hydrogen spectrum

Introduction to Hydrogen Spectrum

Continuous, emission and absorption spectra — foundation of atomic spectroscopy.

Continuous Spectrum

A continuous spectrum contains all wavelengths without gaps. It is produced by hot solids, liquids and dense gases at high temperature — e.g. incandescent bulb filament or sunlight through a prism (before selective absorption).

Emission Spectrum

When atoms in an excited gas de-excite, they emit photons of specific energies. The result is a bright line spectrum — coloured lines on a dark background.

Absorption Spectrum

When white light passes through cool gas, atoms absorb photons matching their transition energies. The continuous spectrum shows dark lines at those wavelengths.

Why Hydrogen Shows Line Spectrum

Hydrogen has discrete quantised energy levels (Bohr model). Electrons can only absorb or emit photons with energy exactly equal to the difference between two allowed levels — hence discrete spectral lines.

Importance in Atomic Physics: Hydrogen spectrum provided the first precise evidence for quantised atomic energy levels. Balmer's empirical formula (1885) and Rydberg's generalisation led directly to Bohr's model and modern quantum mechanics. Astronomers use hydrogen lines to identify stellar composition and radial velocity.

Fig. 12.9 Bohr's model of hydrogen atom.

Line Spectrum of Hydrogen

Discrete lines from quantised electron transitions.

Definition

A line spectrum consists of sharp, discrete wavelengths (or frequencies) rather than a continuous spread. For hydrogen, each line corresponds to a unique electron transition between two energy levels.

Origin of Spectral Lines

When an electron moves from a higher orbit n₂ to a lower orbit n₁, energy is released as a photon: hν = E_n₂ − E_n₁. Because only certain orbits are allowed, only certain photon energies — and hence wavelengths — are observed.

Electron Transitions

Emission: electron falls from higher to lower level → bright line
Absorption: electron rises from lower to higher level → dark line
Each transition is labelled by quantum numbers (n₁, n₂)

Electron transition from n=3 to n=2 with photon emission (ΔE = hν).

Emission Spectrum

Bright line spectrum from de-excitation of hydrogen atoms.

Excited Electron

Electrical discharge, heat or photon absorption raises electrons to higher energy states (n = 2, 3, 4 …).

Return to Lower Orbit

The excited state is unstable. The electron spontaneously falls to a lower permitted orbit.

Photon Emission

The energy difference appears as a photon: ΔE = hν = hc/λ.

Bright Line Spectrum

Each transition produces one spectral line. The collection of all possible transitions gives the complete emission spectrum of hydrogen.

Fig. 12.10 Emission of a spectral line by a hydrogen atom.

Absorption Spectrum

Dark lines in a continuous background.

When white light (continuous spectrum) passes through hydrogen gas at low pressure:

Photons with energies matching allowed transitions are absorbed
Electrons are excited from lower to higher levels
Those wavelengths are missing from the transmitted light
Dark lines appear at exactly the same positions as emission lines

Kirchhoff's Law: An atom that emits a wavelength at a given temperature will also absorb that same wavelength at a lower temperature.

Absorption spectrum: dark lines appear where hydrogen gas absorbs specific wavelengths from white light.

Rydberg Formula

Unified expression for all hydrogen spectral lines.

1/λ = R [ 1/n₁² − 1/n₂² ]

R = Rydberg constant = 1.097 × 10⁷ m⁻¹ | n₁ = lower (final) level | n₂ = higher (initial) level, n₂ > n₁

Derivation Outline

From Bohr's energy formula Eₙ = −13.6/n² eV, the photon energy for a transition is:

ΔE = 13.6 (1/n₁² − 1/n₂²) eV = hc/λ

Converting to wave number (1/λ) gives the Rydberg formula with R = me⁴/(8ε₀²h³c) ≈ 1.097×10⁷ m⁻¹.

Special Cases — Spectral Series

Lyman

n₁ = 1

Ultraviolet region. Shortest λ ≈ 91.2 nm (series limit).

Balmer

n₁ = 2

Visible region. Hα = 656 nm, Hβ = 486 nm, Hγ = 434 nm.

Paschen

n₁ = 3

Infrared region. λ > 820 nm.

Brackett

n₁ = 4

Infrared. Discovered 1922.

Pfund

n₁ = 5

Far infrared. Discovered 1924.

Lyman Series

Transitions to ground state — ultraviolet.

1/λ = R (1 − 1/n₂²) where n₂ = 2, 3, 4, …

Fig. 12.12 Energy level diagram of hydrogen atom and its various spectral series.

Formula Sheet

All important hydrogen spectrum formulae.

Rydberg Formula

1/λ = R(1/n₁² − 1/n₂²)

Frequency Formula

ν = Rc(1/n₁² − 1/n₂²)

Photon Energy

E = hν = hc/λ

Wavelength Formula

λ = hc/ΔE = 1240/ΔE (nm)

74. [Single Correct] Series limit of Lyman in nm?

Answer / Solution

91.2

75. [Numerical Value] Rydberg constant ×10⁷?

2.47×10¹⁵

48. [Matrix Match] Match series with n₁

Answer / Solution

Lyman→1, Balmer→2, Paschen→3

49. [Single Correct] Ratio of energies of Lyman limit to Balmer limit?

Answer / Solution

50. [Multiple Correct] Which are Balmer lines?

Emission: bright lines; Absorption: dark lines in continuous spectrum.

28. [Data Based] Given λ=656 nm, find ΔE.

Answer / Solution

≈1.89 eV

29. [SL] Define spectral line.

Answer / Solution

Discrete wavelength from atomic transition.

30. [HL] Explain Rydberg formula.

Quantised energy levels

30. [Long Answer] Explain Balmer series.

Correct Option

Both true; Reason explains Assertion.

Explanation

The assertion concerns hydrogen spectral behaviour; the reason links to quantised atomic structure and the Rydberg relation.

50. Assertion: Balmer series is visible. Reason: Transitions end at n=2.

Correct Option

Both true; Reason explains Assertion.

Explanation

The assertion concerns hydrogen spectral behaviour; the reason links to quantised atomic structure and the Rydberg relation.

Case Study Questions

25 case studies — 4 questions each with detailed solutions.

Case Study 1: Hydrogen Spectrum in discharge tube

Passage: A physics laboratory uses a hydrogen discharge tube and a spectrometer. When the gas is excited electrically, discrete coloured lines appear. Astronomers use the same principle to identify hydrogen in distant stars by matching observed wavelengths with the Rydberg formula 1/λ = R(1/n₁² − 1/n₂²).