Question

Assumptions:

1. Ensemble with Translational, Vibrational and Rotational degree of freedoms.
2. Not classical, but discrete energy states.

I can't find an clear explanation in kinetic theory of thermodynamics what the measurable temperature is. Some sources state, it's only the average kinetic energy of translation, while other state it's the average kinetic energy of all degree of freedoms.

If we look at a diatomic gas at T=0, there is still a Zero-Point rotatitional energy h×v/2
. If the measurable temperature would be caused by all average kinetic energys, it couldn't be 0 because there would still be K.E. of rotation (even if it's very very small,... well as small as it would ever get).
If measurable temperature would not just be due to average K.E. energy of translation, I assume we wouldn't observe the changes in specific heat capacity with rising temperature. Rising temperature causes rotational and vibrational states to unfreeze, I assume those are only internal ways of storing energy, which doesn't show up in temperature.
Those were 2 reasons I tend to believe that measurable Temperature is only due to average translational energy. But imagine the following:

Reaching the point of energy states, where the gaps of possible translational states is greater than that of rotation (translation is dense at the beginning and tends to separate further apart, while rotation is far apart at the beginning but comes closer together).
If we now a pump a little more energy in that system, we couldn't measure any temperature change for a small interval, because now the energy first populates rotational states which doesn't show up in temperature. Temperature first starts to change again, if the the gap is bridged where the next higher translational state could be populated. So wouldn't that mean, that we can't determine internal energy in that Intervall with temperature measurement? There could be a small variation of internal energy with no change in temperature at all.

To explain, why I added solids in my title, one might argue, that a solid can't have translational energy and thus it's a proof, that also other kind of degree of freedoms contribute to temperature. I think, there is also translational movement in a solid, which is different to the common vibrational degree of freedom. For me, vibration is the movement due to bond length variation of the molecule within (the mass point remains the same), while translation occurs in a more restricted way: Moleculs as a whole vibrate around a point and thus there is a variation of the mass point.

Edit: I mean, in some textbooks they define an extra vibrational temperature or rotational temperature and compare it to normal temperature but what should normal even mean? Average K.E. of translational?

Answer 1

Temperature in thermodynamics is typically linked to the average translational kinetic energy of molecules, yet in systems with internal degrees of freedom like diatomic gases and solids, rotational and vibrational energies also contribute, especially at higher temperatures.

Step-by-step explanation:

In the kinetic theory of thermodynamics, the measurable temperature of a gas is often associated with the average translational kinetic energy of its molecules, which can be described by the equation KE = mv² = 3/2 kT. This relationship indicates that temperature is directly proportional to the translational kinetic energy. However, for systems with molecules that have internal degrees of freedom, such as diatomic gases or solids, other forms of energy like vibrational and rotational kinetic energies also come into play, especially at higher temperatures.

For diatomic gases, at very low temperatures (below 60 K), translational energy is the predominant form, and the specific heat capacity (Cv) resembles that of a monatomic gas. As the temperature rises, rotational degrees of freedom become thermally accessible, with vibrational ones following at even higher temperatures. In a solid, although there is no macroscopic translational motion of the molecules, the thermal vibrations of atoms about their equilibrium positions contribute to the solid's thermal energy and, consequently, its temperature.

Thus, while the average translational kinetic energy is a critical component of temperature, other forms of molecular energy must be considered, particularly at the temperature ranges where rotational and vibrational modes are excited. For ideal gases, equations have been generalized to account for these additional degrees of freedom, allowing the calculation of the internal energy and the subsequent attribution of a temperature to these complex systems.