Physical Chemistry , 1st ed.

(Note that the limits on the integral stay the same.) The integral in equation

18.30 has a known solution; it has the form 

0 eaxdx1/a. Using this so-

lution, we get

qrot



T

r



8

h

(^2) I
2
kT
 (18.31)
where the definition ofhas been used. Equation 18.31 is applicable only if
the temperature of the gas is well above the rotational temperature of the mol-
ecule. If this is so, then the rotational partition function of the heteronuclear
diatomic gas is easy to calculate. Table 18.3 lists several rvalues for some het-
eronuclear diatomic molecules. If the temperature is not obviously higher than
r, then equation 18.31 is not applicable, and an explicit summation using
equation 18.26 is necessary to calculate qrot. Given the low values ofrfor most
molecules (H 2 is a notable exception), the high-temperature expression for qr
can be used at most temperatures.
Example 18.6
Given that the moment of inertia of hydrogen iodide, HI, is 4.269
10 ^47 kgm^2 ,
calculate rand qrotat 310 K. Comment on the units ofr.
Solution
Using the definition ofr, we have
r
The (2 )^2 term comes from the ^2 term in the numerator. One of the joule
units cancels directly, and using the fact that J kgm^2 /s^2 , we see that the
second joule unit cancels the s^2 unit in the numerator and the kgm^2 unit in
the denominator. The only remaining unit, K in the denominator of the de-
nominator, comes up to the numerator. We have
r9.431 K
The final unit of kelvins is appropriate for a rotational temperature. The qrot
is therefore
qrot
9

3

.4

1

3

0

1

K

K

32.9

Again, notice that the partition function is a pure number without units, as

it is supposed to be.

For a homonucleardiatomic molecule, there are additional concerns because

we now have identical nuclei. These concerns have to do with the Pauli prin-

ciple, which was introduced in Chapter 12. Recall that the strict form of the

Pauli principle required that a wavefunction of fermions must be antisym-

metric with respect to exchange of two identical particles, and that a wave-

function of bosons must be symmetric with respect to exchange of two iden-

tical particles. In Chapter 12, we were considering only the electronic

wavefunction, so the antisymmetry requirement was stressed. However, in our

consideration of an overall molecular wavefunction, we are now including

other particles: the nuclei of the atoms. Therefore, we must apply the Pauli

principle to other parts of the (homonuclear diatomic) molecule and see how

(6.626 

10 ^34 Js)^2



(2 )^2  2 (4.269 

10 ^47 kgm^2 )(1.381 

10 ^23 J/K)

630 CHAPTER 18 More Statistical Thermodynamics

Table 18.3 Rotational temperatures rfor

some diatomic molecules

Molecule r(K)

H 2 85.4

N 2 2.86

O 2 2.07

Cl 2 0.346

Br 2 0.116

HCl 15.2

HBr 12.1

HI 9.0

CO 2.77

NO 2.42