Untitled - Aerobib - Universidad Politécnica de Madrid

20 CHAPTER 1. THERMOCHEMISTRY
Two cases can arise in the rotational motion of a molecule:
1) If the molecule is linear, that is, if all its atoms lie on a straight line, the moment
of inertia with respect to the axis of the molecule is zero and it cannot store
energy in such a degree of freedom. Hence, the rotational degrees of freedom of
the molecule reduce to two.
2) Whereas if the molecule is non-linear the number of its rotational degrees of
freedom is 3.
If the rotational coordinates are selected accordingly, 9 the energies corresponding
to the rotational degrees of freedom are also separable. Then, except at temperatures
under ambient, the distribution Q rj from the rotational degree of freedom j to
the rotational partition function Q r is
√
2kT Ij
Q rj = 2π , (1.80)
h
where I j is the moment of inertia of the molecule for such degree. Hence, we have:
a) For linear molecules
Q r = 8π2 kT I
h 2 , (1.81)
δ
where I is the moment of inertia of the molecule respect to an axis passing
through its center of gravity and normal to the axis of the molecule.
b) For non-linear molecules
Q r = 1 δ
( 8π 2 ) 3
2
kT √Ix
h 2 I y I z . (1.82)
Here I x , I y and I z are the three moments of inertia of the molecule with respect
to three orthogonal axis at its center of gravity.
In (1.81) and (1.82) δ is a symmetry factor, that accounts for undisguised stable
states due to symmetries of the molecule. For example, for linear molecules δ = 2.
Thus, Q r is determined when the moments of inertia of the molecule are known. Such
moments are obtained from spectroscopic analysis.
As for vibration energy a direct determination of its levels through spectroscopic
analysis is best. Independent contributions of the vibrational degrees of freedom
can, otherwise, be obtained by assuming that their corresponding energies are
separable. Which happens, for example, when vibration amplitudes are small enough
so that the potential energy of the molecule can be approximated by a quadratic function
of the deviations of the atoms respect to their equilibrium positions. For this
9 For which the principal axis of inertia of the molecule must be taken.