## The Bohr Atom

We often picture atoms as a miniature Solar System, with electrons orbiting around the central nucleus of protons and neutrons. This picture was first proposed in 1911 by the English physicist Ernest Rutherford, but it suffered from a major theoretical problem. When an object moves along a curved path, such as an electron does in orbit around the nucleus, it experiences an acceleration. (Recall that if an object moves along anynon-straight line it experiences an acceleration). The difficulty this posses is that experiment and theory both show that a charge radiates when it accelerates and that the radiation carries energy away from the particle. Thus, an orbiting electron should gradually lose energy and spiral into the nucleus.

In 1912, the Danish physicist Neils Bohr proposed a solution to this . His solution had three special parts.

• First, electron orbits are quantized (as we discussed in chapter 3.). That is an electron can not orbit at any arbitrary distance from the nucleus, but at only certain prescribed distances.
• Second, when an electron moves in such an orbit, the laws of classical radiation do not apply and the electron does not radiate.
• Third, an atom emits light only when an electron drops from an upper to a lower orbit.

Bohr then went on from these assumptions to explain the spectrum of the hydrogen atom. His simple calculation of its structure agreed superbly with the observed spectrum of hydrogen. That is, he was able to explain why the hydrogen spectrum consists of radiation at only certain discrete wavelengths. Moreover he showed how to calculate those wavelengths with a relatively simple formula.

Bohrs' formula for the wavelengths &lambda of the hydrogen lines is

1/&lambda = R[1/(nl)2 -1/n(u)2],

where the n's are the "quantum" numbers of the orbits nl is the "quantum" number of the lower orbit and nu is the quantum number of the upper orbit. The constant R, known as the Rydberg constant has a value in metric units of 1.097x107 m-1.

Bohr further showed that the value depends only on fundamental constants of nature such as the electron charge, pi, etc.

For example, if an electron in a hydrogen atom drops from level 4 to level 3, the formula for the wavelength of the emitted light is

1/&lambda = 1.097x107 m-1x(1/42 -1/32),

1/&lambda = 1.097x107x(1/16-1/9)m-1.

Flipping the equation over so that the wavelength is on top, we get

&lambda = 4.86x10-7 meters = 486 nanometers,

which is precisely the wavelength of the blue spectrum line of hydrogen.

For his work, Bohr was awarded the Nobel Prize in Physics in 1922.

The Bohr model of the atom is not only a scientific triumph, but it also illustrates superbly how science works.

Scientists at the end of the 19th century had a model that was unable to explain the observed spectrum of hydrogen. That is, it disagreed with experiment. Bohr proposed revisions to the model. The changes led to a new model that agreed extremely well with the experiments. To go a step further, we might note that advances in the 20th century showed that the Bohr model itself needed revisions. Those have been made and yield even better agreement with experiment. Nonetheless, the Bohr model is still a useful way to picture what is happening in atoms, much as the celestial sphere is a useful way to picture the position of stars and planets on the sky.