You cannot say when one atom will decay — yet a whole sample is utterly predictable. Each nucleus here is given the same tiny chance to decay every instant, and they wink out one by one at random. Follow the count and it traces a smooth exponential curve, N = N₀·2^(−t/t½): after one half-life half are gone, after two a quarter remain, after three an eighth — no matter how many you start with. Set the half-life and the sample size and watch order rise out of pure randomness. It all runs on your device.
Each glowing dot is an undecayed nucleus — watch them wink out at random
Nuclei remaining200
Decayed0
Elapsed time0.0 s
Half-lives elapsed0.00
Fraction remaining100%
Random one by one, exact in the aggregate. No nucleus has a clock — each simply has a fixed probability of decaying per unit time, and which one goes next is pure chance. Yet because the number of decays each second is proportional to how many are left (dN/dt = −λN), the total obeys N = N₀·2^(−t/t½). The half-life t½ is the time for half to vanish, and it is the same whether you start with forty nuclei or forty billion. That is why the dots blink out unpredictably while the curve glides smoothly down.
How It Works
A fixed chance per nucleus, an exponential in the aggregate, and a half-life that never changes.
1
Each nucleus takes its chance
Every undecayed nucleus is given the same small probability of decaying in each moment, independent of the others and of how long it has already survived. So the dots wink out at random, and you genuinely cannot predict which one goes next — decay has no memory and no schedule.
2
The exponential appears
Because the number decaying each second is proportional to how many remain, the survivor count follows dN/dt = −λN, whose solution is the exponential N = N₀·e^(−λt). The curve overlays that ideal exponential on the real, jittery count, so you can see the randomness scatter around the smooth law — and how a bigger sample hugs it more tightly.
3
Half-life, again and again
The half-life t½ = ln2/λ is the time for half the sample to decay, and it repeats: half gone after one half-life, three-quarters after two, seven-eighths after three. The dashed gridlines mark those points, and the count lands near N₀/2, N₀/4 and N₀/8 — regardless of the starting number, the signature of exponential decay.
What is radioactive decay?
Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation, transforming into a different nucleus. Each nucleus decays at a random moment, entirely independent of the others and of how long it has already existed. You cannot predict when any single nucleus will go, but with a large population the fraction that survives falls in a smooth, predictable way. This simulator shows a population of nuclei decaying one by one and the curve that emerges.
What is half-life?
The half-life is the time it takes for half of the nuclei in a sample to decay. It is a fixed property of each isotope, from fractions of a second to billions of years. Crucially it does not depend on how much you start with: after one half-life half remain, after two half-lives a quarter remain, after three an eighth, and so on. Set the half-life in this tool and watch the count fall to N₀/2, N₀/4 and N₀/8 at one, two and three half-lives on the curve.
Why is radioactive decay exponential?
Because the number of decays in any interval is proportional to how many undecayed nuclei are present: twice as many nuclei means twice as many decays per second. That relationship, dN/dt = −λN, has the exponential as its solution, N = N₀·e^(−λt), equivalently N = N₀·2^(−t/t½) with t½ = ln2/λ. In this simulator each nucleus is given the same small chance to decay each step, and the survivor count traces that exponential — with a little statistical scatter, because real decay is random.
Can you predict when a single atom will decay?
No. The decay of an individual nucleus is genuinely random — there is no internal clock and no way to tell which nucleus will go next or when. All that is defined is the probability of decay per unit time, the same for every nucleus of that isotope. Only in the aggregate, across many nuclei, does the smooth exponential and its well-defined half-life appear. This is why the little dots in the simulator wink out unpredictably while the overall curve stays smooth.