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Bohr atom

Rydberg formula

The Rydberg formula gives every hydrogen line: 1/λ = R Z² (1/n₁² - 1/n₂²), with R = 1.097 × 10⁷ per metre. For the jump from 2 to 1 (Lyman alpha) it gives λ = 121.6 nm.

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.
Worked lines

The first line of each series

Line1/n₁² - 1/n₂²1/λ (per m)λ
Lyman α0.758.226 × 10⁶121.6 nm
Lyman β0.88899.749 × 10⁶102.6 nm
Balmer α0.13891.523 × 10⁶656.3 nm
Balmer β0.18752.056 × 10⁶486.1 nm
Paschen α0.048615.332 × 10⁵1875.1 nm
Derivation

Why it works

  1. An electron falling from level n₂ to n₁ gives a photon of energy hν = Eₙ₂ - Eₙ₁.
  2. With Eₙ = -13.6 Z² / n² eV this is hc / λ = 13.6 Z² (1/n₁² - 1/n₂²) eV.
  3. Dividing by hc gives 1/λ = R Z² (1/n₁² - 1/n₂²), the Rydberg formula.
  4. R = 1.097 × 10⁷ per metre. For hydrogen, whose nucleus also moves a little, R = 1.0968 × 10⁷ per metre.
  5. Each n₁ gives a series: 1 Lyman, 2 Balmer, 3 Paschen, 4 Brackett, 5 Pfund.
Questions

Rydberg formula: quick answers

What is the Rydberg constant?

R = 1.097 × 10⁷ per metre. For hydrogen, allowing for the motion of the nucleus, it is 1.0968 × 10⁷ per metre, which matches the measured lines.

How is the Rydberg formula used for He⁺ and Li²⁺?

Multiply by Z²: 1/λ = R Z² (1/n₁² - 1/n₂²). He⁺ (Z = 2) has every line four times the wavenumber of the matching hydrogen line.

How does the Rydberg formula follow from the Bohr model?

The photon carries the energy difference, hc / λ = 13.6 Z² (1/n₁² - 1/n₂²) eV. Dividing by hc gives the formula, with R = 13.6 eV / hc.

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