## Kinetic energy on a molecular level

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Aikanaro
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### Kinetic energy on a molecular level

So, what I'm going off of here is probably one of those fallacies that they teach you in school, simply because it "functions" for most of the problems you're likely to encounter. A couple of weird thoughts stem from this. If any point of this stream of consciousness is in error, feel free to cut it off there, please.

First of all, in school, we're always taught that the temperature of an object (or the thermal energy, or whatever you choose to call it) is basically the speed/energy of the molecules bouncing around that make up the object. This begets the question then, of a lone particle: Can a single atom of a noble gas, for example, loose in a vacuum, have a temperature? It has no other objects around itself to be bouncing relative TO, so does it effectively have no thermal energy? Is it defined by the speed of its electrons, etc., orbiting it? What about a lone neutron, then? Etc., etc..

I broached this question to a friend of mine, and his response is that the temperature is then defined by the velocity of the lone atom. This is a bit odd to me, because since velocity is relative, it means that the thermal energy and, thus, the temperature of an object is also relative, while I though that it was more an intrinsic quality of an object, regardless of where it was or how it was moving.

Further discussion left me with the impression that, ultimately, what we think of as temperature (and/or thermal energy) is really just kinetic energy, on the molecular level. Odd, but I can kind of see it, I guess. But then there's another level of oddness, if you go deeper.

If I think of two rubber balls in space, hurtling towards eachother, when they hit, a large amount of the energy is conserved as "standard" kinetic energy (i.e., the balls bounce away from eachother). Some of the energy is absorbed/degraded, however, into thermal energy. In other words, the balls bounce away, slightly slower than they were first going, but slightly warmer for it. I forget the precise terminology for it, but I know there's some quality along the lines of...elasticity?....that determines how much energy is conserved thusly. Rubber balls will bounce away nearly as fast, while something like foam or cement will pretty much just stop.

So what happens if we apply this on the molecular level? If two atoms are launched at eachother and collide (assuming they are such that they don't become a single molecule), is there perfect conservation of "kinetic" energy in that case? Do they bounce away at the same speed as they were initially going? And if not, what form does the new energy take instead of kinetic?

Yayy for trying to apply high-school physics after you've learned a lot of it is convenient lies....
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farnsworth
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### Re: Kinetic energy on a molecular level

Aikanaro wrote:Can a single atom of a noble gas, for example, loose in a vacuum, have a temperature?

I have absolutely no physics or thermodynamics education, but here's what I think: Assuming an infinite expanse of vacuum, the temperature would quite literally be infinitesimal.

The flying ice cube problem is tangentially related.

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### Re: Kinetic energy on a molecular level

A single molecule alone in its universe can't even have a velocity. One in an approximately empty universe can still have a meaningful kinetic energy, but it doesn't make much sense to talk about the temperature of one or a few molecules.

The usual thermodynamic temperature for gases that arises from statistical mechanics does depend on the mean velocity (more correctly, the kinetic energy) of the constituent molecules, but also on other degrees of freedom, such as how they are tumbling or how the atoms within the molecule are oscillating. And, as "statistical" implies, it really only works for large numbers of molecules. Most concepts in thermodynamics are hard to apply to small systems, actually.

The term you are looking for is elastic. Collisions that conserve kinetic energy are elastic, and ones that don't are inelastic. It's a little counter-intuitive, since you think of elastic substances as being the rubbery ones Collisions among molecules distribute the energy among all the degrees of freedom, such as internal oscillations, but that still gets added up in the temperature. Atoms don't have any extra degrees of freedom, so the collisions are elastic.

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### Re: Kinetic energy on a molecular level

Also, temperatures become weird when you have energy distribution that does not follow the Boltzman curve, you can even get negative temperature and all.
http://en.wikipedia.org/wiki/Negative_temperature

When two particles "collide", some energy can be spent as photons, especially with charged particles. So, no, collisions are not perfectly elastic.
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Aikanaro
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### Re: Kinetic energy on a molecular level

Ah, so THAT'S where the "lost" energy goes! Also, I never thought about, well, the molecules themselves acting like a larger object does with regards to its own molecules (the smaller components bouncing around within the system). Thanks!

