Lecture 7: Quantum Numbers and
Electron Configurations
Dr Harris
9/10/12
HW: Ch 5: 2, 5, 7, 17, 19, 24, 43, 44
Bohr’s Theory Thrown Out
• In chapter 4, we used Bohr’s model of the atom to describe atomic
behavior
• Unfortunately, Bohr’s mathematical interpretation fails when an atom
has more than 1 electron (nonetheless, it still serves as a useful visual
representation of an atom)
• Also, Bohr has no explanation of why electrons simply don’t fall into the
positively charged nucleus.
• This failure is due to violation of the Uncertainty principle.
2πr = nλ
n= 1, 2, 3…
−2.1799 𝑥 10−18 𝐽
𝐸𝑛 =
𝑛2
Bohr’s model
Heisenburg’s Uncertainty Principle
• The uncertainty principle is a cornerstone of quantum theory
• It asserts that
“You can NOT measure accurately both the position and momentum of
an electron simultaneously, and this uncertainty is a fundamental
property of the act of measurement itself”
• This limitation is a direct consequence of the wave-nature of
electrons
Heisenburg’s Uncertainty Principle
• Consider an electron
• If you wish to locate the electron within a distance Δx, we must use a
photon with a wavelength which is equal to or less than Δx
• When the electron and photon interact, there is a change in momentum
of the electron due to collision with the photon
• Thus, the act of measuring the position results in a change in its
momentum
λ’
λ
Heisenburg’s Uncertainty Principle
• If the minimum uncertainty (precision) in x is defined as Δx, and that of p
is Δp:
(Δx)(Δp) ≥
ℎ
4𝜋
• This indicates the limit of the accuracy of trying to simultaneously
determine both an electron’s position and momentum
• This uncertainty is very significant when applied to atomic and subatomic
particles
Example
• An electron is moving at a speed of 5.0 x 106 m/s. You want to
measure the position of an electron within 5 x 10-11 m. Estimate
the uncertainty in the momentum, then the velocity.
(∆𝑥)(∆𝑝) ≥
(∆𝑝) ≥
ℎ
4𝜋
ℎ
4𝜋 ∆𝑥
(6.626 𝑥 10−34 𝐽𝑠)
∆𝑝 ≥
4𝜋(5 𝑥 10−11 𝑚)
∆𝑝 ≥ 1 𝑥 10−24 𝑘𝑔 𝑚 𝑠 −1
We assume the mass of the
electron to be exact (m= 9.109 x
10-31 kg), so the uncertainty is
only in the velocity
∆𝑝 = 𝑚(∆𝑉)
𝑚 ∆𝑉 ≥ 1 𝑥 10−24 𝑘𝑔 𝑚 𝑠 −1
Very large uncertainty (20% of actual speed)
∆𝑉 ≥ 1 𝑥 106 𝑚𝑠 −1
Why does Bohr’s Model Fail?
• Bohr’s model conflicts with the uncertainty principle because if the
electron is set within a confined orbit, you know both its
momentum and position at a given moment. Therefore, it can not
hold true.
Schrodinger Equation and Wave
Mechanics
• From the uncertainty principle came the Schrodinger equation,
which is now the central equation of a branch of physics known as
quantum mechanics
𝐻Ψ = 𝐸Ψ
• This equation is consistent with both the wave and particle nature of
electrons
• Schrondinger treats the electron as a circular wave around the
nucleus
• You don’t need to know how to use this equation, but you should
understand the importance of Ψ (wavefunction), which we will
discuss in a later slide
Contrast Between Bohr’s Theory and
Quantum Mechanics
• The primary differences between Bohr’s theory and quantum
mechanics are:
• Bohr restricts the motion of electrons to exact, well-defined
orbits
• In quantum mechanics, the location of the electron is not known.
Instead, wavefunctions Ψ(x,y,z) are used to describe an electron’s
wave characteristics
• The square of the wavefunction, Ψ2, gives the PROBABILTY
DENSITY, or the probability that an electron will be found in some
region of volume around the nucleus.
