Lesson 13: Nuclear Chemistry and Radioactivity

Radioactivity is chemistry's reminder that nuclei have their own agenda. An unstable nucleus rearranges itself, emitting alpha particles (⁴₂He nuclei), beta particles (electrons from neutron conversion) or gamma rays (pure electromagnetic energy), and the element itself changes identity. The process is random for any single nucleus yet beautifully predictable for a large sample: the half-life pattern of Lesson 12's first-order kinetics, now with consequences from medical imaging to carbon dating.

The three radiations and decay equations

Alpha decay reduces A by 4 and Z by 2: ²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He. Alpha particles are bulky and highly ionising but stopped by paper or skin. Beta decay converts a neutron to a proton, so A is unchanged and Z rises by 1: ¹⁴₆C → ¹⁴₇N + ⁰₋₁e. Beta particles penetrate a few millimetres of tissue and are stopped by aluminium sheet. Gamma emission usually accompanies the other two, carrying away leftover energy without changing A or Z; gamma rays need lead or thick concrete to attenuate. In every decay equation the totals of A and of Z balance across the arrow — that balance is how unknown products are identified.

Worked example 1: half-life and remaining mass

Iodine-131 (t½ = 8.02 days) is used to treat thyroid conditions. A dose contains 80.0 mg. How much remains after 24.06 days? Number of half-lives elapsed = 24.06 ÷ 8.02 = 3.00. Each half-life halves the amount: 80.0 → 40.0 → 20.0 → 10.0 mg. 10.0 mg remains. For non-integer times, use the fraction form: fraction remaining = (½)^(t ÷ t½). After 20.0 days: (½)^(20.0 ÷ 8.02) = (½)^2.49 = 0.178, so 80.0 × 0.178 = 14.2 mg — the method generalises to any clock time, which is what hospital physicists actually need when scheduling dose administrations.

Worked example 2: carbon-14 dating

Living wood has a characteristic ¹⁴C activity. A wooden artefact shows 25.0% of that activity. Since ¹⁴C has t½ = 5730 years, 25.0% = (½)² means exactly two half-lives have passed: age = 2 × 5730 = 11,460 years. A sample at 70% activity gives a non-integer number of half-lives: n = log(0.70) ÷ log(0.50) = 0.515, so age = 0.515 × 5730 = 2950 years. Two assumptions deserve scrutiny, because they are where dating errors genuinely arise: the atmospheric ¹⁴C level is assumed constant over history (it is not quite — calibration curves from tree rings correct for its wobbles), and the sample is assumed uncontaminated by modern carbon (a fingerprint's worth of modern material can make an old sample date centuries too young).

Fission, fusion and binding energy

Heavy nuclei can split (fission): a neutron striking ²³⁵U can produce ¹⁴¹Ba, ⁹²Kr and three more neutrons, each able to trigger another fission — the chain reaction harnessed in reactors (controlled) and weapons (uncontrolled). Light nuclei can join (fusion): the Sun fuses hydrogen into helium at 15 million °C. Both release energy because the products sit closer to the peak of the binding energy curve (iron-56): mass is converted to energy by E = mc². A typical fission event converts about 0.1% of the uranium's mass — small fraction, enormous energy, because c² is so large.

In the lab: half-life without the hazard

Schools model decay with dice or coins: 100 dice rolled together, every six removed as "decayed", plot the survivors. The class data trace the same exponential curve as real nuclei — which is precisely the point, and the uncertainty lesson is statistical honesty. With 100 dice the count in any roll scatters by roughly √N ≈ 10 around the expectation, so the plotted half-life from a single run might be 2.7 or 3.4 throws rather than the theoretical 3.80 (for a probability of 1/6 per throw, t½ = ln2 ÷ ln(6/5) = 3.80 throws). Students often report this as the model "being wrong"; in fact the scatter *is* the physics — radioactive decay is genuinely random, and real radiochemists face the same Poisson noise, beaten only by counting more events for longer. Contrast that with a systematic error in the same practical: if tired students start missing sixes during removal, that biases the half-life one way only, and no amount of extra data repairs it. Random scatter shrinks with sample size; systematic bias does not.

Chemistry in the real world

Technetium-99m, with a 6-hour half-life, is medicine's favourite tracer: long enough to image a heart, short enough to leave the patient's body quickly — its half-life is chosen, not endured. Smoke detectors exploit americium-241's alpha emission (t½ = 432 years) to ionise air steadily for decades.

Common mistakes

• Halving the time instead of the amount. Half-life divides the quantity of nuclei, repeatedly; the clock runs forward normally.
• Beta decay changing A. A stays put; a neutron becomes a proton, so Z rises by 1.
• Applying carbon dating to rocks. ¹⁴C is gone after ~50,000 years; rocks need U–Pb or K–Ar clocks (themselves half-life problems with longer t½).
• "The sample is half as radioactive after one half-life, so it is safe after two." After two half-lives a quarter remains — decay is asymptotic, never finished.

Exam Experience

1. The radiation stopped by a sheet of paper is: A) gamma B) beta C) alpha D) all three equally Answer: C. Alpha particles are large and doubly charged; they ionise intensely over a very short range and paper suffices. Beta needs aluminium; gamma needs lead or concrete.

