Numford
Bohr atom

Spectral series of hydrogen: Lyman to Pfund

Hydrogen lines fall into series by the level the electron ends on: Lyman (n₁ = 1, ultraviolet), Balmer (n₁ = 2, visible), Paschen (3), Brackett (4) and Pfund (5), all infrared. The first Balmer line, H alpha, is 656.3 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.
Five series

First line and series limit

SeriesEnds onFirst lineLimitRegion
Lymann = 1121.6 nm91.2 nmultraviolet
Balmern = 2656.3 nm364.6 nmvisible
Paschenn = 31875.1 nm820.4 nminfrared
Brackettn = 44051.2 nm1458.4 nminfrared
Pfundn = 57457.8 nm2278.8 nminfrared
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

Hydrogen spectrum series: quick answers

What are the series of the hydrogen spectrum?

Lyman (to n = 1), Balmer (to n = 2), Paschen (to n = 3), Brackett (to n = 4) and Pfund (to n = 5). Each is given by 1/λ = R (1/n₁² - 1/n₂²).

Which series lie in which region?

Lyman is ultraviolet, Balmer is visible (with its last lines in the near ultraviolet), and Paschen, Brackett and Pfund are infrared.

What is the series limit?

The shortest wavelength of a series, when the electron falls from n = ∞. For Balmer it is 364.6 nm; for Lyman 91.2 nm.

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