Half life explained: radioactive decay simulator and carbon dating • The Science 360

By The Science 360 · Updated October 2026 · 9 min read

What is half-life, and how does radioactive decay work?

The half-life of a radioactive isotope is the time it takes for half of its atoms to decay. After one half-life half the sample is left, after two a quarter, after three an eighth, following N = N₀ × 2^(−t ÷ T½). Which individual atom goes next is pure chance, but in a large sample the total follows the curve almost exactly. The simulation above shows both at once: 400 atoms with random decay times, next to the smooth curve.

The half-life is fixed for each isotope and runs from fractions of a second to billions of years: 164 microseconds for polonium-214, 8.02 days for iodine-131, 5,730 years for carbon-14 and 4.47 billion years for uranium-238. That is why carbon-14 can date a mummy while uranium-238 dates rocks. Hit play and each half-life takes 2 seconds on screen, a labelled time-lapse that stands in for any real time scale. Related reading: how Oppenheimer made the first nuke and the Manhattan Project.

Key results from the simulation

The halving rule: after k half-lives a fraction 1 ÷ 2^k is left: 50% after 1, 25% after 2, 12.5% after 3, 0.39% after 8.
Chance in the small, law in the large: after 1 half-life the 400-atom sample should have 200 atoms left. In the starting sample it has 198, within the typical scatter of about ±10. A sample of a million atoms would sit within about ±500 of the half.
Carbon dating: a sample with 53% of its carbon-14 left is 5,272 y old, roughly the age of Ötzi the Iceman.
Short half-life, high activity: a gram of pure polonium-214 would give 11.9 YBq, a gram of uranium-238 only 12.4 kBq.

Five isotopes, five time scales

Each section picks an isotope and a time, shows the numbers the simulation gives, and has a button that loads it. The time slider is in half-lives, so the same picture works for all of them. Only the units on the time axis change.

After one half-life: about half are left

Carbon-14, 5,730 years

Slide the time to 1 half-life. For carbon-14 that is 5,730 y. The law says 200 of the 400 atoms should remain, and in the starting sample 198 do. The gap is not an error: each atom has the same chance of decaying in any moment, so the count scatters about the average by roughly ±10.

Press "New random atoms" a few times and watch the number change while the smooth curve stays put. The decay is random per atom, and the average is not. The mean life, the average time an atom survives, is T½ ÷ ln 2 = 1.443 half-lives, so after one mean life 37% are left.

Elapsed 5,730 y Expected left 200 (50%) This sample 198 of 400 Typical scatter ±10

Carbon dating: reading the fraction as an age

Carbon-14, 53% left

Living things take in carbon-14 made high in the air when cosmic rays hit nitrogen, so their carbon has a steady, tiny share of it: about 1 atom in a trillion, giving roughly 0.23 becquerels per gram of carbon, about 14 decays a minute. When the organism dies the supply stops and the clock starts. Measure the fraction left and invert the law: t = −T½ log₂(N ÷ N₀).

At 0.92 half-lives 53% is left, which reads as 5,272 y, about the age of Ötzi the Iceman, the body found in the Alps in 1991. The simulation's own count is a little different from the expected value, and the "Age from fraction" readout, which uses the sample count, drifts by the same amount. That scatter is why real labs count a huge number of atoms and quote an uncertainty.

Fraction left 53% Age 5,272 y This sample 210 of 400 Sample reads 5,327 y

Uranium-238: a half-life as old as the Earth

4.47 billion years

Uranium-238 has a half-life of 4.47 billion years, close to the age of the Earth (about 4.54 billion years). At 1.02 half-lives, 4.56 bn y, 49% is left: roughly half the uranium-238 that the Earth was born with is still here. A gram of it is only about 12.4 kBq, which is weak next to the other isotopes here.

Natural uranium is about 99.3% uranium-238 and 0.7% uranium-235, the lighter isotope whose fission matters for reactors and weapons. Decay and fission are different processes: decay is spontaneous and random, fission in a bomb is a neutron-driven chain reaction. For the history, see how Oppenheimer made the first nuke and this much energy could blow up the entire Earth.

Half-life 4.47 billion years Elapsed 4.56 bn y Fraction left 49% Activity per g 12.4 kBq

Iodine-131: gone in a few weeks

8.02 days

Iodine-131 has a half-life of 8.02 days. After 3 half-lives, 24.1 d, an eighth is left (50 of 400 expected, 53 in the starting sample). After 8 half-lives, 64.2 d, only 0.39% remains, which is why it stops being a worry within a few months.

The thyroid gland collects iodine, so iodine-131 is used in small doses to treat thyroid disease, and it is also one of the radioactive products released in reactor accidents, where it is a health concern for the same reason. It decays fast, so it is intense: a gram of the pure isotope would give 4.6 PBq.

Elapsed 24.1 d Fraction left 13% This sample 53 of 400 Activity per g 4.6 PBq

Technetium-99m: a medical tracer that fades in a day

6.0 hours

Technetium-99m is the most widely used radioactive tracer in nuclear medicine imaging. Its half-life is 6.0 hours, long enough to scan a patient and short enough that the dose is mostly gone within a day: after 4 half-lives, 24.0 h, 6.3% remains (27 of 400 in the starting sample, 25 expected).

It gives off gamma rays that a camera outside the body can see, and because the half-life is short the radiation dose to the patient stays low. A gram of the pure isotope would give 195 PBq, so real doses use a tiny amount. Because it cannot be stored for long, hospitals make it on site from a longer-lived parent.

