Hydrogen Spectral Lines

Introduction

Hydrogen's emission spectrum contains several distinct lines, not one pink colour. Each line corresponds to a photon energy produced when a hydrogen atom relaxes between allowed states. Transitions ending at the same level form a spectral series. Comparing their gaps shows why one element can emit several wavelengths while retaining a consistent energy-level pattern.

Core explanation

In Bohr's hydrogen model, allowed energy levels are labelled n = 1, 2, 3 and so on, with Eₙ ≈ −2.18 × 10⁻¹⁸ J/n². A downward transition from n high to n low emits a photon with positive energy E high − E low. Different starting levels give different energy gaps, so they give different frequencies and wavelengths even when the final level is the same. The line pattern is evidence for a family of allowed energies.

The Balmer series consists of transitions ending at n = 2. Its visible lines include approximately 656 nm for n = 3 → 2 and 486 nm for n = 4 → 2. The 4 → 2 gap is larger than the 3 → 2 gap, so its photon has higher energy and shorter wavelength. These are rounded representative wavelengths; measurements can be made more precisely. Further Balmer lines cluster toward shorter wavelengths as starting n increases.

The Lyman series consists of transitions ending at n = 1. Those gaps are larger because the ground level is much lower in energy, so the photons lie in the ultraviolet. For example, n = 2 → 1 gives a wavelength near 122 nm. A student seeing only visible light cannot conclude that hydrogen has no n = 1 transitions; the detector must cover the appropriate wavelength region. Other series, including transitions ending at n = 3, occur in the infrared.

The series concept organises many lines without claiming that all transitions occur in every sample with equal intensity. To observe a line, atoms must reach the initial excited state, the transition must occur and photons must reach the detector. Electrical discharge can populate excited hydrogen states. A lower-pressure atomic gas and a spectrometer help reveal separated lines; an undispersed lamp can look like one blended colour.

Absorption uses the same level differences in the upward direction. A hydrogen atom in n = 2 can absorb a photon matching n = 2 → 3, whose wavelength corresponds to the reverse of the 3 → 2 emission. But if nearly all atoms are in n = 1, absorption from n = 2 may be weak because few atoms start there. Thus matching energy gaps do not guarantee equal emission and absorption intensities in a real sample.

There are several possible transition pathways from a high state. An atom at n = 4 might emit a single photon going to n = 2, or it might pass through n = 3 and release other photons. The set of possible lines reflects many atoms and many events, not one atom simultaneously emitting all listed lines in one step. Energy conservation still holds for each path.

Hydrogen's simple one-electron structure made the Bohr model particularly successful. The same fixed-orbit approach does not correctly reproduce the full spectra of neutral multi-electron atoms. Hydrogen lines are therefore both evidence supporting quantised levels and a reminder of the model's scope. Modern quantum mechanics predicts the level structure with a better account of electron states.

Step-by-step reasoning

1. Identify the initial n and final n for an emission line; the initial level is higher. 2. Compare the energy gap with other transitions using Eₙ or an energy-level diagram. 3. Use E = hc/λ to infer that a larger gap has a shorter wavelength. 4. Group transitions by common final n and check whether the wavelength region is visible, ultraviolet or infrared.

Visual explanation

Draw n = 1, 2, 3 and 4 energy lines. Use two downward arrows ending at n = 2, labelling 3 → 2 as about 656 nm and 4 → 2 as about 486 nm. Use another arrow 2 → 1 labelled ultraviolet, about 122 nm.

Real-world analogy

A stairwell has several upper floors, each at a different height above floor two. Descents from those floors to floor two release different energy amounts. Balmer lines likewise share a final level while their different starting levels make different photon energies.

Real-world example

Astronomers can identify hydrogen in a stellar spectrum by a pattern of line wavelengths, including Balmer features. The line strengths and shapes carry additional information about the source, so identification uses a calibrated pattern rather than one apparent colour.

Why?

Why is the 4 → 2 line shorter in wavelength than the 3 → 2 line? The n = 4 state lies higher than n = 3. Falling from n = 4 to the same n = 2 endpoint releases a larger energy gap, and E = hc/λ makes its wavelength smaller.

Common misconception

“Hydrogen has one electron, so it can emit only one spectral line.” One electron can occupy different allowed excited levels at different times, and many transitions are possible. A sample contains many hydrogen atoms undergoing different events.

Worked example

Compare n = 3 → 2 and n = 4 → 2 in hydrogen. Both end at n = 2, so both are Balmer lines. Because E₄ is higher than E₃, E₄ − E₂ exceeds E₃ − E₂. Therefore the 4 → 2 photon has higher frequency and shorter wavelength, approximately 486 nm versus 656 nm for the 3 → 2 photon.

Quick check

1. Which final level defines the Balmer series of hydrogen emission lines? Answer: All Balmer-series emission transitions end at n = 2.

Exam focus

Give both initial and final n values and label the direction of the arrow. Use larger gap → higher frequency → shorter wavelength. Do not infer that only visible lines exist because the eye cannot see ultraviolet or infrared.

Advanced insight

Hydrogen spectral series converge as the initial n becomes large because Eₙ approaches the zero-energy ionisation limit. Fine structure and other small shifts can split or move lines beyond the simple Bohr prediction, giving sensitive tests of modern quantum theory.

Summary

Hydrogen's multiple line wavelengths reflect transitions between different allowed levels. Balmer transitions end at n = 2 and include visible lines near 656 and 486 nm; Lyman transitions end at n = 1 in the ultraviolet. A larger energy gap gives a shorter wavelength.

Practice questions

1. Which transition has the shorter wavelength: 3 → 2 or 4 → 2? Answer: 4 → 2, because it releases more energy to the same final level. 2. What final level defines the Lyman series? Answer: n = 1. 3. Why can hydrogen emit several lines despite having one electron per atom? Answer: The electron can occupy different excited states, and a sample contains many atoms taking different transitions. 4. Why might a line not be visible to the human eye? Answer: Its wavelength may lie in ultraviolet or infrared rather than the visible range.