I Thought Experiment on the Second Law of Thermodynamics

1. Oct 6, 2016

MikeW

Hi,

I have a thought experiment which seems inconsistent with the 2nd Law of Thermodynamics, so there is probably a flaw in the idea somewhere.

Imagine a single molecule of a heavy gas (such as SF6) in a pipe 2.5 metres high and 100mm diametre. The pipe is upright and under the normal influence of earth gravity (9.8N per Kg). The pipe is thermally insulated. The SF6 molecule has a room temperature (293K) velocity of 134 m/s and therefore translational kinetic energy of 134^2 * 0.5 * (146/NA/1000). If the molecule is on a path where it bounces off the bottom of the pipe and is heading for the top of of the pipe, without touching the walls, the translational kinetic energy will be reduced by gravity by 9.8 * 2.5m * (146/NA/1000) at the top. The molecule then collides with the pipe top, speeds up as it takes some energy from collision with atoms at the pipe top (on average). Then gains some speed on the way down due to gravity, then gives the extra energy (on average) to the pipe bottom atoms. My rough calculation is that this is 0.3K drop in temperature at the top of the pipe, from the bottom of the pipe, which would be measured with probe thermometers.

The photons emitted by electron energy state changes during collisions may or may not be significant.

Energy is conserved, but entropy is reduced. If there is a heat exchange (at top and bottom of pipe) connected to a convection loop, work can be extracted.

So:
1) What is the general flaw in the idea?
2) Will this be valid when bouncing off the walls of the pipe is part of the experiment?
3) Will this be valid when the pipe is full of SF6, and the macro effect of molecules colliding all the way up is included? (That is, will the speed of molecules, on average, reduce towards the top)

2. Oct 6, 2016

andrewkirk

Why do you expect the collision at the top to transfer energy from container to molecule and the opposite to happen at the bottom?

3. Oct 6, 2016

MikeW

Hi Andrew, thanks for your question.

I assume when molecules/atoms collide, there is a random redistribution of energy. I assume that the system of the collision is the sum of both energies, and on average each gets half the sum. If on average the gas molecule is a little slower (at the top), then half the sum (on average) is greater than the current translational kinetic energy of the gas molecule. And at the bottom the oppersite (the gas molecule is a little faster).

4. Oct 6, 2016

andrewkirk

It doesn't work that way. In an elastic collision both momentum and kinetic energy are conserved. If one molecule is free and the other is bound as part of a macroscopic solid, that means that almost all the energy will be retained by the free molecule, with an incredibly small bit of energy being transferred to the solid in which the other one is bound (which in the end is probably the entire Earth). That energy transfer will be so tiny that it cannot have any relevance given the unavoidable imperfections of the experimental setup, like the less than perfect heat-shielding of the side.

5. Oct 6, 2016

MikeW

You are right, my two molecule surface system is a poor model, if all the atoms in container top/bottom are being considered. I assume you agree that a cold gas and warm container would affect each other; and that after a time the gas and container would reach the same temperature? That is: even with elastic collision, the gas is gaining some kinetic energy, and the container losing some. Re: "perfect heat-shielding", in the end I do not think this would be a problem, probably desired.

Last edited: Oct 6, 2016
6. Oct 6, 2016

Staff: Mentor

An atom moving from the top of the pipe to the bottom does gain a very small amount of energy while an atom moving upwards loses the same amount. However, that doesn't lead to a temperature difference between the top and the bottom; at equilibrium the temperature is the same throughout while the pressure is greater (very slightly, unless we're working with a very tall pipe) at the bottom.

You can't extract useful work from that configuration, just as you can't extract work from the atmospheric pressure difference between sea level and a mountaintop.

7. Oct 6, 2016

MikeW

Hi Nugatory, Thank you for your comment.

My assumption was that temperature of a gas molecule was the combination of translational kinetic energy, vibration kinetic energy and rotational kinetic energy. If the translational kinetic energy has reduced due to the deceleration of gravity and the other kinetic energies have not increase, then the temperature of the gas molecule has dropped at the top (if only a little, may around 0.3K for SF6) before it touches the top. The thought experiment starts with a single molecule. With your mountain example the temperature has dropped along with the pressure.

8. Oct 6, 2016

Staff: Mentor

That's because, unlike your hypothetical pipe, the earth's atmosphere is not thermally insulated; the upper layers are radiating heat off into space. Put the entire mountain underneath an insulating dome, eliminate the effects of stray air currents, and that temperature gradient will go away.

9. Oct 6, 2016

jartsa

Trillion times a second all kinetic energy is removed from a pipe molecule in a similar process as some kinetic energy is removed from the uphill climbing SF6 molecule. If you take this into account, I believe the problem will be solved.

I'm being cryptic so that you can think about it.

10. Oct 7, 2016

Khashishi

The flaw is that you assume the molecule can just travel up and down the pipe without bumping into anything. There is a pressure gradient in the pipe. This exerts a force on the molecule as it moves up and down the pipe which counteracts gravity.

11. Oct 7, 2016

MikeW

I agree that the earth atmosphere has many factors applying to it, which is why I was concentrating on the sealed pipe example, to establish the flaws in that idea with out having to to take too many things into account.

