Picture a big bowl of popcorn kernels in hot oil. You can't say which kernel pops next — but you know that after a while, about half of them will have popped. A radioactive sample works the same way. Each unstable nucleus is a kernel waiting to "pop" (decay), with no warning and no schedule. It simply has a fixed chance of going off at any moment.
Put numbers on it
Two numbers tell almost the whole story. $N$ is how many nuclei haven't decayed yet, and the half-life $T_{1/2}$ is the time it takes for half of them to go. Every half-life chops the survivors in half, no matter how many you started with. Start with 1000 atoms and a 2-second half-life: after 2 s you have 500, after 4 s about 250, after 6 s about 125. The decay constant $\lambda$ is just the chance per second that any single nucleus pops — a bigger pile pops faster simply because it has more kernels.
The precise law
Written exactly, the number of decays per second is proportional to how many nuclei are left, $\frac{dN}{dt}=-\lambda N$, which integrates to $N(t)=N_0\,e^{-\lambda t}$. Half-life and decay constant are tied together by $T_{1/2}=\ln 2/\lambda$, and the activity (decays per second, measured in becquerels) is $A=\lambda N$. The process is "memoryless": a 4-billion-year-old uranium atom is exactly as likely to decay in the next second as a freshly made one. Nuclei shed their excess in three ways — $\alpha$ (a helium nucleus tunnels out), $\beta^-$ ($n\to p+e^-+\bar{\nu}_e$), and $\gamma$ (a photon from an excited nucleus). The sliders map straight onto the math: $N_0$ sets the starting dots, $T_{1/2}$ steepens or flattens the curve, while $Z$ and $E_\alpha$ feed the $\alpha$-tunneling view.
Try this in the sim above
Set $T_{1/2}$ low and watch the population collapse. Then switch the graph to "ln N" and see the curve straighten into a line — the unmistakable fingerprint of exponential decay. Open "β Spectrum" to see the decay energy smeared across a continuous range (the hidden neutrino carries away the rest), then open "α Gamow" and raise $E_\alpha$ to watch a higher-energy particle punch through the Coulomb barrier far more easily.
Section 03
Equations & Derivation
Radioactivity is the spontaneous transformation of an unstable nucleus, releasing energy as particles or photons. The number of decays per unit time is proportional to the number of nuclei present — a memoryless quantum process.
Radioactive Decay Law
$$\frac{dN}{dt} = -\lambda N \quad\Longrightarrow\quad N(t) = N_0 e^{-\lambda t}$$
Each red dot is a parent nucleus. At every time step, each surviving nucleus has probability $\lambda\,dt$ of decaying. The population follows $N_0 e^{-\lambda t}$ on average, with $\sqrt{N}$ Poisson fluctuations.
Smoke detectors (Am-241), radiocarbon dating, medical imaging (Tc-99m, F-18 PET), cancer therapy (Co-60, I-131), nuclear power, and natural background (~3 mSv/year from K-40, U/Th).
Decay originates in the nucleus where binding energies are MeV-scale, while chemical/thermal energies are eV-scale (millionfold smaller). Even at $10^6$ K, the nucleus is essentially unaffected.
It doesn't — decay is fundamentally random. There's no internal clock. The constant probability per unit time gives an exponential waiting-time distribution: 'memoryless'. A 1000-year-old atom is statistically identical to a fresh one.
α-decay is two-body — kinematics fix the α energy. β-decay is three-body ($n \to p + e + \bar{\nu}$); energy splits among three particles. The 'missing energy' puzzle led Pauli to predict the neutrino in 1930.
Each K-40 decay releases ~1.3 MeV, but with $T_{1/2} = 1.25\times10^9$ years, only ~4400 decays/sec occur in your body. This is far too dim to see; we're transparent to most γ-rays at this rate.
$\log T_{1/2} = a + b/\sqrt{E_\alpha}$. Higher α-energy means easier tunneling through the Coulomb barrier and dramatically shorter half-life. Across 24 orders of magnitude this empirical law holds, explained by Gamow (1928).
Resources: Khan Academy; HyperPhysics; MIT OpenCourseWare; Paul's Physics Notes.
Section 05
Common Misconceptions
❌ Misconception: Half-life is the time for one nucleus to decay.
✅ Correction: Half-life is the time for HALF of a population to decay. Individual nuclei have no half-life — their decay times follow an exponential distribution.
❌ Misconception: Radioactive substances eventually decay completely.
✅ Correction: Mathematically $N(t) = N_0 e^{-\lambda t}$ never reaches zero. The last atom decays at a Poisson-distributed random time.
📖 Reference: HRW 10th Ed., §42-3.
❌ Misconception: All radiation is equally dangerous.
✅ Correction: α, β, and γ have very different biological effects. α-particles are blocked by paper but devastating internally (e.g., Po-210). γ-rays penetrate deeply but ionize less per unit length.
📖 Reference: Cember & Johnson — Introduction to Health Physics, 4th Ed., Ch. 5; ICRP 103.
❌ Misconception: Old atoms are 'tired' and more likely to decay than new ones.
✅ Correction: Decay is memoryless: $P(\text{decay in dt}) = \lambda\,dt$ regardless of history. A 4-Byr-old U-238 atom is exactly as likely to decay next second as a freshly-made one.
📖 Reference: Krane §6.3; Feynman Lectures Vol. III, Ch. 2.
❌ Misconception: Beta decay violates energy conservation.
✅ Correction: It appears to, until you include the neutrino. Pauli predicted it in 1930; Cowan & Reines detected it in 1956. With the neutrino included, energy and momentum are conserved.
❌ Misconception: Radioactive isotopes look different chemically.
✅ Correction: Isotopes of an element are chemically and visually identical — same electron structure. Only specialized detectors (Geiger-Müller, scintillator) distinguish them.