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Statistical Mechanics
Uncertainty Principle Demonstration
Any experiment designed to
observe the electron results in
detection of a single electron
particle and no interference
pattern.
Determinacy vs. Indeterminacy
According to classical physics, particles move in a path determined
by the particle’s velocity, position, and forces acting on it.
Because we cannot know both the position and velocity of an electron,
we cannot predict the path it will follow.
The best we can do is to describe the probability an electron will be
found in a particular region using statistical functions.
Electron energy and position are complementary because
KE = ½mv2.
Statistical
Mechanics
For an electron with a given energy, the best we can do is
describe a region in the atom of high probability of
finding it.
Trajectory vs. Probability
Schrödinger’s Equation
Erwin Schrödinger (1887–1961)
Schödinger’s Equation allows us to calculate
the probability of finding an electron with a
particular amount of energy at a particular
location in the atom.
Solutions to Schödinger’s Equation produce
many wave functions, Ψ.
A plot of distance vs. Ψ2 represents an orbital,
a probability distribution map of a region
where the electron is likely to be found.
Solutions to the Wave Function, Ψ
Calculations show that the size, shape, and
orientation in space of an orbital are
determined by three integer terms in the
wave function.
These integers are called quantum numbers.
principal quantum number, n
angular momentum quantum number, l
magnetic quantum number, ml
The Quantum Numbers
Principal Quantum Number, n
Corresponds to Bohr’s energy level.
The principal quantum number n can be any integer ≥1.
The larger the value of n, the more energy the orbital has.
Energies are defined as being negative. (An electron
would have E = 0 when it just escapes the atom.)
The larger the value of n, the larger the orbital.
As n gets larger, the amount of energy between orbitals
gets smaller.
Principal Energy Levels in Hydrogen
Angular Momentum Quantum Number, l
The angular momentum quantum number determines
the shape of the orbital.
l can have integer values from 0 to (n – 1).
Each value of l is called by a particular letter that
designates the shape of the orbital.
s orbitals (l =0) are spherical
p orbitals (l =1) are like two balloons tied at the knots
d orbitals (l =2) are mainly like four balloons tied at the knot
f orbitals (l = 3) are mainly like eight balloons tied at the knot
Magnetic Quantum Number, ml
The magnetic quantum number is an integer that
specifies the orientation of the orbital.
Values are integers from −l to +l including zero.
Gives the number of orbitals of a particular shape.
Ex: when l = 2, the values of ml are −2, −1, 0, +1, +2; which
means there are five orbitals with l = 2
The Hierarchy of Quantum Numbers
for Atomic Orbitals
Name, Symbol
(Property)
Allowed Values
Quantum Numbers
Principal, n
Positive integer
(size, energy)
(1, 2, 3, ...)
1
Angular
momentum, l
0 to n-1
(shape)
0
0
0
0
Magnetic, ml
-l,…,0,…,+l
(orientation)
2
3
1
0
1
2
0
-1 0 +1
-1 0 +1
-2
-1
0
+1 +2
Describing Orbitals
Describing an Orbital
Each set of n, l, and ml describes one orbital.
(the “position” of one electron)
Orbitals with the same value of n are in the same
principal energy level (or principal shell).
Orbitals with the same values of n and l are said to be
in the same sublevel (or subshell).
What are the quantum numbers and names of the orbitals
in the n = 1 principal level? How many orbitals exist?
n=1
l:0
n =1, l = 0 (s)
ml : 0
1 orbital
1s
total orbitals: 1 = 12 : n2
What are the quantum numbers and names of the orbitals
in the n = 2 principal level? How many orbitals exist?
n=2
l : 0,1
n = 2, l = 0 (s)
ml : 0
n = 2, l = 1 (p)
ml : −1,0,+1
1 orbital
2s
3 orbitals
2p
total of 4 orbitals: 1 + 3 = 22 : n2
What are the quantum numbers and names of the orbitals
in the n = 3 principal level? How many orbitals exist?
n=3
l : 0,1,2
n = 3, l = 0 (s)
ml : 0
1 orbital
3s
n = 3, l = 1 (p)
ml : −1,0,+1
3 orbitals
3p
n = 3, l = 2 (d)
ml : −2,−1,0,+1,+2
5 orbitals
3d
total of 9 orbitals: 1 + 3 + 5 = 32 : n2
What are the quantum numbers and names of the orbitals
in the n = 4 principal level? How many orbitals exist?
n=4
l : 0, 1, 2, 3
n = 4, l = 0 (s) n = 4, l = 1 (p) n = 4, l = 2 (d)
n = 4, l = 3 (f)
ml : −1,0,+1 ml : −2,−1,0,+1,+2 ml : −3,−2,−1,0,+1,+2,+3
ml : 0
1 orbital
4s
3 orbitals
4p
5 orbitals
4d
7 orbitals
4f
total of 16 orbitals: 1 + 3 + 5 + 7 = 42 : n2
Electron Arrangement of Elements
2 6 10 14
Quantum Mechanical Explanation of Atomic Spectra
Each wavelength in the spectrum of an atom corresponds to an
electron transition between orbitals.
