Hydrogen Spectrum and the Rydberg Relation

Introduction

Hydrogen's line spectrum forms orderly families rather than a random collection of colors. A reciprocal-wavelength relation links each line to two positive integer level labels. This pattern was central evidence that hydrogen has discrete allowed energy states with measurable differences between them.

Core explanation

For an emission transition from an upper level nᵤ to a lower level nₗ in ordinary hydrogen, 1/λ ≈ R H(1/nₗ² − 1/nᵤ²), where nᵤ > nₗ and R H is the hydrogen Rydberg constant. The right side is positive because the lower-level reciprocal square is larger. Each integer pair yields a particular wavelength. The relation is a spectroscopy result and can also be derived approximately from Bohr's hydrogen energy expression.

Transitions ending at nₗ = 1 form the Lyman series, mostly ultraviolet. Those ending at nₗ = 2 form the Balmer series, with several visible lines. The Paschen series ends at nₗ = 3 and lies in the infrared. The shared lower level defines the series; the upper level changes from one line to another. As nᵤ increases without bound, the line positions converge toward a series limit corresponding to removing the electron from that lower level in the idealized model.

Reciprocal wavelength, also called wavenumber in a suitable unit convention, is useful because it is proportional to photon energy: E = hc/λ. The Rydberg expression therefore encodes differences in hydrogen energy levels proportional to 1/n². The series pattern is much more specific than simply saying “hydrogen emits some colors.”

Use units carefully. A common approximate Rydberg constant is 1.097 × 10⁷ m⁻¹ for hydrogen calculations. Multiplying it by the dimensionless parentheses gives reciprocal metres. Inverting gives metres, often converted to nm. Actual measured hydrogen wavelengths may differ slightly from a simplified infinite-nuclear-mass constant; ordinary textbook exercises use a stated constant with appropriate precision.

The formula is for a one-electron hydrogen system. It cannot be applied unchanged to a multi-electron atom, where electron-electron repulsion and shielding alter energy levels. Hydrogen-like ions with one electron have related formulas involving nuclear charge and reduced mass, but require their own stated form and constants.

Step-by-step reasoning

1. Identify lower and upper positive integers, with upper greater than lower. 2. Calculate the positive reciprocal-square difference. 3. Multiply by the stated Rydberg constant to get 1/λ. 4. Invert, convert units, and identify the lower-level spectral series.

Visual explanation

Draw horizontal levels n = 1, 2, 3, and 4. Several arrows terminating at level 2 represent Balmer lines of different wavelengths; another group terminating at 1 represents Lyman lines.

Real-world analogy

A building has numbered floors. Many different downward elevator trips can end on floor two; grouping all those trips by destination resembles grouping transitions into one spectral series.

Real-world example

The hydrogen Balmer lines can be measured in a discharge tube with a spectrometer. Their ordered wavelengths give an experimental check on discrete hydrogen-state predictions.

Why?

Why do lines within a series converge? As the upper index grows, 1/nᵤ² approaches zero, so successive reciprocal-wavelength differences get progressively smaller and approach a definite limit.

Common misconception

“A Balmer line starts at n = 2.” An emission Balmer line ends at n = 2; the starting upper level is a larger integer.

Worked example

Estimate the wavelength for hydrogen's n = 3 → n = 2 transition. The bracket is 1/2² − 1/3² = 1/4 − 1/9 = 5/36. Thus 1/λ ≈ (1.097 × 10⁷)(5/36) ≈ 1.524 × 10⁶ m⁻¹. Inverting gives λ ≈ 6.56 × 10⁻⁷ m, or 656 nm, a red Balmer line.

Quick check

1. Which lower level defines the Balmer series? Answer: nₗ = 2; emission transitions from higher levels end there.

Exam focus

Keep upper and lower level order clear so reciprocal wavelength stays positive. Convert the final metres to nanometres only after inversion.

Advanced insight

The most precise line predictions include reduced-mass effects because the proton also moves around the atom's center of mass. Fine and hyperfine structure add further small splittings beyond the simple Rydberg pattern.

Summary

Hydrogen spectral wavelengths follow an integer-indexed reciprocal-wavelength relation. Lines sharing a lower state form a series, and the pattern reveals quantized hydrogen energies.

Practice questions

1. Does n = 4 → n = 2 belong to Lyman, Balmer or Paschen? Answer: Balmer, because the lower level is n = 2. 2. Is 1/λ positive for an emission transition when nᵤ > nₗ? Answer: Yes; 1/nₗ² exceeds 1/nᵤ². 3. What happens to line spacing near a series limit? Answer: Lines converge as increasingly large upper levels produce smaller changes in the reciprocal-square term.