Atoms— Notes, Formulas & Revision

Rutherford's alpha-particle scattering experiment (Geiger-Marsden, 1909): bombarded thin gold foil with α-particles, observed scattering pattern.

Most α's passed through nearly undeflected ⇒ atom is mostly EMPTY space.

A small fraction scattered at LARGE angles (some > 90°) ⇒ atom has a tiny, dense, positively charged core (nucleus).

Result: nuclear (planetary) model. Positive charge and almost all mass concentrated in nucleus (~10⁻¹⁵ m); electrons orbit at ~10⁻¹⁰ m.

Distance of closest approach for head-on collision: r_min = (1/4πε₀) · (2Ze²/K_α). For 5 MeV α on gold: r_min ~ 3×10⁻¹⁴ m.

Impact parameter b vs scattering angle: cot(θ/2) = (4πε₀·2K·b)/(2Ze²).

Limitations of Rutherford's model: orbiting electrons would radiate, spiral in, atoms would collapse — needs quantum fix (Bohr).

Bohr's three postulates for hydrogen-like atoms (1913): (i) electrons orbit only in certain ALLOWED orbits — no radiation in these; (ii) angular momentum is quantised, L = nℏ; (iii) photons emitted/absorbed when electron transitions between orbits, E_photon = E_i − E_f.

Orbit radius: r_n = (n²/Z) · a₀, where a₀ = 0.529 Å is the Bohr radius.

Energy in n-th level: E_n = −(13.6 eV/n²)·Z² for hydrogen-like atoms.

Ground state of H (n=1): E₁ = −13.6 eV, r₁ = 0.529 Å.

Velocity in n-th orbit: v_n = (Z/n) × αc, where α ≈ 1/137 is the fine-structure constant. For H, v₁ ≈ c/137.

Bohr's model is exact for one-electron atoms (H, He⁺, Li²⁺) but fails for multi-electron atoms.

Reduced to a wave-picture by de Broglie's standing-wave condition: 2πr_n = nλ_n — natural consequence of matter waves.

Atomic energy levels are the discrete energies an electron can have in an atom: E_n = −13.6 Z²/n² eV for hydrogen-like.

Levels are NEGATIVE for bound states (relative to a free electron at rest, E = 0).

Larger n: shallower (less negative) — closer to ionisation. Smaller n: deeper (more negative).

Ground state: n = 1 (lowest energy, most stable). Excited states: n = 2, 3, …

Differences between levels give photon energies in transitions: hf = E_i − E_f.

Energy-level diagram: horizontal lines labelled with n; arrows showing transitions.

Hydrogen's first excited state (n=2) is −3.4 eV; spacing E₁ − E₂ = 10.2 eV (Lyman-α).

Multi-electron atoms have more complex level structure (different sub-shells s, p, d, f within same n).

Atoms emit and absorb light only at DISCRETE wavelengths (spectral lines) — direct evidence of quantised energy levels.

Each line corresponds to a transition between two specific levels: hf = E_i − E_f.

Hydrogen line series (named by final state n_f): Lyman (n_f=1, UV), Balmer (n_f=2, visible), Paschen (n_f=3, IR), Brackett (n_f=4, IR), Pfund (n_f=5, IR).

Lyman-α (n=2 → 1): 121.6 nm (UV). Balmer-α (n=3 → 2): 656.3 nm (red, Hα).

Series limit: shortest wavelength in a series — corresponds to n_i = ∞. λ_lim(Lyman) = 91.2 nm. λ_lim(Balmer) = 364.6 nm.

Emission spectrum (hot atoms emit lines) vs absorption spectrum (cool atoms in front of bright continuum — dark lines).

Fraunhofer lines in the solar spectrum are absorption lines from cooler gas in the Sun's atmosphere.

Hydrogen's optical spectrum was the most-studied — its simplicity (one electron) makes it a textbook of quantum mechanics.

Five named series, each ending on a specific lower level n_f:

Lyman (n_f = 1): UV, lines at 121.6, 102.6, 97.3, … nm. Series limit at 91.2 nm.

Balmer (n_f = 2): visible, Hα 656.3, Hβ 486.1, Hγ 434.0, Hδ 410.2 nm. Limit at 364.6 nm.

Paschen (n_f = 3): near-IR, longest 1875 nm. Limit at 820 nm.

Brackett (n_f = 4): IR, around 4 μm. Limit at 1458 nm.

Pfund (n_f = 5): far-IR, around 7.5 μm. Limit at 2280 nm.

Universal pattern: 1/λ = R · (1/n_f² − 1/n_i²) — works for ALL series.

An atomic transition is the jump of an electron between two energy levels, with absorption or emission of a photon.

Energy conservation: hf = |E_i − E_f|. Frequency uniquely set by the level pair.

Emission (downward): electron drops from higher to lower level, EMITS a photon.

Absorption (upward): electron jumps from lower to higher level after ABSORBING a photon of matching energy.

Selection rules (full QM): Δl = ±1, Δm = 0, ±1. Not all transitions are allowed.

Lifetime of typical excited states: ~10⁻⁸ s (allowed) to ~10⁻³ s or longer (forbidden metastable).

Spontaneous emission: an isolated excited atom decays randomly; lifetime obeys exponential decay.

Stimulated emission: a photon of matching energy stimulates an excited atom to emit a coherent photon — basis of LASERS.