Bohr Model: Energy Levels, Radius & Hydrogen Spectrum

The Bohr model says an electron in a hydrogen atom can only occupy certain fixed orbits, each with a definite energy. The lowest-energy orbit (n = 1) is the ground state; higher orbits (n = 2, 3, 4, …) are excited states with higher, less negative energies. An electron is never found between two allowed orbits.

This matters because the exact energies of these levels predict the exact colours hydrogen emits and absorbs. The hydrogen spectrum — the coloured lines seen when hydrogen gas glows — was a mystery until Bohr's formula matched every line, making it the simplest example of energy quantisation in science.

Key Concepts

• Energy level (En): the fixed energy of the n-th orbit. For hydrogen, En = −13.6 eV / n² = −2.18 × 10⁻¹⁸ J / n². The negative sign means the electron is bound; zero energy means it has escaped the atom.
• Bohr radius (a0): the radius of the smallest orbit, a0 = 5.29 × 10⁻¹¹ m = 0.529 Å. The n-th orbit has radius rn = a0·n²/Z.
• Ground state vs excited state: n = 1 is the ground state; any n > 1 is an excited state that can decay by emitting a photon.
• Spectral series: groups of lines ending on the same level — Lyman (n1 = 1, ultraviolet), Balmer (n1 = 2, visible), Paschen (n1 = 3, infrared), Brackett (n1 = 4) and Pfund (n1 = 5).
• Rydberg constant: R = 1.097 × 10⁷ m⁻¹, used in the wavelength formula 1/λ = R·Z²·(1/n1² − 1/n2²).

Formulas

• Energy of level n: En = −RH·Z²/n², with RH = 2.18 × 10⁻¹⁸ J = 13.6 eV. Units: joules or electronvolts; Z is the atomic number (Z = 1 for hydrogen).
• Photon energy for a jump: ΔE = RH·Z²·(1/n1² − 1/n2²), where n2 > n1 for emission.
• Orbit radius: rn = (0.529 Å)·n²/Z. For n = 2 in hydrogen: r = 0.529 × 4 = 2.12 Å.

These formulas apply to hydrogen-like species with one electron only (H, He⁺, Li²⁺).

Worked Example

Find the wavelength of the red Balmer line, the jump from n = 3 to n = 2 in hydrogen.

1. Use 1/λ = R(1/n1² − 1/n2²) with n1 = 2, n2 = 3, R = 1.097 × 10⁷ m⁻¹.
2. Bracket: 1/4 − 1/9 = 9/36 − 4/36 = 5/36 = 0.1389.
3. Multiply: 1/λ = 1.097 × 10⁷ × 0.1389 = 1.524 × 10⁶ m⁻¹.
4. Invert: λ = 1 / 1.524 × 10⁶ = 6.56 × 10⁻⁷ m = 656 nm.

The wavelength is 656 nm, the familiar red line of glowing hydrogen.

Common Misconceptions

• Higher orbits have lower energy. The energies are negative, so n = 3 (−1.51 eV) is higher energy than n = 1 (−13.6 eV). Larger n means a less tightly bound electron.
• An electron spirals gradually between orbits. No — jumps are instantaneous, and the photon carries exactly the energy difference, no more and no less.
• The Bohr radius is the size of every atom. It is only the n = 1 radius of hydrogen; multi-electron atoms are described by orbitals, not Bohr orbits.

Real-World Applications

• Astronomers identify hydrogen in stars and nebulae from Balmer and Lyman lines.
• Hydrogen discharge lamps and neon signs glow through electron jumps between levels.
• Lasers work on the same principle: stimulated jumps between fixed energy levels.
• Atomic clocks and spectroscopy standards rely on exact, quantised energy differences.

Bohr's fixed orbits give energies En = −13.6 eV/n² and radii rn = 0.529 Å·n², and the gap between any two levels fixes the colour of light absorbed or emitted. That one idea explains the whole hydrogen spectrum — and opened the door to quantum mechanics.