Specific heat for a triatomic gas

[Titan97]

Homework Statement

Using equipartition law, find specific heat of gas containing triatomic linear molecules. Will the result be different if the molecule was non- linear? In what way?

Homework Equations

According to equipartion theorem, each degree of freedom gets (1/2)kT kinetic energy and (1/2)kT potential energy.

The Attempt at a Solution

For a linear arrangement,

• number of translational degrees of freedom is $3$
• number of rotational degrees of freedom is $2$ (or is it 6 because the atom can rotate about the central atom or about one of the atoms at the end)
• number of vibrational degrees of freedom is $1$

At room T, i will neglect point 3.

My second doubt is, the internal energy is $\frac{1}{2}NkT\times f$

Why is it only 1/2 the value of kT (which is the sum of kinetic and potential energies as my prof says in one of the slides which i have attached below)?

In the slide, $$U=\frac{h\nu}{e^{\frac{h\nu}{kT}}-1}$$

Why isn't it $$U=\frac{h\nu}{e^{\frac{h\nu}{\frac{1}{2}\times kT}}-1}?$$

 

[Bystander]

Total number of degrees of freedom is 3n.

 

[Titan97]

How did you get that expression?