Rydberg Formula

The Rydberg formula is a mathematical expression used to predict the wavelengths of spectral lines of many chemical elements. It is particularly known for its application to the hydrogen atom, where it explains the wavelengths of the light emitted or absorbed when an electron transitions between energy levels. By linking these transitions to specific wavelengths, the Rydberg formula allows us to precisely describe the emission spectra of hydrogen, laying the groundwork for quantum mechanics.

This formula was developed by the Swedish physicist Johannes Rydberg in the late 19th century and is expressed as:

\[ \frac{1}{\lambda } = R_H\left( \frac{1}{{n_1}^2} – \frac{1}{{n_2}^2} \right) \]

Where:

– λ is the wavelength of the emitted light

– R_H is the Rydberg constant (= 1.097 x 10⁷ m⁻¹)

– n₁ and n₂ are the initial and final energy levels of the electron, respectively

Rydberg Formula for Hydrogen

When an electron in a hydrogen atom transitions from a higher energy level (n₂) to a lower energy level (n₁), it emits a photon with a specific wavelength. This wavelength corresponds to the spectral line observed in the atomic spectrum. Therefore, the Rydberg formula allows us to calculate these wavelengths precisely and explain the observed spectral lines, as shown in the table below.

n1	n2	Wavelength converges towards (nm)	Position in the electromagnetic spectrum	Name of the Series
1	2 → ∞	91.13	Ultraviolet	Lyman series
2	3 → ∞	364.51	Visible	Balmer series
3	4 → ∞	820.14	Infrared	Paschen series
4	5 → ∞	1458.03	Far infrared	Brackett series
5	6 → ∞	2278.17	Far infrared	Pfund series
6	7 → ∞	3280.56	Far infrared	Humphreys series

Rydberg Formula for Other Elements

The Rydberg formula was originally derived to describe the spectral lines of hydrogen, the simplest atom with just one electron. However, with some modifications and caution, it can be extended to other elements, particularly those with hydrogen-like characteristics, meaning they have only one electron in their outer shell.

Hydrogen-like Ions

Hydrogen-like ions are ions that have a single electron, similar to hydrogen, but with a different nuclear charge. Examples include He⁺ (helium with one electron) and Li²⁺ (lithium with one electron). These ions still follow a similar pattern of electron transitions, but the nuclear charge affects the energy levels.

For these ions, the Rydberg formula is modified by including the nuclear charge Z:

\[ \frac{1}{\lambda} = R_H Z^2 \left( \frac{1}{{n_1}^2} – \frac{1}{{n_2}^2} \right) \]

Here, Z is the atomic number of the ion, representing the number of protons in the nucleus. The Z² factor accounts for the stronger attraction between the nucleus and the electron in these ions compared to hydrogen.

Multi-electron Atoms

For elements with more than one electron (multi-electron atoms), the situation becomes more complex. The presence of additional electrons leads to electron-electron interactions, which alter the energy levels and, consequently, the spectral lines.

While the Rydberg formula in its original form does not directly apply to these elements, it can be used with modifications in specific situations, particularly when dealing with the outermost electron transitions in alkali metals (like sodium or potassium) where one electron behaves similarly to a hydrogen-like electron.

Example Problems with Solutions

Problem 1: Calculate the wavelength of light emitted when an electron in a hydrogen atom transitions from the n₂ = 3 energy level to the n₁ = 2 energy level. Identify the series to which this transition belongs and the region of the electromagnetic spectrum where the emitted light lies.

Solution

Given

n₁ = 2

n₂ = 3

Rydberg constant (R_H) = 1.097 x 10⁷ m^-1

Rydberg formula is

\[ \frac{1}{\lambda} = R_H\left( \frac{1}{n_{1}^2} – \frac{1}{n_{2}^2} \right) \]

Substituting the values into the formula:

\[ \frac{1}{\lambda} = 1.097 \times 10^7 \hspace{0.1 cm} m^{-1} \left( \frac{1}{2^2} – \frac{1}{3^2} \right) \\ => \lambda = 655 \text{ nm}\]

The calculated wavelength of 655 nm corresponds to the red part of the visible spectrum, which corresponds to the Balmer series.

References
- 1. Rydberg Formula – Farside.ph.utexas.edu
  2. Rydberg Formula and the Hydrogen Atomic Spectrum – Chem.libretexts.org
  3. Rydberg Formula – Ch301.cm.utexas.edu
  4. What Is the Rydberg Formula and How Does It Work? – Thoughtco.com