# Some questions about heat and temperature

Hi guys,

I am currently working through a book about thermodynamics and statistical mechanics as I was not so great in these course during my undergrad studies.

First question:
The book introduces heat as the kind of energy that terminates the temperature of a system. In other words: Give heat to a system, the temperature will increase. A system with higher temperature holds more heat.

The difference between temperature and other energy (the author calls it "work", so the energy that can be extracted as physical work) can be understood microscopically. Heat is energy that is statistically distributed over all particles, whereas work is other energy, e.g stored in directed motions of particles. E.g. if all particles flow in the same direction, this flow can be used as work.

From the equipartition theorem we know that the average kinetic energy is proportional to the temperature.
From that point of view, if we had N particles flowing in the same direction the temperature of the system would increase with flow velocity. Meaning the heat in the system would, in contrast to what the author claimed, be stored in more and more directed motion.

I went back to check the conditions under which the equipartition theorem holds: He says, it holds in thermodynamic equilibrium. However, in the very beginning of the book, he uses "thermodynamic equilibrium" and "thermal equilibrium" interchangeably as a state where T=const over the whole system (or between different systems when they are in equilibrium with each other).
That would imply that a flowing system with uniform temperature T was in thermal/thermodynamic equilibrium and the equipartition theorem would hold which would lead to the strange conclusion that higher flow velocity => higher T (and to the fact that his microscopical explanation of heat was wrong).

Google shows thermal equilibrium is just PART of thermodynamic equilibrium. In the latter, there has to be chemical and mechanical equilibrium as well. Is a flowing system in mechanical equilibrium? Probably not, because if I were to put a wall in the flow, there would be a force on this wall. So the flowing system is not in thermodynamic equilibrium. If the equipartition theorem was legit only in thermodynamic equilibrium, then everything would be fine (the author simply should have stressed the difference between the two equilibrium terms). But on Wiki(https://en.wikipedia.org/wiki/Equipartition_theorem), it says that the equipartition theorem holds for thermal equilibrium, not thermodynamic equilibrium.
So is Wiki wrong and the equipartition theorem holds only for thermodynamic equilibrium?
Or is the explanation of heat wrong in my book?

Thanks for following my thoughts^^

## Answers and Replies

256bits
Gold Member
Hello Mr Lion,
Some good thoughts about thermo, but just a few comments if you will.

https://en.wikipedia.org/wiki/Heat
the average everyday usage of heat is not what is meant by the word used for thermodynamics
As wiki says heat is a property of a process not of a system.
As well, work is a property of a process.

You may want to re-think, and maybe re-try this statement so that it aligns with the terminology that thermodynamic people use and are accustomed to so that mis-understandings do not occur in conversations. You may know what you are saying, but the recipient will probably not understand nor comprehend the actual meaning.
Temperature, by the way, is not energy, but a measure of the energy within a system, of a conglomeration of matter and not assigned to an individual particle.
The difference between temperature and other energy (the author calls it "work", so the energy that can be extracted as physical work) can be understood microscopically. Heat is energy that is statistically distributed over all particles, whereas work is other energy, e.g stored in directed motions of particles. E.g. if all particles flow in the same direction, this flow can be used as work.

From that point of view, if we had N particles flowing in the same direction the temperature of the system would increase with flow velocity. Meaning the heat in the system would, in contrast to what the author claimed, be stored in more and more directed motion.
Again, heat is not a property of a system.
I'll ask you a question - when you are in a fast moving train does the air temperature change due to the motion? and then change again when it comes to a stop?
Temperature is a measure of the random motion of the particles. Since velocity is relative to some other frame, there can be difficulties assigning a true temperature to a moving gas.

Is a flowing system in mechanical equilibrium? Probably not, because if I were to put a wall in the flow, there would be a force on this wall. So the flowing system is not in thermodynamic equilibrium
It could be in a steady state, where localized state properties do not change temporally.
There is such a thing as non-equilibrium thermodynamics, of which a whole lot of systems are of this category.

as for equipartition, I will refrain from discussing, as there would be better commentary from in house gurus.

