Why does a half-life never reach zero?

Because each half-life removes half of what is left rather than a fixed amount. Halving repeatedly gets arbitrarily close to zero and never arrives, which is what makes decay exponential. In practice the substance reaches a quantity too small to detect, or a single atom decays and there is nothing left — the smooth curve is a description of large numbers.

What is the decay constant?

The probability per unit time that any given atom decays, written λ. It is ln 2 divided by the half-life, about 0.693 over it, and it is what appears in the exponential form N = N₀e^(−λt). The half-life is easier to say; the decay constant is easier to compute with.

Why is the mean lifetime longer than the half-life?

Because the survivors last a long time. The mean is 1/λ, which is the half-life divided by 0.693 — about 44% longer. Half the atoms are gone by the half-life, but the ones remaining have no memory of having survived, so their expected remaining life is the same as it was at the start.

Does this work for medication?

For elimination that follows first-order kinetics, which most drugs do at ordinary doses, yes. The usual clinical rule is that five half-lives clears a drug — about 97% gone — and that it takes the same five to reach a steady level on repeated dosing. Drugs eliminated at a fixed rate rather than a fixed fraction, alcohol among them, do not follow this at all.

Half-Life Calculator

With starting amount 100, half-life 6 hours, time elapsed 24 hours, half-life comes to 6.250000 — amount left. It is reached in 7 steps, the last of which is 100 * pow(0.5, 4), and each one is printed on the page with its numbers filled in. The formula is the one published by NIST Digital Library of Mathematical Functions §4.2, not an approximation fitted to it.

How much of a substance is left after a given time, from its half-life — with the decay constant and the mean lifetime.

Formula and sources checked 27 August 2026 · How we check

Starting amount 100, Half-life 6 hours, Time elapsed 24 hours

6.250000

Amount left for the example below. Editing a field recomputes the calculator below; this figure holds the answer the page was loaded with.

It is written into the HTML rather than drawn by a script, so a search engine reading this page without running JavaScript still finds an answer.

Starting amountHalf-lifehoursIn whatever unit you use for the elapsed time below — hours, years, anything.Time elapsedhours

Amount left

6.250000

Half-lives elapsed: 24 / 64
Amount left: 100 * pow(0.5, 4)6.25
Amount gone: 100 - 6.2593.75
Percentage left: 6.25 / 100 * 1006.25 %
Decay constant, λ: log(2) / 60.116
Mean lifetime, 1/λ: 1 / 0.11552458.656
Time until 1% is left: log(100) / 0.115524539.863

Ask about this in the chat

Worked example

Four half-lives at six hours each leaves 6.25 of the original 100 — a half, a quarter, an eighth, a sixteenth. The mean lifetime is 8.66 hours, longer than the half-life, because the tail of an exponential runs a long way.

How to work it out yourself

1.Use the same time unit for the half-life and the elapsed time. The calculator never converts between them, so hours with hours and years with years.
2.Count in half-lives rather than in time when you want an intuition: after three, an eighth is left, and after ten, about a thousandth.
3.For a drug, five half-lives is the usual rule for "effectively cleared" — about 3% remains, which the last line puts a number on.

What is left after n hours, at a six-hour half-life

What is left after n hours, at a six-hour half-life
Time elapsed (hours)	Amount left	Percentage left
0	100.000000	100 %
3	70.710678	70.71 %
6	50.000000	50 %
12	25.000000	25 %
18	12.500000	12.5 %
24	6.250000	6.25 %
30	3.125000	3.13 %
36	1.562500	1.56 %
48	0.390625	0.39 %
60	0.097656	0.1 %
72	0.024414	0.02 %

Starting from 100 units.

The formula

Half-lives elapsed24 / 6
Amount left100 * pow(0.5, 4)
Amount gone100 - 6.25
Percentage left6.25 / 100 * 100
Decay constant, λlog(2) / 6
Mean lifetime, 1/λ1 / 0.1155245
Time until 1% is leftlog(100) / 0.1155245

Source: NIST Digital Library of Mathematical Functions §4.2 — exponential and logarithmic functions

Questions people actually ask

Why does a half-life never reach zero?: Because each half-life removes half of what is left rather than a fixed amount. Halving repeatedly gets arbitrarily close to zero and never arrives, which is what makes decay exponential. In practice the substance reaches a quantity too small to detect, or a single atom decays and there is nothing left — the smooth curve is a description of large numbers.
What is the decay constant?: The probability per unit time that any given atom decays, written λ. It is ln 2 divided by the half-life, about 0.693 over it, and it is what appears in the exponential form N = N₀e^(−λt). The half-life is easier to say; the decay constant is easier to compute with.
Why is the mean lifetime longer than the half-life?: Because the survivors last a long time. The mean is 1/λ, which is the half-life divided by 0.693 — about 44% longer. Half the atoms are gone by the half-life, but the ones remaining have no memory of having survived, so their expected remaining life is the same as it was at the start.
Does this work for medication?: For elimination that follows first-order kinetics, which most drugs do at ordinary doses, yes. The usual clinical rule is that five half-lives clears a drug — about 97% gone — and that it takes the same five to reach a steady level on repeated dosing. Drugs eliminated at a fixed rate rather than a fixed fraction, alcohol among them, do not follow this at all.

Put this calculator on your site

Free, no attribution required beyond the link.

<iframe src="https://rulecalculators.com/embed/half-life" width="100%" height="420" style="border:1px solid #e7e4de;border-radius:12px" title="Half-Life Calculator"></iframe>

Did this answer your question?