Bohr's ladder and hydrogen's lines.

• Bohr (1913) postulates: the electron moves in fixed circular orbits (stationary states) of fixed energy; energy is absorbed or emitted only in a jump between orbits, with ν = ΔE/h = (E₂ − E₁)/h (Bohr's frequency rule).
• Angular momentum is quantised: mₑvr = nh/2π with n = 1, 2, 3 … (the principal quantum number), so only certain orbits are allowed.
• Radii: rₙ = n²a₀ with a₀ = 52.9 pm, the radius of the first (Bohr) orbit, where the hydrogen electron normally sits.
• Energies: Eₙ = −2.18 × 10⁻¹⁸ (1/n²) J, so E₁ = −2.18 × 10⁻¹⁸ J and E₂ = −0.545 × 10⁻¹⁸ J. The free electron at rest (n = ∞) is zero; the negative sign means the bound electron is more stable than a free one, and n = 1 is the ground state.
• Hydrogen-like ions (He⁺, Li²⁺, Be³⁺, one electron each): Eₙ = −2.18 × 10⁻¹⁸ (Z²/n²) J and rₙ = 52.9 n²/Z pm. Larger Z binds the electron more tightly in a smaller orbit; the electron's speed rises with Z and falls as n rises.
• For a jump from nᵢ to n_f: ΔE = 2.18 × 10⁻¹⁸ (1/nᵢ² − 1/n_f²) J and ν = 3.29 × 10¹⁵ (1/nᵢ² − 1/n_f²) Hz; in wavenumbers this is Rydberg's formula with 109,677 cm⁻¹. Positive ΔE is absorption (n_f > nᵢ), negative is emission.
• Many atoms making many different jumps give the many lines; a line's brightness depends on how many photons of that wavelength are emitted or absorbed.
• Where Bohr fails: it cannot account for the closely spaced doublets seen in hydrogen's lines under fine instruments, for the spectrum of helium or any atom with more than one electron, for lines splitting in a magnetic field (Zeeman effect) or in an electric field (Stark effect), or for the way atoms join by chemical bonds.

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