Stavros Kiri and davenn
Hello 256bits,

thanks for your response.
256bits said:
https://en.wikipedia.org/wiki/Heat
the average everyday usage of heat is not what is meant by the word used for thermodynamics
As wiki says heat is a property of a process not of a system.
As well, work is a property of a process.

You may want to re-think, and maybe re-try this statement so that it aligns with the terminology that thermodynamic people use and are accustomed to so that mis-understandings do not occur in conversations. You may know what you are saying, but the recipient will probably not understand nor comprehend the actual meaning.
I am aware, I am using the terminology of a standard thermodynamics book. I want to know wether my book is wrong.

It says that heat is that kind of energy put into a system (or extracted from it) that raises (reduces) the temperature.
And (I translate from German) the author states:
W. Greiner said:
The decisive qualitative difference between work and heat is easily explained in the microscopic picture: Heat is energy that is statistically distributed over all particles. If we considered a few particles moving with parallel momentum the kinetic energy of these particles could easily be completely extracted from the system and transferred to other kinds of energy, e.g through decelerating them with a force. If the particles, however, moved completely and statistically disordered, it is impossible to extract the kinetic energy of all the particles (at least if the number of particles is high).
In this interpretation a part of the energy inside of the system is "heat", namely the part that is responsible for the disordered motion of particles.

I'll ask you a question - when you are in a fast moving train does the air temperature change due to the motion? and then change again when it comes to a stop?
I know that this does not make sense. The very reason for this thread is figuring out at which point the author (and Wiki) is wrong.
According to the equipartition theorem, what you said would hold. The faster the flow of a gas, the higher the temperature. UNLESS the equipartition theorem required thermodynamic equilibrium (not only thermal equilibrium) and a flowing system was not in thermal equilibrium.

*push push*
=(

vanhees71
Gold Member
I think Greiner's explanation is correct. Formally you can understand it from the 1st and 2nd Law, according to which the internal energy of a canonical system (in the non-relativistic case a fixed number of particles coupled to a heat-bath, exchanging energy with it),
$$\mathrm{d}U=T \mathrm{d} S-p \mathrm{d}V.$$
Here ##\mathrm{d} Q=T \mathrm{d} S## is the change of heat energy, i.e., the part of the energy put into the system that leads to changes in entropy and ##\mathrm{d} W=-p \mathrm{d} V## is the mechanical energy put into the system by mechanical work.

Temperature is indeed defined (and that's utmost important particularly in relativistic physics) via the energy (density) of the system in the (local) restframe of the system.

Stavros Kiri
I think Greiner's explanation is correct. Formally you can understand it from the 1st and 2nd Law, according to which the internal energy of a canonical system (in the non-relativistic case a fixed number of particles coupled to a heat-bath, exchanging energy with it),
$$\mathrm{d}U=T \mathrm{d} S-p \mathrm{d}V.$$
Here ##\mathrm{d} Q=T \mathrm{d} S## is the change of heat energy, i.e., the part of the energy put into the system that leads to changes in entropy and ##\mathrm{d} W=-p \mathrm{d} V## is the mechanical energy put into the system by mechanical work.
You mean the part where he says "heat is energy that is statistically distributed over all particles"?

vanhees71 said:
Temperature is indeed defined (and that's utmost important particularly in relativistic physics) via the energy (density) of the system in the (local) restframe of the system.
So the equipartition theorem holds only within the rest frame of the system? It is not applicable to the kinetic energy of a macroscopic flow and the prerequisite for the equipartition theorem is thermodynamic equilibrium, not simply thermal equilibrium?

Chestermiller
Mentor
Why don't you get yourself a decent Thermodynamics book like Fundamentals of Engineering Thermodynamics by Moran et al? When you read this, all your ambiguity will vanish.

vanhees71