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Bohr atom

Balmer series of hydrogen

The Balmer series is the light hydrogen gives when its electron falls to n = 2. Its lines are H alpha 656.3 nm (red), H beta 486.1 nm, H gamma 434.1 nm and H delta 410.2 nm, closing on the limit at 364.6 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
1234
∞0n=1-13.6n=2-3.4n=3-1.51eVLymanBalmerPaschenBrackettPfund
Radius r₃4.76 Å4.76 × 10⁻¹⁰ m
Speed v₃7.29 × 10⁵m/s
Energy E₃-1.51 eV12.09 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. r₃ = 0.529 n² / Z Å = 0.529 × 9 / 1 = 4.76 Å = 4.76 × 10⁻¹⁰ m
  2. v₃ = 2.19 × 10⁶ × Z / n = 7.29 × 10⁵ m/s
  3. E₃ = -13.6 Z² / n² eV = -13.6 / 9 = -1.51 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. r₃ = 0.529 n² / Z Å = 0.529 × 9 / 1 = 4.76 Å = 4.76 × 10⁻¹⁰ m
  2. v₃ = 2.19 × 10⁶ × Z / n = 7.29 × 10⁵ m/s
  3. E₃ = -13.6 Z² / n² eV = -13.6 / 9 = -1.51 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.
The visible lines

Balmer lines

LineJumpWavelengthEnergy
H alpha3 to 2656.3 nm1.89 eV
H beta4 to 2486.1 nm2.55 eV
H gamma5 to 2434.1 nm2.86 eV
H delta6 to 2410.2 nm3.02 eV
Limit∞ to 2364.6 nm3.4 eV
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

Balmer series: quick answers

What is the Balmer series?

The lines of hydrogen in which the electron falls to the second level, n₁ = 2. They are given by 1/λ = R (1/2² - 1/n²) with n = 3, 4, 5 and so on.

What is the wavelength of H alpha?

656.3 nm in air (656.5 nm in vacuum), a deep red. It comes from the jump n = 3 to n = 2.

Why was the Balmer series found first?

It is the only hydrogen series in visible light, so it could be seen and measured with ordinary prisms and gratings, in 1885.

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