Atoms and Nuclei
Bohr model, hydrogen spectrum, nuclear binding energy, radioactivity, fission, fusion.
Bohr model and hydrogen spectrum
Quantized orbits, Rydberg formula, Lyman/Balmer/Paschen.
Bohr (1913) extended the Rutherford planetary model with quantum postulates:
Postulate 1: Quantized orbits. Electrons orbit only in specific allowed orbits where angular momentum is quantized:
mvr = n · h/(2π), n = 1, 2, 3, ...
Postulate 2: No radiation in stationary orbits. Despite being accelerated, electrons in allowed orbits don't radiate (this overrides classical electrodynamics).
Postulate 3: Photon emission/absorption. Transitions between orbits emit or absorb a photon of energy:
E_photon = E_n − E_m = hν
Bohr radius (smallest allowed orbit, n=1): a₀ = 0.529 Å = 5.29 × 10⁻¹¹ m.
Radius of nth orbit: r_n = n² · a₀ / Z (for hydrogen-like ion with Z protons).
Energy of nth level (hydrogen):
E_n = −13.6 / n² eV
(For Z, multiply by Z².)
Hydrogen spectrum series:
| Series | Final n | Region |
|---|---|---|
| Lyman | n = 1 | UV |
| Balmer | n = 2 | Visible |
| Paschen | n = 3 | IR |
| Brackett | n = 4 | IR |
| Pfund | n = 5 | IR |
Rydberg formula:
1/λ = R · (1/n₁² − 1/n₂²), R = 1.097 × 10⁷ m⁻¹ (for hydrogen).
Worked example. Wavelength of the first Balmer line (n=3 → n=2)?
1/λ = R(1/4 − 1/9) = R · 5/36 = 1.097e7 × 0.1389 = 1.524e6.
λ = 1/1.524e6 = 6.56e-7 m = 656 nm — this is the famous H-α red line.
Bohr's limitations: can't explain (a) hydrogen-like ions imperfectly, (b) fine structure (relativistic corrections), (c) Zeeman effect (B-field splitting), (d) any multi-electron atom. Modern quantum mechanics (Schrödinger equation) replaced Bohr but keeps the energy levels for hydrogen.
Nuclear binding energy
Mass defect, BE per nucleon, fission/fusion.
Mass defect (Δm): the mass of a nucleus is less than the sum of its constituent nucleons. The missing mass is converted to binding energy.
Binding energy: E_B = Δm · c²
The mass defect Δm = (Z m_p + N m_n) − m_nucleus.
Using atomic mass units: 1 u = 931.5 MeV/c². So if Δm is in u, E_B (in MeV) = Δm × 931.5.
Binding energy per nucleon = E_B / A. This is what really matters for stability.
- Peak at A ≈ 56 (Fe-56) — most stable nucleus.
- Light nuclei (A < 56): fusion releases energy (smaller → larger increases BE/nucleon).
- Heavy nuclei (A > 56): fission releases energy (larger → smaller increases BE/nucleon).
This explains why both stars (fusion) and reactors (fission) release energy.
Radioactive decay — random, statistical process.
Decay law: N(t) = N₀ e^(−λt), where λ = decay constant.
Activity (decays per second): A = λN = A₀ e^(−λt).
Half-life (time for N to halve):
T₁/₂ = ln 2 / λ = 0.693 / λ
Mean life (average lifetime of a nucleus): τ = 1/λ = T₁/₂ / 0.693 = 1.44 T₁/₂.
After n half-lives: N = N₀ / 2ⁿ. Fraction remaining after 10 half-lives: 1/1024 ≈ 0.1%.
Three types of decay:
| Decay | Particle emitted | What changes | Conservation |
|---|---|---|---|
| α | ⁴₂He | A: −4, Z: −2 | Mass, charge |
| β⁻ | e⁻ + ν̄_e | A: 0, Z: +1 (neutron→proton) | A unchanged |
| β⁺ | e⁺ + ν_e | A: 0, Z: −1 (proton→neutron) | A unchanged |
| γ | photon | none (just energy release) | Mass, charge unchanged |
Mnemonic:
- α: a heavy thing (helium nucleus) flies out.
- β⁻: electron flies out, neutron turned into proton (gain Z).
- γ: gamma — just photons, nucleus settles to lower energy.
Fission (U-235 + slow neutron):
²³⁵U + n → ²³⁶U* → fission fragments + 2 or 3 neutrons + ~200 MeV.
Each fission releases enough neutrons to potentially trigger more → chain reaction. Controlled (reactor) vs uncontrolled (bomb).
Critical mass = minimum mass to sustain chain reaction. ~50 kg for U-235, less for Pu-239.
Fusion (Sun, hydrogen bomb):
4 H → He + 2 e⁺ + 2 ν + ~26 MeV.
The proton-proton chain in the Sun's core. Requires temperatures > 10⁷ K (overcome Coulomb repulsion). Fusion energy per kg is ~3.5× that of fission.
ITER (in France, under construction) aims for sustained fusion via tokamak (magnetic confinement of plasma).
Worked example. A radioactive isotope has half-life 10 days. What fraction of nuclei remains after 30 days?
n = 30/10 = 3 half-lives. Fraction = 1/2³ = 1/8 = 12.5%.
Radioactivity
α, β, γ decay; half-life; decay law.