Physical Chemistry , 1st ed.

Similarly, the other two vibrational temperatures are 5163 K and 2287 K. All

three vvalues are larger than the stated temperature (298 K), so the high-

temperature expression for qvibshould not be used. We thus calculate the

vibrational partition function using equation 18.24. For this molecule,

3 N 6 3, so there will be three terms in the product for qvib:

qvib

qvib(0.0001263)(0.0001729)(0.02156) 4.7081 

10 ^10

In the above example, the individual vibration’s contribution to the qvibof

the molecule was given to illustrate a point. Note that the lower vvalue’s

contribution to qvibis two orders of magnitude larger than that of the two

higher vvalues. Because of the negative exponential definition ofqvib,

lower-frequency vibrations have a proportionately larger effect on qvib. One

way to consider this in terms of a Boltzmann distribution is that the lower-

frequency (and therefore lower-energy) vibrations are more easily popu-

lated thermally.

18.5 Diatomic Molecules: Rotations

Gas molecules also rotate in three-dimensional space, and quantum mechan-

ics says that rotational energies are also quantized. Therefore, we can also con-

sider a qrotpart of the complete partition function of a molecule. Since this is

the last kind of partition function we define, we will suggest that there is a

complete molecular partition function Qdefined as

Qqtransqelectqvibqrotqnuc (18.25)

in which the complete molecular partition function is represented by the cap-

ital letter Qand the individually defined partition functions are represented by

the lowercase q’s. We will consider Qfurther in the next section.

Before we can consider the complete molecular partition function Q,we

need to know what qrotis. By now you may realize that the basic definition of

qrotwould be

qrot 



ifirst

giei/kT (18.26)

rotational level

in which giis the degeneracy of the ith rotational level and iis the energy of

the ith rotational level.

The energies of rotation of gas-phase molecules can be approximated very

well by the three-dimensional rigid rotor ideal system. In fact, we used this

approximation in Chapter 14 to derive a basic understanding of rotational

spectroscopy. We can apply that understanding to the rotational energy levels

and the rotational partition function. It is easiest to start with the simplest

molecule, a diatomic molecule. Furthermore, we will assume that the molecule

is heteronuclear; that is, there are two different atoms in the molecule. (We will

consider homonuclear diatomic molecules separately, as there are some subtle

but interesting differences.)

exp

2

2



2

2

8

9

7

8

K

K





1 exp

2

2

2

9

8

8

7

K

K



exp

2

5



1

2

6

9

3

8

K

K





1 exp

5

2

1

9

6

8

3

K

K



exp

2

5



3

2

5

9

0

8

K

K





1 exp

5

2

3

9

5

8

0

K

K



628 CHAPTER 18 More Statistical Thermodynamics