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Bohr atom

Bohr radius and the radius of the nth orbit

The Bohr radius, the radius of the first orbit of hydrogen, is a₀ = 0.529 Å = 5.29 × 10⁻¹¹ m. The nth orbit has rₙ = 0.529 n² / Z Å: four times bigger for n = 2, nine times for n = 3.

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.
Radius of the nth orbit

rₙ = 0.529 n² / Z Å

nHHe⁺Li²⁺
10.529 Å0.265 Å0.176 Å
22.12 Å1.06 Å0.705 Å
34.76 Å2.38 Å1.59 Å
48.46 Å4.23 Å2.82 Å
513.2 Å6.61 Å4.41 Å
619 Å9.52 Å6.35 Å
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 radius: quick answers

What is the Bohr radius?

The radius of the smallest orbit of the hydrogen atom in the Bohr model: a₀ = 0.529 Å = 5.29 × 10⁻¹¹ m.

How does the radius depend on n and Z?

rₙ = 0.529 n² / Z Å. It grows as the square of n, and a bigger nuclear charge pulls the orbits in.

What is the radius of the third orbit of He⁺?

r₃ = 0.529 × 9 / 2 = 2.38 Å.

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