Negative temperature still confuses me, though, inasmuch as, if it's something we can achieve, and it's defined as "infinite" temperature, I would think it would break things a little when you try and use it in a power source. That part of my thermo class always lost me a bit, but then, the professor didn't speak English very well....
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### Re: Kinetic energy on a molecular level

Aikanaro wrote:Negative temperature still confuses me, though, inasmuch as, if it's something we can achieve, and it's defined as "infinite" temperature, I would think it would break things a little when you try and use it in a power source. That part of my thermo class always lost me a bit, but then, the professor didn't speak English very well....

"Negative temperature" means that if you fit a Boltzmann distribution (which is the distribution of energies that a system in equilibrium will have) to the actual energy distribution, the best fit is a Boltzmann distribution with negative temperature. The kinetic energy is still positive (of course), and if you put it in contact with a really cold thing it will heat the cold thing up, so "negative temperature" is a very misleading name, and whoever coined it was probably a jerk.
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### Re: Kinetic energy on a molecular level

I'm a layman in physics as well, but I think I remember a Professor in Physics saying that at some point it is useless to talk about the "temperature" of particles, and you start talking about the kinetic energy or movement. I think it was this video.
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### Re: Kinetic energy on a molecular level

And if two molecules collide, you will transfer some energy into internal vibrational modes, and some into center of mass motion. You can model each molecule (super crudely) as a bunch of atoms connected by springs of appropriate k, and when the molecules collide, off those springs go.
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Aikanaro
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### Re: Kinetic energy on a molecular level

Yeah, I saw one of the GIFs someone posted, which helped me with the concept, though you can still strip it further down still, to individual atoms (or however far down you have to go), and there's still the question of what happens to the energy from impacts on (insert smallest known particle scale here).
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### Re: Kinetic energy on a molecular level

There starts to be another problem if you go down to far where "impacts" starts to be a bad word to use. For many sub atomic particles they have more complex interactions than bouncing around like tennis balls.
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mfb
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### Re: Kinetic energy on a molecular level

Charlie! wrote:
Aikanaro wrote:Negative temperature still confuses me, though, inasmuch as, if it's something we can achieve, and it's defined as "infinite" temperature, I would think it would break things a little when you try and use it in a power source. That part of my thermo class always lost me a bit, but then, the professor didn't speak English very well....

"Negative temperature" means that if you fit a Boltzmann distribution (which is the distribution of energies that a system in equilibrium will have) to the actual energy distribution, the best fit is a Boltzmann distribution with negative temperature. The kinetic energy is still positive (of course), and if you put it in contact with a really cold thing it will heat the cold thing up, so "negative temperature" is a very misleading name, and whoever coined it was probably a jerk.

If you introduce temperature via the more general concept of entropy, there are systems with negative temperature which increase their entropy if the energy gets lower. You don't need exotic systems far off an equilibrium, a fixed maximum energy is enough. These systems are hotter than every system with a positive temperature. So what?

Zamfir
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### Re: Kinetic energy on a molecular level

Aikanaro wrote:Yeah, I saw one of the GIFs someone posted, which helped me with the concept, though you can still strip it further down still, to individual atoms (or however far down you have to go), and there's still the question of what happens to the energy from impacts on (insert smallest known particle scale here).

Hmm, someone with more expertise than me should verify my story. But I think that the movement of individual atoms is roughly the smallest scale to which energy dissipates to become 'heat' in many of our day to day situations.

The smaller scales are governed by quantum effects, like the Pauli exclusion principle. The states of particles at these small levels change in discrete chunks, and the chunk sizes required to make changes to the parts of an atom are very large, compared to typical energy level of the degrees of freedom associated with our temperatures. So energy transfers to and from these states are very rare, compared to transfers between moving atoms.

That means the movement of atoms (in a monatomic gas, or within springy molecules and solids) are part of one big coupled system all the way up to our macroscopic degrees of freedom. If I drop a ball, I first put kinetic energy in a single degree of freedom of the ball, then couple that degree of freedom to the lower-level degrees when it hits the floor. Then that energy eventually dissipates to all the interconnected degrees of freedom of the individually moving atoms. But it mostly doesn't dissipate further down, because you need more concentrated energy levels to make changes at the lower scale.
Last edited by Zamfir on Fri Aug 12, 2011 3:36 pm UTC, edited 1 time in total.

doogly
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### Re: Kinetic energy on a molecular level

Maybe you want to get down to something called a KMS state. These are states of thermodynamic equilibrium within quantum field theory.
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### Re: Kinetic energy on a molecular level

Would that give very different results from a somewhat sophisticated version of the atoms-as-balls-on-springs model? I thought it was a pretty good assumption that the extra heat you put in ordinary matter gets distributed over the movements, vibrations and rotations in it.

