Radioactive Decay
A nucleus with no memory, no schedule and no way to be hurried, and a jar of them that keeps reliable time.
Introduction
A single nucleus is the least predictable object in this course: nothing about it tells you when it will decay. Put a few hundred in a jar, though, and the jar keeps perfect time.
Carbon-14 has a half-life of 5700 years. You start with 256 carbon-14 nuclei and wait 11,400 years, two half-lives. How many are left?
Half of what is left
Every nucleus of a given isotope carries the same constant probability per second of decaying, . That is the whole physical input. The simulation rolls each nucleus separately, every step, against that probability, and the count is whatever the dice leave behind.
Why the curve appears at all
When each survivor has the same chance of going, the number of decays is proportional to the number left. Halving what is left, again and again, is an exponential. Its half-life sets its pace, and the activity, in decays per second, follows the same curve:
A nucleus does not age
Take the nuclei still around after many half-lives. They have survived a very long time. Are they the tough ones? Are they due?
A carbon-14 nucleus has survived 20,000 years. Compared with one made yesterday, what is its chance of decaying in the next 5700 years, one half-life?
"Half-life is half of a nucleus's life." A carbon-14 nucleus has a probability, and the half-life describes how a crowd of them thins out. Nuclei decay at every point on the timeline, some in the first tenth of a half-life, some after nine. The half-life is the median of that spread.
Nothing you can do to it makes any difference
Chemistry is electrons, temperature is motion, pressure is crowding. All three act on the outside of the atom, tens of thousands of nuclear radii from the nucleus, with energies of a few electron-volts against a nuclear scale of millions.
You have a sample of iodine-131 decaying on the bench. You want it gone faster, so you heat it to 1200 K. What happens to the count?
Parent, daughter, and a population that rises
A decayed nucleus does not disappear. It becomes a nucleus of a different element (carbon-14 becomes nitrogen-14), still in your sample. Sometimes the new nucleus is radioactive too, and the sample holds two clocks at once.
Reading a clock backwards
So far you knew , waited, and predicted . The useful direction is the other one. Half remaining is one half-life of age; a quarter is two:
Two labs date the same 11,400-year-old sample, one from 256 carbon-14 atoms and one from 4096. Both use the correct half-life. How do their answers differ?
What comes next
A law with no mechanism underneath it, no wearing out and no countdown, makes one of the most reliable timepieces in science, purely by counting. Why a nucleus has a probability rather than a schedule is quantum mechanical. For alpha decay it is tunnelling: a particle held by a barrier it lacks the energy to cross leaks through anyway, at a rate that depends steeply on the barrier, and alpha half-lives span more than thirty orders of magnitude. Beta decay, carbon-14's kind, comes from the weak interaction instead, with the same absence of a schedule.
Ideas this lesson took on
Each of these is a common intuition the lesson put to the test. If one still feels right, the section that tests it is worth another look.
- “After two half-lives nothing is left”
- “An old nucleus is closer to decaying than a fresh one”
- “Heating a sample makes it decay faster”
- “Radioactivity spreads to whatever is sitting next to it”
Finished with this lesson?
Progress is stored on this device only. There is no account and nothing is sent anywhere.
Research behind this lesson
- Hull et al. 2026: Surveys of Austrian high-school and US college students found the reading that half of a nucleus decays in one half-life to be common but context-dependent: cluster analysis of response patterns showed reasoning that shifts with the question rather than one stable misconception, supporting a knowledge-in-pieces account.
- Millar 1994: A diagnostic test given to 144 sixteen-year-olds found many holding a conception of how radiation is absorbed that differs significantly from the accepted account, along with a widespread conflation of radioactive material with the radiation it emits.
- Prather & Harrington 2001: Interviews with college students identified specific difficulties with ionizing radiation and radioactivity, and mapped the ideas students bring to the topic before any instruction on it.
- Prather 2005: An investigation of what introductory students believe individual atoms are doing during radioactive decay and across a half-life. Their accounts rested on inaccurate mental models of atomic behaviour: the same models behind the widely reported failure to distinguish irradiation from contamination.
- White & Gunstone 1992: The Predict–Observe–Explain cycle: committing to a prediction before observing makes a conflicting result impossible to ignore, which is what forces a belief to be revised rather than merely supplemented.