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Recap – Last Lecture
• Bohr model of the atom: electrons occupy
orbits of certain energies.
• Evidence of this from atomic spectra in which
wavelength of light is related to energy
difference between orbits.
E = hν = hc/λ
E = -2.18 x 10-18 J (1/n2final - 1/n2initial) Z2
• Theory and experiment agree closely…for
hydrogen.
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The Bohr model is too simple
Most atomic spectra are much more
complex than expected from a Bohr
model of electron arrangements.
(http://chemistry.bd.psu.edu/jircitano/periodic4.html)
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Wave – mechanical model
• Light has a dual nature and the de Broglie
equation relates wavelength to momentum
 = h/mv
• Heisenberg Uncertainty Principle – ‘fuzziness’
x v ≥ h/4m
• Schrödinger Equation – energy of electron waves
Ĥ =
E
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Wave – mechanical model
• This can only be solved if various boundary
conditions are applied. That is, the waves
must be standing waves that are
– continuous
– single valued
– multiples of a whole number of half wavelengths
• There are then discrete solutions that
represent the energy of each electron orbital.
The orbitals are described by quantum
numbers.
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Quantum Numbers
• The Principal Quantum Number : n
n = 1, 2, 3 …
Energy
E=0
n=3
• Describes the size of the orbital
n=2
• The larger the value of n, the
bigger & the higher energy the
orbital
n =1
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Quantum Numbers
• The Angular Momentum Quantum Number : l
l = 0, 1, 2 … (n -1)
l=2
Energy
l =1
n=3
l=0
l =1
n=2
l=0
n =1
l=0
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Quantum Numbers
l describes the shape of the orbital
d orbital
Quantum no
Orbital description
l=0
s orbital
l=1
p orbital
l=2
d orbital
l=3
f orbital
p orbital
s orbital
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Quantum Numbers
• The Magnetic Quantum Number : ml
ml = -l, -(l -1) … 0 … (l -1), l
l=2
Energy
ml = -2,-1,0,+1,+2
l =1
ml = -1,0,+1
l=0
ml = 0
l =1
ml = -1,0,+1
n=2
l=0
ml = 0
n =1
l=0
ml = 0
n=3
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Quantum Numbers
ml describes the orientation of the orbital
if l = 0; ml = 0
if l = 1; ml = -1, 0, +1
(1 x s orbital)
(3 x p orbitals)
if l = 2; ml = -2, -1, 0, +1, +2
(5 x d orbitals)
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Quantum Numbers
• The Spin Quantum Number : ms
ms = + 1/2 , — ½
• Describes the spin of the electron
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Quantum Numbers
• Each orbital, uniquely described by n, l and
ml, may contain a maximum of two electrons,
one spin + 1/2, the other spin -1/2 .
l=2
Energy
n=3
ml = -2,-1,0,+1,+2
10e -
l =1
ml = -1,0,+1
6e-
l=0
ml = 0
2e-
l =1
ml = -1,0,+1
6e-
n=2
l=0
ml = 0
2e-
n =1
l=0
ml = 0
2e-
18e -
8e-
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Shell
Sub-shell
Orbital
Electrons
Applications
• Bohr model results in a periodic table with a 2, 8, 18 pattern
H
He
Li
Be
B
C
N
O
F
Ne
Na Mg Al
Si
P
S
Cl
Ne Sc
Ti
V
Cr
M
e
Fe Co
Ni
Cu Zn
• Actual periodic table needs to be explained!
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Learning Outcomes:
•
By the end of this lecture, you should be able to:
− Explain the meaning of the orbital quantum
numbers, n l ml ms.
− Understand the designation of orbitals such as
1s, 3d, 4p, 4f.
− Recognise the shapes of s, p and d atomic
orbitals.
− Determine the number of electrons in an
orbital/sub-shell/shell.
− be able to complete the worksheet (if you
haven’t already done so…)
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Questions to complete for next lecture:
1. Provide a valid set of quantum numbers, n, l and ml, of an
electron in a 4p orbital? (Question form 2015 exam)
2. Which of the following is a valid set(s) of quantum numbers
and identify the incorrect number in the other set(s)?
A.
B.
C.
D.
E.
n=1
n=3
n=2
n=2
n=3
l=1
l=1
l=0
l=2
l=2
ml = 0
ml = +1
ml = -1
ml = +2
ml = 0
ms
ms
ms
ms
ms
=
=
=
=
=
+1/2
+1/2
-1/2
-1/2
0
3. Sketch the shape of a px orbital. How many electrons can it
accommodate?
4. Which quantum number describes the shape of an orbital?
5. How many sub-shells are there in the n = 3 shell?
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