Half Life Calculator

How to use this tool

Choose what you want to find: the amount left, the time taken, the starting amount or the half life. Pick an isotope from the list to fill in its half life, or type your own for a drug, a chemical reaction or anything else that decays at a steady proportional rate. Enter the other values and the answer appears with the number of half lives, the percentage left, a decay curve and a table of each halving. For carbon 14 dating, choose time taken and enter 100 as the starting amount and the percentage of carbon 14 left as the amount left.

The input people most often get wrong is the time unit. The half life and the elapsed time can be in different units, days and years for example, and the calculator converts between them; just make sure each box has the unit it was measured in. Amounts can be in any unit, grams, atoms, becquerels or percent, as long as both amounts use the same one.

What the calculation is doing

Radioactive decay is random at the level of a single atom but perfectly regular in bulk. Every atom has the same small chance of decaying in each second, so the number that decay is always proportional to the number left. That gives exponential decay: a fixed fraction disappears in each equal stretch of time, and the half life is the stretch in which that fraction is one half.

So the amount left after time t is N0 × (1/2) to the power t ÷ T. Rearranged, the time to fall from N0 to N is T × log2(N0 ÷ N), and a measured drop gives the half life as t × ln 2 ÷ ln(N0 ÷ N). The decay constant λ = ln 2 ÷ T is the fraction decaying per unit time when the fraction is small, and the mean life τ = T ÷ ln 2, about 1.44 half lives, is the average time an atom survives.

Carbon dating, and half lives outside physics

Carbon 14 is made continually in the upper atmosphere and mixed into living things, which hold roughly the same share as the air while they are alive. At death the intake stops and the clock starts. Because the half life is about 5,700 years, the method works well for material from a few hundred years old back to around 50,000 years, when too little carbon 14 is left to measure. Beyond that, geologists use isotopes with far longer half lives, such as potassium 40 and uranium 238, to date rocks.

Radiocarbon laboratories report a conventional age calculated with the original Libby half life of 5,568 years and counted back from AD 1950, then convert it to a calendar date with a calibration curve such as IntCal20. Calibration is needed because the carbon 14 level in the air has varied over time. This calculator gives the uncalibrated age; choose the Libby value to match a laboratory’s conventional age.

The same maths describes many things besides radioactivity. A drug cleared from the blood at a steady proportional rate has an elimination half life, and after about 5 half lives roughly 97% of a dose is gone. In nuclear medicine the physical and biological half lives combine: 1 ÷ effective half life = 1 ÷ physical + 1 ÷ biological, so the dose fades faster than either alone.

Half lives of common isotopes

Evaluated values from the IAEA Live Chart of Nuclides, with what each is best known for.

A worked example you can check on paper

100 grams of carbon 14 after 10,000 years. The number of half lives is 10,000 ÷ 5,700 = 1.7544. One half to the power 1.7544 is e to the power −1.7544 × 0.693147 = e to the power −1.2161 = 0.2964, so 29.64 grams are left and 70.36 grams have decayed.

Going the other way, a sample with 25% of its carbon 14 left has been through log2(100 ÷ 25) = 2 half lives, so it is 2 × 5,700 = 11,400 years old. And if something falls from 100 to 25 in 10 days, its half life is 10 × 0.693147 ÷ ln 4 = 10 × 0.693147 ÷ 1.386294 = 5 days.

The decay constant for carbon 14 is 0.693147 ÷ 5,700 = 0.0001216 per year, so about 0.012% of the carbon 14 in a sample decays each year, and the mean life is 5,700 ÷ 0.693147 = 8,223 years.

Mistakes that catch people out

Thinking two half lives means all gone. The first half life removes half, the second removes half of what is left. After two, a quarter remains.

Mixing time units. A half life in days with a time in years gives an answer 365 times out. Set each unit box to match its number.

Using the wrong carbon 14 value. 5,700, 5,730 and 5,568 years give ages that differ by a few percent. Use the Libby value only to reproduce a conventional radiocarbon age.

Treating a decay product as the parent. Some isotopes decay into others that are also radioactive, like uranium to radium to radon. Each step has its own half life, and the amount of a daughter can rise before it falls.

Applying a drug half life to a single person. Published figures are averages. Age, kidney and liver function and other medicines can change how quickly the body clears a drug.

Popular calculations

Each opens the calculator with those values filled in, and they can be changed once it is open.

Carbon 14 dating

Carbon 14 left after a time

How much of an isotope is left

Time to decay to a small fraction

Half life from a measured decay

Starting amount from what is left

How this was checked

The isotope half lives were taken from the IAEA Live Chart of Nuclides, which carries the evaluated nuclear structure data, and a year is taken as 365.2422 days, as in that data. Each of the four solving directions was checked against the others: the amount left after a time, fed back in, returns the same time, and the half life found from a measured drop reproduces the drop. Exact cases were checked by hand, such as 2 half lives leaving 25% and 10 half lives leaving 0.0977%, along with the carbon 14 ages in the table above.

Every address in the lists above was then opened and checked for a distinct title, the right result, a decay curve that renders, and clean CSV, image and print output containing only the values entered, at desktop, tablet and phone widths.

Common questions

What is a half life?

The time it takes for half of a quantity to decay or be removed. For a radioactive isotope it is the time after which half of the atoms have decayed, whatever amount you start with. After two half lives a quarter is left, after three an eighth, and so on.

How do you calculate the amount left?

Multiply the starting amount by one half raised to the number of half lives that have passed: N = N0 × (1/2) to the power t ÷ T. After 10,000 years, 100 grams of carbon 14 with a half life of 5,700 years becomes 100 × 0.5 to the power 1.754 = 29.64 grams.

How long until something is completely gone?

In theory never, because each half life only halves what is left. In practice, after 10 half lives less than 0.1% remains, and after 20 less than a millionth. For a drug, about 5 half lives removes 97% of a dose, which is why that figure is often used as the time to clear it.

How does carbon 14 dating work?

Living things take in carbon, including a small fixed share of carbon 14, from the air and food. When they die the intake stops and the carbon 14 decays with a half life of about 5,700 years, so measuring the fraction left gives the time since death. A sample with 25% of the living level is about 11,400 years old, before calibration.

Which half life should I use for carbon 14: 5,700, 5,730 or 5,568 years?

The best current measurement is 5,700 ± 30 years, the value in the IAEA nuclear data. 5,730 years is an older measurement still found in many textbooks. Laboratories report conventional radiocarbon ages using the original Libby value of 5,568 years by convention and then calibrate them, so pick the Libby option to reproduce a published radiocarbon age.

Can a half life change?

For radioactive decay, no in any practical sense: heat, pressure and chemistry leave it essentially unchanged, which is what makes radiometric dating possible. Biological half lives are different. How fast the body clears a drug depends on the person, their kidneys and liver and other medicines, so a published drug half life is an average.

Related Calculators and Reference

• Log Calculator: the base 2 and natural logarithms behind every half life rearrangement.
• Scientific Calculator: powers and exponentials for checking the working by hand.
• Scientific Notation Converter: write billion year half lives and tiny decay constants cleanly.
• Percentage Calculator: percentage decayed between two measurements.
• Caffeine Half Life Calculator: the same decay law applied to caffeine, with your drinks and bedtime.
• IAEA Live Chart of Nuclides: evaluated half lives and decay data for every known isotope.