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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

Molar heat capacities at constant pressure

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?

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

where the experimental data is from Rogers and Mayhew Steam Tables (5th edition). I recall that we explained the derivation for high temperatures with initially increased deviation caused by the anharmonicity of the vibrational modes, and then later on the onset of CO2 dissociation which increases the DOF up to another maximum.

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