Numford
Bohr atom

Bohr's postulates

Bohr's three postulates: the electron moves in certain orbits without radiating; in them its angular momentum is mvr = nh / 2π; and light is given out only in a jump, as one photon with hν = Eᵢ - E_f. In orbit n = 2 of hydrogen, mvr = 2.11 × 10⁻³⁴ J s = 2 × h / 2π.

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
1234
∞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, 3 to 21.89 eVBalmer α
Wavelength656.3 nmvisible

Hydrogen: the jump from n = 3 to n = 2 gives a 656.3 nm photon in the Balmer 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 3 to 2: ΔE = 13.6 × (1/4 - 1/9) = 1.89 eV
  5. 1/λ = R Z² (1/n₁² - 1/n₂²) = 1.0968 × 10⁷ × (1/4 - 1/9) = 1.523 × 10⁶ per m
  6. λ = 656.5 nm in vacuum, 656.3 nm in air: the Balmer series, visible.
Working

Step by step

  1. Postulate 2 in numbers for n = 2: r₂ = 0.529 × 4 Å = 2.12 × 10⁻¹⁰ m, v₂ = 2.19 × 10⁶ / 2 = 1.09 × 10⁶ m/s.
  2. mvr = 9.11 × 10⁻³¹ × 1.09 × 10⁶ × 2.12 × 10⁻¹⁰ = 2.11 × 10⁻³⁴ J s.
  3. nh / 2π = 2 × 6.63 × 10⁻³⁴ / 2π = 2.11 × 10⁻³⁴ J s. They match.
  4. Postulate 3: the jump 3 to 2 gives hν = E₃ - E₂ = -1.51 eV - (-3.4 eV) = 1.89 eV.
  5. That photon has λ = 656.3 nm: the red H alpha line.
  6. de Broglie's view: mvr = nh / 2π is the same as 2πr = nλ, a whole number of electron waves fitting round the orbit.
Derivation

Why it works

  1. First postulate: the electron moves in certain circular orbits without radiating energy. These are the stationary states.
  2. Second postulate: in these orbits the angular momentum is a whole number of h / 2π, mvr = nh / 2π.
  3. Third postulate: the atom gives out or takes in light only when the electron jumps between orbits, as one photon with hν = Eᵢ - E_f.
  4. Put mvr = nh / 2π into mv² / r = ke² / r² and the radius comes out as rₙ = 0.529 n² Å for hydrogen.
  5. The energy follows as Eₙ = -13.6 / n² eV: negative because the electron is bound.
Questions

Bohr's postulates: quick answers

What are Bohr's postulates?

1. The electron revolves in certain stable orbits without radiating energy. 2. In these orbits its angular momentum is an integral multiple of h / 2π. 3. It gives out or takes in a photon only when it jumps between orbits, with hν = Eᵢ - E_f.

Why did Bohr need the first postulate?

By classical physics an orbiting electron accelerates and should radiate, spiralling into the nucleus. Bohr simply declared the allowed orbits stable.

How does de Broglie explain the second postulate?

If the electron is a wave, a whole number of wavelengths must fit round the orbit: 2πr = nλ. With λ = h / mv this is mvr = nh / 2π.

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