2. After four half-lives, the fraction of a radioisotope remaining is: A) 1/4 B) 1/8 C) 1/16 D) 1/32 Answer: C. (½)⁴ = 1/16. A common slip answers 1/8 by halving once too few.

3. A nucleus emits a beta particle. Its atomic number: A) falls by 1 B) is unchanged C) rises by 1 D) rises by 2 Answer: C. A neutron converts to a proton (n → p + e⁻), so Z increases by 1 while A is unchanged.

Constructed response (5 marks). Radon-222 has a half-life of 3.82 days and decays by alpha emission (Z = 86). (a) Write the balanced nuclear equation for the decay, identifying the product's symbol and numbers. (b) A sealed source contains 4.80 mg of radon-222. Calculate the mass remaining after 11.46 days. (c) Explain why the sealed source's mass barely changes even as the radon decays. Scoring guide: (a) 1 mark for the alpha particle ⁴₂He; 1 mark for the product with A = 218 and Z = 84 (polonium, ²¹⁸₈₄Po) — totals must balance. (b) 1 mark: 11.46 ÷ 3.82 = 3.00 half-lives; 1 mark: 4.80 × (½)³ = 0.600 mg. (c) 1 mark: the alpha particles and polonium nuclei remain trapped in the sealed source, so the mass is conserved apart from the negligible mass converted to emitted energy (E = mc² scale) — the radon *identity* disappears, the atoms do not. Answers claiming the mass leaves as "radiation" earn nothing.

Worked example 3: binding energy of helium-4

Weigh a helium-4 nucleus against its parts. Two hydrogen atoms (each a proton plus an electron, mass 1.00783 u) plus two neutrons (1.00867 u) total 2(1.00783) + 2(1.00867) = 4.03300 u. A helium-4 atom weighs 4.00260 u. The mass defect is 4.03300 − 4.00260 = 0.03040 u — mass that vanished when the nucleus formed, converted to binding energy at the exchange rate 1 u = 931.5 MeV: E = 0.03040 × 931.5 = 28.3 MeV per nucleus, or 7.1 MeV per nucleon. Per mole of helium, that is 28.3 × 10⁶ × 1.602 × 10⁻¹⁹ × 6.022 × 10²³ ≈ 2.7 × 10¹² J — roughly the energy released by burning a hundred tonnes of coal, locked into 4 grams of helium by four particles holding on tight. Fusion's promise (and the Sun's power) is this arithmetic applied to hydrogen; fission's is the same curve read from the heavy end, where splitting uranium moves nucleons *toward* iron's 8.8 MeV-per-nucleon peak from uranium's 7.6.

Radiocarbon's cousins and medical isotopes

Every isotope clock is chosen for a half-life matched to its question: ¹⁴C (5730 yr) for archaeology, K–Ar (1.25 × 10⁹ yr) for rocks, ⁹⁹ᵐTc (6 h) for imaging, ¹³¹I (8 d) for thyroid therapy. The medical choices balance physics against logistics: an imaging isotope must survive transport from reactor to hospital yet clear the patient within days; cobalt-60 (5.27 yr) irradiates tumours from outside the body, so longevity is a virtue instead. Positron emission tomography (PET) uses fluorine-18 (t½ = 110 min) attached to a glucose analogue: tumours' elevated glucose uptake lights up as the positrons annihilate — half-life so short that many hospitals make it on-site in a cyclotron each morning. Nuclear chemistry is half-life engineering, and the decay equations of this lesson are its design language.

Background radiation and honest dose numbers

Every measurement in this lesson sits on a background: cosmic rays, potassium-40 in your own body (about 50 Bq in your tissues), radon seeping from granite. A Geiger counter in an empty room clicks roughly 30 times a minute, and any source measurement must subtract that baseline — a sample reading 90 counts/min is a 60 counts/min source, and forgetting the subtraction inflates its activity by 50%, the largest single error in school radioactivity work. Scale keeps the subject sane: the annual average dose is about 2.7 mSv, dominated by natural radon and medical imaging; a chest CT adds roughly 7 mSv; the banana's famous potassium contributes 0.1 µSv. Radioactivity is neither magic nor trivially safe — it is a measurable quantity with a background, a unit (the becquerel, one decay per second) and uncertainties that behave exactly as Lesson 13's dice said they should.

Becquerels, grays and sieverts: three units, three questions

Radiation measurement separates three ideas students often merge. Activity (becquerel, Bq) counts decays per second in the source — a property of the material. Absorbed dose (gray, Gy) measures energy deposited per kilogram of tissue — physics at the target. Equivalent dose (sievert, Sv) weights the absorbed dose by the radiation's biological effectiveness — alpha particles count twenty times gamma rays per gray, because their dense ionisation (Lesson 13) damages DNA in clusters that repair enzymes cannot easily fix. A chest X-ray (~0.02 mSv) and a year of natural background (~2.7 mSv) are sievert statements; the radon source in a school laboratory is a becquerel statement; confusing the units is like confusing a car's speed with its journey time — related, and dangerously not the same.