Download - Lambda Absorption Lecture
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A Bit More on the Absorption of Light
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How do we associate the energy of a particle in a 1-Dbox with the wavelength of the particle?
velocity nu
velocitynu
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http://en.wikipedia.org/wiki/Image:Particle_in_a_box_wavefunctions.png#filea
We require that λ return to zero at the ends of the box
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in general where n = 1, 2, 3,…
since
Particle has no potential energy, so the only energy we can associate with the particle is kinetic energy:
We require also that the particle always be moving
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Apply to “real world”, i.e., π electrons in a ~linear,conjugated hydrocarbon, e.g. butadiene
CH2=CHCH=CH2
Length of “box” is: 2 × 135 pm + 1 × 154 pm + 2 ×77 pm = 578 pm
How many π electrons?
n = 1
n = 2
n = 3
• •• •
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At what λ will absorption occur?
h = 6.626 × 10−34 J s; (J = kg m2/s2); c = 3.0 × 108 m/s; me = 9.1039 × 10−31 kg
How does this estimate compare with the experimentally determined λ? (λmax experimentalis 217 nm).
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For more complicated molecules, our simple theoretical model breaks down, as expected. Someof the salient features, however, can be retained, especially in a qualitative sense.
For example, we saw that:
So, the energy gap is inversely proportional to square of the length of the box.
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If we equate our expression for the energy gap (∆E)with a modified form of Planck’s relation we can solve for the wavelength necessary to bridge thatenergy gap:
And we see that λ is directly proportional to the square of the length of the box (or the size of theconjugated molecule).
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And, all things being equal, a longer, conjugated molecule should absorb at a longer wavelength. Ifthe molecule is long enough, it should absorb at a long enough wavelength that we might even seethe transmitted light with our eyes. Consider themolecule β-carotene.
http://www.chm.bris.ac.uk/motm/carotene/beta-carotene_structure.html
This molecule is a relatively long, conjugated “box”.
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Accordingly, β-carotene has a λmax of 455 nm, whichis in the “blue” region of the visible portion of the electromagnetic spectrum, and thus it appears orange to our eyes.
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700 nm
400 nm
http://en.wikipedia.org/wiki/Visible_spectrum