Nuclei and Radioactivity — Formula Sheet

R = r₀ A^(1/3), where r₀ = 1.2 × 10⁻¹⁵ m (1.2 fm)

N = A − Z

Nuclear density d = 3m_p / (4π r₀³) ≈ 2.3 × 10¹⁷ kg m⁻³

Δm = [Z·mₚ + (A−Z)·mₙ] − M

BE = Δm × c² (joules) OR BE = Δm × 931.3 MeV (when Δm in u)

B = BE / A = [Z·mₚ + (A−Z)·mₙ − M] × 931.3 / A MeV per nucleon

dN/dt = −λN(t)

N(t) = N₀ exp(−λt)

T₁/₂ = 0.693 / λ (since ln 2 = 0.693)

λ = −(1/N) × dN/dt

After n half-lives: N = N₀ / 2ⁿ

By Einstein's mass-energy equivalence (E = mc²), this lost mass appears as the energy that binds the nucleons together — called BINDING ENERGY (BE).

Nuclear radius formula

R = r₀A^(1/3), where r₀ = 1.2 fm = 1.2 × 10⁻¹⁵ m

Nuclear volume ∝ A (mass number) → nuclear density is constant for all nuclei

Nuclear density ≈ 2.3 × 10¹⁷ kg/m³ — extremely high, about 10¹⁴ times denser than water

Binding Energy BE = Δm × c² = Δm × 931.3 MeV (when Δm in u)

Binding energy per nucleon B = BE/A — measures stability of nucleus

The radius of a nucleus is given by the empirical formula

R = r₀A^(1/3), where r₀ = 1.2 fermi = 1.2 × 10⁻¹⁵ m (1 fermi = 10⁻¹⁵ m).