Quantization of Energy. Calculating the Wavelength of a Line in the Hydrogen Spectrum. Wave Mechanics. Spectroscopy. Ionization energy of Hydrogen

1. The Bohr Atom
Dr. K. Shahzad Baig
Memorial University of Newfoundland
(MUN)
Canada
Petrucci, et al. 2011. General Chemistry: Principles and Modern Applications. Pearson Canada Inc., Toronto, Ontario.
Tro, N.J. 2010. Principles of Chemistry. : a molecular approach. Pearson Education, Inc.

2. The Bohr Atom
The Rutherford model of a nuclear atom does not indicate
How electrons are arranged outside the nucleus of an atom.
According to classical physics, stationary, negatively charged electrons would be pulled
into the positively charged nucleus. This suggests that the electrons in an atom must be in
motion, like the planets orbiting the sun.
Orbiting electrons should be constantly accelerating and should radiate energy. By losing
energy, the electrons would be drawn ever closer to the nucleus and soon spiral into it.

3. • The electron moves in circular orbits about the nucleus with the motion described
by classical physics.
• An electron can pass only from one allowed orbit to another. In such transitions,
fixed discrete quantities of energy (quanta) are involved either absorbed or emitted.
• The electron has only a fixed set of allowed orbits, called stationary states.
• The allowed orbitals have a discrete set of vlues, is called the angular momentum.
Its possible values are nh/2п,
where n must be an integer.
Thus the quantum numbers progress: n = 1 for the first orbit; for the second orbit;
n=2 and so on.

4. The Bohr theory predicts the radii of the allowed orbits in a hydrogen atom.
𝑟𝑛 = 𝑛2 𝑎0 Where n= 1,2,3 ….
When the electron is free of the nucleus, by convention, it is said to be at a 0 of energy.
When a free electron is attracted to the nucleus and confined to the orbit n, the
electron energy becomes negative, with its value lowered to RH , is a numerical
constant with a value of 2.179 x 10-18 J.
𝐸 𝑛 =
−𝑅 𝐻
𝑛2
… 8.5

5. According to the Bohr model, lower energy
orbits are closer to the nucleus and electrons
associated with low energy orbits must absorb
more energy to be removed from the atom.
Normally, the electron in a hydrogen atom is found in the
orbit closest to the nucleus (n=1). This is the lowest
allowed energy, or the ground state. When the electron
gains a quantum of energy, it moves to a higher level
(n=2,3 and so on) and the atom is in an excited state.
When the electron drops from a higher to a
lower numbered orbit, a unique quantity of
energy is emitted the difference in energy
between the two levels.
∆𝐸 = 𝐸𝑓 − 𝐸𝑖
=
−𝑅 𝐻
𝑛 𝑓
2 −
−𝑅 𝐻
𝑛 𝑖
2
= 𝑅 𝐻
1
𝑛𝑖
2
−
1
𝑛 𝑓
2
= 2.179 𝑥 10−18 𝐽
1
𝑛𝑖
2
−
1
𝑛 𝑓
2
… 8.6

7. Understanding the Meaning of Quantization of Energy
Example
Is it likely that there is an energy level for the hydrogen atom En = -1.00 x 10-20 J
Solution
Let us rearrange equation (8.5), solve for and then for n2 and then for n.
𝑛2 =
−𝑅 𝐻
𝐸 𝑛
=
−2.179 𝑥 10−18
−1.00 𝑥 10−20
= −2.179 𝑥 102 𝐽 = 217.9 𝐽
𝑛 = 217.9 = 14.7 6
Because the value of n is not an integer, this is not an allowed energy level for the
hydrogen atom.

8. Calculating the Wavelength of a Line in the Hydrogen Spectrum
Example
Determine the wavelength of the line in the Balmer series of hydrogen corresponding to the
transition from n=5 to n=2
Solution
∆𝐸 = 2.179 𝑥 10−18 𝐽
1
𝑛𝑖
2
−
1
𝑛 𝑓
2
… 8.6
∆𝐸 = 2.179 𝑥 10−18 𝐽
1
52
−
1
22
∆𝐸 = 2.179 𝑥 10−18 𝑥 (0.004000 − 0.25000)
∆𝐸 = −4.576 𝑥 10−19
𝐽

9. Rearranging 𝐸 𝑝ℎ𝑜𝑡𝑜𝑛 = 𝐸 = ℎ𝜈 gives the frequency
𝜈 =
𝐸 𝑝ℎ𝑜𝑡𝑜𝑛
ℎ
=
4.576 𝑥 10−19
6.626 𝑥 10−10 𝐽 𝑆 𝑝ℎ𝑜𝑡𝑜𝑛
= 6.90𝐽 𝑝ℎ𝑜𝑡𝑜𝑛6 𝑥 1011 𝑆−1
Rearranging ‘c = λ ν’ for the wavelength gives the following result:
𝜆 =
𝑐
𝜈
=
2.998 𝑥 108 𝑚 𝑆−1
6.906 𝑥 1014 𝑆−1
=
Type equation here.
= 4.341 𝑥 10−7
𝑚 = 434.10 𝑛𝑚

