How would you explain the hydrogen spectrum?
I outline the nature of Bohr's work and how it accounted for the hydrogen spectrum below...
By the time Neils Bohr tackled the issue in 1913, the line spectrum of hydrogen was well known, and its limited number of discrete colour lines had been studied in detail.
Bohr basically decided to "build" a hydrogen atom that would enable him to account for these spectral lines. Whatever it took, his new model of the atom would be one which could produce the line spectrum.
So, you could say he knew the answer was the spectrum; now, he had to assemble the question - how did hydrogen do it?
He started with the idea that electrons orbited Rutherford's nucleus in circular paths, so that he could apply the concepts of centripetal force and a central coulomb potential to the electron.
However, the brilliant stroke that made it work was his assertion that the angular momentum of the electron had to be quantized - only specific values permitted. It is a long story to detail how he got from there to the permitted energy levels of the atom, but none the less, this did ultimately lead him to the famous result that changed the atom into a particle ruled by the quantum world - electrons in atom could possess only certain allowed levels of energy. To Bohr, this meant certain allowed orbits with specific radii.
With only certain allowed energy levels available for electrons, Bohr's next step was to state that this meant the hydrogen atom could only absorb energy in certain amounts, by absorbing a photon of a certain frequency. If white light were to pass through a sample of hydrogen gas, certain specific colours, corresponding to these allowed frequencies would be absorbed and not pass through the gas. When this happened, according to Bohr, it meant that electrons were being raised to higher energy orbits within the atom. This accounted for the absorption spectrum.
Conversely, if a sample of hydrogen was energized (by subjecting it to high voltage for example) the atoms would absorb energy, and the electrons would jump into high-energy orbits as mentioned. It was when these electrons returned to lower energy orbits that photons of light would be emitted.
When Bohr calculated the specific energies that his hydrogen atom should be able to emit, he got a near-perfect match to the colours seen in the visible portion of the spectrum. He showed that these visible lines resulted when electrons dropped from higher levels into the n=2 orbit (the second-lowest energy).
Of course, other jumps were possible as well, and Bohr predicted ultraviolet and infrared spectra that were soon "discovered". This was a strong case for the quantum atom, and gave support to Bohr's assumptions.