# At what size is kinetic energy no longer related to temperature?

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## Summary:

I've read so many times that "temperature is a measure of the average kinetic energy of the molecules in a substance," or sometimes "particles" to encompass atoms and ions too. But how big can "molecules/particles" be before their kinetic energy is no longer relevant to temperature?
Hi all,

I've read so many times that "temperature is a measure of the average kinetic energy of the molecules in a substance," or sometimes "particles" to encompass atoms and ions too. But how big can "molecules/particles" be before their kinetic energy is no longer relevant to temperature?

If a crowd of people are milling about in a room, the people are far too big to be considered "molecules" and their average kinetic energy isn't considered when talking about the temperature of the room (just as well - going by KE = 1/2mv^2 even the kinetic energy of a slow human would raise the average by an insane amount and lead to a ridiculously high temperature) but the average kinetic energy of oxygen and nitrogen molecules in the air is.

But what about larger molecules in the air? Sugars or amino acids? Polymers? Ribosomes? Viruses? Bacteria? Specks of dust? I have a feeling that the average kinetic energy of specks of dust in the air has nothing to do with temperature, so where do we draw the line between "molecules" and "big things," and say temperature falls on one side of the line while kinetic energy falls on the other?

Delta2

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anorlunda
Staff Emeritus
How about this? When the number of molecules (not size, but number) is small enough that you can track each one individually, then you don't need temperature any more.

When I say track, I mean each collision and scattering event.

How about this? When the number of molecules (not size, but number) is small enough that you can track each one individually, then you don't need temperature any more.

When I say track, I mean each collision and scattering event.
So what if you have, hypothetically, a ball pit filled with numbers of balls too big to count, a swarm of millions of flies, a collection of millions of rocks or dust particles, or a crowd of millions of humans? Would the average kinetic energy of these really big "molecules" contribute to temperature too?

hutchphd
Or one can look from the other side at this question. The average thermal energy per degree of freedom ## \approx kT## which is ##\frac {1} {40}eV = 4x 10 ^{-21}J## at room T. For a macroscopic object of even small mass (described by a single coordinate) this corresponds to negligible speeds (left to interested reader to calculate).

anorlunda
Staff Emeritus
So what if you have, hypothetically, a ball pit filled with numbers of balls too big to count, a swarm of millions of flies, a collection of millions of rocks or dust particles, or a crowd of millions of humans? Would the average kinetic energy of these really big "molecules" contribute to temperature too?
A rock or a human is not a molecule.

But words aside, I could stand next to a hot rock or a hot girl and have my temperature raised.

There's another factor too. The kinetic energy must not come from all particle having motion in the same direction. Throwing a baseball does not make its temperature increase. But warming a baseball in an oven, gives its atoms and molecules higher average K.E.

hutchphd
A collection of millions of rocks will necessarilly contain ##10^{23}## x (a million) internal degrees of freedom all with energy distibuted according to the equipartition theorem. The big chunks just don't materially contribute to the party when we talk about thermal issues.

A rock or a human is not a molecule.

But words aside, I could stand next to a hot rock or a hot girl and have my temperature raised.

There's another factor too. The kinetic energy must not come from all particle having motion in the same direction. Throwing a baseball does not make its temperature increase. But warming a baseball in an oven, gives its atoms and molecules higher average K.E.
Let's try and get as close to molecular motion as we can on a macroscopic scale, then - instead of a single baseball being thrown, or even a billion baseballs being thrown in the same direction, how about a collection of a billion baseballs in a giant room all moving in different directions? Let's say the room itself isn't moving anywhere (so the baseballs aren't all moving in the same direction), but baseballs are bouncing off of the walls and off of each other, so many of them moving in so many different directions that you have to use statistical mechanics and not be able to track them individually. Let's say this room is in deep space, so no gravity, and let's pretend the collisions are elastic so there's no loss of kinetic energy. This seems like a good analogy to molecules in a gas. (I suspect the baseballs' energy would even follow a Maxwell-Boltzmann distribution, no?) Would the average kinetic energy of those baseballs be related to the temperature of the room, then? If we put a big block of ice next to it, would the temperature difference between our room and the ice block that we'd use in energy flow calculations take the baseballs' random motion into account?

And of course a human or a rock isn't a molecule, but that's what I'm trying to get at - where do you draw the dividing line where "molecules" stop for temperature purposes? Why is the kinetic energy of a huge number of oxygen molecules moving randomly part of temperature, but (I assume) the kinetic energy of a huge number of baseballs moving randomly not?

Delta2
gleem
Another consideration should be that the kinetic energy of the bodies should not be high enough to destroy them.

hutchphd
For your baseball analogy to work you would need a temperature high enough to cause the thermal KE of the baseballs to be significant......say 0.1 Joule . This would require a temperature of ##10^{20}## Kelvin, and a vessel at that T to hold the baseballs. And some pretty stout balls. Not an easy experiment to perform!!
So we need not worry about these effects in our usual world.

Staff Emeritus
2019 Award
This sounds a lot like the paradox of the heap. It's 2400 years old, and the resolution is that there is no bright line where N elements have a temperature and N-1 do not.

hutchphd
anorlunda
Staff Emeritus
Your baseball analogy is flawed as @hutchphd said. But you're on the right track.

Instead of an analogy, why not just think of molecules in a gas? It also works for vibrations in the lattice of a solid, but it's much easier to think about in a gas.

