# Equal average energies in translational and rotational degrees of freedom

In, An Introduction to Thermal Physics, Schroeder states

It’s not obvious why a rotational degree of freedom should have exactly the same average energy as a translational degree of freedom. However, if you imagine gas molecules knocking around inside a container, colliding with each other and with the walls, you can see how the average rotational energy should eventually reach some equilibrium value that is larger if the molecules are moving fast (high temperature) and smaller if the molecules are moving slow (low temperature). In any particular collision, rotational energy might be converted to translational energy or vice versa, but on average these processes should balance out.

I am not sure how one can conclude from this that the average energy of each degree of freedom must be the same. I agree that it is plausible to assume that average energies might be proportional to each other, but exact equivalence seems strange to me.

• I mean, Schroeder's just trying to make it plausible that there should be some sort of equilibrium due to the randomization of the energy into different "places" due to collisions. Obviously, this argument can't explain why the average energies should be the same. At its heart, (I think) equipartition (the general case of what you're talking about here) relies on the assumption of ergodicity. From there, it can be proven mathematically. But my understanding is that the assumption of ergodicity cannot always be sustained, and I'm pushing up on the limits of my understanding. Commented Feb 13 at 16:28
• @march Looks like an answer to me. Comments are for suggesting post improvements or asking clarifying questions Commented Feb 13 at 16:31
• @BioPhysicist I'm not comfortable writing an answer for this one. My comment was actually meant to be a clarifying one (even if there's not quite a question in there, exactly). Commented Feb 13 at 16:34

I am not sure how one can conclude from this that the average energy of each degree of freedom must be the same.

We can conclude it from the models of constant-volume specific heat based on equipartition of energy, which you can find here:

http://hyperphysics.phy-astr.gsu.edu/hbase/Kinetic/shegas.html#c5

Since, in the absence of work and phase change, specific heat is proportional to the change in temperature for a given amount of heat transfer, and since temperature is a measure of the average kinetic energy of the molecules, we can conclude that the average kinetic energy of the molecules is proportional to the number of degrees of freedom of the individual molecules.

Hope this helps.

I will first give some general remarks, and then I will specifically address what I think your concern is.

Part of the confidence in the equal (average) energy property, I think, is that there is good experimental corroboration for it.

From a book available on Libretexts.org: Thermodynamics, Electricity, and Magnetism

Section 2.4: Heat Capacity and Equipartition of Energy

If the assumption of equipartition of energy is justified then the expectation is, as formulated by Maxwell: the molar heat capacity of an ideal gas is proportional to its number of degrees of freedom

A specific instance of heat flowing out from one degree of freedom: Heating water in a microwave oven:
The microwave energy bumps water molecules to a rotational state. That kinetic energy then flows out to other degrees of freedom.

By nature there is freedom to interconvert in both directions. From rotational to translational, and from translational to rotational. It's just that if there is initially very little translational motion then when a spinning water molecule bumps into another water molecule there is a bias towards transfer of kinetic energy from rotational form to translational form.

As we know: a state of dynamic equilibrium is reached when (on average) in both directions there is the same rate of transfer.

I surmise that your concern is somewhat as follows: what if there are classes of molecules such that transfer in one direction is favored over the opposite direction?

And yeah, there are cases where a selective effect is at play. Example: osmotic effect elicited with a semi-permeable membrane. Water molecules can dissolve in the membrane, allowing water molecules to migrate across the membrane, large molecules cannot migrate; as a consequence an osmotic pressure builds up.

I expect that any form of selective effect is not possible in the realm of transfer of kinetic energy in collisions between molecules.

Rotational kinetic energy or translational kinetic energy, it is both kinetic energy.

If in the population of molecules a state of rotational kinetic energy is over-represented then transformation from rotational to translatinal is the more propable outcome, leading to equalization.

If in the population of molecules rotational kinetic energy is under-represented then transformation from translational to rotational is the more propable outcome,leading to equalization.

So it is to be expected that it's not possible for any degree of freedom to accumulate more kinetic energy than another degree of freedom.

In order for kinetic energy to accumulate in one degree of freedom, at the expense of (an)other degree(s) of freedom, the conversion would have to be, in some way, not bi-directional.

But a mechanism, at molecular level, that would result in selective conversion is very problematic. (Related: an attempt by Maxwell in another context, called Maxwell's demon)