12. Oct 7, 2016

MikeW

Hi Khashishi, thanks for the comment. If you look at what I wrote, the thought experiment starts with a single molecule of a heavy gas that happens to be travelling straight up the pipe. So in that scope there is nothing to bump into and no pressure gradient. I was hoping the flaws in that could be isolated before moving onto the macro scope (where bumping and pressure gradients will be applicable).

13. Oct 7, 2016

MikeW

Hi Jartsa, thank you for your comment.

I assume that you are meaning that the collision pipe molecule has transferred excess kinetic energy to its neighbouring pipe molecules via atomic bond vibration? Sorry if I am a little slow on this, does this mean you think there will be some kinetic energy/temperature reduction on the upward route, but it will not affect the top surface because it will dissipate very quickly?

14. Oct 7, 2016

MikeW

I find this a really interesting idea. To put this in the context of my thought experiment, there is only one molecule of nitrogen in the entire earth atmosphere and no solar radiation or solar wind. Would you expect the molecule to reach escape velocity? or would you expected it to reach a maximum altitude where it would have no velocity (no kinetic energy/temperature) and then to fall back to earth?

15. Oct 7, 2016

Staff: Mentor

It depends on the initial kinetic energy (and hence speed) of the molecule. If its initial speed is greater than escape velocity the molecule will escape and if it is less the molecule will fall back, just like a cannonball.

This might be a good time to point out that the concept of temperature only applies to a large number of particles; it's meaningless to talk of the temperature of a single molecule. Given a volume of gas at a given temperature I can calculate an average kinetic energy of the particles in the gas, but it doesn't work the other way. There's no temperature associated with a single particle with a given kinetic energy.

16. Oct 7, 2016

Staff: Mentor

The kinetic energy of the single molecule is higher when it is at the bottom of the pipe than when it is at the top. We can start with one molecule at rest at the top, let it fall to the bottom and collide inelastically with the bottom and some of that energy will indeed be transferred to the bottom of the pipe, increasing its temperature. So far, so good. However (and this is likely the flaw in your thinking), at that point the system is not in equilibrium because the bottom of the pipe is hotter than the top. If we wait a while the temperature difference between top and bottom will go away as the warmer bottom of the pipe transfers heat to the cooler top of the pipe.

Now we can move onto the multi-molecule case. We imagine starting with the molecules uniformly distributed throughout the pipe so that the density is uniform and there is no pressure gradient; achieving this state is a bit tricky but it can be done in principle. Because of the gravitational field, this is not a stable configuration. The molecules are going to move downwards under the influence of gravity, losing gravitational potential energy and gaining kinetic energy as they do, and eventually the system will reach a stable configuration with a slight pressure gradient from top to bottom and a net loss of potential energy (the center of mass of the system is lower because the density is no longer uniform).

The net downwards movement of the molecules will heat the bottom of the pipe, just as in the single molecules case. But also as in the single molecule case, the system is not in thermal equilibrium until the temperature equalizes throughout the entire volume. When it does, you end up with gas and pipe at a constant temperature slightly higher (because we did turn some potential energy into kinetic energy and then heat) than when we started.

(There's a simpler argument: You have a closed and thermally isolated system with neither heat sources nor heat sinks. Such a system cannot have a temperature gradient at equilibrium).

17. Oct 7, 2016

jartsa

Not quite. I mean the collisions are random.

Let's consider your scenario, but without gravity: It's quite simple: Gas molecule bounces around randomly, moving sometimes slowly, sometimes fast.

Now with gravity: Same as above, plus a downwards acceleration of the gas molecule.

And taking into account that randomness will make everything sensible. I promise.

So let me try it:

Let's say the gas molecule loses energy at the floor, then it loses energy at the ceiling, then it again loses energy at the floor, then it can't reach the ceiling. This is possible, so it happens sometimes.

But this is impossible: The gas molecule loses energy at the floor, then it loses energy at the ceiling, then it can't reach the floor.

So, if a gas molecule has low kinetic energy, it will most likely touch the floor next.

Last edited: Oct 8, 2016
18. Oct 8, 2016

MikeW

Fair comment; I will use 'kinetic energy' for the single molecule case, instead of temperature. I do not think it changes the concept of the single molecule thought experiment though.

19. Oct 8, 2016

MikeW

I think it would gain kinetic energy at the ceiling, do you think it would lose kinetic energy?

Last edited: Oct 8, 2016
20. Oct 8, 2016

MikeW

I assume the pipe walls are the means for the warmer bottom to transfer to the top of the pipe, is there an another means? The thought experiment does not require pipe walls, the pipe was used to bring it back to a real world experiment. If you look at it in the single molecule of nitrogen/planet example and having the top at such an altitude that the molecule has lost half its vertical velocity (relative to the planet) by the time the top surface is reached (would it on average, gain an increase in kinetic energy as it bounced off the top surface?), this is the equivalent of the pipe without walls.

I agree this would happen at a given time rate in the pipe; but meanwhile the single molecule is bouncing up and down repeating the process of increasing kinetic energy at the bottom and reducing kinetic energy at the top. I think the rate of temperature transference in the pipe wall would affect the ability to observe something, rather than there would not be a kinetic energy difference.

Sorry for staying on just the single molecule case, but I would like to find the flaw in that case, as I have a thought process for scaling it up to the macro, that would be redundant if there is a flaw in the single molecule assumptions.