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.
Orange = parent nucleus, blue = daughter — watch parents wink out at random
Half-life t½5,730 yr
Decay constant λ1.21e-4 /yr
Nuclei remaining N200
Decayed (daughter)0
Activity A = λN2.42e-2 /yr
Elapsed time0 yr
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.
Reading the simulation
What the winking dots, the counts and the two colours are telling you.
1
Decay is random per nucleus, predictable in bulk
Each orange parent nucleus is given the same tiny chance to decay every instant, with no memory of how long it has lasted — so which dot goes next is pure chance and you cannot predict it. Yet watch the on-canvas Parent and Daughter counts and the whole population marches down a smooth curve. Order out of randomness: individually unknowable, collectively exact.
2
Each half-life halves what remains
When a parent decays it flips to a blue daughter. The lower graph plots the surviving count N against time measured in half-lives, with dashed gridlines at 1t½, 2t½, 3t½. The moving dot lands near N₀/2, N₀/4, N₀/8 at those marks — half, then a quarter, then an eighth — no matter whether you start with 40 nuclei or 400.
3
Activity falls as the sample shrinks
The activity A = λN is how many decays happen per unit time, so it is largest at the start and fades as parents run out — the readout drops in step with N. Switch isotopes and the pattern is identical; only the clock changes: a long half-life like Uranium-238's means a tiny λ and a nearly flat, barely-active sample, while Iodine-131 empties in days.
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 and its rate
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 decay constant λ = ln2/t½ sets the pace, and the sample's activity A = λN is its decays per unit time. The curve overlays that ideal exponential on the real, jittery count, so you can see the 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. Pick a real isotope and the same maths is relabelled into years or days — Carbon-14's 5,730 years for dating, Iodine-131's 8.02 days in medicine, Uranium-238's 4.47 billion years in the rocks.
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 t½ 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 — this tool lets you pick real ones such as Carbon-14 (5,730 years), Iodine-131 (8.02 days), Cobalt-60 (5.27 years) and Uranium-238 (4.47 billion years), or set a custom value. Crucially it does not depend on how much you start with: after one half-life half remain, after two a quarter, after three an eighth. Watch the count fall to N₀/2, N₀/4 and N₀/8 at one, two and three half-lives on the curve, while the activity A = λN fades as the sample shrinks.
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 — you cannot predict when a single atom will decay: 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.
A ready-to-assign lab using the simulation above. Free to use — nothing to install or sign up for, and no student data leaves the browser.
High school, College10–20 minInteractive simulation
Learning objectives
Define half-life and apply it.
Describe decay as an exponential process.
Relate activity to the number of remaining nuclei.
Variables to change
Half-life
Initial number of nuclei
Procedure
Start with a full sample and step forward one half-life at a time.
Record the fraction remaining after 1, 2, 3, and 4 half-lives.
Observe how the activity (decays per second) changes over time.
Observations
Record the fraction of nuclei remaining after each half-life and how activity falls.
Questions
What is a half-life?
What fraction remains after 3 half-lives?
Write the decay law.
How does activity change as the sample decays?
Can you predict when a specific atom will decay?
Explanation
Radioactive decay is exponential: each half-life halves the remaining nuclei, so N = N₀·(½)^(t/T½). Activity falls in step with the number remaining. Individual decays are random, but the half-life makes large samples predictable — the basis of radiometric dating.
Answer key (for instructors)
1. The time for half of the radioactive nuclei in a sample to decay.
2. One eighth (½ × ½ × ½).
3. N = N₀·(½)^(t/T½), equivalently N = N₀·e^(−λt).
4. It decreases over time, proportional to the number of nuclei remaining.
5. No — individual decays are random; only the statistical half-life of a large sample is predictable.
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