Physical Chemistry , 1st ed.

Example 11.17

The carbon molecule buckminsterfullerene, C 60 , can be approximated as a

sphere, and the electrons of the molecule can be thought of as being confined

to the surface of a sphere. If one of the absorptions of C 60 corresponds to an

electron going from the 4 state to the 5 state, what wavelength of

light would cause this change? Use r3.50 Å to calculate the moment of

inertia. (A transition in the C 60 spectrum is seen at 404 nm).

Solution

The moment of inertia of an electron in this system is

I(9.109 

10 ^31 kg)(3.50 

10 ^10 m)^2 1.12 

10 ^49 kg m^2

The energy of the 4 state is therefore

E(4) 9.93 

10 ^19 J

and the energy for the 5 state is

E(5) 5(5 1)(6.626^10 1.49 

10 ^18 J

 (^34) J s) 2

2 1.12
10 ^49 kg m^2 (2)^2
4(4 1)(6.626^10 ^34 J^ s)^2
2 1.12
10 ^49 kg m^2 (2)^2
346 CHAPTER 11 Quantum Mechanics: Model Systems and the Hydrogen Atom
m  0
2-D rotation
m  1
m  2
m  3
m  4
m  5
m  6
3-D rotation
Energy
 0
 1
 2
 3
 4
 5
 6
Figure 11.13 A comparison of the 2-D and 3-D rotations’ quantized energies. The drawing is
to scale vertically. 3-D rotations have slightly more energy than 2-D rotations of the same quan-
tum number.