Nuclear Chemistry

Nuclear chemistry studies reactions that change the composition of atomic nuclei. A nucleus is characterized by its atomic number (Z, the number of protons) and its mass number (A, the total number of protons and neutrons). Protons and neutrons are collectively called nucleons.

Not all combinations of protons and neutrons produce stable nuclei. The band of stability is the region on a plot of neutron number versus proton number where stable nuclides exist. Light stable nuclei have roughly equal numbers of protons and neutrons (n:p ≈ 1), but heavier stable nuclei require increasingly more neutrons than protons (n:p up to ∼1.5) because additional neutrons help offset the electrostatic repulsion between protons.

Nuclei that fall outside the band of stability are radioactive — they undergo spontaneous decay to move toward a more stable neutron-to-proton ratio.

Radioactive decay is the spontaneous emission of particles or energy from an unstable nucleus. The main types are:

Alpha and beta decay change the identity of the element. Gamma emission releases energy without changing the nuclide’s composition. Positron emission and electron capture both convert a proton to a neutron, lowering Z by 1.

In nuclear equations, both mass number (A) and atomic number (Z) must be conserved — the sums on each side of the arrow must be equal.

Example of alpha decay: ²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He. Check: A = 238 = 234 + 4 ✓, Z = 92 = 90 + 2 ✓.

Example of beta decay: ¹⁴₆C → ¹⁴₇N + ⁰₋₁e. A neutron converts to a proton, emitting an electron. A stays at 14, Z goes from 6 to 7.

To predict the product of a nuclear reaction, use conservation rules to find the missing A and Z values, then identify the element from the periodic table. This is the key skill: if you know the parent nuclide and the type of decay, you can always determine the daughter nuclide.

Radioactive decay is a first-order kinetic process. The half-life (t_1/2) is the time required for half the radioactive atoms in a sample to decay. After n half-lives, the fraction remaining is (1/2)ⁿ.

The quantitative relationship: N = N₀ · (1/2)^t/t_1/2, or equivalently N = N₀ · e^−λt, where the decay constant λ = 0.693 / t_1/2.

Half-lives vary enormously: ^99mTc (used in medical imaging) has t_1/2 = 6 hours, ¹⁴C (used in radiocarbon dating) has t_1/2 = 5730 years, and ²³⁸U has t_1/2 = 4.5 billion years. The half-life is a fixed property of each nuclide — it cannot be changed by temperature, pressure, or chemical environment.

Radiocarbon dating uses the known half-life of ¹⁴C (5730 years) to determine the age of organic materials. Living organisms continuously exchange carbon with their environment, maintaining a constant ¹⁴C/¹²C ratio. Once an organism dies, ¹⁴C decays without being replenished, so the ratio decreases predictably over time.

By measuring the remaining ¹⁴C/¹²C ratio and comparing it to the ratio in living organisms, you can calculate how many half-lives have elapsed and thus the age of the sample. This technique is reliable for materials up to about 50,000 years old (roughly 9 half-lives).

Other dating methods use isotopes with longer half-lives for geological timescales: uranium-lead dating (²³⁸U → ²⁰⁶Pb, t_1/2 = 4.5 × 10⁹ years) can date rocks billions of years old.

Nuclear fission is the splitting of a heavy nucleus into two lighter fragments, triggered by absorption of a neutron. The process releases enormous energy and additional neutrons.

The classic example: ²³⁵U absorbs a neutron and splits into two mid-mass nuclei (such as ⁹²Kr and ¹⁴¹Ba) plus 2–3 additional neutrons. These released neutrons can trigger further fission events, creating a chain reaction.

A chain reaction becomes self-sustaining when the critical mass of fissile material is present — enough material that, on average, each fission event triggers at least one more. In nuclear power plants, the chain reaction is controlled using neutron-absorbing rods. The energy released per fission event is roughly a million times greater than the energy released by a typical chemical reaction.

Nuclear fusion is the combining of light nuclei to form a heavier nucleus, releasing even more energy per gram than fission. The Sun is powered by fusion, primarily the conversion of hydrogen into helium.

Fusion requires extremely high temperatures (~10⁷–10⁸ K) to overcome the electrostatic repulsion between positively charged nuclei. This is why controlled fusion for power generation remains a major engineering challenge.

Both fission and fusion release energy because the products have higher nuclear binding energy per nucleon than the reactants. Binding energy is calculated from the mass defect (Δm) — the difference between the mass of the individual nucleons and the mass of the assembled nucleus — using Einstein’s equation E = Δmc². The binding-energy-per-nucleon maximum lies in the iron–nickel region: nickel-62 is the measured maximum, while iron-56 lies very near the top. Nuclear processes release energy when their products move upward toward this region of the curve.

The binding-energy calculation runs in three steps: mass defect → energy → per-nucleon comparison.

1. Mass defect. Δm = (mass of the separated pieces) − (mass of the assembled atom). Working from an atomic mass, count Z protons, (A − Z) neutrons, and Z electrons: Δm = [Z·m_p + (A−Z)·m_n + Z·m_e] − m_atom, with m_p = 1.0073, m_n = 1.0087, and m_e = 0.00055 amu. Example (F-19, atomic mass 18.99840): Δm = [9(1.0073) + 10(1.0087) + 9(0.00055)] − 18.99840 = 0.15925 amu.

