BioPHYSICAL chemistry

and the transition is from the second to the third level (Figure 10.4) and
the difference in the energy is:

(10.35)

Research direction: carotenoids

For larger systems the calculation is the same
but there are more electrons. Carotenoids are
conjugated molecules that are found widely in
biological systems. In some systems, their func-
tion is to serve as a pigment giving color to birds,
fish, or lobsters (the red color of lobsters when
cooked arises from changes in the carotenoid).
Carotenoids are often good choices for coloration
as they absorb light very strongly. In humans,
carotenoids are necessary precursors for the
formation of vitamin A, or retinal, which is an
important molecule in vision (Chapter 17). In
photosynthetic organisms, carotenoids play crit-
ical roles in the conversion of light energy into
chemical energy. The ability of carotenoids to
strongly absorb light also serves them well in performing critical roles
in energy transfer; that is, absorbing light energy and transferring it to
other molecules which can convert the energy into chemical energy. There
are many chemically distinct carotenoids; however, all carotenoids have
extended conjugated systems with delocalized πelectrons.
Let us consider the specific example of one of the common carotenoids,
β-carotene, which is a linear polyene with 11 double bonds alternating
with 10 single bonds (Figure 10.5). The electrons can move within the total
length of the conjugation so this will represent the length Lof the box.
Typical lengths for single and double bonds between carbon atoms are 1.46
and 1.35 Å respectively. By adding up the 11 double bonds and 10 single
bonds we estimate the length to be 29 Å. There are 22 πelectrons, so
the 11 lowest-energy levels are filled with electrons. The first transition
between a filled level and an unfilled level is between the eleventh and
twelfth levels, yielding:

(10.36)

(10.37)

λ==

8

23

1207

mcL^2

h

e nm

ΔEE E

h

mL

h

m

=−= 12 11 ()− =

2

2

22

2

8

12 11

23

8 LL

hc

2 = λ

ΔEE E

h

mL

h

mL

=−= 32 ()− =

2

2

22

2

8 32 2

5

8

CHAPTER 10 PARTICLE IN A BOX AND TUNNELING 205

n  10

n  11

hν

n  12

CH 3

CH 3 CH^3 CH^3

CH 3

CH 3 CH 3 H^3 C

H 3 C

Energy

Figure 10.5The

chemical structure

of a carotenoid,

β-carotene, and the

transition for the

particle-in-a-box

model of the

conjugated system.