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Quantum Numbers
A Short Tutorial
Bohr Model of Hydrogen Atom
• an e-’s is found in specific
energy levels.
• These levels represent a
fixed distance from the
• A quantum of energy is the
amount of energy needed
to move one electron.
• Neils Bohr was also a GK
for the Danish National
Team !
The Bohr Model
• Like the rungs of this ladder,
the energy levels in an atom are
not equally spaced.
• The higher the energy level of
an e-, the less energy it takes to
move the e- to the next higher
energy level.
• Moving requires a quantum
Modern View of Atoms
1. Energy of e-’s is quantized.
2. e-’s sometimes act like
3. Position of e- is based on
• Modern view is the
model of the atom.
Location of Electrons Based on Probability
• e-’s are mathematically
predicted to exist in
electron clouds.
• 90% chance that e-’s
exist in this shape.
• e- clouds have high
electron density.
• An Atom contains
4 quantum
• Schrodinger was
always very
serious about
Principal Quantum Number
• Symbol is n.
• Distance from nucleus.
• Values are n = 1, 2, 3, 4, 5, 6, 7, ….
Clearly, the further the distance from the
nucleus, the bigger the volume of the
energy level is.
Orbital Quantum Number
 Symbol is l.
 It corresponds to the type of orbital,
either s, p, d or f.
 It has values of 0, 1, 2, 3, ….
 Check the next several slides for a
look at the different orbitals.
The s - orbital
The s-orbital is
shaped like a sphere
with its center at
the origin of a 3
dimensional grid.
The axes on the grid
are the x-axis, the yaxis, and the z-axis.
The p - orbital
The p – orbitals are dumbbell shaped and lie across
each axis, so they are 90° from each other. There
are always 3 p – orbitals for each energy level.
The p - orbital
Another look at p – orbitals and how they
orientate themselves to each other.
The d - orbitals
The d – orbitals are also dumbbell shaped and
generally lie between the axes. There are always
5 d – orbitals.
The f - orbitals
Here is a list of the 7 f – orbitals.
fxyz, fx3, fy3, fz3, fz(x2-y2), fy(z2-x2), fx(z2-y2)
They have complex shapes.
Magnetic Quantum Number
 Symbol is m though sometimes lm.
 It corresponds to the orientation of the
orbital around the axis.
 It has values of - l, … 0, …. + l
 You have seen these on earlier slides.
Check the next slide in the presentation
to look at the p – orbitals again.
Magnetic Quantum Numbers
• Notice the x, y, and z part of the orbital
notation below ?
• This is telling us how the orbital is orientated
around the axes.
• This is the magnetic quantum number.
Magnetic Quantum Number
• There are as many orientations as there are
• Check the table:
Orbital type
# of orbitals
Magnetic QN
-1 0 + 1
-2 -1 0 +1 +2
-3 -2 -1 0 +1 +2 +3
Magnetic Quantum Number
• Now let’s apply this to the p – orbitals.
• Recall: There are three p – orbitals and they are the
px , py , and pz .
• From the slide above we see that the magnetic
quantum numbers for the p – orbitals are as
follows: -1 , 0, +1.
• Therefore the px corresponds to -1, the py
corresponds to 0, and the pz corresponds to + 1.
• The other orbitals work in the same way – there are
as many orbitals as magnetic quantum numbers.
Spin Quantum Number (ms)
• Symbol is s, though
sometimes ms
• It tells us that the two
electrons in the orbital have
opposite spins.
• It’s values are either
+ ½ or – ½ .
• Remember that there can be
no more than 2 electrons per
Is there a pattern here?
for n = 1 : 1orbital
for n = 2 : 4 orbitals (one s and three p orbitals)
for n = 3 : 9 orbitals (one s, three p, and five d
For a given energy level n,
there are n2 orbitals
Since there can be 2 electrons per orbital, the
number of electrons in any energy level is
described as 2 n2
Lets put all the info together
Lets put all the info together
Lets put all the info together
• Remember that the quantum numbers are the
solutions to the Schrodinger equations.
• They are actually numbers but it would be
confusing to have 4 numbers right next to
each other so some are given letter
designations to make it easier to read.
• For example: For orbital quantum numbers
the s – orbital = 0. The p – orbital = 1. The d –
orbital = 2 etc.
• Let’s look at a certain electron in an Iron atom.
We will look at the last electron to be filled in
its’ electron configuration which would be 3d6
Principal QN = 3 It is in the 3rd energy level.
Orbital QN = 2 It is a d – orbital
Magnetic QN = -2 It is filling up the d – orbital
Spin QN = - ½
It has an opposite spin from
the electron already there.