Download Electrons, Quantum Numbers, and Energy Levels

CHEM 107, Fall 2016
Electrons &
Quantum Mechanics
Class #16
Electrons,
Quantum Numbers,
and Energy Levels
CHEM 107
L.S. Brown
Texas A&M University
Uncertainty Principle
•  Curious result of wave interpretation
•  “Position and energy (or momentum)
cannot be specified simultaneously”
•  Mathematically:
•  Experiments prove that energy levels
exist
•  How can we explain this?
•  “Quantum Mechanics”
Application of wave concepts to
atoms & electrons
Electrons
•  Electrons in atoms can only have
certain quantized energies.
•  If we measure electron energy, the
position CAN’T be specified
(Δx)(Δp) > h/4π
Electron Diffraction
Electrons
•  Electrons show properties of waves
AND particles. (Like light does!)
•  Electrons in atoms best described as
“delocalized waves”
Aluminum
Graphite
(single crystal)
© 2016, L.S. Brown
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CHEM 107, Fall 2016
Standing Waves
n=3
Ψ
n=2
0
L
n=1
Wavefunctions & Intensity
Ψ2
n=3
n=2
n=1
Equation
•  We can write a general equation for the
allowed standing waves:
Ψ = sin
nπx
L
Where n = 1, 2, …
n is called a quantum number
Schrödinger Equation
HΨ=EΨ
•  This is a differential equation in disguise
•  H is an “operator,” Ψ is a “wave
function,” and EΨ is an energy.
•  H includes kinetic and potential
energies
Intensity at any point is proportional to
the square of the amplitude of the wave.
Schrödinger Equation
For a hydrogen atom:
∂2 Ψ ∂ 2 Ψ ∂ 2 Ψ e 2
+
+
Ψ = EΨ
∂x 2 ∂y 2 ∂z 2 4πε0 r
Schrödinger Equation
•  Solving equation
gives Ψ and E for
allowed states.
•  Use spherical
coordinates:
Ψ = Ψ(r, θ, φ )
•  Solution contains 3
different quantum
numbers
© 2016, L.S. Brown
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CHEM 107, Fall 2016
Orbitals &
Quantum Numbers
•  Q. numbers: n, ℓ, and mℓ
•  Also called “principal,” “azimuthal,”
“magnetic”
•  A set of the 3 numbers defines an
orbital
•  Orbital = the wave representation of an
electron in an atom
Quantum Numbers
•  n - principal quantum number
influences energy and size of the
orbital
●  n = 1, 2, 3, ...
● 
•  ℓ - azimuthal quantum number
● 
● 
shape of orbital (mainly)
ℓ = 0, 1, 2, ..., (n – 1)
Allowed combinations
Quantum Numbers
•  n = 1, 2, 3, …
•  ℓ = 0, 1, 2, ..., (n – 1)
•  mℓ - magnetic q. number
orientation of orbital (mainly)
●  mℓ = –ℓ, ..., 0, ... +ℓ
n
l
ml
# of
orbitals
type of
orbitals
1
0
0
1
1s
2
0
1
0
-1,0,+1
1
3
2s
2p
3
0
1
2
0
-1,0,+1
-2,-1, 0,+1,+2
1
3
5
3s
3p
3d
● 
Combining these rules lets us find the
allowed orbitals
Some Orbital Wavefunctions
(DON’T try to memorize these!)
r
1
Ψ2s = (2a)-3 / 2 (2 - ) exp(-r / 2a)
a
4π
1
r
3
Ψ2 pz =
(2a)-3 / 2 exp(-r / 2a)
cos θ
3
a
4π
1
r
3
Ψ2 px =
(2a)-3 / 2 exp(-r / 2a)
sin θ cos φ
3
a
4π
1
r
3
Ψ2 py =
(2a)-3 / 2 exp(-r / 2a)
sin θ sin φ
3
a
4π
The meaning of Ψ
•  Orbitals are wavefunctions, defined
mathematically
•  Physical interpretation?
•  Ψ 2 is the probability of finding the
electron at some point in space
•  “Pictures” of orbital shapes are graphs
of Ψ 2
© 2016, L.S. Brown
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CHEM 107, Fall 2016
Representing Orbitals
Electron probability
•  We depict orbitals in different ways
•  Graph of Ψ2 (or “electron density”)
•  Pictorial representations
1s orbital
Most probable distance:
Bohr radius = 53 pm
Distance from nucleus
1s orbital
•  s orbital wavefunctions contain no
angular terms
☛ Spherical orbitals
p orbitals
•  Set of 3 p orbitals for each n-value
(2p, 3p, ...)
•  All same shape
“dumbbell” or “hourglass”
•  All mutually perpendicular
© 2016, L.S. Brown
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