# Heat, Temperature and Scale

Temperature and heat are normally considered as manifestations of molecular motion as in the Kinetic theory. But the concept is scale dependent. By that I mean that there are two scales...a micro scale where certain definitions apply and a macro scale where they don't.

The kinetic energy due to small and fast motions count towards "heat". But large and slow motions e.g. the motion of a car is not considered as "heat". My question is what defines the cutoff for what motions count towards "heat" and which ones don't. Also is there a scale independent definition of heat where even bulk motions could be thought of as heat?

Let me first say that this is such a nice question. Maybe many people might say it's obvious or whatever, but it's a question we all should have when hearing for the first time that "heat is related to internal movement".

The answer is really complex. I'll try to make it short. You have a system under consideration. That system will have a total mean energy $U$. Heat are the variations of that energy minus the work done: $Q=\Delta U - W$.

So the way to distinguish between "normal movement" and "heat" is if they increase the internal energy. A consequence for this is that such movement cannot be globally translational or rotational. This is easily visible from the point of view of "translational invariance" and "galilean/relativistic" transformations.

Let's go to the point: a global movement will not contribute to heat. (Global refers to "affecting to your entire system under consideration").

So for example, an elastic rubber can be moved and that will not increase its heat (in a frictionless environment). However, if the movement is not global, you are probably changing it's temperature.

And any of these movements, which are "not uniform", can be described as a sum of the normal modes of oscillation of your system.

In conclusion: heat is excitation of normal modes of vibration.

The higher the harmonic is, the higher motion it causes, so the higher heat you get.

This is because these movements are probably changing the number of available microstates of the system. That changes the entropy, and

$\frac{\partial S}{\partial U}=\frac{1}{T}$, so you are changing temperature.

• I think this answer, while strong, could be improved. You say that "heat is excitation of normal modes of vibration". What about when I heat monatomic He, which doesn't vibrate and only translates? The missing element seems to be that heating a system increases the dispersion or width of energy distributions; in contrast, doing work on a system (such as imparting a uniform speed, as in a box of gas moving rapidly) elevates the energy level of each particle uniformly. I think this is what you reference when you refer to "global" movement. – Chemomechanics Apr 22 '18 at 1:54
• Thanks a lot. I didn't know how to explain it well. All suggestions are greatly appreciated. Great comment. – FGSUZ Apr 22 '18 at 10:40

It's not primarily the scale that matters; it's the numbers. There are approximately $2.5 \times 10^{16}$ molecules in just $1\ \mathrm{mm}^3$ of air (source). That's quite a huge number.

A lot of thermodynamics is about statistics and probability. For example, the second law of thermodynamics could be formulated as "energy distribution will as time flows only change into more probable distributions".

This question is rather unique - I've never seen a textbook discuss the matter. However, I would say that your "scale" for when molecular motion can count as heat would be similar to the assumptions for when the ideal gas equation is applicable, or nearly applicable:

1. The mass of the objects bouncing around is negligible
2. The collisions are elastic or nearly so
3. The size of the objects bouncing around is negligible in comparison with their mean-distance-between collision.
4. Gravitational acceleration or weight is negligible.

With regard to your "scale independence", I feel like I know what you're talking about. But I also think that the definition above is ultimately scale independent. The assumptions either produce results that are approximate to your satisfaction or they don't - scale independence doesn't really enter into the analysis at all.