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The Bohr Model of the Atom
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By 1911 Ernest Rutherford had solved the problem of alpha scattering investigated by Geiger and
Marsden by developing a picture of the atom in which there is a small, dense positively charged nucleus
surrounded by mostly empty space in which were sprinkled some electrons.
In the spring of 1912, … a young Danish visitor named Neils Bohr had turned up a Rutherford's
laboratory. He had just taken his Ph.D. in physics in Copenhagen and now he was finishing up a year
of study abroad. Geiger and Marsden were still busy with the last long tests of the scattering formula,
but Bohr found that in Manchester Rutherford's atom-model was already taken for granted. Everyone
knew, that is to say, that the atom was as good as empty, that around its outer edge were electrons
which moved in the electric field of a very tiny, very massive, positively charged scattering-center,
and that it's electric field fell of with the square of the distance outward. Bohr slipped easily into the
life of the laboratory, talked, had ideas, made suggestions, listened with both ears, wrote a theoretical
paper on the passage of alpha particles through matter and after four months returned to Copenhagen.
As his visit came to its end, the positive scattering-center at the heart of the atom casually picked up a
name, "the nucleus."
It was an impossible task from the start (talking about how Bohr could incorporate the electron into
Rutherford's view of the atom), for there was no proper way to build-in the electrons. If Bohr
imagined them spotted in fixed positions around the nucleus, there could be no stability. If the
repulsion of all their negative charges did not explode them outwards, the attraction of the positive
nucleus must take over to collapse them inwards. If he thought of them as circling like the planets, the
repulsions (which do not appear in the solar system) still came in to complicate the mathematics.
Worse than this, if an electron circled it had to accelerate; if a charged body accelerated, it radiated its
energy in electromagnetic waves; if a circling electron lost energy, it must spiral down into the
nucleus, and once more the atom would collapse.
Quotes from The Restless Atom - The Awakening of Nuclear Physics by Alfred Romer; pp 163 - 165
By 1913 Bohr had gone on to develop an atomic model that incorporated Rutherford's nucleus and
adequately explained the emission spectrum of hydrogen. He did so by simply ignoring some rather
important physics of moving charged particles, and by incorporating the radical ideas of Max Planck
called the Quantum Theory.
Bohr explained the hydrogen spectrum by saying that electrons could exit only at certain distances from
the nucleus, which corresponded to certain allowed energies. An electron in its "ground state", closest to
the nucleus, would have the lowest allowed energy. If the electron absorbed energy, it became "excited"
and moved to an allowed state farther away from the nucleus. An electron in an excited state would be
unstable and the electron would return
to the ground state by shedding its
"extra" energy as a photon of
electromagnetic energy. In other
words, the electron would give off the
acquired energy as light, light either in
the infrared, visible or ultraviolet
portions of the electromagnetic
spectrum.
http://www.lbl.gov/images/MicroWorlds/EMSpec.gif
Transitions to the
n = 2 level produce
visible light.
Transitions to the
ground state where
n = 1 produce UV
light.
1
2
3
4
5
Transitions to the
n = 3 and n = 4 levels
produce IR light.
The Bohr model explains the hydrogen spectrum.
Despite the model usually being represented by concentric circles, implying circular "orbits" taken by the
electron, the Bohr model doesn't really describe a two-dimensional plane occupied by electrons. Despite
the fact that atoms are indeed three-dimensional, the Bohr model has only one dimension, n, which is the
distance from the nucleus to the electron which is proportional to the energy of the electron. The value n
can also be used to describe the allowed energy levels.
According to Bohr, electrons can exist only in discrete energy levels, and no where else in the atom.
Electrons can move from one energy level to another only by gaining or losing energy.
The energy of the nth energy level is given by: En = -RH / n2
RH is the Rydberg constant for hydrogen, and has the value 2.18x10-18 J.
The energy of the emitted photon, and hence its wavelength, depends on the difference in energy of the
two energy levels between which the electron moves. For visible light, the color corresponds to the
wavelength.
Example: Compute the wavelength of light emitted as an electron moves from the n=4 energy level to
the n=2 energy level.
n = 4 … En = -RH / n2 … E = -2.18x10-18J / 22 = -1.36x10-19J
An electron moves from a higher energy level to a lower energy level
and emits energy which is the difference between the two levels.
n = 2 … En = -RH / n2 … E = -2.18x10-18J / 22 = -5.45x10-19J
∆E = Efinal - Einitial
∆E = -5.45x10-19J - (-1.36x10-19J)
∆E = -4.09x10-19J …. The energy difference between the two energy levels.
The negative sign indicates that energy is being given off.
∆E = hc/λ ... The energy of the light is inversely proportional to its wavelength, λ.
λ = hc/∆E ….h is Planck's constant, c is the speed of light
λ = (6.63x10-34 Js) x (3.00x108 m/s) / (−4.09x10-19J) = -4.86 x107m
The negative sign indicates that energy is being given off.
λ = 486 nm … This is the wavelength of the light emitted as an electron moves
from the n=4 energy level to the n=2 energy level. (In this case we ignore the
negative sign that indicates that energy is being given off, and simply look at the
fact that you measured a wavelength of approximately 486 nm when you looked
at the hydrogen spectrum.
400 nm
500 nm
600 nm
The hydrogen spectrum (© Mike Jones 2004)
Postulates of the Bohr model
1. Despite the nucleus (positively charged) and electron (negatively changed) being oppositely charged
they do not attract and collapse together. The electrons are stable in their energy levels and are not
pulled towards the nucleus. This violates classical physics. Also an electron that is stable and doesn’t
change energy level will not radiate energy.
2. If an electron moves down an energy level then quantised energy in the form of a photon will be
emitted. Similarly if an electron moves to a higher energy level then the atom must absorb some
quanta of energy. This explains the existence of spectral lines.
3. Angular momentum = nh/2π, where n is an integer, h is Planck's constant. This means that the angular
momentum of an electron can only take on certain values and is quantised. The concept of quantized
energy comes from Planck's quantum theory.
The Bohr model has some severe limitations.
1. The calculations only work for hydrogen or a 1-electron ion. It does not work atoms with two or more
electrons.
2. It does not predict variations in the intensity of spectral lines.
3. It does not explain the "hyperfine" spectral lines. Hyperfine lines refer to some spectral lines which
actually consist of two or more closely spaced lines.
In addition, there are additional, more complicated, characteristics of atom spectra that cannot be
explained with the Bohr model. This tells us that the Bohr model, despite its simplicity, is inadequate for
a modern-day atomic model.
Electromagnetic energy and the Bohr model - Questions
1. Consider electromagnetic energy, and describe the relationship between the energy of the wave and the
wavelength.
2. Which has more energy, UV light or IR light? ___________________
3. Which has a shorter wavelength, yellow light or blue light? ___________________
4.
Which is more energetic, red light or green light? ____________________
5. Compute the energy of an electron in the third energy level in a hydrogen atom.
6. Determine the wavelength of light (in nanometers) emitted as an electron moves from the third energy level to
the second energy level. (Show your work.)
7. Explain why light is emitted when a high voltage is applied to a tube containing hydrogen gas.
8. Explain why there are only certain specific lines in the hydrogen spectrum. Compare that to the spectrum of
sunlight and why we see a continuous spectrum despite the fact that the sun is composed of hydrogen. (It
might help to know that the hydrogen in the sun is a plasma. (Feel free to look up plasma, the fourth state of
matter.)
9. If the Bohr model is so bad - for instance, it only works for hydrogen - then why do we include it in our study
of atomic theory and the arrangement of electrons in atoms?