Page 56 - Modern physical chemistry

3.6 Translational, Rotational, Vibrational Energies 45

Each separate disjoint mode of motion corresponds to a degree of freedom. Molecu-
lar collisions distribute the available energy among the molecules and their degrees of
freedom as randomly as possible. How much anyone degree of freedom holds on the
average is presumably determined by the mathematical expression for its energy.
The energy associated with translation of the ith molecule in the ±x direction is

[3.38]

For N molecules, we have the energy

IN ~
E x=- mx [3.39]
2
associated with this degree of freedom. Equations (3.8) and (3.24) transform this to

1
Ex=-nRT. [3.40]
2
Similarly, the energy associated with translation of the molecules in the ±y direction is

1
Ey =-nRT, [3.41 ]
2
and that for translation in the ±z direction,

1
Ez =-nRT. [3.42]
2
The energy of rotation of the ith molecule at angular velocity O·around an axis is
(SO)i =i18 2 [3.43]

where 1 is the moment of inertia about the axis. Since the expression for (eO)i is like that
for (eJ we expect that the N identical molecules would have a rotational energy after
randomnization like that for translation; thus
1
Eo =-nRT. [3.44]
2
For a linear molecule, we have two disjoint rotational modes. Then the rotational
energy is
E rot =nRT. [3.45]
A nonlinear molecule has three disjoint rotational modes; then

3
E rot =-nRT. [3.46]
2
In classical mechanics, the energy associated with a simple harmonic vibration is

)
( S"b = - 1 .2 1 2 [3.47]
lin" + - kq" .
~ i 2~~ 2 ~
Here )1 is the reduced mass, k the force constant, and qj the generalized coordinate
for the mode. In the motion, the average potential energy over a cycle equals the average