CLASS 12 · CHAPTER 13 · MODERN PHYSICS
Nuclei
Drag time forward on a radioactive sample and watch it 'run out', except it never quite does. Whatever is left right now takes exactly the same fixed 10 years to halve again, whether that's most of the sample or almost none of it.
Watch it happen
At t = 15.0 yr, 35.355% of the original sample is still undecayed (1.50 half-lives elapsed). Whatever amount remains right now, the next 10 years will cut it exactly in half again, to 17.678% — no faster and no slower than the very first halving from 100% to 50%. The half-life never changes, only the shrinking amount it’s applied to.
Push the slider all the way from t = 0 to t = 50 years. Every time the clock ticks past another multiple of the half-life, the fill level lands exactly on the next dashed reference line, 50%, then 25%, then 12.5%, then 6.25%, then 3.125%, each transition taking precisely the same 10 years as the one before it.
Where the formula comes from
Radioactivity is a purely random process at the level of a single nucleus, but with a huge number of identical nuclei, the number of decays per unit time is proportional to the number of undecayed nuclei present, with a fixed per-nucleus probability of decay per unit time, the decay constant :
Solving this first-order equation (separate variables and integrate from at ) gives the radioactive decay law:
The half-life is defined as the time at which exactly half the sample remains, at . Substituting:
Taking the natural log of both sides and rearranging gives the link between the decay constant and the half-life:
Feeding this back into the decay law lets you write it directly in terms of the half-life, exactly the form the simulation above uses:
Where the shortcut stops working
The single idea this whole chapter turns on: the half-life is a fixed constant of the nuclide, completely independent of how much sample you started with or how much is left right now. It is tempting to reason that a sample with “only a little bit left” should decay differently, faster because there’s less competition for whatever is causing the decay, or slower because there’s less material driving it. Neither is true.
What actually stays constant is the fractional decay rate, . The number of decays per second, , does fall as falls, so a shrinking sample does get quieter in an absolute sense. But the time for whatever remains to halve again never changes: 100% → 50% takes , and so does 50% → 25%, and so does 25% → 12.5%, forever. Every halving is identical, because each one is really the same calculation applied to a new starting amount.
A second, subtler point: because is a smooth exponential, it has no finite root, gets arbitrarily small but never reaches exactly zero at any finite time. There is no last tick of the clock at which “the sample is officially gone.” In a real sample this stops mattering once only a few atoms remain (the continuous formula was only ever an approximation for huge ), but the idea that decay has no sharp finish line is itself a common exam misconception worth holding onto.
Apply it under exam conditions
Q1. A radioactive sample has a half-life of 5 days. What fraction of the original sample remains after 15 days?
15 days is exactly 3 half-lives (), so:
Q2. A radioisotope has a half-life of 4 years. How long does it take for only 20% of the original sample to remain?
Set and solve for using logarithms:
Sanity check: 9.29 years is a little under 2.33 half-lives (), and , consistent with the target fraction.
Quick answers
Does a sample with only a little material left decay slower, since there's less of it to decay?+
No, and this is the chapter's central trap. The number of decays per second does fall as the sample shrinks, since rate = λN and N is smaller, but the fractional decay rate λ never changes, it's a fixed property of the nuclide. That's exactly why the half-life stays the same 10 years whether you start with a kilogram or a milligram: each halving takes the same fixed time, all the way down.
Does the sample ever actually reach zero undecayed nuclei?+
Mathematically, no. N = N0e^(−λt) is a pure exponential, it approaches zero but never touches it for any finite t, there's no sharp moment where 'it's all gone.' In reality, once only a handful of atoms remain, the smooth continuous formula (built for huge N) stops being a good description, and you're left with a few discrete, randomly-timed decays rather than a smoothly shrinking curve, but there's still no exact zero you can point to.
What's the difference between half-life and mean life?+
Half-life T½ is the time for N/N0 to fall to exactly 1/2. Mean life τ = 1/λ is the average lifetime of an individual nucleus before it decays, a slightly different average over all the nuclei's individual (random) decay times. Since λ = ln2/T½, the two are related by τ = T½/ln2 ≈ 1.44 × T½, so the mean life is always a bit longer than the half-life.
How does mass defect connect to the energy released in nuclear reactions?+
A nucleus's mass is always slightly less than the sum of the masses of its separated protons and neutrons, that missing mass Δm is the mass defect, and it was converted into the binding energy holding the nucleus together, via Einstein's ΔE = Δmc². Because c² is enormous, even a tiny Δm (a small fraction of one nucleon mass) corresponds to a huge binding energy, which is why nuclear reactions release so much more energy per reaction than chemical ones.
Related concepts
Physics doesn’t stay inside chapter boundaries. Neither should you.