Hydrogen Spectrum

When a hydrogen gas discharge tube is illuminated and its light passed through a prism or diffraction grating, a series of sharp, discrete colored lines appear against a dark background — this is the hydrogen emission spectrum. Unlike the continuous rainbow produced by a hot solid, these lines occur only at specific wavelengths, revealing that hydrogen atoms can only emit or absorb light at particular, quantized energies. The existence of these spectral lines was one of the great puzzles of 19th-century physics, and their mathematical regularity eventually led directly to quantum mechanics. Niels Bohr's 1913 model of the hydrogen atom provided the first successful theoretical explanation for these spectral lines. Bohr proposed that the electron in hydrogen can only occupy specific circular orbits, each with a fixed energy, labeled by a principal quantum number n = 1, 2, 3, … When an electron transitions from a higher energy orbit (n₂) to a lower energy orbit (n₁), it releases a photon whose energy exactly equals the difference between the two orbital energies. Conversely, absorbing a photon of precisely the right energy promotes the electron to a higher orbit. These fixed energy differences explain why only discrete wavelengths are observed. The spectral lines of hydrogen are organized into named series, each corresponding to transitions that end on a particular lower energy level. The Lyman series (ultraviolet) involves transitions ending on n = 1; the Balmer series (visible) ends on n = 2; the Paschen series (near-infrared) ends on n = 3; higher series (Brackett, Pfund) fall in the infrared. The quantitative formula governing all these lines — the Rydberg formula — allows precise calculation of the wavenumber (and hence wavelength or frequency) of any hydrogen spectral line, making it one of the most accurate empirical relationships in all of physics.

Key takeaways
  • Bohr's Energy Levels
  • Photon Energy and the Transition
  • The Rydberg Formula and Wavenumber
  • Identifying the Series
  • Limitations of the Bohr Model

Bohr's Energy Levels

Bohr showed that the allowed energies of the hydrogen electron are given by Eₙ = −13.6 eV / n², where n is a positive integer. The lowest energy state (n = 1, the ground state) is most negative (most stable). As n increases, the energy becomes less negative, approaching zero at n = ∞ (ionization). The energy differences between levels decrease rapidly as n increases, which is why spectral lines within each series crowd together toward shorter wavelengths (called the series limit).

Photon Energy and the Transition

When an electron drops from level n₂ to level n₁ (n₂ > n₁), the photon energy released is ΔE = E(n₁) − E(n₂) = 13.6 eV × (1/n₁² − 1/n₂²). Using E = hf = hc/λ, this links directly to the photon's wavelength. Larger energy gaps produce shorter-wavelength (higher-energy) photons; this is why Lyman series lines (ending at n = 1, the largest possible gap) appear in the ultraviolet, while Balmer lines (ending at n = 2) are less energetic and fall in the visible region.

The Rydberg Formula and Wavenumber

Spectroscopists often express spectral line positions as wavenumbers (ν̃), defined as the number of wave cycles per centimetre: ν̃ = 1/λ. The Rydberg formula states: ν̃ = R_H × (1/n₁² − 1/n₂²), where R_H = 1.097 × 10⁷ m⁻¹ (or 1.097 × 10⁵ cm⁻¹) is the Rydberg constant for hydrogen. This single formula, with the appropriate choice of n₁ and n₂, reproduces every observed hydrogen spectral line to extraordinary precision — a remarkable triumph of Bohr's model.

Identifying the Series

For the Lyman series: n₁ = 1, n₂ = 2, 3, 4, … (ultraviolet, 91–122 nm). For the Balmer series: n₁ = 2, n₂ = 3, 4, 5, … The first four Balmer lines (Hα at 656 nm red, Hβ at 486 nm blue-green, Hγ at 434 nm violet, Hδ at 410 nm deep violet) are visible to the naked eye. For the Paschen series: n₁ = 3, n₂ = 4, 5, 6, … (near-infrared, 820 nm–1875 nm). The series limit (when n₂ → ∞) gives the maximum wavenumber/minimum wavelength of each series: ν̃_limit = R_H / n₁².

