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Bohr atom

Bohr model of the hydrogen atom

In the Bohr model the hydrogen electron moves in allowed orbits of radius rₙ = 0.529 n² Å and energy Eₙ = -13.6 / n² eV. The ground state has r₁ = 5.29 × 10⁻¹¹ m and E₁ = -13.6 eV; falling from n = 2 to 1 gives a 121.6 nm ultraviolet photon.

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 r₁0.529 Å5.29 × 10⁻¹¹ m
Speed v₁2.19 × 10⁶m/s
Energy E₁-13.6 eVground state
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. r₁ = 0.529 n² / Z Å = 0.529 × 1 / 1 = 0.529 Å = 5.29 × 10⁻¹¹ m
  2. v₁ = 2.19 × 10⁶ × Z / n = 2.19 × 10⁶ m/s
  3. E₁ = -13.6 Z² / n² eV = -13.6 / 1 = -13.6 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. r₁ = 0.529 n² / Z Å = 0.529 × 1 / 1 = 0.529 Å = 5.29 × 10⁻¹¹ m
  2. v₁ = 2.19 × 10⁶ × Z / n = 2.19 × 10⁶ m/s
  3. E₁ = -13.6 Z² / n² eV = -13.6 / 1 = -13.6 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

Bohr model of the hydrogen atom: quick answers

What are the main results of the Bohr model?

For a hydrogen-like atom of atomic number Z: radius rₙ = 0.529 n² / Z Å, speed vₙ = 2.19 × 10⁶ Z / n m/s and energy Eₙ = -13.6 Z² / n² eV.

Why is the energy negative?

Zero is the energy of an electron that is free and at rest, far from the nucleus. A bound electron has less than that, so its energy is negative. 13.6 eV must be given to free it from the ground state.

What are the limits of the Bohr model?

It works for one electron only (H, He⁺, Li²⁺). It cannot explain the relative brightness of lines, their fine structure, or atoms with many electrons.

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