Physics › Nuclear physics › Radioactive decay and half-life
Radioactive decay and half-life
No nucleus knows its age and none can be hurried, yet a mole of them keeps time better than any clock. One constant, lambda, sets the odds; from it come the exponential, the half-life, and a straight-line log plot you have met once before.
Builds on The time constant and exponential decay and Stable and unstable nuclei.
IN THIS TOPIC
- Use λ, A = λN and the exponential decay equations, converting a mass to a number of nuclei when needed.
- Determine a half-life from decay curves and from log graphs.
- Predict the decay mode of a nuclide from its position on the N against Z graph and write the decay equation.
WHAT YOU PROBABLY THINK
After two half-lives, a sample has all gone.
Random for one, dependable for a mole
Radioactive decay is random: no measurement can tell you when a given nucleus will go, and nothing you do to it, heat, pressure, chemistry, changes its chances. What each nucleus has instead is a fixed decay constant λ, the probability of decaying per second. Model it with dice: each die has no memory, yet a large enough handful loses very nearly one sixth per throw.
With N undecayed nuclei each carrying probability λ per second, the expected rate of loss is
and the size of that rate is the activity A, in becquerels, one decay per second:
Counting nuclei is chemistry's job. A mass m of a nuclide with molar mass M holds m/M moles, so N = (m/M) × NA nuclei, with the Avogadro constant from the data booklet. Exam questions lean on this chain often.
The exponential and its half-life
A rate of loss proportional to the amount remaining is the recipe you met with capacitor discharge, and it has the same solution:
The half-life is the time for N to halve. Set N = N0/2 and take logs: e-λT = 1/2 gives λT = ln 2, so
A big decay constant means a short half-life and a fierce source. Because activity is proportional to N, it decays with the same clock,
which is the form the detector sees. And the lie above answers itself: two half-lives leave a quarter, three an eighth. Halving forever never reaches nothing.
Reading decay data
From a decay curve, read the half-life directly: find the time for the activity to halve, then check it again from any other starting point; a genuine exponential halves in the same time everywhere along its length.
For scattered data the better tool is the promised twin of the capacitor practical. Taking natural logs gives ln N = ln N0 − λt, a straight line of ln N against t with gradient −λ. Straightness is the evidence the decay is exponential, and the gradient hands you λ, from which T½ = ln 2/λ follows. Symbol for symbol, this is the RP9 analysis with λt in place of t/RC.
The half-life sets the applications. Radioactive waste must be stored securely for many half-lives of its longest-lived components, which for some isotopes means centuries. Dating runs the clock backwards: living things hold a known fraction of carbon-14, so the fraction remaining in a bone, with T½ = 5730 years, dates its death.
Which nuclei decay, and how
Plot every stable nuclide as a point with proton number Z across and neutron number N up, and they hug a narrow stable band: N roughly equal to Z for light nuclei, bending neutron-rich as Z grows, because extra neutrons add strong-force glue without adding proton repulsion.
Position on the map predicts the decay. Neutron-rich nuclei, above the band, undergo β⁻ decay: a neutron becomes a proton, so N falls by 1 and Z rises by 1. ⁹⁰Sr becomes ⁹⁰Y (Z: 38 to 39) plus e⁻ and an antineutrino e. Proton-rich nuclei, below the band, go the other way by β⁺ decay or electron capture, each turning a proton into a neutron. The heaviest nuclei shed bulk by α decay: ²²⁶Ra becomes ²²²Rn (Z: 88 to 86) plus ⁴He, dropping A by 4 and Z by 2. In every equation, A and Z balance across the arrow.
Decay often leaves the daughter in an excited state: nuclei have energy levels like atoms, drawn as nuclear energy level diagrams, and the drop to the ground state emits a γ photon. Medicine exploits one such state deliberately. Technetium-99m is a long-lived excited state that emits gamma alone, with a six hour half-life: injected as a tracer, its photons escape the body to a camera, and the source is gone by the next day.
THE EXAM BIT
- The word random earns its mark with a definition: constant probability of decay per nucleus per unit time, unaffected by conditions. Say probability, not chance it might happen.
- Convert mass to nuclei via moles: divide by molar mass, multiply by the Avogadro constant. Grams and kilograms trip this chain; the booklet's Nₐ is per mole.
- Half-life from a log graph: the gradient is −λ, so read its magnitude and use T½ = ln 2/λ. Quote the straightness of the line as your evidence the decay is exponential.
- Decay equations balance twice: nucleon numbers across the top, proton numbers across the bottom, with the antineutrino written in for β⁻.
- Technetium-99m answers want three properties: gamma only, so it leaves the body without heavy ionisation; a six hour half-life, long enough to image, short enough to clear; and emission from an excited nuclear state.
CHECK YOURSELF
Caesium-137 has a half-life of 30 years. A sealed source contains 3.0 μg of caesium-137 (molar mass 137 g mol⁻¹). Find the decay constant and the activity of the source. Take 1 year = 3.15 × 10⁷ s.
Show a hint
Half-life to λ first; then mass to moles to nuclei; then A = λN.
Show the answer
T½ = 30 × 3.15 × 107 = 9.45 × 108 s, so λ = ln 2 / 9.45 × 108 = 7.3 × 10-10 s-1.
N = (3.0 × 10-6 / 137) × 6.02 × 1023 = 1.3 × 1016 nuclei.
A = λN = 7.3 × 10-10 × 1.3 × 1016 = 9.7 × 106 Bq: ten million decays a second from three millionths of a gram.
One nucleus is a coin toss; a mole of them is a clock.
Every half-life keeps the same fraction, so the count halves forever and never reaches zero.
No animated video for this topic yet; these notes stand alone. InkPhysics on YouTube.