Limitations of the Bohr Model

Despite its success with hydrogen, Bohr's model fails for multi-electron atoms because it ignores electron–electron repulsion and uses classical circular orbits. It cannot predict the relative intensities of spectral lines, fine structure (splitting of lines due to electron spin), or the Zeeman effect (splitting in magnetic fields). These phenomena required full quantum mechanics (Schrödinger equation, 1926). Nevertheless, Bohr's model remains a foundational conceptual step and gives correct energies for any single-electron ion (He⁺, Li²⁺) by replacing R_H with R_H × Z², where Z is the nuclear charge.

The Key Equation

ν~=RH(1n12−1n22)\tilde{\nu} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)
  • \tilde{\nu} — Wavenumber (m⁻¹ (or cm⁻¹)): The number of wavelengths per unit length; reciprocal of wavelength. Often expressed in cm⁻¹ in spectroscopy.
  • R_H — Rydberg constant (hydrogen) (m⁻¹): Empirical constant equal to 1.097 × 10⁷ m⁻¹; encodes the energy scale of hydrogen's electronic transitions.
  • n_1 — Lower principal quantum number (dimensionless): Integer ≥ 1 identifying the final (lower) energy level of the electron after the transition. Defines which series the line belongs to.
  • n_2 — Upper principal quantum number (dimensionless): Integer > n₁ identifying the initial (higher) energy level. The specific value determines which line within a series is produced.

Worked Example

Problem. Calculate the wavenumber and wavelength of the second line of the Balmer series (the Hβ line). Then state which region of the electromagnetic spectrum this line falls in. (R_H = 1.097 × 10⁷ m⁻¹)

ν~=2.057×106 m−1;λ=486 nm\tilde{\nu} = 2.057 \times 10^6 \text{ m}^{-1};\quad \lambda = 486 \text{ nm}

Answer. Wavenumber = 2.057 × 10⁶ m⁻¹; Wavelength = 486 nm; visible (blue-green) region.

Figure
Diagram
Hydrogen Energy Level Diagram and Spectral Seriesn=∞0 eVn=5−0.54 eVn=4−0.85 eVn=3−1.51 eVn=2−3.40 eVn=1−13.6 eVLymanUVBalmervisibleHαHβHγPaschenIRLyman (UV)Balmer (visible)Paschen (IR)
Origins · 1 minJohannes Rydberg and how this was discovered
JR
Johannes Rydberg
1854–1919 · Swedish

Rydberg was a Swedish physicist who, in 1888, derived a general empirical formula relating the wavenumbers of spectral lines in hydrogen and other elements. His formula, ν̃ = R_∞(1/n₁² − 1/n₂²), unified the previously separate empirical formulas of Balmer, Lyman, and Paschen into a single expression. The Rydberg constant (R_H ≈ 1.097 × 10⁷ m⁻¹) bears his name and is today one of the most precisely measured fundamental constants in physics. Rydberg discovered his formula entirely empirically — through careful analysis of spectral data — years before Bohr's model or quantum mechanics existed to explain why it worked. He never received the Nobel Prize despite nominations, partly because he died in 1919 just as his work was gaining its full theoretical significance.

  • Bohr's Model: Introduction — Provides the foundational postulates of quantized orbits and stationary states that underpin the explanation of spectral series.
  • Quantum Numbers and Atomic Orbitals — The principal quantum number n in the Rydberg formula is formally defined and extended in the full quantum mechanical treatment of atomic structure.
  • The Photoelectric Effect — Both phenomena demonstrate the particle (photon) nature of light and the quantization of energy, forming the experimental basis for early quantum theory.
  • The Electromagnetic Spectrum — Understanding where UV, visible, and infrared regions lie is essential for correctly interpreting which spectral series corresponds to which range of wavelengths.
  • Ionization Energy — The series limit of the Lyman series corresponds to complete removal of the electron from n=1, which is equivalent to the first ionization energy of hydrogen (13.6 eV).