Quantum Numbers and
Atomic Structure
Refining Bohr’s Model
What are Quantum Numbers?
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Bohr defined the principal energy levels
(n = 1,2,3,4…)
experimental evidence indicated the need
for changes to this simple system
quantum numbers are quantized values
used to describe electrons in an atom
there are four quantum numbers
represented by the letters n (Bohr’s
number), l, ml and ms
The Principal Quantum Number, n
(Bohr, 1913)
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based on Bohr’s observations of line
spectra for different elements
‘n’ relates to the main energy of an
electron
allowable values: n = 1, 2, 3, 4, …
electrons with higher ‘n’ values have
more energy
The Secondary Quantum Number, l
(Sommerfeld, 1915)
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based on the observation (Michelson,
1891) that lines on line spectra are
actually groups of multiple, thin lines
‘l ’ relates to the shape of the electrons’
orbits
allowable values: l = 0 to l = n - 1
 i.e. for n = 4:
l = 0, 1, 2, or 3
the ‘l ’ values 0, 1, 2, and 3 correspond to
the shapes we will call s, p, d and f,
respectively
The Magnetic Quantum Number, ml
(Sommerfeld and Debye, 1915)
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based on the observation (Zeeman, 1897)
that single lines on line spectra split into new
lines near a strong magnet
‘ml ’ relates to the direction/orientation of the
electrons’ orbits
allowable values: ml = - l to + l
 i.e. for l = 2:
ml = -2, -1, 0, 1, or 2
electrons with the same l value but different
ml values have the same energy but different
orientations
The Spin Quantum Number, ms
(Pauli, 1925)
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based on the observation that magnets
could further split lines in line spectra, and
that some elements exhibit paramagnetism
‘ms ’ relates to the ‘spin’ of an electron
allowable values: ms = - ½ or + ½
 i.e. for any possible set of n, l, and ml
values, there are two possible ms
values
when two electrons of opposite spin are
paired, there is no magnetism observed; an
unparied electron is weakly magnetic
Defining Electrons Using
Quantum Numbers
Let’s look at the energy level n = 2:
 Possible l values: 0, 1
 For l = 0, ml = 0
 For l = 1, ml = -1, 0 or 1
 For every value of ml, there are two
electrons (ms = ½ and ms = - ½)
So, there would be 8 electrons found
in principal energy level 2 and they
would have the following
designations…
Electrons in energy level 2:
Electron
n
l
ml
ms
1
2
0 (or s)
0
½
2
2
0 (or s)
0
-½
3
2
1 (or p)
-1
½
4
2
1 (or p)
-1
-½
5
2
1 (or p)
0
½
6
2
1 (or p)
0
-½
7
2
1 (or p)
1
½
8
2
1 (or p)
1
-½
Orbits vs. Orbitals
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initially, electrons were thought to
travel in orbits (2D, travels around
nucleus at fixed distance in a
circular path, 2n2 electrons per
orbit)
quantum theory describes electrons
as existing in orbitals (3D region,
distance from nucleus varies, no
path, 2 electrons per orbital)
For our purposes:
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primary energy level (n) = ‘shell’
energy sublevel (l) = ‘subshell’
orbitals are named as a combination
of the n and l values
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e.g. an electron may exist in a ‘2p’ orbital
(n = 2, l = 1 or p)
shapes of these orbitals will be
discussed soon
Energy-Level Diagrams
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now we can be more specific
for every ‘n,’ energy increases from
spdf
quantum number restrictions state that
there can only be:
one s orbital (= 2 electrons) for any value of n
 three p orbitals (= 6 electrons) for n = 2,3,4, …
 five d orbitals (= 10 electrons) for n = 3,4,5, …
 seven f orbitals (=14 electrons) for n = 4,5,6, …
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Relative Energies of Electron Orbitals
Ref: http://www.chemistry.mcmaster.ca/esam/Chapter_4/section_3.html
When Placing Electrons in Orbitals…
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aufbau principle: fill lower-energy
orbitals first
Hund’s rule: within the same energy
level, give each orbital one electron
before pairing up electrons
Pauli exclusion principle: two
electrons within the same orbital
must have opposite spins
Aufbau (‘building up’) Diagram
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this diagram will help you remember the proper order for
filling orbitals
7s
6s
5s
4s
3s
2s
1s
7p
6p
5p
4p
3p
2p
7d
6d
5d
4d
3d
7f
6f
5f
4f
Energy-Level Diagram for Vanadium
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vanadium has 23
electrons
read on pages
189 – 190 to
learn how to
draw energylevel diagrams
for ions
The Following is Just Beautiful…
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The quantum theory of the atom
agrees completely with the periodic
table, which had been around for 30
years and was developed without
any knowledge of electron
arrangements….
Wait for it…
Relationship between the first two quantum numbers and
the periodic table:
Referring to quantum theory and the
periodic table of the elements:
“The unity of these concepts is a triumph
of scientific achievement that is
unparalleled in the past of present.”
- Text, pg. 185
Read more on pp. 194 – 195 in your text!
Electron Configurations
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More concise than energy-level
diagrams but provide same
information
e.g. for vanadium:
V: 1s2 2s2 2p6 3s2 3p6 4s2 3d3
Try chlorine right now…
Cl: 1s2 2s2 2p6 3s2 3p5
Shorthand Electron Configurations
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use noble gases as a starting point
e.g. for vanadium:
V: [Ar] 4s2 3d3
for chlorine: Cl: [Ne] 3s2 3p5
The Power of What You Now Know
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You have seen that the periodic table is
explained for you as never before
Charges of ions can be explained
e.g. lead Pb: 6s2 4f14 5d10 6p2
Pb2+ ion: remove two electrons from 6p
Pb4+ ion: remove two electrons from 6p
and two electrons from 6s
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Magnetism is explained (pp. 195-196)