Half-Life Calculator

About the Half-Life Calculator

The half-life of a substance is the time it takes for half of it to disappear. It is the defining feature of exponential decay: after one half-life, half remains; after two, a quarter; after three, an eighth — and so on, never quite reaching zero. Half-lives describe radioactive isotopes, the clearance of medicines from the body, the discharge of capacitors and many chemical reactions.

This half-life calculator solves the decay equation three ways. It finds the amount remaining after a given time, the half-life from two measurements, or the time needed to decay to a chosen amount. It also gives the percentage decayed, the number of half-lives elapsed, the decay constant and the mean lifetime, with a table of the amount left after 1 to 10 half-lives.

How to Use the Half-Life Calculator

Choose Find amount remaining, Find half-life or Find time.

Enter the initial amount in any unit — grams, becquerels, milligrams or percent.

Enter the two other known values.

Choose the time unit; the half-life and elapsed time must use the same unit.

The Formulas

  N = N₀ × (½)^(t ÷ T½)

  half-life      T½ = t × ln 2 ÷ ln(N₀ ÷ N)
  time           t  = T½ × log₂(N₀ ÷ N)
  decay constant λ  = ln 2 ÷ T½
  mean lifetime  τ  = 1 ÷ λ = T½ ÷ ln 2

Step-by-Step Examples

Carbon-14 after 11,460 years (half-life 5,730 years).

  half-lives  11,460 ÷ 5,730 = 2
  remaining   100 × (½)² = 25

25 percent remains and 75 percent has decayed.

Time for carbon-14 to fall to 10 percent.

  t = 5,730 × log₂(100 ÷ 10) = 5,730 × 3.3219 = 19,035 years

Iodine-131 after 24 days (half-life 8 days): three half-lives, so 100 mg becomes 12.5 mg.

Half-life from measurements: 80 units fall to 20 in 10 hours — a quarter remains, two half-lives — so T½ = 5 hours.

Half-Lives of Some Isotopes

  Isotope        Half-life          Use
  fluorine-18    about 110 minutes  PET scans
  technetium-99m about 6 hours      medical imaging
  iodine-131     about 8 days       thyroid treatment
  cobalt-60      about 5.27 years   radiotherapy, sterilisation
  caesium-137    about 30 years     industrial gauges
  carbon-14      about 5,730 years  radiocarbon dating
  uranium-238    about 4.5 billion years  dating rocks

Radiocarbon Dating

Living things constantly take in carbon, including a tiny, steady proportion of radioactive carbon-14. When an organism dies, the intake stops and the carbon-14 begins to decay with its half-life of about 5,730 years. By measuring how much remains compared with living material, scientists estimate the time since death. A sample with 25 percent of the original carbon-14 is about 11,460 years old. The method works for organic material up to roughly 50,000 years old, and real dates are calibrated against tree rings and other records, because atmospheric carbon-14 levels have varied.

Half-Life in Medicine

Doctors use the half-life of a drug to decide how often it must be taken. After about four to five half-lives, a drug is largely cleared from the body — about 94 to 97 percent gone — and with regular doses, levels settle into a steady state after a similar time. A medicine with a half-life of 6 hours is mostly cleared in a day or so, while one with a half-life of several days builds up slowly and lingers long after the last dose.

Half-Life, Decay Constant and Mean Lifetime

The decay constant λ is the probability per unit time that any one atom will decay; for carbon-14 it is about 0.00012097 per year. The mean lifetime, 1 ÷ λ, is the average time an atom survives — about 8,266.6 years for carbon-14. It is always longer than the half-life, by a factor of 1 ÷ ln 2, about 1.4427. Each describes the same exponential curve, just from a different angle.

Why Decay Is Exponential

Radioactive decay is random at the level of individual atoms: each nucleus has the same chance of decaying in the next second, regardless of how long it has already existed or what its neighbours do. With enormous numbers of atoms, that constant chance produces a smooth, predictable curve. Because the number of decays in any interval is proportional to the number of atoms still present, the amount falls by the same fraction in every equal time step — the defining property of exponential decay. That is why half-lives are constant: a sample always takes the same time to halve, whether it starts with a gram or a microgram. Temperature, pressure and chemical form have essentially no effect on radioactive half-lives, which is what makes them reliable clocks for dating rocks and ancient remains.

Activity and Safety

The activity of a radioactive sample, measured in becquerels, falls with the same half-life as the amount. Short half-lives mean intense activity that fades quickly, which is why medical tracers such as technetium-99m are chosen: they give a clear image, then largely disappear within a day or two. Isotopes with very long half-lives, such as uranium-238, decay so slowly that their activity per gram is low. Waste management plans depend on half-lives: a common rule of thumb is that after about ten half-lives, less than a thousandth of the original amount remains.

Understanding Your Result

The headline is the value you asked for: the amount remaining, the half-life or the time.

The decayed line gives the percentage remaining and decayed, and the number of half-lives.

The decay constant line gives λ and the mean lifetime.

The half-life table shows how much remains after 1, 2, 3, 4, 5 and 10 half-lives.

When Should You Use This Calculator?

Use it for physics and chemistry homework on radioactivity.

Use it to understand radiocarbon dating.

Use it to estimate how long a medicine or tracer stays in the body.

Use it for any first-order exponential decay.

Common Mistakes

Mixing time units. The half-life and time must use the same unit.

Assuming decay is linear. After two half-lives, a quarter remains, not none.

Expecting it ever to reach zero. Exponential decay approaches zero but never gets there.

Confusing half-life with mean lifetime. The mean lifetime is about 1.44 times longer.