Atoms - Part 4 - The Bohr Atom
In 1913 the Danish physicist Niels Bohr proposed that the structure of the hydrogen atom can be approached by assuming that the electron is held in its orbit by equating the force required to accelerate it into a circular orbit with the coulombic attraction between nucleus and electron. (Opposite charges attract.)
where m is the mass of the electron, v the velocity, e the charge on the electron, and r the radius of the orbit. The term on the LHS is the centripetal force. The term on the RHS is the coulombic attractive force. The total energy of the electron is given by the sum of the kinetic and potential energies.
As mev2=e2/r, then, by substitution,
This will give the energy E of the electron as a function of the distance r from the nucleus. The problem here is that the radius r and thus the energy E could have any value. This is the planetary model in which the energy would continuously decrease until r goes to zero. To accommodate the known behavior of the atom, Bohr postulated that the angular momentum of the electron could have only certain values. He proposed that the angular momentum mevr was an integral multiple n of h/2p where n is a positive integer ( 1, 2, 3, ... ).
The resultant formula for the Bohr radius is
where n is the quantum number. This model explained the spectral lines of the hydrogen atom as transitions DEH of the electron from one energy level I to another II.
DEH is the difference in energy between the initial state I and the final state II . Spectroscopic measurements are usually measured in wave numbers. The wave number is the inverse of the wave length l. The unit is inverse centimeters cm-1.
The wave number of a transition is given by
Below is the absorption spectrum of hydrogen when the gas is exposed to light containing all the wavelengths (Figure 6). This corresponds to transitions from n=1 to n=2,3,4,.... with the wave numbers and energy increasing to the left.
<---- energy
Figure 6.
The right-most line in Figure 6 represents the absorption of the energy required to excite an electron from the ground state E0 to the next highest level E1 at 82,259cm-1. The next line to the left corresponds to the transition from the ground state E0 to E2 at 97,492cm-1. The lines get closer and closer to the maximum wave number of 109,678cm-1, at which point the electron has left the atom.
The proposal that the electrons could only exist in discrete stationary states with energies E0, E1, E2, ... allowed for an interpretation of the observed emission and absorption lines in the hydrogen spectrum.
The frequency of a spectral line corresponds to a certain energy. That energy corresponds to a transition of an electron from one discrete energy level to another. An atom can only absorb a photon with an energy that matches the difference between two energy levels in the atom.
Given enery levels E0, E1, E2, ..., the transition from E0 to E1 would result from absortion of a photon with energy hn1 and the transition from E1 to E0 would emit a photon with energy hn1, the transition from E0 to E2 would result from absortion of a photon with energy hn2 and the transition from E2 to E0 would emit a photon with energy hn2. Once an electron has absorbed a photon with the energy to excite it from the ground state E0 to energy level E3, there would be several possible ways in which it gives up its excess energy. It can fall all the way back to E0 and emit hn30, (see Figure 6 below) or it can go to E2 and then to E0, emitting hn32 and then hn20, and so forth. (Subscript 31 means the transition from energy level 3 to 1.)
Figure 7.
Then hn30 should equal hn31 + hn10, and hn32 + hn20 and hn32, + hn21 + hn10. This is the pattern that is observed. This explains why there are more emission lines than absorption lines.
The Lyman series is the emission from excited electrons falling to the ground state n=1 (Figure 8).
Figure 8.
The Balmer series and Paschen series result from excited electrons falling to the n=2 and n=3 states.
The Bohr theory was very successful but it was not the whole story. For example it could not predict the behavior of atoms in an electric field. Nor could it explain the spectral details of atoms larger than hydrogen. Such things required the methods of wave mechanics.