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### Re: Kinetic energy on a molecular level

Oh yeah you would not want to use a QFT description for an atom. They are not very relativistic, no need for this sort of thing. Balls on springs are great.
Some people just ain't satisfied till they have some QFT though. It is pretty dope, after all.
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Charlie!
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### Re: Kinetic energy on a molecular level

mfb wrote:
Charlie! wrote:
Aikanaro wrote:Negative temperature still confuses me, though, inasmuch as, if it's something we can achieve, and it's defined as "infinite" temperature, I would think it would break things a little when you try and use it in a power source. That part of my thermo class always lost me a bit, but then, the professor didn't speak English very well....

"Negative temperature" means that if you fit a Boltzmann distribution (which is the distribution of energies that a system in equilibrium will have) to the actual energy distribution, the best fit is a Boltzmann distribution with negative temperature. The kinetic energy is still positive (of course), and if you put it in contact with a really cold thing it will heat the cold thing up, so "negative temperature" is a very misleading name, and whoever coined it was probably a jerk.

If you introduce temperature via the more general concept of entropy, there are systems with negative temperature which increase their entropy if the energy gets lower. You don't need exotic systems far off an equilibrium, a fixed maximum energy is enough. These systems are hotter than every system with a positive temperature. So what?

Temperature is a group of properties in thing-space, and one of those properties is an ordering on "what heats up what." And so if something has some but not all of those properties (e.g. entropy decreases as you add energy, but it will heat up cold things if you put them in contact), calling it a "temperature" is misleading.
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### Re: Kinetic energy on a molecular level

If you pick one property to define a concept, then the concept will probably "behave" strangely with respect to the other possible properties. I don't see the problem unless you're fine with ambiguous concepts in science, or you somehow have a better definition of "temperature".

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### Re: Kinetic energy on a molecular level

Meh, I guess there's not much else you can call the term in the equation dS = dQ/T aside from a "negative temperature." Upon further thinking, my problem is really with assigning "negative temperature" as a context-independent property of the system, e.g. "this system has negative temperature."
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### Re: Kinetic energy on a molecular level

Zamfir wrote:If I drop a ball, I first put kinetic energy in a single degree of freedom of the ball, then couple that degree of freedom to the lower-level degrees when it hits the floor. Then that energy eventually dissipates to all the interconnected degrees of freedom of the individually moving atoms.
Nice example, Zamfir.

While the ball is falling, the kinetic energy of all the atoms in the ball is increasing, but the temperature of the ball doesn't increase purely by the fact that it's falling (neglecting friction). True, temperature is a measure of molecular kinetic energy, but those molecular motions responsible for what we call temperature are random: the kinetic energies of the atoms or molecules are randomly spread over a range of values, conforming to the Boltzmann distribution mentioned earlier in the thread, and the directions the atoms or molecules are moving in at any given instant are randomized as well.

Consider: while the ball is falling, the kinetic energies of the atoms relative to the centre of mass of the ball don't change. But when the ball hits the floor, things get a bit scrambled, causing vibrations to propagate through the ball in all directions. If the collision is totally inelastic, all of the nice coherent motion of the ball gets totally randomized, altering the atoms' kinetic energies relative to the centre of mass of the ball significantly.

Aikanaro
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### Re: Kinetic energy on a molecular level

PM 2Ring wrote:
Zamfir wrote:If I drop a ball, I first put kinetic energy in a single degree of freedom of the ball, then couple that degree of freedom to the lower-level degrees when it hits the floor. Then that energy eventually dissipates to all the interconnected degrees of freedom of the individually moving atoms.
Nice example, Zamfir.