Probability Density???
• This is directly in line with the uncertainty principle.
• We CAN NOT locate an electron accurately
• We CAN calculate a probability of an electron being in a certain
region of space in the atom
• From Ψ2, we get ORBITALS
• An orbital is a theoretical, 3-D “map” of the places where an
electron could be.
Orbitals and the Quantum Numbers
• An orbital is defined by 4 quantum numbers
n
L
mL
ms
(principle quantum number)
(azimuthal quantum number)
(magnetic quantum number)
(magnetic spin quantum number)
1st quantum number ‘n’: Principal Quantum
Number
• n = 1, 2, 3…..etc These numbers correlate to the distance of an
electron from the nucleus. In Bohr’s model, these corresponded to
the “shell” orbiting the nucleus.
• n determines the energies of the electrons
• n also determines the orbital size. As n increases, the orbital
becomes larger and the electron is more likely to be found farther
from the nucleus
2nd Quantum Number ‘L’: Aziumthal Quantum
Number
• L dictates the orbital shape
• L is restricted to values of 0, 1….(n-1)
• Each value of L has a letter designation. This is how we label orbitals.
Orbital Labeling
• Orbitals are labeled by first writing the
principal quantum number, n, followed
by the letter representation of L
Value of L
0
1
2
3
Orbital type
s
p
d
f
3rd quantum number ‘mL’: Magnetic quantum
number
• The 3rd quantum number, mL, determines the spatial orientation of
an orbital
• mL can assume all integer values between –L and +L
• Number of possible values of mL gives the number of each orbital
type in a given shell
Magnetic Quantum Number
S-orbitals
• When n=1, the wavefunction that describes this
state (Ψ1) only depends on r, the distance from
the nucleus.
• Because the probability of finding an electron
only depends on r and not the direction, the
probability density is spherically symmetric
(same in all 3 dimensions). The wavefunction is
denoted Ψ1s .
• n=1 can only be an s -orbital
Since L= 0 for s orbital, mL can only be 0. Hence,
each “shell” only has one s-orbital
z
z
1s
z
3s
2s
y
y
x
x
y
x
Spheres shown above represent the volume of space around the nucleus in
which an electron may be found. Images below are cross sections with
probabilities plotted versus distance from the nucleus. Regions where an
electron has 0% probability of existing is called a NODE.
node
2 nodes
P-orbitals
pz
• P orbitals exist in all shells where n> 2.
• For a p orbital, L =1.
• Therefore, mL = -1, 0, 1. Hence, there are 3
arrangements of p-orbitals in each shell n> 2
py
px
• Unlike s-orbitals, p-orbitals are not spherically
symmetric. Instead, electron density is
concentrated in lobes around the nucleus along
either the x, y, or z axis.
D-orbitals
• D-orbitals exist in all shells
where n>3
• L= 2, so mL can be any of
the following: -2,-1,0,1,2
• Thus, there are five dorbitals in every shell where
n>3
• 4 of them have a ‘four-leaf
clover’ shape. The fifth is
represented as two lobes
along the z axis with a
doughnut shaped region of
density around the nucleus
Questions:
• Can a 2d orbital exist?
• Can a 1p orbital exist?
• Can a 4s orbital exist?
The 4th and final quantum number ‘ms’:
Magnetic Spin Number
Stern- Gerlach experiment
• A beam of Ag atoms was
passed through an uneven
magnetic field. Some of the
atoms were pulled toward the
curved pole, others were
repelled.
• All of the atoms are the
same, and have the same
charge. Why does this
happen?
The 4th and final quantum number ‘ms’:
Magnetic Spin Number
• Spinning electrons have magnetic fields. The direction of spin
changes the direction of the field. If the field of the electron does
not align with the magnetic, it is repelled.
