A Bit More on the Absorption of Light

How do we associate the energy of a particle in a 1-Dbox with the wavelength of the particle?

velocity nu

velocitynu

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

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

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

• •• •

•

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).

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.

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).

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”.

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.

R

O

Y

G

BI

V

700 nm

400 nm

http://en.wikipedia.org/wiki/Visible_spectrum