While the ball is falling, the kinetic energy of all the atoms in the ball is increasing, but the temperature of the ball doesn't increase purely by the fact that it's falling (neglecting friction). True, temperature is a measure of molecular kinetic energy, but those molecular motions responsible for what we call temperature are random: the kinetic energies of the atoms or molecules are randomly spread over a range of values, conforming to the Boltzmann distribution mentioned earlier in the thread, and the directions the atoms or molecules are moving in at any given instant are randomized as well.

Consider: while the ball is falling, the kinetic energies of the atoms relative to the centre of mass of the ball don't change. But when the ball hits the floor, things get a bit scrambled, causing vibrations to propagate through the ball in all directions. If the collision is totally inelastic, all of the nice coherent motion of the ball gets totally randomized, altering the atoms' kinetic energies relative to the centre of mass of the ball significantly.

Hmm, but don't things like relative motion, etc., depend on where you define the borders of your closed system? For example, what if we define the "object" in question to be both the ball, AND the earth, in space? The movement of the molecules/atoms of the ball is different relative to the center of mass of the newly defined system. Furthermore, that means that when you stand on the earth and throw a ball, you're logically increasing the "temperature" of the system as a whole, not just by the friction, impact, waste heat, etc., but by the simple fact of molecules being put into motion relative to the system.

I think I'm missing a step or two in the definition of temperature again here, since I'm reading this right after getting off work, before bed. Sorry.....
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Zamfir
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### Re: Kinetic energy on a molecular level

@Aikanaro, to some extent you're right. Thermodynamic properties like temperature and entropy are defined as a properties of a system (or part of a system), and their values depend somewhat on the way we describe the system.

A way to think about it is through 'microstates' and 'macrostates'. A microstate is one specific situation of your system, based on all the variables that are part of the system. So the location and velocity of every atom, for example. A macrostate is a group of such states that we consider "the same".

In case of a ball: we can decide that al the internal vibrations of the ball don't matter for the macrostates, as long as their total energy is the same, and as long as the centre of mass of the ball is (very near) the same location and has the same speed. But moving the centre of mass of the ball to a dfferent location changes the macrostate, a higher the velocity of the centre of mass is a different macrostate, and a higher total energy in the vibrations is also a different macrostate.

That's a bit arbitrary. If you're playing golf, you do not care about the exact energy in the vibrations within the ball at all, so only the velocity and location of the centre of mass count to determine which macrostate it is in. If the ball is melting, you have to look at the of bonds between molecules as well, and say that a macrostate only includes all microstates with the same amount of solid bonds, and that a ball with more broken bonds is different even if it has the same total vibrational energy and location and velocity.

In practice, it is often reasonably clear how to define useful macrostates. If you want to assign a statistical concept like temperature, you need a subsystem that meets enough criteria to allow that. The exact details of those criteria are beyond my paygrade, but it roughly means that you need lots of variables in the subsystem (to allow averaging), and so much coupling between them that the microstate at one moment of time can be seen as uncorrelated to the microstate at an earlier time, where the time step in between is somewhat up to you.

The velocity of the centre of mass alone is not a subsystem with enough variables. But if you put the velocity of the ball together with all internal vibration modes in one subsystem, there is not enough coupling to assign a temperature to the subsystem as a whole. On our timescales of interest, the velocity of the ball is highly correlated to its earlier velocity, and hardly at all to the other variables. So it makes sense to separate the system into a subsystem that includes the velocity (this subsystem has no meaningful temperature) and a subsystem with all the internal vibrations (which can have a meaningful single temperature).

Now for fun, imagine we put lots of these moving balls in an enormous box. The balls are perfectly elastic, so the balls form a supersized model of a perfect gas. Now it is possible to have a subsystem with all the centres of mass of all the balls in it (isolated from the internal modes of the balls), and this subsystem has a temparature when considered on long enough time scales , just like a gas. The temperature will be a ridiculously high value. If we zoom to a specific ball for a short time (without collisions), then the time scale is not long enough to talk about temperatures.

If we make the balls slightly inelastic, there is now coupling between the subsystem of the centres of mass and all the subsystems of the internal vibrations of the balls. Energy will now flow from the "high temperature" subsystem of centres of mass, to the "low temperature" subsystems of the internal vibrations. Eventually so much energy will flow that the centres of mass have no noticable movement left, since the average energy per degree of freedom here is now of the same order (not necessarily the same) as those of each internal vibration mode.

EDIT: holy shit I was bored. Still a useful story, I hope