This small essay concisely outlines how Classical mechanics was deemed unacceptable when describing the motions of electrons within an atom through the observations made by hydrogen spectra, and how this lead to a revolution in atomic theory. Included is a brief overview of how Bohr arrived at his model through applying quantum mechanics. Written for; First year Undergraduate study, Materials Science and Engineering, The University of Sheffield Graded at 78%

1. MAT1610 – Thomas Howard Oulton
How the Bohr model of the atom accounts for limitations with
classical mechanics through observations of hydrogen spectra.
It was in 1911 when Ernest Rutherford put forward his theory for the structure of the atom [1],
which was based on results from his gold leaf experiment. The experiment (as shown in figure
1) consisted of a thin film of gold leaf positioned a set distance away from a radioactive source
which emitted α-particles during decay. A detector screen was positioned around the
experiment where three observations about the trajectories of the α-particles were made. It was
noted that; most of the particles passed through with little or no alteration to the angle of their
path, some of the particles were deflected, and very few particles were reflected (i.e. high
angles of deflection) back towards the direction of the α-emitter. From this, Rutherford
overturned previous theories of atomic structures such as J. J. Thomson’s plumb pudding
theory and the cubical atom theory.
Rutherford suggested that an atom was composed of electrons
which orbited the centre of the atom, and that the centre, called
the nucleus, is positive in charge and accounts for the majority
of an atom’s mass. Yet despite Rutherford’s postulation, there
were still certain phenomena such as Hydrogen spectra and the
stability of atoms which could not be accounted for.
Hydrogen spectra can be observed when bright light is emitted
from a glass tube, caused by a large potential difference at both
ends of the glass envelope which are completely sealed and
contain a hydrogen atmosphere at low pressure [2]. The
emitted light passes through the atmosphere, allowing the light
waves to interact with the hydrogen atoms which alter the
wavelength of the light. This now altered light is then passed through a prism to split it up into
its constituent wavelengths (red at 656.2nm, blue-green at 486.1nm, blue-violet at 434.0nm
and violet at 410.0nm) so that they can be observed (Figure 2). The results perplexed scientists,
in that all constituent wavelengths of the altered light were discrete - i.e. there was no smearing
of the wavelengths. This proved the inadequacy of classical mechanics which Rutherford
constructed his model upon. A new set
of governing principles was needed in
order for a better atomic model to be
theorised – one that could best describe
the results seen.
Bohr understood that the application of
classical mechanics could not be used to
describe the motion of electrons within
atoms. Under classical laws, orbiting
point-like charges, such as electrons,
emit radiation within the
electromagnetic (EM) spectrum [3]. The
continual emission of EM waves causes a loss of energy and therefore a reduction in orbiting
radius, until eventually the atom collapses on itself. Since this was not the case and most atoms
are stable, Bohr also concluded that the orbits which electrons occupy must have fixed
Figure 1: Schematic diagram of Ernest
Rutherford's gold leaf experiment and
the observed Alpha particle
trajectories. [5]
Figure 2: Schematic diagram of how Hydrogen spectra is conducted,
and its observed results [2].

