Assuming that the forces between atoms that form molecules can be approximated by elastic restoring forces, count the number of energy degrees of freedom, including those for translations and rotations, for (a) O2 molecules, (b) H2O molecules, (c) the ammonia molecule NH3. Use your results to calculate the molecular heat capacities of gasses of these molecules in the classical limit.

(a) Three translational degrees of freedom, 2 rotational degrees. 5 total

Cv = 5/2 R; Cp = 7/2 R

(b) and (c) 6 total degrees of freedom (3 translational, 3 rotational)

Cv = 3 R; Cp = 4R

Explanation:

(a) O₂

Oxygen being a diatomic molecule has three translational degrees of freedom and two rotational degrees of freedom since it can move in the three axis and can rotate around two.

(b) H₂O

This is a polyatomic molecule and it has three translational and three rotational degrees of freedom.

(c) Same as water it has three translational degrees of freedom and three rotational degrees of freedom

To calculate the heat capacities we have to make use of the equipartition theorem which tell us that for each degree of freedom imparts 1/2 R to the heat capacity at constant volume.

(a)

5 total degrees of freedom ⇒ Cv = 5/2 R

Cp (heat capacity at constant pressure) is determined from the relation

Cp - Cv = R

Cp = 7/2 R for O2 molecule

(b) and (c)

Total degrees of freedom 6

Cv = 3 R

Cp = 4 R

Here we are ignoring any contribution of the vibrational modes to the contribution of the heat capacities