This document discusses the absorption of light by particles in a 1D box. It explains that the energy of the particle is associated with its wavelength based on requiring the particle's wavefunction to be zero at the ends of the box. For a conjugated hydrocarbon like butadiene, the length of the box is estimated to be 578 pm. The wavelength where absorption will occur can be calculated using Planck's relation and equating the energy gap to hν. A longer conjugated molecule like beta-carotene, which appears orange, absorbs visible light with a maximum wavelength of 455 nm due to its extended conjugated system acting as a longer 1D box.

1. A Bit More on the Absorption of
Light

2. velocity nu
nu velocity
How do we associate the energy of a particle in a 1-D
box with the wavelength of the particle?

3. We require that l return to zero at the ends of the box
a
http://en.wikipedia.org/wiki/Image:Particle_in_a_box_wavefunctions.png#file

4. We require also that the particle always be moving
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:

5. Apply to “real world”, i.e., p electrons in a ~linear,
conjugated hydrocarbon, e.g. butadiene
CH2=CHCH=CH2
How many p electrons?
Length of “box” is: 2 ´ 135 pm + 1 ´ 154 pm + 2 ´
77 pm = 578 pm
n = 3
n = 2
n = 1
·
· ·
· ·

6. At what l 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 l? (lmax experimental
is 217 nm).

7. For more complicated molecules, our simple
theoretical model breaks down, as expected. Some
of 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.

8. If we equate our expression for the energy gap (DE)
with a modified form of Planck’s relation we can
solve for the wavelength necessary to bridge that
energy gap:
And we see that l is directly proportional to the
square of the length of the box (or the size of the
conjugated molecule).

9. And, all things being equal, a longer, conjugated
molecule should absorb at a longer wavelength. If
the molecule is long enough, it should absorb at a
long enough wavelength that we might even see
the transmitted light with our eyes. Consider the
molecule b-carotene.
This molecule is a relatively long, conjugated “box”.
http://www.chm.bris.ac.uk/motm/carotene/beta-carotene_structure.html

10. Accordingly, b-carotene has a lmax of 455 nm, which
is in the “blue” region of the visible portion of the
electromagnetic spectrum, and thus it appears
orange to our eyes.
R
O
Y
G
B I
700 nm
400 nm
V
http://en.wikipedia.org/wiki/Visible_spectrum