2. MAT1610 – Thomas Howard Oulton
(discrete) energy values, and the occupying electrons do not radiate EM waves as they orbit.
However, the atom must still be able to emit visible light so as to follow the observations made
with Hydrogen spectra. It was known that the energy of the observed wavelengths from
hydrogen spectra must have the equivalent energy as what is emitted from the electrons within
hydrogen atoms, and Bohr was aware of the work Max Planck carried out in earlier years
stating that the Energy of light waves is equal to a constant multiplied by the frequency of
observed light:
𝐸 = ℎ𝜈
Since a change in energy of the electrons causes the emission of EM waves, to emit light the
electrons have to change their angular momentum (L) [4], which can be done through changing
the size of the radius (r) of the occupied orbits. Bohr stipulated that the angular momentum of
these different orbits must be spaced equally apart, and that electrons would only radiate EM
waves by lowering their energy - accomplished by moving from higher orbitals to lower ones.
From this, Bohr related the time period for an electron to complete one orbit to the radius of
that orbit using Kepler’s equation: (𝑇 ∝ 𝑟
3
2⁄
), which could be substituted into Planck’s relation
giving:
∆𝐸 =
ℎ
𝑟3 2⁄
With further manipulation, Bohr found that with quantised values of angular momentum of the
electrons, L, the spacing between each orbit was a multiple of ħ:
𝐿 =
𝑛ℎ
2𝜋
= 𝑛ħ
where n represents the principal
quantum number and h is Planck’s
constant. From this, using the angular
momentum equation (L=mvr), the
distance of each energy level (n-value)
from the nucleus could be calculated,
and in turn, the minimum energy
required by each electron to exist in the
different energy orbitals.
Building on Rutherford’s model and
his own work, the Rutherford-Bohr
model, or Bohr model for short (Figure
3a), states that an atom is composed of a small, positively charged
centre, called the nucleus, which accounts for the majority of the
atom’s mass. Orbiting this nucleus are the electrons which sit in
orbitals of fixed/quantised energy, all of which have equally
spaced angular momentum, L, and are multiples of ħ (equation 3).
Each orbital can house a fixed number of electrons, which, once full, start to fill up the next
orbital. Electrons can jump from lower to higher energy orbitals by absorbing energy
(absorbing light energy for example). When returning to their original orbitals/ground state,
the electrons emit energy within the Electromagnetic spectrum. This energy is emitted as
(1)
(2)
(3)
Figure 3(a): Bohr model of the
atom. The orbital shells are labled
from 1 upwards [6].
Figure 3(b): shows how the
quantum jump made from excited
electrons moving from higher
energy orbitals to lower energy
orbitals produces different
colours of spectral lines, and
where these relate in order of
increasing wavelengths. [7]

3. MAT1610 – Thomas Howard Oulton
specific wavelengths, often within the visible light spectrum, as observed with Hydrogen
spectra. The energy of the incident light depends on the magnitude of the quantum jump made
by any given electron – e.g. from the fourth shell down to the first, as indicated in figure 3(b).
In summary, the Bohr model was able to account for the observed phenomena with hydrogen
spectra which previous atomic models could not explain. This was backed up by Bohr’s
mathematical workings, acknowledging that classical mechanics has certain limitations, and
working with new systems and Physics that is now part of Quantum mechanics. By applying
these quantum rules first, Bohr could describe the behaviour of an atom’s electrons using
classical physics. More to Bohr’s calculations, predictions of which wavelengths might be
present in spectroscopy using different gases could be theoretically made, and then supported
with empirical data, thus justifying and supporting Bohr’s model.

4. MAT1610 – Thomas Howard Oulton
References
[1] L. Akhlesh and E. E. Salpeter, Models and Modelers of Hydrogen, vol. 65, -: American Journal of
Physics, 1997, pp. 933-934.
[2] Bodner Research Web, "Development of Current Atomic Theory," ChemEd, [Online]. Available:
http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch6/bohr.php. [Accessed 06 April
2016].
[3] Wikipedia, "Larmor Formula," [Online]. Available:
https://en.wikipedia.org/wiki/Larmor_formula. [Accessed 04 April 2016].
[4] Wikipedia, "The Bohr Model," [Online]. Available: https://en.wikipedia.org/wiki/Bohr_model.
[Accessed 02 April 2016].
[5] Glenco Online Learning Centre, "Matter and Change; The atom and sub-atomic particles,"
2016. [Online]. Available:
http://glencoe.mheducation.com/sites/007874637x/student_view0/chapter4/section2/self_ch
eck_quizzes.html. [Accessed 06 April 2016].
[6] W. D. Callister, Jr. and D. D. Rethwisch, "Electrons in Atoms," in Materials Science and
Engineering, New Jersy, Wiley, 2015, p. 20.
[7] butane.cham, "Lecture 09," [Online]. Available:
http://butane.chem.uiuc.edu/cyerkes/chem102ae_fa08/homepage/Chem102AEFa07/Lecture_
Notes_102/Lecture%209.htm. [Accessed 04 April 2016].