption the half-life is a property of the substance and never changes — the second half-life takes exactly as long as the first. IUPAC is careful about this. Its entry for the half life of a reaction states that the term has meaning only in special cases, chiefly first-order kinetics; for a second-order reaction the half-life depends on the starting concentration, and each successive half-life takes twice as long as the one before. If you find yourself computing a half-life for a process whose rate does not track its own size, the number you get describes one particular starting point and nothing else.

The same care applies when two independent removal routes act at once. A radionuclide inside a living organism is leaving by two mechanisms — it decays, and the body excretes it — and the two rate constants add: 1 / t_eff = 1 / t_phys + 1 / t_bio. Because reciprocals add, the effective half-life is always shorter than either of the two on its own. Take a physical half-life of 8 days and a biological half-life of 24 days: the reciprocals give 0.125 + 0.041667 = 0.166667, so t_eff = 6 days, less than the shorter of the pair. Adding the half-lives, or averaging them, gets this badly wrong in the direction that overstates how long the substance sticks around.

Frequently asked questions

How do I work out a half-life from two measurements?

Rearrange the decay law: t½ = t × ln 2 / ln(N0 / N), where N0 is the first reading, N the second and t the time between them. Say 30% is left after 12 hours. Then ln(1 / 0.30) = 1.2040, and t½ = 12 × 0.6931 / 1.2040 = 6.91 hours. Check it by going forward again: (1/2)^(12 / 6.91) = 0.300. Two readings are the minimum, but they are also the least robust estimate you can make — the ratio N0 / N sits inside a logarithm, so a few percent of measurement noise moves the answer noticeably. Where the data allow it, fit a straight line to ln N against t across many points and take the slope as −λ.

Is the half-life the same as the average lifetime?

No, and the average is the longer of the two. The mean lifetime is τ = 1 / λ = t½ / ln 2, which is 1.4427 times the half-life — so an isotope with a half-life of 10 days has a mean lifetime of 14.4 days. The half-life is the median of the distribution of individual lifetimes, the point where half the entities have gone; the mean is pulled to the right by the small number that survive several half-lives. Mixing the two up is a 44% error, and it always errs in the same direction, so it is worth checking which quantity a published figure actually is.

Does every quantity that decreases have a half-life?

Only if it decays exponentially, meaning the rate of loss is proportional to the amount still present. That covers radioactive decay, first-order chemical reactions, a capacitor discharging through a resistor, and elimination processes that are not saturated. It does not cover a quantity that falls by a fixed amount per unit time: a tank draining at a steady 10 litres a minute takes 5 minutes to go from 100 to 50 litres and only 2.5 minutes to go from 50 to 25, so the halving time keeps shrinking and there is no single half-life to quote. The quick test is to check whether successive halvings take the same time. If they do not, use the underlying rate law instead of forcing a half-life onto it.