Transcript Lecture 8

1
Particle in a Box
• The wavefunctions for the
particle are identical to the
displacements of a stretched
string as it vibrates.
2 1/ 2 nx 
n (x)    sin 
 where n=1,2,3,…
L 
 L 

n is the quantum number
•It defines a state

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Particle In a Box
2
• Now we know that the allowable energies are
:
n 2h 2
En 
8mL2
•
Where n=1,2,3,…
This tells us that:
1. The energy levels for heavier particles are less than

those of lighter particles.
2. As the length b/w the walls decreases, the ‘distance’
b/w energy levels increases.
3. The energy levels are Quantized.
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
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Zero Point Energy
• A particle in a container CANNOT have zero
energy
– A container could be an atom, a box, etc.
• The lowest energy (when n=1) is:
h2
En 
8mL2
Zero Point Energy
•This is in agreement with the Uncertainty Principle:
•∆p and ∆x are never zero, therefore the particle is

always moving
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Wavefunctions and Probability
Densities
•
•
•
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Examine the 2
lowest energy
functions n=1 and
n=2
We see from the
shading that when
n=1, 2 is at a
maximum @ the
center of the box.
When n=2, we see
that 2 is at a
maximum on
either side of the
center of the box
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Wavefunction Summary
• The probability density for a particle at a location is
proportional to the square of the wavefunction at the
point
• The wavefunction is found by solving the SchrÖdinger
equation for the particle.
• When the equation is solved to the appropriate
boundary conditions, it is found that the particle can
only posses certain discrete energies.
• The wavefunctions and their associated energies
placed electrons in defined orbitals whose size and
shape is determined by the quantum numbers of the
particle.
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Types of Orbitals
s orbital
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p orbital
d orbital
Orbitals
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• No more than 2 e- assigned to an orbital
• Orbitals grouped in s, p, d (and f)
subshells
s orbitals
d orbitals
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p orbitals
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s orbitals
p orbitals
d orbitals
s orbitals
p orbitals
No.
orbs.
1
3
5
No.
e-
2
6
10
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d orbitals
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Subshells & Shells
• Subshells grouped in shells.
• Each shell has a number called
the PRINCIPAL QUANTUM
NUMBER, n
• The principal quantum number
of the shell is the number of the
period or row of the periodic
table where that shell begins.
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Subshells & Shells
n=1
n=2
n=3
n=4
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QUANTUM NUMBERS
The shape, size, and energy of each orbital is a function of 3
quantum numbers:
n (major; Principal)
l
shell
Determines the energy and size of the orbital
– The bigger the number, the higher the energy and the larger
the orbital radius
(angular or Azimuthal) subshell
l=n – 1
Determines the shape of the orbital
ml (magnetic) designates an orbital within a subshell
ml = l, l - 1, …, -l
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QUANTUM NUMBERS
Symbol
Values
Description
n (major)
1, 2, 3, ..
Orbital size and
energy where
E = -R(1/n2)
l (angular)
0, 1, 2, .. n-1
Subshell
(Orbital Shape)
ml (magnetic)
Orbital
orientation
# of orbitals in subshell = 2l + 1
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-l..0..+l
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s Orbitals— Always Spherical
Dot picture of
electron
cloud in 1s
orbital.
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Surface
density
4πr2y versus
distance
Surface of
90%
probability
sphere
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p Orbitals
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When n = 2, then l = 0 and 1
Therefore, in n = 2 shell there
are 2 types of orbitals — 2
subshells
For l = 0
ml = 0
this is a s subshell
For l = 1
ml = -1, 0, +1
this is a p subshell
with 3 orbitals
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When s = 1, there is
a PLANAR
NODE thru the
nucleus.
p Orbitals
The three p orbitals lie 90o apart in space
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d Orbitals
When n = 3, what are the values of l?
