# Why does Hydrogen molar heat capacity reach 7/2 R?

If a diatomic gas like Hydrogen has 6 maximum degrees of freedom why its molar heat capacity reaches at high temperatures $C_V = \frac{7}{2} R$ and not $C_V = \frac{6}{2} R= 3R$?

molar heat capacity of hydrogen

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The degrees of freedom of a diatomic gas are as follows:

• 3 translational: The molecule can move in x, y and z-direction.
• 2 rotational: The molecule can in principle rotate around each axis. But consider rotations around the molecular axis (connecting the H atoms): in this case, the physics doesn't change. Another way of thinking about it is that the axial rotation mode only can store a vanishing amount of energy, compared to the others.

The rotational modes are only available at higher temeratures, since the molecule has a moment of inertia that has to be overcome to start rotating.

• 2 vibrational: The atoms can wiggle together and apart, which is one degree of freedom. But there is also another one, which is harder to see: Think of the molecule as an harmonic oszillator with kinetic and potential energy.

If both atoms are displaced towards the middle, the molecule has a higher potential energy. The equations of the harmonic oscillator would normally fix the kinetic energy of the atoms in this case. But in a gas with its random kinematics it is totally possible that they have the "wrong" kinetic energies for their relative displacement. So, for the purposes of statistical mechanics, these are 2 further DOFs, making seven.

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A lot of words for a simple fact: Vibration is two degrees, textbooks explain that with kinetic and potential energy. I found that rather strange as a student too, but textbooks are textbooks. I think mistnim is asking on the same problem. Your explanation isn't a explanation to me. –  Georg Apr 23 '13 at 14:26
@Georg: I found it in need of an explanation that the 7th degree of freedom is really a separate DOF. Some simple arguments suggest otherwise: Say #6 is the wiggling, or $\dot p$ of the HO, and #7 is the motion of the center of mass, or $\dot x$. Then a) one variable is determined by the other through the HO DGL, thus they are not independent and b) isn't the movement of the center just a translation? –  jdm Apr 23 '13 at 14:37