Download Energy Levels and Light Absorption

Energy Levels and Light Absorption
Lecture 19
www.physics.uoguelph.ca/~pgarrett/Teaching.html
Review of L-18
• For π-electron in linear conjugated molecule of length L
Ψn ( x ) =
2
nπx
sin
L
L
• π electrons more loosely bound that σ electrons
• C − C bond length is ~ 0.15 nm
• L = (#conjugated atoms-1) × 0.15 nm
Energy of π electrons
• In a linear molecule, π electron has energy (we ignore potential energy)
E = 12 me v 2
• Using the particle wavelength relation
h
h
λ= =
p
2me E
and that the wavelength is related to the length of the box via
L=
nλ
2
λ=
2L
n
h
2L
=
n
2me E
n 2h 2
En =
8me L2
Energy levels in linear molecules
n 2h 2
• Energy levels in linear molecules En =
8me L2
25h 2
8me L2
n=5
16h 2
8me L2
n=4
9h 2
8me L2
n=3
4h 2
8me L2
n=2
n=1
h2
8me L2
• How many electrons can each energy level hold?
Pauli Exclusion Principle
• Particles have a quantity, called spin, that behaves like
angular momentum
– Particles act like spinning tops
– Spin is quantized ( like energy, angular
momentum comes in discrete units) in
units of
– Particles like electrons have ½ integer spin – fermions
– Particles like photons have integer spin – bosons
• A fundamental principle in quantum mechanics – the Pauli
exclusion principle – is that no two fermions can be in
exactly the same state
• Any number of bosons can be in the same state
Pauli Exclusion Principle
• Consider an electron in an atom with quantum number n = n1.
How many electrons in this atom can have the quantum
number n1?
n1
• Pauli exclusion says no two electrons can be in exactly the
same state
– But we also have spin to consider
– Electrons can be in a “spin up” or a “spin down” state
– These are distinct quantum states, so 2 electrons can have the same
value n = n1, as long as one has spin up, and the other spin down
Electronic configurations
• Electrons have two spin states – spin “up” (counterclockwise)
and spin “down” (clockwise)
• Electrons are put into levels from the “bottom up”
n=5
16h 2
8me L2
n=4
9h 2
8me L2
n=3
4h 2
8me L2
n=2
n=1
h2
8me L2
This is the
“ground state”
configuration
ex. 8 π
electrons
– octene
25h 2
8me L2
Excited states
• Molecules can form excited states by promoting an electron
from one configuration into a higher configuration
25h 2
8me L2
n=5
16h 2
8me L2
n=4
9h 2
8me L2
n=3
4h 2
8me L2
n=2
n=1
h2
8me L2
1’st excited
state
ground state
Excited state energies
• The energy of the excited state is Eex = Ei − E f
• Since most of the electrons are in common, the difference is
Eex = En =5 − En =4
9h 2
=
8me L2
9h
9 × (6.63 × 10 )
=
8me L2 8 × 9.11 × 10−31 × (7 × 0.15 × 10−9 )2
− 34 2
2
= 4.92 × 10−19 J = 3.07eV
25h 2
8me L2
n=5
16h 2
8me L2
n=4
9h 2
8me L2
n=3
4h 2
8me L2
h2
8me L2
n=2
n=1
Light absorption
• If photons of the right energy are incident on a material, they
can cause the promotion of electrons – excited states
– The photons are absorbed by the molecules
– If the sample is thick enough, the particular wavelengths can be
completely absorbed
– If white light is used, the absorption of the particular wavelengths can
cause a “gap” to appear in the transmitted light spectrum
• Transmission or absorption spectroscopy
• ex. If white light is incident on octene, what wavelengths
would be absorbed?
– selection rule → photons can only promote electrons between orbitals
with ∆n = ±1
Light absorption
• Already worked out first excited state E = 3.07 eV, so λ=hc/E
= 403.8 nm
25h 2
8me L2
n=5
16h 2
8me L2
n=4
9h 2
8me L2
n=3
4h 2
8me L2
h2
8me L2
n=2
n=1
Light absorption
• What about the second excited state?
= 6.01 × 10
−19
J = 3.75eV
• But this energy is relative to
the first excited state…
25h 2
8me L2
n=5
16h 2
8me L2
n=4
9h 2
8me L2
n=3
h2
8me L2
4h 2
8me L2
n=6
n=2
n=1
11h
11 × (6.63 × 10 )
=
8me L2 8 × 9.11 × 10−31 × (7 × 0.15 × 10−9 )2
−34 2
2
36h 2
8me L2
Energy level diagrams
• We can redefine the ground state to be zero energy, and give
the energies of all excited states relative to it
13.3 eV
4’th excited state
8.2 eV
3’rd excited state
6.9 eV
2’nd excited state
3.1 eV
1’st excited state
ground state
0 eV