10. Wave Particle Duality
How precisely the behavior of subatomic particles can be determined.
The two variables that must be measured are the position of the particle and its
momentum.∆𝑥 ∆𝑝 ≥
ℎ
4𝜋
The conclusion they reached is that there must always be uncertainties in measurement
such that the product of the uncertainty in position, ∆x, and the uncertainty in momentum
∆p.
The de Broglie relationship (implies that for a wavelength the momentum of the
associated particle is precisely known. However, to get around the inability to locate the
particle, we can combine several waves of different wavelengths into a wave packet to
produce an interference pattern that tends to localize the wave the momentum becomes
more and more uncertain. And vice versa.
we understand that the consequence of wave particle duality is the uncertainty
principle

11. Wave Mechanics
Standing wave
Waves which appear to be vibrating
vertically without traveling horizontally.
Created from waves with identical
frequency and amplitude interfering with
one another while traveling in opposite
directions.
Node
Positions on a standing wave where the wave stays in a fixed position over time because
of destructive interference
Antinode
Positions on a standing wave where the wave vibrates with maximum amplitude.

12. The Bohr Theory and Spectroscopy
• When atoms are given energy, their electrons jump up levels. The energy that is needed
to cause the electrons to jump energy levels is specific.
• The atoms are “excited”, meaning they gain energy.
• After the atom electrons have been promoted, they get demoted again, that is they move
back down the energy levels. When being demoted, the atoms emit the specific
amounts of energy.
• This emitted energy is in the form of light. When the light is viewed through a
spectroscope, it splits up into an emission spectrum.
Spectroscopy
The spectrum consists of a series of lines;
The colour of these lines is specific to the wavelength.
The frequency is related to the energy:

13. For absorption of a photon to take place, the energy of the photon must exactly match the
energy difference between the final and initial states,
that is
Emission spectra are generally more complicated than absorption spectra.
An excited sample will contain atoms in a variety of states, each being able to drop
down to any of several lower states.
An absorbing sample generally is cool and
transitions are possible only from the ground
state.
𝜈 =
𝐸𝑓 − 𝐸𝑖
ℎ
The hydrogen Balmer lines are not seen, in absorption from cold hydrogen atoms

14. •Bohr’s theory uses the idea of quantisation of energy. The main points of Bohr’s theory
were:
1. The electron in the H atom is only allowed to exist in certain definitive energy
levels.
2. A photon of light is emitted or absorbed when an electron changes from one
energy level to another.
3. The energy of the photon is equal to the difference between the two energy levels.
4. The frequency of the emitted or absorbed light is related to the energy by: E=hv.
5. The uv emission spectrum of hydrogen can be related to the Lyman series.
As the separated electrons demote, the energies are emitted. The spectrum lines become
closer together the further from the nucleus. This is because the energy levels are closer
together.
Bohr’s theory of the Hydrogen Atom

15. The Bohr Theory and the Ionization Energy of Hydrogen
The Bohr model of the atom helps to clarify the mechanism of formation of cations
When an electron is freed, the atom is ionized, the quantity of energy is called the
ionization energy of the hydrogen atom and the energy of the free electron is zero.
ℎ𝜈 𝑝ℎ𝑜𝑡𝑜𝑛 = 𝐸𝑖 = −𝐸1
If ni= 1 and nf = ∞ are substituted in the Bohr expression for an electron initially in the
ground state of an H atom, then
ℎ𝜈 𝑝ℎ𝑜𝑡𝑜𝑛 = 𝐸𝑖 = −𝐸1 =
𝑅 𝐻
12 = 𝑅 𝐻 =
For ionic species, the nuclear charge (atomic number) appears in the energy-level expression. That is,
𝐸 𝑛 =
−𝑍2 𝑅 𝐻
𝑛2
= 𝑅 𝐻
1
𝑛𝑖
2
−
1
𝑛𝑖𝑓
2

16. Inadequacies of the Bohr Model
From an experimental point of view, the theory cannot explain:
• the emission spectra of atoms and ions with more than one electron.
• the effect of magnetic fields on emission spectra
The Bohr theory is an uneasy mixture of classical and nonclassical physics.
A fundamental error of the Bohr model was to constrain an electron to a one-
dimensional orbit 1-D. The electron cannot move off a circular path of a fixed radius.