Question

Describe the steps and assumptions that simplify the rotational partition function to a formula without a sum.

Answer 1

Rotational partition function

The length of the bond in oxygen molecule is 1.2074 ̊A. Determine the rotational partition function for oxygen at 300 K. THe rotational partition function for a diatomic is

zrot=8π2IkT= T , σh2 σθrot

Note that for a polyatomic the rotational partition function is a product of three terms corre- sponding to rotation about three perpendicular axes. Returning to the problem at hand, the moment of inertia of oxygen molecule is

kg −102
2 mR2 0.016mol (1.207410 m) −46 2

I=μR = 2 = 2 6.023×1023molecules =1.937×10 kgm , mol

from which we determine the rotational temperature to be

h2
θrot = 8π2Ik = 2.08 K

At room temperature, T/θrot ≈ 150, which indicates that the thermal energy is almost 150 times higher than the rotational energy and we expect the rotational partition function to have extensive contribution from the excited states. As expeted, we find that the rotational partition to be zrot = T/(σθrot) = 72.

where I is the moment of inertia μR2, σ is a symmetry factor which accounts for equivalent orientations of the molecule and is 1 for a heteronuclear diatomic and 2 for a homonuclear molecule. For a polyatomic molecule, zrot becomes

8π2IkT 3/2 πIaIbIc zrot= h2 σ .