# What is the high temperature molar heat capacity of $\mathrm{CO}_2$ (carbon dioxide)?

In determining the heat capacity of a substance we just count the number of quadratic degrees of freedom in the Hamiltonian, right? Using that logic, for the linear molecule $\mathrm{CO}_2$ in a gaseous state I get:

1. 3 degrees of freedom from translation of the center of mass,
2. 2 from rotations (linearity means rotations around the small axis are frozen out),
3. 4 from longitudinal normal modes (symmetric and anti-symmetric), and
4. 4 from transverse normal modes (one bending mode, two polarizations).

So, that's a molar heat capacity of $13R/2$ for $C_V$ and $15R/2$ for $C_p$. I'm not 100% confident in the above accounting for two reasons. First, this plot originally from http://direns.mines-paristech.fr/Sites/Thopt/en/co/gaz-ideaux.html, has $C_p$ exceeding $15R/2\approx62.4 \operatorname{kJ}\operatorname{kmol}^{-1}\operatorname{K}^{-1}$ at about $2671\operatorname{K}$ and more likely asymptoting to $8R\approx66.5\operatorname{kJ}\operatorname{kmol}^{-1}\operatorname{K}^{-1}$ as though it had 14 degrees of freedom.

The only way that makes sense to me is from the second consideration - the bending mode excited states spread the mass out away from the ground state's axis of symmetry and so could have a non-frozen rotational excitations. The counter-argument to that is that a $90^\circ$ out of phase superposition of the bending modes looks like a rotation plus bending mode, so the bending modes should cover it.

Long story short: is the plot correct or the counting argument? If the counting argument is wrong, what alters the picture?

I got the same results by applying the counting argument. From quite a while ago I found this plot 