Transcript Chapter 7 The Quantum-Mechanical Model of the Atom

Chapter 7
The Quantum-Mechanical Model
of the Atom
Electron Energy
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electron energy and position are complimentary
◦ because KE = ½mv2
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 – called an orbital
◦ a probability distribution map of a region where the electron is
likely to be found
◦ distance vs. y2 (wave function)
many of the properties of atoms are related to the energies of the
electrons
Η ψ  Eψ
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Wave Function, y
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calculations show that the size, shape and orientation in space of an
orbital are determined be three integer terms in the wave function
◦ added to quantize the energy of the electron
these integers are called quantum numbers
◦ principal quantum number, n
◦ angular momentum quantum number, l
◦ magnetic quantum number, ml
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Principal Quantum Number, n
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characterizes the energy of the electron in a particular orbital
◦ corresponds to Bohr’s energy level
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
E n  -2.18  10
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-18
 1
J 2 
n 
for an electron in H
–The negative sign means that the energy of the electron bound to the
nucleus is lower than it would be if the electron were at an infinite
distance (n = ∞) from the nucleus, where there is no interaction.
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Principal Energy Levels in Hydrogen
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Electron Transitions
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in order to transition to a higher energy state, the electron must gain the
correct amount of energy corresponding to the difference in energy
between the final and initial states
electrons in high energy states are unstable and tend to lose energy and
transition to lower energy states
◦ energy released as a photon of light
each line in the emission spectrum corresponds to the difference in
energy between two energy states
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Hydrogen Energy Transitions
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Predicting the Spectrum of Hydrogen
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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
Eatom  E final  E initial 
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
 1  
 1  
18 

18

 
  2.18 10 J
   2.18 10 J
2
2
 

n

n

final
initial

 

 
Since the energy must be conserved, the exact amount of energy
emitted by the atom is carried away by the photon
ΔEatom = - ΔEphoton
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Example
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Determine the wavelength of light emitted when an electron in a hydrogen
atom makes a transition from an orbital in n= 6 to an orbital in n=5
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As electron in the n=6 level of the hydrogen atom relaxes to a lower energy
level, emitting light of λ= 93.8 nm. Find the principle level to which the
electron relaxed
The angular Momentum Quantum
number (l)
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Is an integer that determines the shape of the orbital.
Quantum number
n
(shell)
Value of l
Letter designation
(subshell)
n=1
l=0
s
n=2
l= 1
p
n=3
l=2
d
n=4
l=3
f
the energy of the subshell increases with l (s < p < d < f).
The magnetic quantum number (ml)
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Specifies the orientation in space of an orbital of a given energy (n) and
shape (l).
This number divides the subshell into individual orbitals which hold the
electrons; there are 2l+1 orbitals in each subshell. Thus the s subshell has
only one orbital, the p subshell has three orbitals, and so on
n
l
# Orbitals ml
1
0
0
2
0, 1
-1, 0, 1
3
0, 1, 2
-2, -1, 0, 1, 2
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0, 1, 2, 3
-3, -2, -1, 0, 1, 2, 3, 4
Examples
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Give the possible combination of quantum numbers for the following
orbitals
3s orbital
2 p orbitals
Give orbital notations for electrons in orbitals with the following quantum
numbers:
n = 2, l = 1 ml = 1
n = 3, l = 2, ml = -1
Probability Density Function
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The Shapes of Atomic Orbitals
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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 4 balloons tied at the knot
◦ f orbitals are mainly like 8 balloons tied at the knot
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l = 0, the s orbital
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each principal energy state has 1 s
orbital
lowest energy orbital in a principal
energy state
spherical
number of nodes = (n – 1)
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2s and 3s
2s
n = 2,
l=0
3s
n = 3,
l=0
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l = 1, p orbitals
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each principal energy state above n = 1 has 3 p
orbitals
◦ ml = -1, 0, +1
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each of the 3 orbitals point along a different axis
◦ px, py, pz
2nd lowest energy orbitals in a principal energy state
 two-lobed
 node at the nucleus, total of n nodes
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Tro, Chemistry: A Molecular
Approach
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p orbitals
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l = 2, d orbitals
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each principal energy state above n = 2 has 5 d orbitals
◦ ml = -2, -1, 0, +1, +2
4 of the 5 orbitals are aligned in a different plane
◦ the fifth is aligned with the z axis, dz squared
◦ dxy, dyz, dxz, dx squared – y squared
3rd lowest energy orbitals in a principal energy state
mainly 4-lobed
◦ one is two-lobed with a toroid
planar nodes
◦ higher principal levels also have spherical nodes
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d orbitals
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l = 3, f orbitals
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each principal energy state above n = 3 has 7 d orbitals
◦ ml = -3, -2, -1, 0, +1, +2, +3
4th lowest energy orbitals in a principal energy state
mainly 8-lobed
◦ some 2-lobed with a toroid
planar nodes
◦ higher principal levels also have spherical nodes
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f orbitals
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Now we know why atoms are
spherical
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