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Half-Life Calculator
Use this half-life calculator to find how much of a substance is left after a given time, or how long it takes to fall to a target amount. It works for radioactive decay, medicines clearing from the body, caffeine wearing off, and any process where the amount halves over a fixed period.
The half-life is the time for half of the substance to decay or be eliminated. Mix units freely — the calculator converts them.
How it works
A half-life is the time it takes for half of something to disappear. It shows up in surprising places: radioactive atoms breaking apart, a drug being cleared from the bloodstream, and caffeine fading from your body. The key feature is that the same fraction is lost in each equal stretch of time. If 100 mg falls to 50 mg in 5 hours, it will fall from 50 mg to 25 mg in the next 5 hours, then to 12.5 mg in the 5 hours after that.
To find the remaining amount, first count how many half-lives have passed. Divide the elapsed time by the half-life. If 15 hours have passed and the half-life is 5 hours, that is 15 ÷ 5 = 3 half-lives. Then multiply the starting amount by one half raised to that number of half-lives. That is the whole calculation, and it works for fractional half-lives too: after 2.5 half-lives, the remaining fraction is (1/2)2.5, which is about 0.177, or 17.7 percent.
The inputs. The initial amount is how much you start with, in whatever unit suits you — milligrams for a medicine, grams for a chemical sample, or just plain units. The half-life is the time for half of it to go, and you can enter it in seconds, minutes, hours, days, or years. The elapsed time is how long has passed since the start, in any of the same units. The calculator converts the units behind the scenes, so you can mix them: a half-life in hours and an elapsed time in days is no problem at all.
The second mode works the other way round. Instead of asking what is left after a time, it asks how long it takes to fall to a target amount you choose. The maths is the reverse of the first mode: the number of half-lives needed is the base-2 logarithm of the starting amount divided by the target amount, and multiplying that by the half-life gives the time. For example, to fall from 100 mg to 12.5 mg is a factor of 8, and log₂(8) = 3, so it takes 3 half-lives. This mode is handy for questions like "when will this drug be mostly out of my system?" or "when will this sample be safe to handle?"
Why the curve flattens. The diagram shows the classic exponential decay curve: steep at first, then levelling off. In the first half-life you lose half of everything you started with. In the second you lose half of what is left, only a quarter of the original. Each step loses less, so the curve flattens while never quite touching zero. After 10 half-lives, only about 0.1 percent remains, which for most purposes counts as gone.
Real half-lives. Familiar examples show the huge range of half-lives in nature. Caffeine has a half-life of about 5 hours in most adults, which is why a morning coffee is mostly gone by bedtime. Ibuprofen has a half-life of about 2 hours. Carbon-14, used to date ancient bones and wood, has a half-life of about 5,730 years, while uranium-238 has one of about 4.5 billion years. Whatever the scale, the same maths applies.
Worked example
A patient takes a 100 mg dose of a medicine. The medicine has a half-life of 5 hours. How much is left in the body after 15 hours?
Step 1: Find how many half-lives have passed. Divide the elapsed time by the half-life: 15 hours ÷ 5 hours = 3 half-lives.
Step 2: Work out the fraction remaining. Each half-life leaves half, so after 3 half-lives the fraction is (1/2)3 = 1/2 × 1/2 × 1/2 = 1/8 = 0.125.
Step 3: Multiply the starting amount by that fraction: 100 mg × 0.125 = 12.5 mg.
Step 4: Check it by halving step by step. After 5 hours (1 half-life): 50 mg. After 10 hours (2 half-lives): 25 mg. After 15 hours (3 half-lives): 12.5 mg.
Try it yourself: choose "Remaining amount after a time", enter an initial amount of 100 mg, a half-life of 5 hours, and an elapsed time of 15 hours, then press Calculate. The tool gives exactly the same answer — 12.5 mg remaining, after 3 half-lives. You can also try the second mode: enter a target of 12.5 mg and it tells you it takes 15 hours to get there.
FAQ
What is a half-life?
A half-life is the time it takes for half of a substance to disappear through decay or elimination. After one half-life, half of the original amount remains. After two half-lives, a quarter remains. After three half-lives, an eighth remains. The key point is that the same fraction is lost in each equal time period, so the amount falls by half every half-life, no matter how much you start with.
How do you calculate the remaining amount after a half-life?
First work out how many half-lives have passed by dividing the elapsed time by the half-life. Then multiply the starting amount by one half raised to that number. For example, 100 mg with a half-life of 5 hours, after 15 hours, has been through 3 half-lives, so 100 times one half to the power of 3 equals 12.5 mg. This calculator does both steps for you and also shows the number of half-lives that have passed.
Can a half-life be a fraction, like 2.5 half-lives?
Yes. The elapsed time does not have to be a whole number of half-lives, and the formula works for any value. After 2.5 half-lives, the remaining fraction is one half to the power of 2.5, which is about 0.177, so about 17.7 percent of the starting amount is left. In practice this matters a lot: medicines are usually measured in fractional half-lives because doses are taken at times that do not line up exactly with the half-life.
Does the amount ever reach exactly zero?
Mathematically, no. Halving an amount again and again makes it smaller and smaller, but it never reaches exactly zero — after 10 half-lives about 0.1 percent is left, and after 20 half-lives about 0.0001 percent is left. In practice, though, the amount eventually becomes too small to measure or too small to matter, and at that point it is treated as gone. For a medicine, doctors usually consider about 5 half-lives, when around 97 percent is gone, as effectively cleared.
What is the difference between half-life and decay constant?
They describe the same decay in two different ways. The half-life is a time: how long until half is gone. The decay constant, usually written as the Greek letter lambda, is a rate: how fast the substance decays per unit of time. They are linked by the formula lambda equals the natural log of 2 divided by the half-life, so lambda equals about 0.693 divided by the half-life. A short half-life means a large decay constant and fast decay; a long half-life means a small decay constant and slow decay.