Electronic Structure of Atoms

Bohr's model of the hydrogen atom


In 1913 Niels Bohr developed a theoretical explanation for a phenomenon known as line spectra.

 
Bohr's Model of the Hydrogen Atom

Line Spectra

Lasers emit radiation which is composed of a single wavelength. However, most common sources of emitted radiation (i.e. the sun, a lightbulb) produce radiation containing many different wavelengths.

When the different wavelengths of radiation are separated from such a source a spectrum is produced.

A rainbow represents the spectrum of wavelengths of light contained in the light emitted by the sun

Not all radiation sources emit a continuous spectrum of wavelengths of light When the spectrum emitted by hydrogen gas was passed through a prism and separated into its constituent wavelengths four lines appeared at characteristic wavelengths

In 1885 a Swiss school teacher figured out that the frequencies of the light corresponding to these wavelengths fit a relatively simple mathematical formula:

where C = 3.29 x 1015 s-1 (not the 'c' used for the speed of light)

However, the physical basis for this relationship was unknown.

Bohr's Model

If the orbits of the electron are restricted, the energies that the electron can possess are likewise restricted and are defined by the equation:

Where RH is a constant called the Rydberg constant and has the value

2.18 x 10-18 J

'n' is an integer, called the principle quantum number and corresponds to the different allowed orbits for the electron. Thus, an electron in the first allowed orbit (closest to the nucleus) has n=1, an electron in the next allowed orbit further from the nuclei has n=2, and so on.

Thus, the relative energies of these allowed orbits for the electrons can be diagrammed as follows:

All the relative energies are negative

Bohr also assumed that the electron can change from one allowed orbit to another
DE = Ef - Ei

Substituting in for the previously defined energy equation:

When an electron "falls" from a higher orbit to a lower one the energy difference is a defined amount and results in emitted electromagnetic radiation of a defined energy (DE)

Note: Revisiting Balmer's equation:

In 1885 a Swiss school teacher figured out that the frequencies of the light corresponding to these wavelengths fit a relatively simple mathematical formula:

where C = 3.29 x 1015 s-1 (not the 'c' used for the speed of light)

Since energy lost by the electrons is energy "gained" by the emitted EM energy, the EM energy from Bohr's equation would be:

Thus, Balmer's constant 'C' = (RH/h) (Rydberg constant divided by Planck's constant), and nf = 2. Thus, the only emitted energies which fall in the visible spectrum are from those electrons which fell down to the second quantum orbital. Those which fell down to the first orbital have a higher energy (frequency) than can be seen in the visible spectrum.

Calculate the wavelength of light that corresponds to the transition of the electron from the n=4 to the n=2 state of the hydrogen atom. Is the light absorbed or emitted by the atom?

Since the electron is "falling" from level 4 down to level 2, energy will be given up and manifested as emitted electromagnetic radiation:

DE = (2.18 x 10-18 J)((1/16)-(1/4)) = -4.09 x 10-19 J (light is emitted)
4.09 x 10-19 J = (6.63 x 10-34 Js) * (n)
6.17 x 1014 s-1 = n
l = (3.00 x 108 m s-1)/ (6.17 x 1014 s-1) = 4.87 x 10-7m = 487 nm

Bohr's model of the atom was important because it introduced quantized energy states for the electrons. However, as a model it was only useful for predicting the behavior of atoms with a single electron (H, He+, and Li2+ ions). Thus, a different model of the atom eventually replaced Bohr's model. However, we will retain the concept of quantized energy states


1996 Michael Blaber