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7.4.5. Hydrogen Spectrum

Table of Contents

Why Hydrogen Is Special

Hydrogen is the simplest atom. It has one proton in the nucleus and one electron around it. Because of this simple structure, its light spectrum is much easier to study than the spectra of more complicated atoms.

When hydrogen is excited, its electron can move to a higher energy level. When the electron falls back to a lower level, the atom emits light. This light does not form a continuous rainbow. Instead, it appears as separate lines at specific wavelengths. These lines make up the hydrogen spectrum.

The hydrogen spectrum became one of the most important clues in atomic physics because it showed that atomic energies are not arbitrary. They come in fixed allowed values.

Spectral Lines of Hydrogen

If white light passes through a prism, it spreads into a continuous spectrum. Hydrogen does not usually produce a continuous spectrum. Instead, it gives bright lines at certain colors when it emits light, or dark lines at those same positions when it absorbs light.

Each spectral line corresponds to a transition of the electron between two allowed energy levels.

If the electron moves from a higher level $n_i$ to a lower level $n_f$, the emitted photon has energy

$$
E_{\gamma} = E_{i} - E_{f}
$$

and since photon energy is related to frequency by

$$
E_{\gamma} = hf
$$

the wavelength is determined by

$$
E_{\gamma} = \frac{hc}{\lambda}
$$

So every allowed transition gives one particular wavelength.

For hydrogen, each spectral line is produced by an electron transition between two quantized energy levels.
$$
\Delta E = hf = \frac{hc}{\lambda}
$$

The Balmer Series and Visible Light

The most famous part of the hydrogen spectrum is the Balmer series. These are the lines produced when the electron falls to the level $n = 2$ from higher levels.

These lines lie in the visible region, so they can be seen with ordinary optical instruments. Some important Balmer lines are shown below.

TransitionCommon nameApproximate wavelengthColor
$3 \to 2$H$\alpha$$656.3 \, \text{nm}$Red
$4 \to 2$H$\beta$$486.1 \, \text{nm}$Blue-green
$5 \to 2$H$\gamma$$434.0 \, \text{nm}$Violet
$6 \to 2$H$\delta$$410.2 \, \text{nm}$Violet

As the starting level gets larger, the lines get closer together. This happens because the energy levels themselves get closer together at high $n$.

Visible Balmer lines of hydrogen

Other Hydrogen Series

Hydrogen has several spectral series, depending on the final energy level.

The Lyman series corresponds to transitions ending at $n = 1$. These lines are in the ultraviolet region.

The Balmer series corresponds to transitions ending at $n = 2$. These are mainly visible.

The Paschen series corresponds to transitions ending at $n = 3$. These are in the infrared region.

There are also higher series such as Brackett and Pfund, which lie deeper in the infrared.

SeriesFinal levelRegion of spectrum
Lyman$n_f = 1$Ultraviolet
Balmer$n_f = 2$Visible
Paschen$n_f = 3$Infrared
Brackett$n_f = 4$Infrared
Pfund$n_f = 5$Infrared

The Rydberg Formula

The wavelengths of hydrogen spectral lines follow a simple formula called the Rydberg formula:

$$
\frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)
$$

where $R_H$ is the Rydberg constant for hydrogen, $n_i$ is the initial level, and $n_f$ is the final level, with $n_i > n_f$.

For hydrogen,

$$
R_H \approx 1.097 \times 10^7 \, \text{m}^{-1}
$$

This formula successfully describes the observed wavelengths of hydrogen lines.

Hydrogen spectral wavelengths are given by the Rydberg formula
$$
\frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right), \quad n_i > n_f
$$
This shows that hydrogen spectral lines occur only at specific wavelengths.

Connection to Energy Levels

In the hydrogen atom, the allowed energies are

$$
E_n = -\frac{13.6 \, \text{eV}}{n^2}
$$

So the energy difference between two levels is

$$
\Delta E = 13.6 \, \text{eV} \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)
$$

This directly leads to the Rydberg formula through the photon relation $E = hc/\lambda$.

For example, for the Balmer line $3 \to 2$,

$$
\Delta E = 13.6 \left( \frac{1}{2^2} - \frac{1}{3^2} \right)
= 13.6 \left( \frac{1}{4} - \frac{1}{9} \right)
= 13.6 \cdot \frac{5}{36}
\approx 1.89 \, \text{eV}
$$

Then the emitted photon has wavelength

$$
\lambda = \frac{hc}{\Delta E}
$$

which gives the red H$\alpha$ line near $656 \, \text{nm}$.

Emission and Absorption Spectra

Hydrogen can produce both emission and absorption spectra.

In an emission spectrum, excited hydrogen atoms emit bright lines on a dark background.

In an absorption spectrum, light containing many wavelengths passes through cooler hydrogen gas. The hydrogen absorbs photons whose energies match allowed transitions, leaving dark lines in the spectrum.

The positions of the emission and absorption lines are the same, because they correspond to the same energy differences.

Emission and absorption lines in hydrogen

Why the Hydrogen Spectrum Matters

The hydrogen spectrum was a major step toward modern atomic theory. It showed that atoms have discrete energy levels. Classical physics alone could not explain why only certain wavelengths appear.

The regular pattern of hydrogen lines led to formulas such as the Balmer and Rydberg relations, and later helped support the Bohr model and quantum theory.

Hydrogen spectra are also important in astronomy. Since hydrogen is the most abundant element in stars, its spectral lines are used to identify hydrogen in stellar atmospheres and interstellar gas.

The hydrogen spectrum is evidence that atomic energy is quantized. Only certain electron transitions are allowed, so only certain photon wavelengths are emitted or absorbed.

Series Limit

Within any hydrogen series, the lines get closer and closer together as $n_i$ increases. In the limit $n_i \to \infty$, the series reaches a shortest wavelength, called the series limit.

For a given final level $n_f$,

$$
\frac{1}{\lambda_{\text{limit}}} = R_H \left( \frac{1}{n_f^2} \right)
$$

For the Balmer series, where $n_f = 2$,

$$
\frac{1}{\lambda_{\text{limit}}} = \frac{R_H}{4}
$$

so

$$
\lambda_{\text{limit}} = \frac{4}{R_H}
$$

This is the shortest possible wavelength in the Balmer series.

A Simple Energy Level Picture

A helpful way to visualize the hydrogen spectrum is to think of horizontal energy levels and arrows showing electron jumps downward.

Hydrogen energy levels and transitions

This picture shows several transitions ending at $n = 2$, which produce Balmer lines.

Summary

The hydrogen spectrum consists of discrete spectral lines produced when the electron in a hydrogen atom moves between allowed energy levels. Each line corresponds to a photon with energy equal to the difference between two levels. The visible hydrogen lines form the Balmer series, while other series lie in the ultraviolet or infrared. The wavelengths are described by the Rydberg formula, and the existence of these lines is strong evidence for quantized atomic energy levels.

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