Physical Chemistry , 1st ed.

symmetric tops, however, the situation is somewhat more complicated be-

cause there are two different moments of inertia and therefore two series of

rotational spectra. As the molecule is distorted about one rotational axis, it

affects the moment of inertia in another axis. Centrifugal distortions of non-

linear molecules are therefore extremely complicated and will not be consid-

ered here.

14.8 Vibrations in Molecules

For a molecule having Natoms, it is necessary to use an x,a y, and a zcoor-

dinate to describe the positions of each of the atoms, and so such a molecule

requires a total of 3Ncoordinates to describe its position in space. Such a sit-

uation is shown in Figure 14.19.

For changes in position, the number of necessary changes in coordinates is

the same. For a molecule having Natoms, each atom’s change in position can

be broken down into a change in x, a change in y, or a change in zcoordinate.

(These changes are simply written as x,y, and zfor each of the Natoms.)

These changes in coordinates may have different values, so an N-atom mole-

cule needs a total of 3Nchanges in coordinates to describe its motion. Such a

situation is shown in Figure 14.20. Note how this corresponds to Figure 14.19.

Because the atoms are free to move in 3Ndifferent independent ways, we say

that the molecule has 3Ndegrees of freedom.

It turns out that we can always choose the coordinates so that the combined

motion of all atoms for three of the 3Nways corresponds to all of the atoms

moving in either the x, the y, or the zdirection. That is, they describe transla-

tionsof the molecule as a whole. Figure 14.21 shows these three translational

degrees of freedom. Since the entire molecule moves in space, we can describe

these motions as translations of the center of mass of the molecule. In trans-

lational motions, the atoms of the molecule are not moving with respect to

each other.

These same changes in coordinates can simultaneously be chosen so that

two (for a linear molecule) or three (for a nonlinear molecule) of the com-

bined motions of all atoms correspond to rotations of the molecule about a

14.8 Vibrations in Molecules 481

yB

xB

zB yA

xA

zA

yD

xD

zD

D

yC

xC

zC

C

B

A

D

C

B

A

xD, yD, zD

xC, yC, zC

xA, yA, zA

xB, yB, zB

Figure 14.20 For any general motion of a mol-

ecule, 3Nchanges in coordinates are necessary to

describe the motion. Each atom requires a x,a

y, and a zto describe its motion.

Figure 14.19 For a molecule having Natoms,

a total of 3Ncoordinates is necessary to describe

the position of atoms in the molecule. The four

atoms shown here require 3 

4 12 total co-

ordinates. Therefore, we say that this molecule

has 12 degrees of freedom.