Nuclei — Exam Practice

Answer Nuclear radius follows R = R₀A^1/3, so volume ∝ A. Since mass also scales with A (each nucleon contributes roughly the same mass), density = mass/volume stays roughly constant across every nucleus, light or heavy.

Answer Binding energy per nucleon rises sharply for light nuclei, peaks around A ≈ 56 (iron/nickel), then declines slowly for heavier nuclei.

Fission splits a heavy nucleus (on the declining side) into two lighter nuclei closer to the peak — each fragment is more tightly bound per nucleon than the original, releasing the difference as energy.

Fusion combines light nuclei (on the rising side) into a heavier one, also moving toward the peak and releasing energy for the same underlying reason — either direction of movement toward the peak releases energy.

Answer (a) N = N₀e^−λt. At t = T_½, N = N₀/2:
N₀/2 = N₀e^−λT½ → ln(2) = λT_½ → T_½ = 0.693/λ.

(b) 1/8 = (1/2)ⁿ → n = 3 half-lives.
Time = 3 × 20 = 60 days.

Solution Δm = [Zm_p + (A−Z)m_n] − M_nucleus — this "missing" mass has been converted to binding energy holding the nucleus together.

Solution β⁻ decay converts a neutron to a proton within the nucleus — total nucleon count (A) stays the same, while atomic number Z increases by 1.

Solution Effective only over a few femtometres, and acts the same way between proton-proton, proton-neutron, and neutron-neutron pairs alike.

Solution The two (or more) fragments fly apart carrying the bulk of the ~200 MeV released per fission event as kinetic energy, with smaller contributions from gamma rays and neutrons.

Solution E_b = Δm × 931.5 MeV = 0.025 × 931.5 ≈ 23.3 MeV.

Solution An emitted alpha particle is a helium nucleus (2 protons + 2 neutrons), so A decreases by exactly 4, and Z decreases by 2.

Solution T_½ = 0.693/λ = 0.693/0.0231 ≈ 30 days.