Why does $H_2$ have $C_V$=$7/2 R$ at high temperatures, while the total number of degrees of freedom is 6?

Question

The two hydrogen atoms have 6 degrees of freedom in total. Of them, $3$ contribute to translation, $2 $contribute to rotation and $1$ contribute to the vibration.

I know that the vibrations motion is frozen at low temperature due to quantum mechanical effects.

However, then the $C_V$ at high temperature should be $6/2 R$, while experimentally, it is $7/2 R$ (source: Principles of Physics by Walker, Resnick and Halliday)

Edit: The answers reveal that the missing part of specific heat is due to potential energy of vibration. So I am extending the question for clarification. $CO_2$ has total 9 degrees of freedom, of which 3 are translational, 2 are rotational, 4 are vibrational. So, at high temperature, will the $C_V$ of $CO_2$ be $\frac{R}{2} × [3+2+4+4] $? The two 4s are due to kinetic and potential energy of vibrational motion.

Answer 1

quoting from wikipedia on heat capacity:

Each rotational and translational degree of freedom will contribute R/2 in the total molar heat capacity of the gas. Each vibrational mode will contribute R to the total molar heat capacity, however. This is because for each vibrational mode, there is a potential and kinetic energy component. Both the potential and kinetic components will contribute R/2 to the total molar heat capacity of the gas.

For more general gas, wikipedia [general gas] also give you example of how to calculate the number of degree of freedom and how to apply properly the equirepartion theorem:

For example, triatomic nitrous oxide N2O will have only 2 degrees of rotational freedom (since it is a linear molecule) and contains n=3 atoms: thus the number of possible vibrational degrees of freedom will be v = (3⋅3) − 3 − 2 = 4. There are four ways or "modes" in which the three atoms can vibrate, corresponding to 1) A mode in which an atom at each end of the molecule moves away from, or towards, the center atom at the same time, 2) a mode in which either end atom moves asynchronously with regard to the other two, and 3) and 4) two modes in which the molecule bends out of line, from the center, in the two possible planar directions that are orthogonal to its axis. Each vibrational degree of freedom confers TWO total degrees of freedom, since vibrational energy mode partitions into 1 kinetic and 1 potential mode. This would give nitrous oxide 3 translational, 2 rotational, and 4 vibrational modes (but these last giving 8 vibrational degrees of freedom), for storing energy. This is a total of f = 3 + 2 + 8 = 13 total energy-storing degrees of freedom, for N2O.

For a bent molecule like water H2O, a similar calculation gives 9 − 3 − 3 = 3 modes of vibration, and 3 (translational) + 3 (rotational) + 6 (vibrational) = 12 degrees of freedom.

Answer 2

The elastic potential energy of the bond is another degree of freedom which contributes to the heat capacity.