Bohr's Atom Model: UNIZOR.COM - Physics4Teens - Building Blocks of Matter

Notes to a video lecture on http://www.unizor.com

Bohr's Atom Model

In the previous lecture we briefly mentioned the planetary model of an atom, authored by Ernest Rutherford in 1911 and supported by many physicists at that time. We also mentioned two fundamental problems with this model.

Firstly, a purely theoretical problem with planetary model of an atom was related to the fact that an electron rotating around a nucleus, that is going with centripetal acceleration, should produce oscillations of an electromagnetic field and, therefore, is supposed to lose energy. This would cause its falling onto a nucleus, which destroys the fundamental structure of matter.

Secondly, the spectrum of radiation produced by an electron falling onto a nucleus had to be continuous, which contradicted experimental results that showed discrete spectrum.
Experiments showed that gases emit light when exposed to intensive electric field. The electric field supplies energy to electrons, and they increase the radius of their orbits. Then they spontaneously release this energy as visible light (electromagnetic field oscillations in visible spectrum of frequencies) and lower their orbits. This light, going through a prism, produces distinct spectral lines of monochromatic light specific for each gas and independent of an intensity of the electric field applied to it, gas temperature or density.
This consistency of spectral lines had to be explained and planetary model failed to do it.

In 1900 Max Plank, based on his experiments with radiation caused by heat, has suggested that radiation is carried in chunks, and each chunk has an amount of energy E that depends only on the frequency f of this radiation:
E = h·f, where
h=6.62607015·10⁻³⁴ m²·kg/s is Planck's constant

At the same time, analyzing the process of photoelectric emission, Albert Einstein used the idea of quantum character of the electromagnetic oscillations to introduce a photon as an indivisible unit of absorbed or emitted electromagnetic energy.

Combining the theoretical knowledge and results of experiments, Niels Bohr suggested a new atom model that seemed to be capable of explaining all the experimental results on a new theoretical foundation.

The Bohr's atom model was an enhancement of the Rutherford's planetary model and repeats its geometrical configuration of a central positively charged nucleus and orbiting around it negatively charged electrons.
The main modification to that model was using a quantum character of energy carried by electromagnetic field oscillations.

The development of quantum concepts of electromagnetic energy was a collective effort of famous physicists Planck, Lorentz, Einstein, Haas, Nicholson and others. Bohr used these concepts, applying them to formulate his model of an atom.

Bohr's model of an atom is based on these main principles:

I. For each type of an atom there are certain stable electron orbits (or shells), called stationary, where electron, as long as it stays on such an orbit, emits no energy.
This proposition contradicts the classical theory of electromagnetism, which states that accelerated electron must emit energy.

II. Each stationary orbit is associated with certain level of energy. The larger the radius of an orbit of an electron - the higher energy it possesses.
Energy is absorbed by an electron, when it jumps from a lower energy shell to a higher energy one.
Energy is emitted by an electron, when it jumps from a higher energy shell to a lower energy one.
The amount of energy absorbed or emitted by an electron when it jumps from one shell to another equals exactly the difference in energy levels of these shells.

III. When electron jumps from an orbit of higher energy level E_hi to an orbit of lower energy level E_lo, it emits electromagnetic radiation of frequency f, determined by an equation
E_hi − E_lo = h·f
where h is Planck's constant.
Obviously, to jump from an orbit of lower energy level E_lo to an orbit of higher energy level E_hi, electron absorbs this amount of energy from outside.

IV. The next principle is more complicated and was formulated by Bohr as follows.
The angular momentum of an electron rotating on a stationary orbit equals to an integer multiple of reduced Planck constant ħ=h/(2π) (Latin letter h with a horizontal stroke).
This principle is known as quantization of angular momentum.
As we know, a momentum of a body of mass m moving along a straight line with speed v equal to m·v.
An angular momentum L of a body of mass m uniformly rotating with linear speed v along a circular trajectory of radius r equals to m·v·r.
So, this principle of Bohr's model can be expressed in an equation
L = m·v·r = n·ħ
where n is a positive integer number and ħ is a reduced Planck's constant.

This was Bohr's hypothesis given based on some experimental facts and certain theoretical derivations from them.
At the same time it corresponded to experimentally obtained formula for radii of electron orbits of an atom of hydrogen suggested by Rydberg (see previous lecture on Rydberg Formula).

The theoretical explanation of this last Bohr's principle that quantizes the angular momentum was suggested later on by de Broglie in 1924.

Contemporary explanation, based on duality of a particle and a wave, can be shortened to the following.

The full energy E of a particle of mass m, using the Theory of Relativity, can be expressed as
E = m·c²
where c is the speed of light.

From the Quantum Theory the energy E of a quantum of light (photon) of frequency f and wave length λ is
E = h·f = h·c/λ
where h is Planck's constant.
Therefore,
m·c² = h·c/λ
m·c = h/λ

Expression p=m·c is a momentum of a particle of mass m moving with speed c.
Therefore,
p = h/λ

When an object of mass m rotates with linear speed v along an orbit of radius r. its angular momentum is
L = p·r = h·r/λ

Electron, rotating around a nucleus, from the wave theory, is analogous to a string fixed at both ends, like on a guitar. The wave length of a sound this string produces must fill the length of a string integer number of times, otherwise it will interfere with itself.
Using this principle, the wave length of an electron λ and a radius of its orbit must be in a relation
2π·r = n·λ
where n any positive integer number.

Using the above, we obtain
L = h·r/λ =
= h·r·n/(2π·r) =
= h·n/(2π) = n·ħ
where
n is any positive integer number and
ħ is a reduced Planck constant.