the_emi_guy said:
Perhaps this is just semantics, but I was taught that all of the thermal energy was kinetic. The motion of the center of mass being the translational kinetic energy, plus the internal kinetic energy in the form of lattice vibrations/rotations etc. Quantum mechanics was required in order to get the right count of degrees of freedom, but that it was all kinetic energy.
Vibrational DoF includes kinetic and potential contributions. That's where the double-count of these DoF comes from. Rotational is purely kinetic, yes, but usually when people talk about "average kinetic energy," they are referring to translational part only. Otherwise, it becomes unclear what you mean. Including kinetic term, but not potential term from vibrations would be just silly, for example. And looking at just translational and rotational doesn't make much sense either.
There is sense in considering just translational kinetic energy, however. For example, when you consider pressure of an ideal gas, only translational kinetic energy is relevant. So the formula is exactly the same for all gasses. This is the typical context in which the term "average kinetic energy" is most frequently used. And if you look at Wikipedia article on
kinetic theory, you'll notice that it's most frequently referred to as just "kinetic energy" and twice as "(translational) kinetic energy".
So in context of gas kinematics, "average kinetic energy" will almost always refer to just the translational part.It might be just semantics, but I also wouldn't say QM was necessary to count DoF. Just to figure out why not all of them contribute, and why some of them contribute only partially. That, of course, has to do with quantization of rotational and vibrational energies.
Leoragon said:
So, in a nut shell, temperature is proportional to the kinetic energy? And there's that formula that shows the average kinetic energy of a monoatomic gas. For diatomic, its 5/2?
If you talk only about translational kinetic energy, which is what is usually meant by "average kinetic energy", then it's always 3/2.
If you look at total mechanical energy of a diatomic gas, you get 3/2 from translational DoF, 2/2 from rotational, and 2/2 from vibrational, of which 6/2 total is kinetic and 1/2 is potential energy. However, some of these will be "frozen out". Specifically, rotational degrees of freedom are usually inaccessible because the quantum of energy is much higher than available amount of energy at room temperatures. So you end up with roughly 5/2 total mechanical energy for diatomic gases.