Energy and temperature are related by Boltzman's constant. If you really want to know how the thinking about temperature and gasses developed, you should study about Boltzman's work.

vanhees71
Gold Member
2019 Award
I think the answer to the question is a question of statistics. First of all one must emphasize that temperature is related to the thermal motion, or more precisely, to the mean energy of a particle (where particle refers to the "relevant degrees of freedom", e.g., atoms or molecules in a gas or quark-gluon like quasiparticles in the QGP phase of strongly interacting matter and hadrons in the hadronic phase, etc.) in the rest frame of the fluid cell. A fluid cell is a macroscopically small volume element of the medium, which can be considered large on a microscopic time scale, i.e., containing many particles or quasiparticles. Of course temperature only makes sense if the system is close to thermal equilibrium.

So you can always define a temperature for a single particle in such a fluid cell. E.g., it makes sense to say a single particle (e.g., a heavy quark within a quark gluon plasma consisting mostly of light quarks and gluons) in close to local thermal equilibrium has a temperature. Then the meaning is statistical, i.e., if you consider a lot of such systems, then on average the momentum distribution of that single particle is described by a thermal distribution. An example are heavy-ion collisions where for a short time a hot and dense fireball is created in a collision of two heavy nuclei (gold or lead at RHIC and LHC) and you investigate the heavy quarks. Per event in a collision at RHIC you have only a few c quarks but measuring many collisions you get a ##p_T## distribution and elliptic flow ##v_2## close to what you expect from heavy quarks being close to local thermal equilbrium with the QGP, i.e., the heavy quarks being "dragged" along with the collective flow of the bulk medium consisting of light quarks and gluons. For details, see, e.g., review:

https://arxiv.org/abs/1506.03981

etotheipi
DrClaude
Mentor
I've read so many times that "temperature is a measure of the average kinetic energy of the molecules in a substance," or sometimes "particles" to encompass atoms and ions too.
I would argue that the statement only strictly applies to a gas. Also, that should be the kinetic energy in the reference frame where the center of mass is fixed. The air in your car is not getting warmer because you are speeding!

etotheipi and vanhees71
etotheipi
Gold Member
2019 Award
I would argue that the statement only strictly applies to a gas.
Though we can also apply similar reasoning to other states, right? For instance, a crystalline solid of N ions consists of 3N effective harmonic oscillators, each with energy ##k_B T## (i.e. ##\frac{1}{2}k_B T## allocated to the potential and kinetic degrees of freedom each by the equipartition theorem). From this we identify the specific heat capacity as the partial derivative of ##E## w.r.t. ##T## i.e. ##C = 3Nk_{B}##, which is a statement of the Dulong-Petit law.

There are probably other examples also

DrClaude
Mentor
Though we can also apply similar reasoning to other states, right? For instance, a crystalline solid of N ions consists of 3N effective harmonic oscillators, each with energy ##k_B T## (i.e. ##\frac{1}{2}k_B T## allocated to the potential and kinetic degrees of freedom each by the equipartition theorem). From this we identify the specific heat capacity as the partial derivative w.r.t. ##T## i.e. ##C = 3Nk_{B}##, which is a statement of the Dulong-Petit law.

There are probably other examples also
Yes, and this is exactly why I would argue that it is incorrect to say that "temperature is a measure of the average kinetic energy of the molecules in a substance." Things get more complicated when more degrees of freedom are involved. I don't think that thinking of temperature as the motion of molecules is a good mental model; it leads to too many wrong conclusions and misunderstandings.

I am okay with the converse: the kinetic energy of the components of a substance increases with temperature. But one has to be careful not to equate temperature with kinetic energy.

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etotheipi
sophiecentaur
Gold Member
But how big can "molecules/particles" be before their kinetic energy is no longer relevant to temperature?
Science constantly tries to get as simple as it can be and there's a convenient line between microscope / statistical behaviour of molecules and 'mechanical' behaviour of 'large' objects. But that line is very broad and fuzzy.

Even when you consider a 'real' gas, the energy you put into it will not all turn up as a rise in Temperature (defined as above). There are internal mechanism inside molecules that store varying amounts of Potential Energy and there are much more significant forces between adjacent molecules in a gas. Both of these will modify the behaviour of a 'gas' so that the simple statistic are changed and the concept of Temperature is not clear cut.

So I would say that what happens in a box of baseballs is not too different from what happens with a box of real gases. Temperature is a quantity which does allow one to predict which way energy will flow from one region to another (hot to cold) but actually, only in some circumstances. If you have a box of cold baseballs, moving and bouncing about fast and a box of hot stationary baseballs, you cannot just use Temperature as the only measure of which way the energy will flow if they are connected by some sort of flexible membrane.

A nuts and bolts comment:
@Amaterasu21 This was a good question, IMO, but don't lose sleep over it because we can nearly always make a suitable choice about which way to go in analysing any situation. On a practical level, there's one advantage we have and that is the Thermal Energy of a situation tends to be a lot greater than the Mechanical Energy. Electric Motors can make a huge difference a situation , compared with Electric Heaters with the equivalent Power rating. It's only when we are talking about Heat Engines that the Mechanical Power out of a good heat engine is a good proportion of the Chemical / Heat Power input. (You need a massive temperature different for high efficiency.) A human can do a lot of useful mechanical work in carrying a load up a hill but warming that load up by rubbing it, will take ages.

etotheipi
Gold Member
2019 Award
Yes, and this is exactly why I would argue that it is incorrect to say that "temperature is a measure of the average kinetic energy of the molecules in a substance." Things get more complicated when more degrees of freedom are involved. I don't think that thinking of temperature as the motion of molecules is a good mental model; it leads to too many wrong conclusions and misunderstandings.

I am okay with the converse: the kinetic energy of the components of a substance increases with temperature. But one has to be careful to equate temperature with kinetic energy.
Ah okay, I understand now. Thank you for clarifying! Yes for the example I quoted we have for any oscillating ion ##\langle E_k \rangle = \langle U \rangle = \frac{3}{2}kT##, i.e. quadratic degrees of freedom are also associated with the potential energy, which we must worry about also

vanhees71