radioactive decay
An unstable atomic nucleus will, sooner or later, spit out a particle and turn into something else — but you can never say when a particular atom will do so. It is genuinely random, like a coin that flips itself at unpredictable moments. Yet when you have trillions of such atoms, the randomness averages out into a beautifully simple and reliable law, and that law is a first-order differential equation.
Because each atom decays independently with the same fixed chance per second, the number that decay in a short interval is proportional to how many are still present. If N(t) is the number of undecayed nuclei, then dN/dt = -lambda N, where lambda (the decay constant) is the probability per unit time that any one nucleus decays. This is exponential decay, so N(t) = N0 e^(-lambda t): the population of nuclei falls off exponentially, quickly at first and then ever more slowly as fewer remain to decay.
This is one of the rare models that is essentially exact — there is no crowding, no resource limit, no feedback to spoil the proportionality, because the atoms ignore one another entirely. That reliability is why radioactive decay underpins dating methods (carbon dating, uranium-lead) and medical tracers. The everyday way people quote the rate is not lambda itself but the half-life, the time for half the sample to decay.
Iodine-131 has decay constant lambda ≈ 0.0866 per day. Starting from N0 atoms, after 8 days about N0 e^(-0.0866·8) ≈ N0 e^(-0.693) ≈ 0.5 N0 remain — half is gone, matching its roughly 8-day half-life.
dN/dt = -lambda N gives exact exponential decay because atoms decay independently.
Decay is exponential only on average over many atoms. For a single atom there is no 'halfway through its half-life' — the law is a statistical statement, not a deterministic clock for one nucleus.