l = 0, 1, 2
and so there are 3 subshells in the shell.
For l = 0, ml = 0
s subshell with single orbital
For l = 1, ml = -1, 0, +1
p subshell with 3 orbitals
For l = 2, ml = -2, -1, 0, +1, +2
d subshell with 5 orbitals
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d Orbitals
s orbitals have no planar
node (l = 0) and so are
spherical.
p orbitals have l = 1, and
have 1 planar node,
and so are “dumbbell”
shaped.
This means d orbitals (with
l = 2) have 2 planar
nodes
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f Orbitals
When n = 4, s = 0, 1, 2, 3 so there are 4 subshells
in the shell.
For l = 0, ml = 0
s subshell with single orbital
For l = 1, ml = -1, 0, +1
p subshell with 3 orbitals
For l = 2, ml = -2, -1, 0, +1, +2
d subshell with 5 orbitals
For l = 3, ml = -3, -2, -1, 0, +1, +2, +3
f subshell with 7 orbitals
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Arrangement of
Electrons in Atoms
Each orbital can be assigned no
more than 2 electrons!
This is tied to the existence of a 4th
quantum number, the electron
spin quantum number, ms.
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Electron Spin Quantum Number,
ms
Can be proved experimentally that electron
has an intrinsic property referred to as
“spin.” Two spin directions are given by
ms where ms = +1/2 and -1/2.
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Electron Spin and Magnetism
•Diamagnetic: NOT attracted to a magnetic
field
•Paramagnetic: substance is attracted to a
magnetic field.
•Substances with unpaired electrons are
paramagnetic.
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QUANTUM NUMBERS
Now there are four!
n
shell
1, 2, 3, 4, ...
l
subshell
0, 1, 2, ... n - 1
ml
orbital in subshell
ms
electron spin
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- l ... 0 ... + l
+1/2 and -1/2
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Pauli Exclusion Principle
No two electrons in the
same atom can have
the same set of 4
quantum numbers.
That is, each electron has a
unique address.
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Hund’s Rule
• You must add electrons to unoccupied
orbitals of a subshell before doubly
occupying any of them.
• Very critical in determining the filling order for
electron shells.
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Arrangement of
Electrons in Atoms
Electrons in atoms are arranged as
SHELLS (n)
SUBSHELLS (l)
ORBITALS (ml)
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Electrons in Atoms
When n = 1, then l = 0
this shell has a single orbital (1s) to which 2ecan be assigned.
When n = 2, then l = 0, 1 and ml = -1, 0, +1
2s orbital
2e-
three 2p orbitals
6e-
TOTAL =
8e-
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Electrons in Atoms
When n = 3, then l = 0, 1, 2 and ml = -2, -1, 0, +1, +2
3s orbital
three 3p orbitals
five 3d orbitals
TOTAL =
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2e6e10e18e-
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Electrons in Atoms
When n = 4, then l = 0, 1, 2, 3 and
ml = -3, -2, -1, 0, +1, +2, +3
4s orbital
2e-
three 4p orbitals
five 4d orbitals
seven 4f orbitals
TOTAL =
6e10e14e32e-
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Assigning Electrons to Atoms
• Electrons generally assigned to orbitals of
successively higher energy.
• For H atoms, E = - C(1/n2). E depends only on n.
• For many-electron atoms, energy depends on
both n and l.
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Electron
Filling
Order
Print this chart out
and use it at home!
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Electron Filling Order
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Effective Nuclear Charge, Zeff
• Zeff is the nuclear charge experienced by the
outermost electrons.
• Explains why E(2s) < E(2p)
• Zeff increases across a period owing to incomplete
shielding by inner electrons.
• Estimate Zeff = [ Z - (no. inner electrons) ]
• Charge felt by 2s e- in Li
Zeff = 3 - 2 = 1
• Be
Zeff = 4 - 2 = 2
• B
Zeff = 5 - 2 = 3
and so on!
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Effective
Nuclear
Charge
Electron cloud
for 1s electrons
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Zeff is the nuclear
charge experienced
by the outermost
electrons.
This will help explain
some of the periodic
properties we’ll
examine later this
chapter.
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Writing Atomic Electron
Configurations
Two ways of
writing configs.
One is called
the spdf
notation.
spdf notation
for H, atomic number = 1
1
1s
value of n