Elapsed 24.0 h Fraction left 6.3% This sample 27 of 400 Activity per g 195 PBq

Polonium-214: a half-life under a millisecond

164 microseconds

Polonium-214 is a link in the uranium-238 decay chain with a half-life of 164 microseconds. Five half-lives take only 820 µs, and by then 3.1% is left (13 of 400 in the starting sample, 13 expected). The curve is identical to carbon-14's: only the units on the time axis change.

Short half-life means very high activity. A gram of the pure isotope would give 11.9 YBq, but it exists only for an instant after being made, so no such sample can be held. It decays by emitting an alpha particle.

Elapsed 820 µs Fraction left 3.1% This sample 13 of 400 Activity per g 11.9 YBq

The five isotopes side by side

Isotope	Half-life	After 3 half-lives	After 8 half-lives	Activity per gram	Where it matters
Carbon-14	5,730 years	17,190 y	45,840 y	165 GBq	Dating remains up to about 50,000 years
Uranium-238	4.47 billion years	13.4 bn y	35.8 bn y	12.4 kBq	Dating rocks; reactor fuel
Iodine-131	8.02 days	24.1 d	64.2 d	4.6 PBq	Thyroid medicine; reactor releases
Polonium-214	164 microseconds	492 µs	1,312 µs	11.9 YBq	Part of the uranium-238 chain
Technetium-99m	6.0 hours	18.0 h	48.0 h	195 PBq	Medical imaging

Activity is for a gram of the pure isotope, from A = λN with λ = ln 2 ÷ T½, which no one could hold for the shortest-lived isotopes. After 8 half-lives 0.39% of the atoms remain.

How the simulation works

Each isotope decays with a fixed chance per unit time, so the number left follows N = N₀ × 2^(−t ÷ T½) = N₀ e^(−λt) with λ = ln 2 ÷ T½. The activity, the number of decays per second, is A = λN. Divide the half-life in seconds into ln 2 and you have λ; multiply by the number of atoms in a gram (Avogadro's number ÷ molar mass) for the activity per gram.

The 400 atoms are not told "half of you must go". Each is given its own decay time from a seeded random number: t = −log₂(u) half-lives for a random u between 0 and 1, which is exactly the distribution a constant chance per moment produces. An atom is drawn as decayed once the clock passes its time, with a short glow as it goes. Because the times are fixed by the seed, moving the slider back and forth always shows the same atoms in the same state, and "New random atoms" draws a new seed. The yellow line is the true count of survivors, the dashed line is the law, and the scatter between them is about √(N p (1 − p)) atoms.

The time-lapse plays 2 seconds per half-life, and the badge on the stage says so. The two rows under the time axis give the half-life count and the real time for the chosen isotope. For carbon dating, the age comes from the fraction left in the sample itself, t = −T½ log₂(N ÷ N₀). With few atoms this is noisy, and the readout shows it honestly. Carbon-14 dating works to about 50,000 years, which is 8.7 half-lives. Beyond that less than 0.3% is left, too little to measure well.

What the model leaves out

Real sample sizes: a gram of any element holds about 10²¹ to 10²³ atoms, not 400. At that size the scatter is a billionth or less and the curve is a straight-edged law.
Decay chains: many isotopes, such as uranium-238, do not decay to a stable atom but to another radioactive one. The simulation shows a single step.
Calibration of carbon dates: the carbon-14 fraction in the air has changed over time, so raw radiocarbon ages are corrected against tree rings and other records before they are quoted as calendar dates.
Types of radiation: alpha, beta and gamma emissions, and their energies and dangers, are not shown. All that is modelled is whether an atom has decayed.
Constant half-lives: the half-lives are treated as constants. They are essentially unaffected by temperature, pressure or chemistry.

Common misconceptions

“After two half-lives everything is gone.” No: after two half-lives a quarter is left, after three an eighth. The amount shrinks by half each time and never reaches zero in a perfectly smooth law, only in a small sample where the last atom finally goes.

“An older atom is more likely to decay.” Atoms do not age. Each has the same chance of decaying in the next second, no matter how long it has already survived. That is why the law is an exponential.

“Radioactive decay is the same as a nuclear explosion.” Decay is spontaneous and slow. A bomb relies on fission, in which neutrons split heavy nuclei in a chain reaction, releasing energy in a fraction of a second.

Frequently asked questions

What is half-life?

The time for half of the radioactive atoms in a sample to decay. After one half-life half remain, after two a quarter, after three an eighth. Carbon-14 has a half-life of 5,730 years, iodine-131 8.02 days.

How does carbon dating work?

Living things take in carbon-14 along with ordinary carbon. When they die, they stop, and the carbon-14 decays with a half-life of 5,730 years. Measuring what fraction is left gives the age through t = −T½ log₂(N ÷ N₀). It works up to about 50,000 years, after which less than 0.3% is left.

Can you predict when a single atom will decay?

No. Each atom has a fixed chance per unit time, and no way is known to tell which will go next. What can be predicted is the total for a large number of atoms, which is why half-lives can be measured so precisely.

Does temperature or pressure change the half-life?

For practical purposes no. Half-lives of most isotopes are the same hot or cold, in a lab or deep underground. A few special decays, which involve an atom capturing one of its own electrons, change by a tiny fraction with chemistry, but not enough to matter for dating.

What is the difference between half-life and activity?

Half-life is how fast the number of atoms halves. Activity is how many decays happen per second, A = λN. For the same number of atoms a shorter half-life gives a higher activity, and as the atoms decay the activity falls with the same half-life.

Is radioactive decay the same as a nuclear bomb?

No. Decay is a slow, spontaneous change in one nucleus. A bomb forces a chain reaction, with each fission releasing neutrons that cause more. The Manhattan Project had to enrich uranium to raise the share of uranium-235 for this reason. See the Manhattan Project and Oppenheimer, father of the nuclear bomb: was he proud?.