When an electron is excited, it transitions from an orbital in a lower
energy level to an orbital in a higher energy level.
When an electron relaxes, it transitions from an orbital in a higher
energy level to an orbital in a lower energy level.
When an electron relaxes, a photon of light is released whose energy
equals the energy difference between the orbitals.
Each line in the emission spectrum corresponds to the difference in
energy between two energy states.
Energy Transitions in Hydrogen
Bohr Model of H Atoms
Quantum Leaps
Predicting the Spectrum of Hydrogen
The wavelengths of lines in the emission spectrum of
hydrogen can be predicted by calculating the
difference in energy between any two states.
For an electron in energy state n, there are (n – 1) energy
states it can transition to, therefore (n – 1) lines it can
generate.
Both the Bohr and Quantum Mechanical Models can
predict these lines very accurately for a 1-electron
system.
Energy Transitions in Hydrogen
The energy of a photon released is equal to the difference in energy
between two energy levels.
ΔEelectron = Efinal state − Einitial state
Eemitted photon = −ΔEelectron
It can be calculated by subtracting the energy of the initial state from
the energy of the final state.
RH
Calculate the wavelength of light emitted when the
hydrogen electron transitions from n = 3 to n = 2
n i, n f
Ephoton
ΔEatom
λ
ΔEatom = −Ephoton
1 1
( 22 - 32 )
-3.03 x 10-19
(1/4-1/9 = 0.139)
−18 J) = 1.64
-19 J J
3.03xx10
10−18
Ephoton = −(−1.64
10-19
-3.03 x 10
3.03 x
10-19
J
6.56
Wavelengths Associated with
Hydrogen Electron Transitions
6.56 x10-7 m
Calculate the wavelength of light emitted when the
hydrogen electron transitions from n = 2 to n = 1
n i, n f
ΔEatom
Ephoton
λ
ΔEatom = −Ephoton
(1-¼ = 0.750)
Ephoton = −(−1.64 x 10−18 J) = 1.64 x 10−18 J
The unit is correct, the wavelength is in the UV, which is
appropriate because it has more energy than 3→2 (in the visible).
Calculate the wavelength of light emitted when the
hydrogen electron transitions from n = 6 to n = 5
n i, n f
ΔEatom
Ephoton
λ
ΔEatom = −Ephoton
Ephoton = −(−2.6644 x 10−20 J) = 2.6644 x 10−20 J
(1/25 -1/36 =
0.122)
The unit is correct, the wavelength is in the infrared, which is
appropriate because it has less energy than 3→2 (in the visible).
Wavelengths Associated with
Hydrogen Electron Transitions
↓
7.46 x10-6 m
6.56 x10-7 m
1.21 x10-7 m
The Shapes of Atomic Orbitals
The Shapes of Atomic Orbitals
The l quantum number primarily determines the shape
of the orbital.
l can have integer values from 0 to (n – 1)
Each value of l is called by a particular letter that
designates the shape of the orbital.
s orbitals are spherical
p orbitals are like two balloons tied at the knots
d orbitals are mainly like four balloons tied at the knot
f orbitals are mainly like eight balloons tied at the knot
l = 0, the s orbital
Each principal energy level
has one s orbital
Lowest energy orbital in a
principal energy state
Spherical
Number of nodes = (n – 1)
1s, 2s and 3s Orbitals
1s
2s
3s
l = 1, p orbitals
Each principal energy state above n = 1 has three p
orbitals
ml = −1, 0, +1
Each of the three orbitals points along a different axis
px, py, pz
2nd lowest energy orbitals in a principal energy state
Two-lobed
One node at the nucleus, total of n nodes
p orbitals
l = 2, d orbitals
Each principal energy state above n = 2 has five d
orbitals
ml = −2, − 1, 0, +1, +2
Four of the five orbitals are aligned in a different plane
the fifth is aligned with the z axis,
d 2, d , d , d , d 2 – 2
z
xy yz xz z
y
3rd lowest energy orbitals in a principal energy level
Mainly four-lobed
Planar nodes
higher principal levels also have spherical nodes
d orbitals
l = 3, f orbitals
Each principal energy state above n = 3 has seven d
orbitals
ml = −3, −2, −1, 0, +1, +2, +3
4th lowest energy orbitals in a principal energy state
Mainly eight-lobed
Planar nodes
higher principal levels also have spherical nodes
f orbitals