What Is the Rydberg Formula and How Does It Work?

The Rydberg formula (or Rydberg equation) is a mathematical formula used to predict the wavelength of light resulting from an electron moving between energy levels of an atom.

When an electron changes from one atomic orbital to another, the electron's energy changes. When the electron changes from an orbital with high energy to a lower energy state, a ​photon of light is created. When the electron moves from low energy to a higher energy state, a photon of light is absorbed by the atom.

Each element has a distinct spectral fingerprint. When an element's gaseous state is heated, it will give off light. When this light is passed through a prism or diffraction grating, bright lines of different colors can be distinguished. Each element is slightly different from other elements. This discovery was the beginning of the study of spectroscopy.

Rydberg Equation

Johannes Rydberg was a Swedish physicist who attempted to find a mathematical relationship between one spectral line and the next of certain elements. He eventually discovered there was an integer relationship between the wavenumbers of successive lines.

His findings were combined with Bohr's model of the atom to create this formula:

1/λ = RZ²(1/n₁² - 1/n₂²)

where

λ is the wavelength of the photon (wavenumber = 1/wavelength)
R = Rydberg's constant (1.0973731568539(55) x 10⁷ m^-1)
Z = atomic number of the atom
n₁ and n₂ are integers where n₂ > n₁.

It was later found that n₂ and n₁ were related to the principal quantum number or energy quantum number. This formula works very well for transitions between energy levels of a hydrogen atom with only one electron. For atoms with multiple electrons, this formula begins to break down and give incorrect results. The reason for the inaccuracy is that the amount of screening for inner electrons or outer electron transitions varies. The equation is too simplistic to compensate for the differences.

The Rydberg formula may be applied to hydrogen to obtain its spectral lines. Setting n₁ to 1 and running n₂ from 2 to infinity yields the Lyman series. Other spectral series may also be determined:

For most problems, you'll deal with hydrogen so you can use the formula:

1/λ = R_H(1/n₁² - 1/n₂²)

where R_H is Rydberg's constant, since the Z of hydrogen is 1.

Rydberg Formula Worked Example Problem

Find the wavelength of the electromagnetic radiation that is emitted from an electron that relaxes from n = 3 to n = 1.

To solve the problem, start with the Rydberg equation:

1/λ = R(1/n₁² - 1/n₂²)

Now plug in the values, where n₁ is 1 and n₂ is 3. Use 1.9074 x 10⁷ m^-1 for Rydberg's constant:

1/λ = (1.0974 x 10⁷)(1/1² - 1/3²)
1/λ = (1.0974 x 10⁷)(1 - 1/9)
1/λ = 9754666.67 m^-1
1 = (9754666.67 m^-1)λ
1 / 9754666.67 m^-1 = λ
λ = 1.025 x 10^-7 m

Note the formula gives a wavelength in meters using this value for Rydberg's constant. You'll often be asked to provide an answer in nanometers or Angstroms.