Physical Chemistry , 1st ed.

Planck’s constant hhas the same units, J s, as the angular momentum,

kg m^2 /s. This is a different unit from that oflinearmomentum, where the unit

is kg m/s. Planck’s constant hhas units that classical mechanics would call

units ofaction. What we will find is that any atomic observable that has units

of action is an angular momentum of a sort, and its value at the atomic level

is related to Planck’s constant. It is relationships like this that reinforce the cen-

tral, irreplaceable role of Planck’s constant in the understanding (indeed, the

very existence) of matter.

Finally, now that we have shown that angular momentum is quantized for

some systems, we bring up an old idea, one that Bohr had when he put forth

his theory of the hydrogen atom. He assumed that the angular momentum was

quantized! By doing so, Bohr was able to theoretically predict the hydrogen

atom spectrum, although the justification of the assumption was highly de-

batable. Quantum mechanics does not assume the quantization of angular

momentum. Rather, quantum mechanics shows that it is inevitable.

Example 11.14

The organic molecule benzene, C 6 H 6 , has a cyclic structure where the carbon

atoms make a hexagon. The electrons in the cyclic molecule can be ap-

proximated as having two-dimensional rotational motion. Calculate the

diameter of this “electron ring” if it is assumed that a transition occurring at

260.0 nm corresponds to an electron going from m3 to m4.

Solution

First, calculate the energy change in J that corresponds to a photon wave-

length of 260.0 nm, which is 2.60 

10 ^7 m:

c



2.9979 

108 m/s(2.60 

10 ^7 m)



1.15 

1015 s^1

Therefore, using Eh :

E(6.626 

10 ^34 J s)(1.15 

1015 s^1 )

E7.64 

10 ^19 J

This energy difference should be equal to the energy difference between the

m4 and m3 energy levels:

E7.64 

10 ^19 J 

2

4

m

(^2) 
r
2
 2   2

3

m

(^2) 
r
2
 2
where mr^2 has been substituted for Iin the denominators. Substituting for h,
2 , and the mass of the electron:
7.64
10 ^19 J

7.64
10 ^19 J(16 9)

6.104

r

1

2

0 ^39 m^2



r^2 5.59 

10 ^20 m^2

r2.36 

10 ^10 m 2.36 Å

32 (6.626 

10 ^34 J s)^2



(2)^2 2(9.109 

10 ^31 kg) r^2

42 (6.626 

10 ^34 J s)^2



(2)^2 2(9.109 

10 ^31 kg) r^2

340 CHAPTER 11 Quantum Mechanics: Model Systems and the Hydrogen Atom