### Lecture 13 (Slides) September 26

```Line Spectra & Quantization
• Last day we stated that, in the absorption and
emission spectra of atoms, only a small
number of frequencies of light (or
wavelengths) are absorbed or emitted. A
reminder slide (emission spectrum) follows.
This slide is consistent only with energy
quantization – an atom possesses, at a given
time, only one energy from a possible set of
allowed energies.
H Atom Electronic Emission Spectrum
FIGURE 8-10
The Balmer series for hydrogen atoms – a line spectrum
General Chemistry: Chapter 8
Slide 2 of 50
Bohr Theory for the Hydrogen Atom
• Bohr used the existing physics of the early 20th
century and a set of quantum numbers to
model the observed absorption and emission
spectra of the H atom.
• Tenets of the Bohr Theory:
• (1) Energies of the H atom/electron are
quantized. Line spectra prove this!!!
• (2) The electron in a H atom moves around
the nucleus in a circular orbit. (Not true!)
Bohr Theory for the H Atom
• (3) The angular momentum of the electron in a H
atom is quantized.
• (4) Energy and angular momentum values for the
electron in a H atom are calculated using a
quantum number, n. Further, n has integer
values n = 1, 2, 3, 4, 5….. infinity.
• Angular momentum (electron) = nh/2π
• Radius of orbit = n2ao = rn (Not true!)
• Electron energy = En = -RH/n2 (Rydberg constant)
H Atom Energies - Bohr
• Electron energy = En = -RH/n2
• Notes: En is always negative. Why?
(Coulombic rationalization?) The quantity ao
is called the Bohr radius and specifies the
radius of the electron orbit for the lowest
energy state of the H atom. (Again, actual H
atoms do not have an e- circling the nucleus at
A Stamp and Bohr’s Model
• The next slide shows a stamp with Bohr’s
picture and the very important and generally
valid equation ∆E = Efinal state – Einitial state = hν.
• The H atom model of Bohr (Newtonian and
deterministic!) is not valid. Electrons do not
“circle” nuclei in regular orbits. However,
energy is required to move an electron away
from the nucleus (and vice versa).
The Bohr Atom
-RH
E= 2
n
RH = 2.179  10-18 J
FIGURE 8-13
•Bohr model of the hydrogen atom
General Chemistry: Chapter 8
Slide 7 of 50
• We could draw a simple energy ladder to
represent the gravitational potential energy of
a can of paint on various steps of a ladder. The
possible potential energy values would be
equally spaced if the ladder steps are equally
spaced. For atoms we can construct similar
“energy ladders”. For one electron atoms the
energy ladders are simple but the energy
levels are not equally spaced. The H atom
“energy ladder” is shown on the next slide.
-RH
ΔE = Ef – Ei =
nf2
1
= RH ( 2
ni
–
–
-RH
ni2
1
) = h = hc/λ
2
nf
FIGURE 8-14
Energy-level diagram for the hydrogen atom
General Chemistry: Chapter 8
Slide 9 of 50
Absorption & Emission Spectra
• Very hot H atoms can exist in electronically
excited states (the single electron is in a high
energy state with n > 1). Such atoms can emit
light as they move to a lower energy state. A
small # of light frequencies are emitted.
“Cold” H atoms can absorb light as they move
to states with larger n values. What is the next
slide telling us?
Emission
Absorption
FIGURE 8-15
Emission and absorption spectroscopy
General Chemistry: Chapter 8
Slide 11 of 50
Coulombic Interactions in the H Atom
• The Coulombic force of attraction/repulsion
between stationary point charges is described
by the familiar Coulomb’s Law
• F = keQ1Q2
r2
• where Q1 and Q2 specify the magnitude and
sign of the two point charges and r is the
distance between them. Since the H atom
nucleus (a proton) and the lone electron have
opposite charges there is a strong attractive
force between these two subatomic particles.
• Coulomb’s Law → to pull the proton and
electron apart we must do work/supply
energy. Conversely, energy must be released if
the proton and electron come closer to each
other. The closer the e- comes to the nucleus
the greater the amount of energy released.
The application of Coulomb’s Law to atomic
structure is not straightforward since electrons
in an atom are not stationary and, in fact, have
wavelike properties! In many electron atoms
the rapidly moving electrons also interact with
each other as well as the nucleus.
•
•
•
•
Calculations with the Bohr Expression
Again, by experiment
En = -RH/n2 for the single e- H atom.
We can use this energy expression to calculate:
(1) Energies for levels with different values of the
quantum number n).
• (2) Ionization energies (energy required for
removal of an electron from an atom!).
• (3) Any ΔE for a transition nFinal ← nInitial.
• (4) the frequency of light absorbed (or emitted)
for a transition nFinal ← nInitial.
H Atom Ionization – Example:
• Example: How much energy is needed to
ionize (a) one H atom and (b) one mole of H
atoms initially in their ground (lowest energy)
state.
• Hint: The problem is easily solved if we use
the Bohr energy expression for H and choose
appropriate initial and final values for n.
• Hint(?): Buzz Lightyear would not be happy!
