Other Gases

It took almost no effort to correctly predict the specific heat of helium gas. What complications do we face when we try to predict the specific heat of other gases? The problem is that other gases form molecules. From one point of view the molecules themselves are the gas particles, so that their average thermal kinetic energy must be 3/2 kT just like the helium atoms. So far so good, but molecules have an internal structure. An oxygen molecule, for example, consists of two oxygen atoms held together by the molecular force, We can fairly accurately picture the molecule as two atoms held together by a spring force as shown here in Figure (1).
If an oxygen molecule collides with another molecule in the gas, one would expect that the molecule would start to vibrate, and perhaps rotate. This vibration and rotation represent forms of internal motion of the molecule that are quite distinct from the motion of the molecule as a whole, distinct from what we would call the center of mass motion.
If the center of mass motion has an average thermal kinetic energy 3/2 kT just like helium atoms, but the molecules can have internal motions and internal energy, you would expect that it would require more energy to heat a mole of oxygen than a mole of helium. For with the oxygen you not only have to supply the kinetic energy of the center of mass motion, but also the internal energy of the molecules. And that is correct. The molar specific heat of oxygen is 20.8 joules/(mole K) , as compared to 12.5 joules/(mole K) for helium.
However it is when we try to calculate how much the internal energy of the molecules contribute to the specific heat, we run into trouble. As far back as 1858, James Clerk Maxwell, who was working on these calculations, repeatedly failed, and suspected that the failure was due to a problem with Newtonian mechanics.

Figure 1; Model of an oxygen molecule.