Why wouldn't the temperatures be different in this scenario

[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?

 

[Bystander]

Think about densities.

 

[PaperProphet]

Think about densities.

Thanks. Explanation?

 

[russ_watters]

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

 

[PaperProphet]

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

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.

 

[Bystander]

Consider all the conditions for equilibrium.

 

[PaperProphet]

Consider all the conditions for equilibrium.

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.

 

[A.T.]

As they randomly move upward against gravity, the molecules lose energy and speed.

Why do you assume their total energy is constant?

 

[PaperProphet]

Why do you assume their total energy is constant?

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.

 

[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.

 

[A.T.]

It just seems that...

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

 

[PaperProphet]

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

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?

 

[PeterDonis]

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.

 

[PeterDonis]

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

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?

 

[PaperProphet]

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?

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?

 

[PeterDonis]

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?

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?

 

[PaperProphet]

If we assume that TTT is constant, the above equations tell you that PPP and ρρ\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?

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.

 

[Bystander]

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

 

[A.T.]

molecule moving downward with gravity should gain additional kinetic energy

Why?

 

[PeterDonis]

I can see you believe you've answered my question.

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.

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.

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?

I'm specifically asking why the temperature *wouldn't* change due to the effect of gravity pulling on the molecules.

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.

 

[jartsa]

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?

The molecules in the tube create a difference in temperature between the top and the bottom of the tube, by falling down and heating up. That seems quite clear to me, particularly if I think about a gas with low density.

When there is difference in temperature between the top and the bottom of the tube we can conduct the heat back up, I don't see a problem there. The bottom of the tube loses the extra heat, and the the top of the tube also gets cooler when we conduct heat away from the bottom. The tube cools when we conduct heat away from it, when it has cooled enough we can not conduct heat away from it anymore.

 

[PaperProphet]

Peter, PV=nRT (and variations) don't constrain temperature. I won't be able to work out any answer based on what you've given me. If you complete your line of reasoning with the change in pressure, I'm confident you'll realize the mistake in your reasoning. I know you genuinely believe you've given me tools to get an answer and I thank you for trying.

 

[PaperProphet]

The molecules in the tube create a difference in temperature between the top and the bottom of the tube, by falling down and heating up. That seems quite clear to me, particularly if I think about a gas with low density.

When there is difference in temperature between the top and the bottom of the tube we can conduct the heat back up, I don't see a problem there. The bottom of the tube loses the extra heat, and the the top of the tube also gets cooler when we conduct heat away from the bottom. The tube cools when we conduct heat away from it, when it has cooled enough we can not conduct heat away from it anymore.

Thanks, Jartsa!

You're the first person here who was able to see (or at least acknowledge) the effect of gravity on the molecules. However keep in mind that if the effect was really there, there would be a potentially useful heat source and heat sink implying free energy. Obviously that can't be the case. It seems like a paradox.

 

[PeterDonis]

The bottom of the tube loses the extra heat, and the the top of the tube also gets cooler when we conduct heat away from the bottom.

In other words, in your model, there is continuous heat flow through the tube, and heat exchange between the tube and the outside environment. My understanding of the OP was that there should be no heat flow or heat exchange--that the tube is to be considered as an isolated system in thermal equilibrium with no heat exchange with the outside.

 

[PeterDonis]

PV=nRT (and variations) don't constrain temperature.

Not by themselves, no. But that wasn't the only assumption I made. Look at item #1 in post #16. What does it say? And does what it says match the scenario you were envisioning? (Also see my previous post in response to jartsa.)

 

[PeterDonis]

You're the first person here who was able to see (or at least acknowledge) the effect of gravity on the molecules.

You are mistaken. Everyone in this thread has acknowledged the effect of gravity on the molecules. The equation for hydrostatic equilibrium that I gave you explicitly includes the effect of gravity on the molecules. If there were no gravity, there would be no pressure gradient.

 

[PaperProphet]

You are mistaken. Everyone in this thread has acknowledged the effect of gravity on the molecules. The equation for hydrostatic equilibrium that I gave you explicitly includes the effect of gravity on the molecules. If there were no gravity, there would be no pressure gradient.

Yes, I can see how you pointed out the change in pressure vs. height. Sure, an isolated system is fine. I'm just saying that there is no constraint on temperature with PV=nRT. There's nothing to prevent a molecule from working its way to the bottom and picking up kinetic energy or vise versa.

Would it be easier for you to think of a single molecule in a tube?? The molecule can freely move to the top, losing kinetic energy, and on the next leg the molecule can move all the way to the bottom of the tube, having kinetic energy added to it. With a molecule in flight, it will lose kinetic energy as it moves against gravity and gain kinetic energy as it moves with gravity.

 

[PeterDonis]

an isolated system is fine.

Ok, good.

I'm just saying that there is no constraint on temperature with PV=nRT.

But that isn't the only equation. You keep on ignoring the other constraints.

There's nothing to prevent a molecule from working its way to the bottom and picking up kinetic energy or vise versa.

Really? Have you actually thought this through? Including what happens when that molecule collides with other molecules?

Would it be easier for you to think of a single molecule in a tube?

A single molecule is not a gas. Can you see what the key difference is between the two? (I implicitly told you what it was just above.) And how does that difference affect your reasoning about what happens to a molecule as it falls or rises within the tube?

 

[PeterDonis]

The molecule can freely move to the top, losing kinetic energy, and on the next leg the molecule can move all the way to the bottom of the tube, having kinetic energy added to it.

Even this example, as unrelated as it is, can still be instructive. What happens to the molecule at the bottom and top of the tube? And how does that square with your assumption that the tube is isolated, exchanging nothing with its environment? (For example, could I extract work from the tube with one molecule in it?)

 

[klimatos]

Under conditions of equilibrium, there are just as many molecules moving upward as downward in any measurable volume of gas; i. e., no net fluid flow. Moreover, statistical mechanics tells us that the speed distribution function is the same for both arms of the Z-axis, even though there are differences in molecular number density throughout the tube due to gravity. Every upward molecule in a given volume that starts with speed v will lose speed Δv in going distance d. Every downward molecule in that same volume that starts with speed v will gain that same speed Δv in going that same distance. Since gas temperatures measure only the kinetic energies of translation of the molecules, no change in the gas temperature will occur due to gravity-induced acceleration or deceleration.