The first decent explanation of the atomic spectrum for hydrogen was made by the Danish scientist Neils Bohr in 1913. While the model contains some dubious assumptions, it was able to closely predict the spectrum of the hydrogen atom.

Bohr assumed that an atom looked something like the solar system: a nucleus at the center, with an electron orbiting the atom, attracted by the positive charge in the nucleus. He then made a rather large assumption; that the electron could have only certain orbits. He denoted these orbits as n, where n could go from 1 to infinity. The energy of an electron in one of these orbits is

E = -R_H/n²

His next assumption was that photons are given off or absorbed when an electron moves from one orbit to another. If an electron moves from a high energy state to a lower energy state, it gives off energy in the form of a photon. To move an electron into a higher energy state requires energy, gotten from absorbing a photon. Since the states have specific energies, only specific energy photons are generated or absorbed when electrons move between these states: thus, an atomic spectrum is made up of distinct wavelengths of light. The frequency of light given off or absorbed when an electron moves between states is

n = -R^H/h * (1/(n_hi)² - 1/(n_low)²)

The Bohr model begins to fail when applied to atoms other than hydrogen: the same model applied to helium has about 5% error, and atoms higher in the periodic table are much worse. The Schrodinger quantum mechanical model explains these atoms and their spectra more correctly.

Example: What is the frequency of light given off when an electron moves from the n=5 level to the n=2 level?

Solution: Use the equation above and simply plug the values in. n_hi= 5, n_low = 2, so

n = -R^H/h * (1/(n_hi)² - 1/(n_low)²)

n = -2.18*10^-18 J/(6.626*10^-34 J*s) *(1/5² -1/2²)

n = 6.91*10¹⁴ 1/s

Information provided by: http://learn.chem.vt.edu