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no. of
electrons
value of l
Writing Atomic Electron
Configurations
Two ways of
writing
configs. Other
is called the
orbital box
notation.
ORBITAL BOX NOTATION
for He, atomic number = 2
Arrows
2
depict
electron
spin
1s
1s
One electron has n = 1, s = 0, ms = 0, ms = + 1/2
Other electron has n = 1, s = 0, ms = 0, ms = - 1/2
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Electron Configurations
and the Periodic Table
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See Active Figure 7.4
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Lithium
Group 1A
Atomic number = 3
1s22s1 f 3 total electrons
3p
3s
2p
2s
1s
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Beryllium
3p
3s
2p
2s
1s
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Group 2A
Atomic number = 4
1s22s2 f 4 total
electrons
Boron
Group 3A
Atomic number = 5
1s2 2s2 2p1 f
3p
3s
2p
2s
1s
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5 total electrons
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Carbon
Group 4A
Atomic number = 6
1s2 2s2 2p2 f
6 total electrons
3p
3s
2p
2s
1s
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Here we see for the first time
HUND’S RULE. When
placing electrons in a set of
orbitals having the same
energy, we place them singly
as long as possible.
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Nitrogen
Group 5A
Atomic number = 7
1s2 2s2 2p3 f
3p
3s
2p
2s
1s
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7 total electrons
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Oxygen
Group 6A
Atomic number = 8
1s2 2s2 2p4 f
3p
3s
2p
2s
1s
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8 total electrons
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Fluorine
Group 7A
Atomic number = 9
1s2 2s2 2p5 f
3p
3s
2p
2s
1s
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9 total electrons
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Neon
Group 8A
Atomic number = 10
1s2 2s2 2p6 f
10 total electrons
3p
3s
2p
2s
1s
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Note that we have
reached the end of
the 2nd period, and
the 2nd shell is full!
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Electron Configurations of
p-Block Elements
PLAY MOVIE
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Sodium
Group 1A
Atomic number = 11
1s2 2s2 2p6 3s1 or
“neon core” + 3s1
[Ne] 3s1 (uses rare gas notation)
Note that we have begun a new period.
All Group 1A elements have
[core]ns1 configurations.
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Aluminum
Group 3A
Atomic number = 13
1s2 2s2 2p6 3s2 3p1
[Ne] 3s2 3p1
All Group 3A elements
have [core] ns2 np1
configurations where n
is the period number.
3p
3s
2p
2s
1s
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Phosphorus
Yellow P
Group 5A
Atomic number = 15
1s2 2s2 2p6 3s2 3p3
[Ne] 3s2 3p3
All Group 5A elements
have [core ] ns2 np3
configurations where n
is the period number.
Red P
3p
3s
2p
2s
1s
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Calcium
Group 2A
Atomic number = 20
1s2 2s2 2p6 3s2 3p6 4s2
[Ar] 4s2
All Group 2A elements have
[core]ns2 configurations where n
is the period number.
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Electron Configurations
and the Periodic Table
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Transition Metals
All 4th period elements have the
configuration [argon] nsx (n - 1)dy
and so are d-block elements.
Chromium
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Iron
Copper
Transition Element
Configurations
3d orbitals used
for Sc-Zn
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Lanthanides and Actinides
All these elements have the configuration
[core] nsx (n - 1)dy (n - 2)fz and so are
f-block elements.
Cerium
[Xe] 6s2 5d1 4f1
Uranium
[Rn] 7s2 6d1 5f3
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Lanthanide Element
Configurations
4f orbitals used for
Ce - Lu and 5f for
Th - Lr
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Ion Configurations
To form cations from elements remove 1 or
more e- from subshell of highest n [or
highest (n + l)].
P [Ne] 3s2 3p3 - 3e- f P3+ [Ne] 3s2 3p0
3p
3p
3s
3s
2p
2p
2s
2s
1s
1s
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Ion Configurations
For transition metals, remove ns electrons and
then (n - 1) electrons.
Fe [Ar] 4s2 3d6
loses 2 electrons Fe2+ [Ar] 4s0 3d6
Fe2+
Fe
4s
3d
To form cations, always
remove electrons of
highest n value first!
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4s
3d
Fe3+
4s
3d
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