Half Life Calculator

1. Enter any three of initial amount, remaining amount, elapsed time and half-life.
2. Use the same unit for both amounts, and the same time unit for elapsed time and half-life.
3. Leave the one you want to find blank.
4. Read the result and how many half-lives have passed.

Here is what each result means:

Half-life is the time it takes for half of a decaying substance to disappear. Start with any amount and after one half-life, half remains; after two half-lives, a quarter; after three, an eighth. This steady halving is called exponential decay, and it is the fingerprint of radioactive materials, but the same maths describes drug clearance, capacitor discharge and many other processes.

The remaining amount equals the initial amount times one half raised to the number of half-lives that have passed. The number of half-lives is simply the elapsed time divided by the half-life. Because the decay is exponential, the amount never quite reaches zero: each half-life removes half of whatever is left, so a tiny trace always remains in principle.

A crucial feature is that the half-life is constant. It does not matter how much you start with or how much has already decayed: the time to halve is always the same. This is what makes half-life so useful for dating ancient objects with carbon-14 and for setting safe handling times for radioactive materials and medicines.

It rearranges the exponential decay law to solve for whichever value you leave blank.

1. For the remaining amount, multiply the initial amount by one half raised to the number of half-lives.
2. For the initial amount, divide the remaining amount by that same factor.
3. For elapsed time or half-life, use logarithms to undo the exponent.

Decay is an exponential process; see the logarithm calculator and exponent calculator.

A substance with a half-life of 5 years starts at 100 grams. How much remains after 15 years?

Calculation: 15 years is 3 half-lives, so remaining = 100 x (1/2)^3 = 12.5 grams. To find the half-life instead, if 100 grams falls to 25 grams in 10 years, that is 2 half-lives, so the half-life is 5 years.

One law, rearranged for whatever you need to find.

All four come from the single decay law, so any three of the values give the fourth. Solving for time or half-life uses logarithms because those quantities sit in the exponent, which is why a scientific calculator is needed to do it by hand.

Each half-life removes half of whatever is left, so the amount falls quickly at first.

After about 10 half-lives less than a thousandth of the original remains, which is often taken as the point where a radioactive source is considered practically gone. The decay never reaches exactly zero, but it becomes negligible.

The Half-life

A shorter half-life means faster decay. The half-life is a fixed property of each isotope or process.

The Elapsed Time

More time means more half-lives have passed and less remains, following the exponential curve.

The Starting Amount

The initial amount scales the whole curve, but not the half-life, which stays the same.

Physics and Chemistry Homework

Solve for the amount, time or half-life in a radioactive decay problem.

Radiocarbon Dating

Estimate the age of an object from how much carbon-14 remains.

Medicine and Pharmacology

Work out how quickly a drug clears from the body over several half-lives.

1. Mismatched Time Units

Elapsed time and half-life must use the same unit, both years or both hours.

2. Different Amount Units

The initial and remaining amounts must be in the same unit for the ratio to be correct.

3. Remaining Above Initial

In decay the remaining amount is always less than the initial amount, so it cannot be larger.

4. Confusing Half-life with Lifetime

Half-life is the time to halve, not the time for the substance to vanish completely.

5. Expecting Linear Decay

Decay is exponential, not steady. The same fraction is lost each half-life, not the same amount.

The relation is exact for exponential decay; only the displayed decimals are rounded, and very large or small results are shown in scientific notation.

Half-life is the time it takes for half of a decaying substance to disappear. After one half-life half remains, after two a quarter, and after three an eighth.

If you know how much remains after a time, count the halvings. If 100 grams falls to 25 grams, that is a quarter, or 2 half-lives, so divide the elapsed time by 2.

One eighth, or 12.5 percent. Each half-life removes half of what is left, so three halvings give one half times one half times one half.

The remaining amount equals the initial amount times one half raised to the number of half-lives, which is the elapsed time divided by the half-life.

No. The half-life is constant regardless of how much you start with. It always takes the same time for the amount to halve.

Carbon-14 has a half-life of about 5,730 years. By measuring how much carbon-14 remains in an object, scientists work back to how long ago it stopped taking in carbon.

Use the same unit for both amounts, and the same time unit for the elapsed time and the half-life. The units cancel, so any consistent choice works.

No. Half-life is the time to halve. The mean lifetime is longer, and the substance never fully disappears, though after about ten half-lives only a trace remains.

No. The calculation runs in your browser and nothing you enter is stored or sent anywhere, unless you choose Save, which keeps the result only on this device.

• Half-life (Wikipedia).
• Radioactive decay (Wikipedia).
• Exponential decay (Wikipedia).