• Thus, because the beam splits two ways, electrons must spin in
TWO opposite directions with equal probability. We label these
“spin-up” and “spin-down”
1
2
1
2
Two possible orientations: ms = + (𝑠𝑝𝑖𝑛 𝑢𝑝), − (𝑠𝑝𝑖𝑛 𝑑𝑜𝑤𝑛)
Example: What are the allowed quantum
numbers of a 2p electron?
1
1
1
1
2
1
n
l
0
1
1
-1
ml
1
2
n
l
ml
ms
2
1
1
+½
2
2
1
1
-½
2
2
1
0
+½
2
1
0
-½
2
1
-1
+½
2
1
-1
-½
2
2
2
ms
Spin up:
+½
Spin down: - ½
-1
0
1
Representation of the 3 p-orbitals
Pauli Exclusion Principle
NO TWO ELECTRONS IN THE SAME ATOM CAN HAVE THE
SAME 4 QUANTUM NUMBERS!!!
1s
Quantum numbers:
1, 0 , 0, + ½
1, 0, 0, – ½
* Allowed
1s
Quantum numbers:
1, 0, 0, + ½
1, 0, 0 , + ½
* Forbidden !!
Example
• List all possible sets of quantum numbers in the n=2 shell?
• n=2
S
• L = 0, 1
• mL = 0
= -1, 0, 1
• ms = +/- ½
(L=0)
(L=1)
P
Electron Configurations
• As previously stated, the
energy of an electron depends
on n.
• Orbitals having the same n,
but different L (like 3s, 3p, 3d)
have different energies.
• When we write the electron
configuration of an atom, we
list the orbitals in order of
energy according to the
diagram shown on the left.
REMEMBER: S-orbitals can hold no more than TWO
electrons. P- orbitals can hold no more than SIX, and
D-orbitals can hold no more than TEN.
Example
• Write the electron configurations of N, Cl, and Ca
𝑁 (7 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠): 1𝑠 2 2𝑠 2 2𝑝3
𝐶𝑙 (17 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠): 1𝑠 2 2𝑠 2 2𝑝6 3𝑠 2 3𝑝5
𝐶𝑎 (20 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠): 1𝑠 2 2𝑠 2 2𝑝6 3𝑠 2 3𝑝6 4𝑠 2
Energy
Example, contd.
• If we drew the orbital representations of N based on the
configuration in the previous slide, we would obtain:
Energy
2p
2s
1s
For any set of orbitals of the
same energy, fill the orbitals
one electron at a time with
parallel spins. (Hund’s Rule)
Excited States
• When we fill orbitals in order, we obtain the ground state (lowest
energy) configuration of an atom.
• What happens to the electron configuration when we excite an
atom?
• Absorbing light with enough energy
will bump a valence electron into
an excited state.
• The electron will move up to the
next available orbital. This is the 1st
excited state.
Example: Multiple Excited states of Li
• Ground state Li:
1s2 2s1
• 1st excited state Li:
1s2 2s0 2p1
• 2nd excited state Li: 1s2 2s0 2p0 3s1
Energy
3s
2nd excited state
2p
1st excited state
2s
Ground state
1s
Valence Electron Configurations
ns2
np6
ns1
1
ns2 ns2 ns2 ns2 ns2
np1 np2 np3 np4 np5
ns2
2
3
4
5
6
7
ns2 (n-1)dx
Noble Gas Configurations
• In chapter 4, we learned how to write Lewis dot configurations.
Now that we can assign orbitals to electrons, we can write proper
valence electron configurations.
Group Examples
• Write noble gas electron configurations for As+
• Write the noble gas electron configuration for the 1st and
2nd excited states of Mg+
Transition Metals
• As you know, the d-orbitals hold a max of 10 electrons
• These d-orbitals, when possible, will assume a half-filled, or fully-filled
configuration by taking an electron from the ns orbital
• This occurs when a transition metal has 4 or 9 valence d electrons
Example: Cr  [Ar] 4s2 3d4
[Ar] 4s1 3d5
3d
3d
Unfavorable
4s
Favorable
4s