Physical Chemistry , 1st ed.

Solution

According to equation 11.39, the values of the angular momenta are  2 ,

 1 ,0,1, and 2. Notice that although the energies are the same for cer-

tain pairs of quantum numbers, the values of the quantized angular mo-

menta are not.

A few comments are necessary about the angular momentum. First, classi-
cal mechanics treats possible angular momentum values as continuous, whereas
quantum mechanics limits angular momentum to discrete, quantized values.
Second, the quantized angular momentum does not depend on mass or mo-
ment of inertia. This is completely counter to the ideas of classical mechanics,
where the mass of a particle is intimately tied to its momentum. This is an-
other example in which quantum mechanics departs from the ideas of classi-
cal mechanics.
Also, because the quantized values of angular momentum depend on mand
not m^2 , every wavefunction has its own characteristic value of the angular
momentum, as mentioned in Example 11.12. The energy levels may be doubly
degenerate, but each state has its own angular momentum. One state has an
angular momentum value ofm, and the other m. Since momentum is a
vector quantity, there is a simple way of rationalizing the differences between
the two states. In one state, the particle is moving in one direction (say, clock-
wise), and in the other, it is moving in the opposite direction (say, counter-
clockwise).
In cases where two masses (say, two atoms) are connected and rotating in a
plane, all of the above equations would apply except that the mass would be
replaced by the reduced massof the two-mass system. This is consistent with
earlier treatments of two masses moving relative to each other. A system of two
(or more) particles rotating in two dimensions is called a 2-D rigid rotor.

Example 11.13

The bond distance in HCl is 1.29 Å. In its lowest rotational state, the molecule

is not rotating, and so the rigid rotor equations indicate that its rotational en-

ergy is zero. What are its energy and its angular momentum when it is in the

first nonzero energy state? Use the atomic weight of Cl as an approximation for

the mass of the Cl atom.

Solution

Using the masses of H and Cl as 1.674 

10 ^27 kg and 5.886 

10 ^26 kg, the

reduced mass of the molecule is 1.628 

10 ^27 kg. The bond distance, in

units of meters, is 1.29 

10 ^10 m. For this case we will not calculate the mo-

ment of inertia as a separate step, but will substitute the numbers into the en-

ergy formula as appropriate. For the first nonzero rotational energy state:

E(m1) 

E(m1) 2.05 

10 ^22 J

Because the molecule can have this energy in the m1 state and the

m1 state, the angular momentum of the molecule can be either 1or

 1 . With the information provided, there is no way to distinguish between

the possibilities.

(1)^2 (6.626^10 ^34 J^ s)^2

2(1.628 

10 ^27 kg)(1.29 

10 ^10 m)^2 (2)^2

11.6 Two-Dimensional Rotations 339