Half Life Calculator

What this calculator does

Half-life is the time it takes for something to drop to half of what it was. It is how we talk about radioactive material, how long a drug stays active in the body, and how archaeologists date ancient bones. This tool works with that halving from any three of four quantities.

Those four are the starting amount, the half-life, the time elapsed, and the amount remaining. Leave one blank and it finds it. It runs right here in the browser.

Using the calculator

1. Fill in any three of the four: initial amount, half-life, time, and remaining amount.
2. Leave the one you want to find blank.
3. Press Calculate.

Keep the half-life and the time in the same unit, so if the half-life is in hours, the time should be in hours too. The tool solves for the empty box. Reset clears them all.

What half-life means

The idea is beautifully simple. Pick any starting amount. The half-life is however long it takes for that amount to fall to one half. Here is the part that surprises people: it takes exactly the same time for the next half to disappear, and the half after that. The amount does not vanish at a steady pace, it keeps halving over equal stretches of time.

And crucially, the half-life does not depend on how much you start with. A gram and a tonne of the same substance have the same half-life. That fixed halving time is what makes half-life such a clean, reliable way to describe decay.

How much is left after each half-life

Count in half-lives and the pattern is easy to follow. Each one leaves half of the previous amount:

A rough rule of thumb: after about ten half-lives, less than a thousandth is left, which is often treated as effectively gone.

The formula

Put into a formula, the amount remaining is:

remaining = initial × (1/2)^{(t ÷ half-life)}

The exponent, t divided by the half-life, is simply the number of half-lives that have passed. If that comes out to 3, you halve three times. If it comes out to 2.5, you are two and a half halvings along, and the formula handles the in-between smoothly. Everything else the tool does, finding the half-life or the elapsed time, is this same relationship rearranged.

A worked example

Start with 80 units, a half-life of 5, and 15 units of time elapsed. Leave the remaining amount blank.

1. Number of half-lives: 15 ÷ 5 = 3.
2. Halve three times: 80 goes to 40, to 20, to 10.
3. By the formula: 80 × (1/2)³ = 80 ÷ 8 = 10. Same answer.

Carbon dating and medicine

Two everyday uses make half-life concrete. In radiocarbon dating, carbon-14 in living things decays with a half-life of about 5,730 years once they die, so measuring how much is left dates the remains. In medicine, a drug's half-life sets how long it keeps working and how often you take it: a short half-life means frequent doses, a long one means the drug lingers. Under the surface, half-life is just another way of stating a continuous decay rate, so if you have a rate rather than a halving time, the exponential decay calculator handles it. The growth counterpart, the time to double rather than halve, is the doubling time calculator.

Questions people ask

What is a half-life?

The time it takes for a quantity to fall to half its value. It stays the same for every successive halving and does not depend on the starting amount.

How much is left after several half-lives?

Halve for each one: after 1 half-life, one half is left; after 2, a quarter; after 3, an eighth. In general, (1/2) to the power of the number of half-lives.

Does the starting amount change the half-life?

No. The halving time is the same whether you start with a gram or a tonne, which is what makes it so useful.

How does half-life relate to a decay rate?

They describe the same decay two ways. A half-life can be converted to a continuous decay rate and back, so you can use whichever the problem gives you.

How does carbon dating use half-life?

Carbon-14 decays with a half-life of about 5,730 years. Measuring the fraction remaining in old organic material tells you roughly how long ago it stopped taking in new carbon.

References

A note on where this comes from. Half-life is the time for a quantity undergoing exponential decay to reduce by half, a constant for a given process. The concept was introduced in the study of radioactivity by Ernest Rutherford in the early 1900s, and it is a standard result of the radioactive decay law. For further reading, see Half-life.

1. Half-life, the constant time for a decaying quantity to fall by half, related to the continuous decay rate through the natural logarithm of 2.
2. Ernest Rutherford, who introduced the concept of half-life in the study of radioactive decay in the early 1900s.