Bohr's model of the hydrogen atom

Bohr (1913) explained the hydrogen spectrum with a new model. His postulates:

• The electron moves round the nucleus in fixed circular orbits, called stationary states, each with a fixed energy. In these orbits it does not radiate energy.
• The electron's angular momentum is quantised: mₑvr = nh/2π, where n = 1, 2, 3 …
• Energy is absorbed or emitted only when the electron jumps from one orbit to another. The light given out or taken in has ΔE = hν, the energy difference between the two orbits.

Bohr's model gives formulas for the size and energy of each orbit, for hydrogen and for ions with one electron (such as He⁺, Li²⁺), with atomic number Z:

• Radius: rₙ = 52.9 n²/Z pm.
• Energy: Eₙ = –2.18 × 10⁻¹⁸ (Z²/n²) J.

For hydrogen's second orbit (n = 2, Z = 1): E₂ = –2.18 × 10⁻¹⁸ ÷ 4 = –5.45 × 10⁻¹⁹ J, and r₂ = 52.9 × 4 = 211.6 pm.

Why is the energy negative? The zero of energy is set at n = ∞, where the electron is completely free of the nucleus. A bound electron has less energy than a free one, so its energy is negative. The more negative, the more tightly it is held.

The ionisation energy is the energy needed to remove the electron completely, from n = 1 to n = ∞. Because E depends on Z², it rises steeply for heavier one-electron ions.

Bohr's model works well for hydrogen, but it fails for:

• atoms with more than one electron;
• the fine doublet lines (pairs of very close lines) seen with better instruments;
• the splitting of lines in a magnetic field (Zeeman effect) and an electric field (Stark effect);
• explaining how atoms form chemical bonds.