E.3 Radioactive decay

UNIT SYLLABUS

SL/HL 7 hours SL + 5 hours HL

Some nuclei are unstable, and their decay is nature's purest randomness: no trigger, no memory, just a fixed probability per unit time. Alpha, beta and gamma emissions each change the nucleus in characteristic ways, with penetrating powers spanning paper to lead. Binding energy — the mass defect via $E = mc^2$ — explains both why decay releases energy and why iron sits at the curve's peak, dividing fusion territory from fission territory. The half-life turns randomness into clockwork at scale: at HL the exponential decay law $N = N_0e^{-\lambda t}$ makes it quantitative, dating everything from archaeological remains to the Earth itself.

Guiding Questions

? Why are some nuclei stable while others decay?
? How can a process that is random for one nucleus be so predictable for many?

Exponential decay: each half-life halves the remaining nuclei. After n half-lives, a fraction 1/2ⁿ remains.

✓ What the IB expects you to master

Define isotopes, nuclear binding energy and mass defect; use $E = mc^2$ in nuclear reactions.
Interpret the binding-energy-per-nucleon curve, including its maximum near iron and the approximate constancy above $A \approx 60$ (HL detail).
Describe the strong nuclear force: short-range, attractive, acting between nucleons.
Explain the random, spontaneous nature of decay, and balance decay equations for $\alpha$ , $\beta^-$ , $\beta^+$ and $\gamma$ emissions, including neutrinos and antineutrinos.
Compare the penetration and ionising power of alpha, beta and gamma radiation.
Use activity, count rate and half-life, with integer numbers of half-lives (SL); account for background radiation.
HL: explain nuclear stability via the neutron–proton ratio, and the evidence for nuclear energy levels (discrete alpha/gamma spectra) and for the neutrino (continuous beta spectrum).
HL: apply the decay law $N = N_0e^{-\lambda t}$ , activity $A = \lambda N = A_0e^{-\lambda t}$ , and $T_{1/2} = \frac{\ln 2}{\lambda}$ .

1 Key Formulas

Mass–energy equivalence

E = mc^{2}

Decay law (HL)

N = N_{0}e^{-\lambda t}

Activity (HL)

A = \lambda N = A_{0}e^{-\lambda t}

Half-life and decay constant (HL)

T_{\frac{1}{2}} = \frac{\ln 2}{\lambda}

2 Exam Preparation & Topic Explanations

Half-life arithmetic and beyond

SL questions stay on integer half-lives: divide by two the right number of times, remembering to strip background first. HL unlocks arbitrary times via $N = N_0e^{-\lambda t}$ — get $\lambda$ from $\ln 2 / T_{1/2}$ and keep units consistent throughout.

Binding-energy calculations follow one recipe: mass defect in u → multiply by 931.5 for MeV. State whether energy is released (products more bound) or required.

Pro Exam Strategy

Alpha: helium nucleus, stopped by paper, heavily ionising. Beta: electron/positron, stopped by aluminium. Gamma: photon, attenuated by lead. Know all nine facts.
Decay is random per nucleus but statistically exponential for large N — be ready to articulate both halves.
HL neutron–proton ratio: light stable nuclei have $N \approx Z$ ; heavy ones need $N > Z$ ; the wrong side of the stability band decides $\beta^-$ vs $\beta^+$ .
Discrete alpha/gamma spectra ↔ nuclear energy levels; continuous beta spectrum ↔ neutrino. Two pieces of evidence, two conclusions.

3 MCQ Practice

Q1. A sample contains $8.0\times10^{20}$ radioactive nuclei with a half-life of 6.0 hours. After one day, the number remaining is:

$4.0\times10^{20}$
$2.0\times10^{20}$
$1.0\times10^{20}$
$0.5\times10^{20}$

Q2. In beta-minus decay, the emitted electrons show a continuous energy spectrum. This was evidence for:

Discrete nuclear energy levels
The existence of the antineutrino, sharing the decay energy
The quantisation of charge
The strong nuclear force

Q3. The binding energy per nucleon curve peaks near iron ( $A \approx 56$ ). This explains why:

Iron is the most abundant element
Fusion releases energy below iron and fission above it
Iron cannot undergo any nuclear reactions
All nuclei decay to iron eventually

4 Short Answer Questions

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