Question

Calculate , using the equipartition theorem, the constant-volume heat capacity of CO_2 in three cases:

a.) no vibrations active (the low-T limit)

b.) all vibrations active (the high-T limit)

c.) with just the lowest-energy vibration active.

Answer 1

For CO2 (i.e; a linear molecule) ,Total degrees of freedom = 3N =(3*3)  =9 [Where N = no of atoms present in the CO2 molecule.] Translational modes of freedom =3 Rotational modes of freedom =2 and Vibrational modes of freedom = (9-3-2) = 4

Using the equipartition theorem, calculating the constant-volume heat capacity(Cv) of CO2 in three cases :-

• No vibrations active (at the low-T limit)-

(Translational modes + Rotational modes) = [{3(1/2)RT+2(1/2)RT}] = (5/2)RT.

Therefore, Cv =(5/2)R.

• All vibrations active (at the high-T limit)-

(Translational modes + Rotational modes+Vibrational modes) = [{3(1/2)RT+2(1/2)RT+4RT}] = (13/2)RT.

Therefore, Cv =(13/2)R.

• With just the lowest-energy vibration active -

Vibrational modes of freedom =(9-3-2) =4 . In which 2 are streaching modes of vibration(i.e; higher energy vibration) and other 2 are bending modes of vibration(i.e; lower energy vibration)

So,

(Translational modes + Rotational modes+Vibrational modes) = [{3(1/2)RT+2(1/2)RT+2RT}] = (9/2)RT.

Therefore, Cv =(9/2)R.