Page 57 - Modern physical chemistry

46 Gases and Collective Properties

kinetic energy over the cycle. So we consider the energy associated with the last term
to equal that associated with the preceding term. But this has the same form as (3.38),
with which we associated 112 nRT of energy. We thus expect the total energy associated
with one molecular vibrational degree of freedom to be
Evib =nRT. [3.48J
Implicit in the derivations of this section is the assumption that a degree of freedom
behaves as if it could absorb or emit any available amount of energy. So any quantiza-
tion present must be small with respect to the average energy in the degree of freedom
under consideration.

3.7 Quantum Restrictions
During an intermolecular collision, a degree of freedom can only be induced to shift
from one quantized level to another. The energy involved equals the difference between
the energy levels.
But from (3.22), the average kinetic energy in a classical degree of freedom at a given
temperature Tis
s=!kT. [3.49J
2
For a degree of freedom to be excited classically, the separations between the pertinent
energy levels must be small compared to this. Otherwise, in many of the encounters with
more energetic molecules, the excess energy would not be large enough to cause exci-
tation, and vice versa
Translational eigenstates lie very close together. As a consequence, they are excited
classically even at very low temperatures. The energy levels in rotational degrees of
freedom are fairly close together; so generally they contribute classically. But at very low
temperatures, the separation becomes significant for a molecule with a low moment of
inertia, like H 2 •
The separation between vibrational levels is generally larger, so that most molecular
vibrations contribute less than nRT to E unless the temperature is high. The separation
between electronic levels is usually so great that they do not contribute to the energy E
at ordinary temperatures.

Example 3.7
Estimate the energy associated with thermal agitation in 1 mole water vapor at 25° C.
The translational energy is given by equation (3.23):

I
Etr =%nRT =%(1.000 mOIX8.3145 J K- mor X298.15 K) = 3718 J.
l
Since the H20 molecule is nonlinear, the rotational energy is given by equation (3.46):
3
E rot = -nRT = 3718 J.
2
The three normal vibrations of H 20 have the wave numbers 1595, 3652, and 3756 cml;
so the least energy needed to excite a vibration is

E min = N Ahv = N AhcO' = (0.119627 J m mol- I X 1595 x 10 2 m- I ) = 19,080 J mol-I.