Physical Chemistry , 1st ed.

Example 10.13

-Carotenes are highly conjugated polyenes found in many vegetables. They

can be oxidized and used to synthesize pigments that play important roles in

the chemistry of mammalian vision. The parent compound,-carotene, has

a maximum absorption of light that occurs at 480 nm. If this transition cor-

responds to an n11 to n12 transition of an electron in a particle-in-a-

box system, what is the approximate length of the molecular “box”?

Solution

First, we should convert the wavelength of the light absorbed into the equiv-

alent energy in joules:

E

h



c

4.14 10 ^19 J

Next, using this value for the change in energy for the transition and the ex-

pression for the particle-in-a-box energy values, we can set up the following

relationship:

EE 12 E 11



8

1

m

22

e

h

a

2

2  (^8)

1

m

12

e

h

a

2

2 (144 121) (^8) m

h
e
2
a^2
 23

8 m
h
e
2
a^2
4.14 10 ^19 J
In the last step, we are equating the energy difference between the two energy
levels with the energy of the light absorbed. We know the value ofhand me,
so we can substitute and solve for a, the length of the molecular “box.” We get
4.14 10 ^19 J
a^2 
All units cancel except for m^2 in the numerator. (You have to decompose the
J unit to get this, however.) Evaluating:
a^2 3.35 10 ^18 m^2
a1.83 10 ^9 m 18.3 Å
Experimentally, we find that a -carotene molecule has a length of about
29 Å—not perfect agreement, but still good enough to be used for qualita-
tive purposes, especially in comparing similar molecules of different conju-
gation lengths.

10.10 Tunneling

We have assumed in the particle-in-a-box model that the potential energy out-

side the box is infinity, so that the particle has absolutely no chance of pene-

trating the wall. The wavefunction is identically zero at any position where the

potential energy is infinity. Suppose the potential energy weren’t infinity, just

some very large value K? What if it weren’t so large after all, just some value

higher than the energy of the particle? If the wall were limited in width (that

is, if some area on the other side had V0 again), how would that affect the

wavefunction?

23 (6.626 10 ^34 J s)^2

8 (9.109 10 ^31 kg) 4.14 10 ^19 J

23 (6.626 10 ^34 J s)^2

8 (9.109 10 ^31 kg) a^2

(6.626 10 ^34 J s)(2.9979 108 m/s)

4.8 10 ^7 m

296 CHAPTER 10 Introduction to Quantum Mechanics