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Bohr atom

Energy levels of the hydrogen atom

The energy levels of hydrogen are Eₙ = -13.6 / n² eV: E₁ = -13.6 eV, E₂ = -3.4 eV, E₃ = -1.51 eV. The first excitation energy is 10.2 eV and the ionisation energy is 13.6 eV.

Bohr model

Orbits grow as n². Energies are -13.6 Z² / n² eV. Wavelengths use the Rydberg constant of each nucleus (1.0968 × 10⁷ per metre for hydrogen); above 200 nm they are given in air, as spectrum tables list them.

Atom
Level n
Jump from n₂
down to n₁
123
∞0n=1-13.6n=2-3.4n=3-1.51eVLymanBalmerPaschenBrackettPfund
Radius r₂2.12 Å2.12 × 10⁻¹⁰ m
Speed v₂1.09 × 10⁶m/s
Energy E₂-3.4 eV10.2 eV above ground
Photon, 2 to 110.2 eVLyman α
Wavelength121.6 nmultraviolet

Hydrogen: the jump from n = 2 to n = 1 gives a 121.6 nm photon in the Lyman series.

Show the working
  1. r₂ = 0.529 n² / Z Å = 0.529 × 4 / 1 = 2.12 Å = 2.12 × 10⁻¹⁰ m
  2. v₂ = 2.19 × 10⁶ × Z / n = 1.09 × 10⁶ m/s
  3. E₂ = -13.6 Z² / n² eV = -13.6 / 4 = -3.4 eV
  4. Jump 2 to 1: ΔE = 13.6 × (1/1 - 1/4) = 10.2 eV
  5. 1/λ = R Z² (1/n₁² - 1/n₂²) = 1.0968 × 10⁷ × (1/1 - 1/4) = 8.226 × 10⁶ per m
  6. λ = 121.6 nm: the Lyman series, ultraviolet.
Working

Step by step

  1. r₂ = 0.529 n² / Z Å = 0.529 × 4 / 1 = 2.12 Å = 2.12 × 10⁻¹⁰ m
  2. v₂ = 2.19 × 10⁶ × Z / n = 1.09 × 10⁶ m/s
  3. E₂ = -13.6 Z² / n² eV = -13.6 / 4 = -3.4 eV
  4. Jump 2 to 1: ΔE = 13.6 × (1/1 - 1/4) = 10.2 eV
  5. 1/λ = R Z² (1/n₁² - 1/n₂²) = 1.0968 × 10⁷ × (1/1 - 1/4) = 8.226 × 10⁶ per m
  6. λ = 121.6 nm: the Lyman series, ultraviolet.
Level by level

Radius, speed and energy

nrₙvₙ (m/s)Eₙ
10.529 Å2.19 × 10⁶-13.6 eV
22.12 Å1.09 × 10⁶-3.4 eV
34.76 Å7.29 × 10⁵-1.51 eV
48.46 Å5.47 × 10⁵-0.85 eV
513.2 Å4.38 × 10⁵-0.54 eV
619 Å3.65 × 10⁵-0.38 eV
Derivation

Why it works

  1. The electron moves in a circle and the Coulomb pull gives the centripetal force: mv² / r = kZe² / r².
  2. Only orbits with angular momentum mvr = nh / 2π are allowed.
  3. Together these give rₙ = 0.529 n² / Z Å and vₙ = 2.19 × 10⁶ Z / n m/s.
  4. The total energy is half the potential energy: Eₙ = -13.6 Z² / n² eV.
  5. Falling from n₂ to n₁ gives one photon with hν = Eₙ₂ - Eₙ₁, which is the Rydberg formula.
Questions

Energy levels of hydrogen: quick answers

What is the first excitation energy of hydrogen?

The energy to lift the electron from n = 1 to n = 2: E₂ - E₁ = -3.4 - (-13.6) = 10.2 eV.

What is the ionisation energy of hydrogen?

The energy to free the electron from the ground state: 0 - (-13.6) = 13.6 eV. The ionisation potential is 13.6 V.

Why do the levels crowd together near the top?

Eₙ goes as 1 / n², so the gaps shrink fast: 10.2 eV from 1 to 2, but only 0.66 eV from 3 to 4. They close in on 0 eV as n grows.

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