Bohr Theory and the
Ionization Energy of Hydrogen
1
ΔE = RH ( 2
ni
–
1
) = h
2
nf
As nf goes to infinity for hydrogen starting in the ground state:
1
h = RH ( 2 ) = RH
ni
This also works for hydrogen-like species such as He+ and Li2+.
-Z2
En = RH ( 2 )
ni
General Chemistry: Chapter 8
Slide 16 of 50
One Electron Monatomic Species
• The modified Bohr energy expression on the
previous slide can be used to calculate
electronic energies for the H atom (atomic
number = Z = 1), He+ (Atomic number = Z = 2),
Li2+ (Z = 3), etc. The next slide shows a few
energy levels calculated using the modified
Bohr equation for both H and He+. Can we
account for the differences by considering
Coulombic forces and potential energies?
H Atom and He+ Energies (kJ∙mol-1)
H Atom (1e-)
0
He+ Ion (1e-)
n=∞
n=∞
n=2
n=4
n=3
n=2
n=1
Energy
(kJ∙mol-1)
-3000
-5000
Experimental
energy
Observed energy
spacings
much
gaps are are
much
larger
+ than
larger
He+for
then
for Hefor
thefor
H
the Hatom.
atom.Why?
Why?
n=1
The Bohr Energy Expression
• The Bohr Energy expression can be used to
calculate energy differences between any two
“levels” in the H atom. The energy differences
can be quoted on a J/atom or kJ/mol basis.
Energy level differences can be calculated for
monatomic one e- ions if we include the
atomic number in the Bohr expression.
• Special case: Ionization
• H(g) → H+(g) + e- or He+(g) → He2+(g) + e-
8 - 20
8 - 21
Wave Particle Duality – Light and
Subatomic Particles
• In high school physics light was treated as
having both wave like and particle like
character. Diffraction and refraction of light
both exemplify its wave like properties.
Particle like properties of light can also be
demonstrated readily. In one such experiment
– Compton effect - light hitting black blades
attached to a “wind mill” cause the wind mill
to spin. This implies, surprisingly, that light
photons have momentum.
Subatomic Particles – Wave Character
• A number of experiments show that small
particles have observable wave like properties.
Such wave like properties become increasingly
important as one moves to particles of smaller
mass. The electron is the most important of
these particles. Interesting diffraction and
refraction experiments have been conducted
with electrons.
Mathematical Description of Electrons
• The fact that electrons exhibit wave like
behavior suggested that equations used to
describe waves, and light waves in particular,
might be modified to describe electrons. We
will see some familiar mathematical functions
used to describe the electron (e.g. cos θ, sin θ,
eiθ). We will use so-called wave functions (Ψ’s)
to gain insight into the behavior of electrons in
atoms.
De Broglie’s Contribution
• De Broglie used results/equations from
classical physics to rationalize experimental
results which proved that subatomic particles
(and some atoms and molecules) had wave
like properties. He proposed, in particular, that
particles with a finite rest mass had a
characteristic wave length – as do light waves.
8-5 Two Ideas Leading to a New
Quantum Mechanics
• Wave-Particle Duality
– Einstein suggested particle-like
properties of light could explain the
photoelectric effect.
–Diffraction patterns suggest photons
are wave-like.
• deBroglie, 1924
– Small particles of matter may at
times display wavelike properties.
General Chemistry: Chapter 8
Louis de Broglie
Nobel Prize 1918
Slide 26 of 50
Wave-Particle Duality
E = mc2
h = mc2
h/c = mc = p
p = h/λ
λ = h/p = h/mu
General Chemistry: Chapter 8
Slide 27 of 50
FIGURE 8-16
Wave properties of electrons demonstrated
General Chemistry: Chapter 8
Slide 28 of 50
Probabalistic Description of Electrons
• Classical physics suggests that we should be
able (given sufficient information) to describe
the behaviour of any body – its velocity,
kinetic energy, potential energy and so on at
any point in time. Classical physics suggests
that all energies are continuously variable – a
result which very clearly is contradicted by
experimental results for atoms and molecules
(line spectra/quantized energies).
Uncertainty Principle
• The quantum mechanical description used for
atoms and molecules suggests that for some
properties only a probabalistic description is
possible. Heisenberg suggested that there is a
fundamental limitation on our ability to
determine precise values for atomic or
molecular properties simultaneously. The
mathematical statement of Heisenberg’s socalled Uncertainty Principle is given on the
next slide.
The Uncertainty Principle
h
Δx Δp ≥
4π
Heisenberg and Bohr
FIGURE 8-17
•The uncertainty principle interpreted graphically
General Chemistry: Chapter 8
Slide 31 of 50
Atomic “Diagrams”
• Many simple diagrams of atoms/atomic
structure have limitations. In class we’ll
consider some limitations of the C atom
diagram shown on the next slide.
• Representations of molecules are even more
challenging – at least if we want to consider
the electrons!
The “Carbon Atom”
• Limitations?
e le c tr o n
n e u tr o n
p r o to n
A Molecular Model
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