# Why wouldn't the temperatures be different in this scenario

1. Feb 9, 2016

### PaperProphet

Imagine a vertical tube filled with a gas. As the gas molecules move to the bottom with gravity, they gain energy and speed. As they randomly move upward against gravity, the molecules lose energy and speed. It seems like that would create a difference in temperature but obviously if the top of the tube was actually cooler than the bottom, it means a spontaneous, potentially useful, difference in energy was created so that can't be.

Why wouldn't the molecules in the tube create a difference in temperature between the top and the bottom of the tube?

2. Feb 9, 2016

3. Feb 9, 2016

### PaperProphet

Thanks. Explanation?

4. Feb 9, 2016

### Staff: Mentor

Can you apply some thought/logic to Bystander's suggestion? We try to make people exercise their brains here...

5. Feb 10, 2016

### PaperProphet

I was actually hoping Bystander could expand upon his own suggestion.

I have a good physics background but I'll admit I don't have the answer to my question. I'm hoping someone can provide a cogent, clear, understandable explanation that even someone as dumb as myself can understand. Logically the molecules can't create a difference in temperature otherwise that would violate the laws of thermodynamics...but I don't know why they don't.

6. Feb 10, 2016

### Bystander

Consider all the conditions for equilibrium.

7. Feb 11, 2016

### PaperProphet

Bystander, I assume you're trying to be helpful but that you don't know the answer either.

No pretenses here--I don't know the answer despite pondering it in depth. This isn't a homework question where I'm looking for hints--it's something I randomly thought about years ago and just can't figure out. I've given up and am hoping someone else can give a clear, unambiguous explanation. If you have (or anyone else has) an explanation, please share. Understandably, this isn't an easy question and you can see nobody else has come up with an answer either.

8. Feb 11, 2016

### A.T.

Why do you assume their total energy is constant?

9. Feb 11, 2016

### PaperProphet

I'm not sure I follow.

It just seems that molecules going upward inside the tube would lose kinetic energy due to gravity and be 'colder' at the top of the tube while gaining kinetic energy due to gravity to be 'warmer' at the bottom of the tube. It would seem, in my mind, that molecules would continuously transfer heat energy from the top of the tube to the bottom of the tube. Obviously that can't be the case otherwise that would violate the second law of thermodynamics by spontaneously creating a heat source and a heat sink. It's a paradox that I can't solve and I'm hoping someone has a good explanation.

10. Feb 11, 2016

### PaperProphet

A little background, I asked this question about the gas-filled tube maybe fifteen years ago on a message board and nobody knew the answer then either. It doesn't appear to be an easy question.

11. Feb 11, 2016

### A.T.

That is not a physical argument. Check the assumptions you are making.

12. Feb 11, 2016

### PaperProphet

Thank you for the hints, A.T.

Can I ask if you know the answer as to why there isn't a difference in temperature?

13. Feb 11, 2016

### Staff: Mentor

This is just a reminder to everyone to keep the discussion focused on the original question, not on who is or isn't being respectful to whom. I have deleted some off topic posts accordingly.

14. Feb 11, 2016

### Staff: Mentor

I suspect he does, but that's not really relevant. You say you have a "good physics background", but that's a general statement. Do you know the equations governing the equilibrium of a gas in a vertical tube in a gravitational field? There are at least two equations that are relevant, involving the temperature, density, and pressure. Have you studied any material of this type?

15. Feb 11, 2016

### PaperProphet

Thanks Peter. The reason I asked is because I would like to know whether the person giving me hints has the answer or not. I don't mind discussing the topic since I'm obviously very interested in the topic...but I just want to make sure that someone trying to 'mentor' me has more answers than I do before I re-explore the angles I've already explored.

I certainly understand that gas pressure increases under gravity but I don't know any formulae which indicate if or how temperature would spontaneously change near the top and bottom of a gas-filled tube. In fact, the second law of thermodynamics dictates that it shouldn't. Still, molecules are affected by gravity and a molecule moving downward with gravity should gain additional kinetic energy and be 'hotter'.

Do you have any thoughts on whether or not molecules at the top of a gas filled tube would be cooler than molecules near the bottom?

16. Feb 11, 2016

### Staff: Mentor

This isn't a matter of "thoughts", it's a matter of physics, and the answer can be worked out from physical principles. That is what the other posters in this thread have been trying to get you to do.

Here are the key physical principles involved. We are assuming that the system is in equilibrium, i.e., nothing is changing with time.

(1) There must be no net heat flow anywhere in the gas. That implies, as you have assumed, that there cannot be any variation of temperature anywhere in the tube; the entire tube must be at the same temperature. The question then is, how does this relate to the effects of gravity? So the next thing we need to look at is, what are the effects of gravity?

(2) The effect of gravity is that the gas inside the tube must be in hydrostatic equilibrium. That means, if we consider a small parcel of gas at a given height $h$ in the tube, with a small thickness $dh$, there must be a pressure difference from the bottom to the top of the parcel, and this pressure difference must be just sufficient to balance the weight of the gas above the small parcel we are looking at. If we work out what this means, taking the limit as the thickness $dh$ goes to zero, we get the equation

$$\frac{dP}{dh} = - \rho g$$

where $dp / dh$ is the rate of change of pressure with height, $\rho$ is the density of the gas, and $g$ is the acceleration due to gravity (we assume this is constant everywhere in the tube). The fact that $dp / dh$ is negative means that pressure decreases with height.

(3) Also, it is important to realize that the pressure, density, and temperature of the gas are not independent; they are related by an equation of state. For this case, we can assume that the equation of state is that of an ideal gas, which is

$$P = \rho R T$$

where $P$ is the pressure, $\rho$ is the density, $R$ is a physical constant called the "gas constant", which will depend on the chemical constitution of the gas, and $T$ is the temperature.

If we assume that $T$ is constant, the above equations tell you that $P$ and $\rho$ will vary with height in the tube. Can you see how they will vary? And can you see how that explains how the temperature can be constant in the presence of gravity?

17. Feb 11, 2016

### PaperProphet

Peter, I can see you believe you've answered my question. I can see you've shown in detail how pressure changes with height given a constant temperature. Keep in mind that your equations don't exclude the possibility that temperature can vary from the top to the bottom. In fact, you could put a heater on the bottom of the hypothetical tube and a similar set of equations will solve for the change in pressure. In other words, you can assume T varies and nothing would break...other than the equation which assumes T is static.

I know I'm not entitled to an answer and I'm not trying to be mean, but I'm specifically asking why the temperature *wouldn't* change due to the effect of gravity pulling on the molecules.

18. Feb 11, 2016

### Bystander

There is a non-zero thermal conductivity; there is zero heat flow at equilibrium. Therefore, zero temperature gradient, QED.

19. Feb 11, 2016

### A.T.

Why?

20. Feb 11, 2016

### Staff: Mentor

Then you see incorrectly, because I didn't quite answer your question. I only stated the key physical principles that lead to the answer. I specifically left out part of the reasoning, in the hope that you would work it out for yourself.

Obviously if there is a heater present, then gravity is not the only effect. So saying that the equations allow $T$ to vary if a heater is present does not mean $T$ will vary if a heater is not present.

One of the assumptions I made actually requires that there is no heater. (One of the things I left out was not explicitly pointing out this fact.) Can you see which one?

You should be able to work that out for yourself with the information you've already been given, just as you worked out the fact that the information I gave you was not quite a complete answer.