2. Energy. E = Δmc². The fast route uses the conversion 1 amu = 931.5 MeV: BE = 0.15925 × 931.5 = 148.3 MeV per atom. The long route converts Δm to kilograms and multiplies by c² = (2.998 × 10⁸ m/s)² — useful when the answer is wanted in J or kJ per mole (multiply the per-atom energy by Avogadro's number). Report a per-nucleus binding energy as a plain energy (MeV or J), not "MeV/nucleus".

3. Per nucleon. BE/A divides the total binding energy by the mass number: 148.3 / 19 = 7.81 MeV/nucleon for F-19. Stability comparisons use BE per nucleon, never total BE. The curve reaches its maximum in the iron–nickel region: nickel-62 is highest, while iron-56 lies near the top at about 8.8 MeV/nucleon.

Decay-energy bookkeeping with atomic masses. The energy released by a decay is the mass lost, but electron bookkeeping depends on the mode: for electron capture, atomic masses already include the captured electron, so Δm = m(parent) − m(daughter) directly. For positron emission, subtract two electron masses: one for the emitted positron and one because the daughter atom keeps one fewer bound electron than the parent — Δm = m(parent) − m(daughter) − 2m_e. Using only one electron mass is the classic error and overstates the energy.

Activity measures how fast a source decays — the number of decays per second — and is a different quantity from the half-life (a time) and the decay constant (a rate per atom). The working equation is

A = λN, with λ = ln 2 / t_1/2 and N = (mass / molar mass) × 6.022 × 10²³

To get A in becquerels (decays per second), λ must be in s⁻¹ — convert the half-life to seconds first (1 y = 3.156 × 10⁷ s). Convert to curies at the end if asked: 1 Ci = 3.7 × 10¹⁰ Bq. Worked pattern (Co-60, t_1/2 = 5.26 y, molar mass 59.9 g/mol, 1.00 mg): λ = ln 2 / (5.26 × 3.156 × 10⁷ s) = 4.18 × 10⁻⁹ s⁻¹; N = (0.00100 / 59.9) × 6.022 × 10²³ = 1.01 × 10¹⁹ atoms; A = 4.20 × 10¹⁰ Bq = 1.14 Ci.

Comparing the activity of equal-mass samples. Both factors matter: A ∝ 1 / (t_1/2 × molar mass). When the molar masses are nearly equal (U-232 vs U-233), equal masses hold equal atom counts and the shorter half-life simply wins. When the molar masses differ greatly, the lighter isotope's larger atom count can dominate: equal masses of Ne-24 (molar mass 24, t_1/2 3.38 min) and Bi-211 (molar mass 211, t_1/2 2.14 min) give Ne-24 about 8.8× more atoms against Bi-211's mere 1.6× faster decay — so Ne-24 has roughly 5.6× the activity despite its longer half-life. Never compare half-lives alone.

A systematic approach to nuclear chemistry problems:

1. Identify the type of decay from the particle emitted: alpha (⁴₂He), beta (⁰₋₁e), positron (⁰₊₁e), or gamma (γ, no mass or charge change).
2. Balance both mass number and atomic number in every nuclear equation. The sum of mass numbers and the sum of atomic numbers must be equal on both sides.
3. For half-life calculations: use the relationship N = N₀ × (½)^t/t½. Count the number of half-lives elapsed, then halve the starting amount that many times.
4. Distinguish nuclear from chemical: nuclear reactions change the identity of the element (transmutation), involve enormous energy changes, and are unaffected by temperature, pressure, or catalysts.

Common mistakes: not balancing both mass number and atomic number independently, confusing beta decay (neutron → proton + electron) with electron capture (proton + electron → neutron), misapplying the half-life formula by using elapsed time without dividing by the half-life period, and thinking that chemical conditions (temperature, bonding) affect nuclear decay rates.

The three common types of radiation differ sharply in their penetrating power, which determines the shielding needed to stop them. Penetrating power increases in the order alpha < beta < gamma:

The large, doubly charged alpha particle interacts so strongly with matter that it is stopped almost immediately — even by paper or skin — so alpha sources are mainly dangerous only if ingested or inhaled. The lighter beta particle penetrates farther but is blocked by a thin metal sheet. High-energy gamma photons are uncharged and very penetrating; only a thick, dense, high-atomic-number shield such as lead effectively attenuates them. Match the shield to the most penetrating radiation present: a gamma source requires the thick lead block, not paper or thin metal.

Several different units describe radiation, and choosing the right one means knowing what each measures:

The becquerel (and the larger curie) describe how fast a source decays — they say nothing about how much energy reaches tissue. The gray measures the energy absorbed per kilogram, but treats all radiation equally. The sievert is the unit for equivalent dose: absorbed dose in grays multiplied by the radiation weighting factor w_R. For alpha particles w_R = 20; for beta particles and gamma rays w_R = 1. Effective dose applies an additional tissue weighting when different organs are involved. These radiation-protection factors are standardized quantities and are distinct from experimentally measured relative biological effectiveness (RBE).