Why wouldn't the temperatures be different in this scenario

[PeterDonis]

any initial temperature gradient will be stable if it is equal to or less than the dry lapse rate

Stable against convection, yes. Not stable against conduction. Conductive heat transfer will occur if there is any temperature gradient at all.

It is true that, because the thermal conductivity of air is very low, conduction is a very slow process, so if you take a vertical tube of gas and start it off with a temperature gradient less than the dry lapse rate, it will take a long time, compared to convective time scales, for the temperature in the tube to equilibrate. But it will equilibrate; the time for the thermal gradient to be equalized is not infinite.

 

[DrStupid]

No. Hydrostatic equilibrium exists even in the absence of convection, i.e., even in the absence of adiabatic expansion.

OK, if you consider the existence of the hydrostatic equilibrium, why do you think that gas in a close tube can't expand adiabatically?

 

[PeterDonis]

if you consider the existence of the hydrostatic equilibrium, why do you think that gas in a close tube can't expand adiabatically?

Because the volume of the gas is constrained by the tube. In the atmosphere, the volume of the air is unconstrained. More precisely, the volume of a given parcel of air in the atmosphere is only constrained by the weight of the air above it. In a tube of gas, the volume is also constrained by the walls of the tube.

 

[spareine]

Stable against convection, yes. Not stable against conduction. Conductive heat transfer will occur if there is any temperature gradient at all.

It is true that, because the thermal conductivity of air is very low, conduction is a very slow process, so if you take a vertical tube of gas and start it off with a temperature gradient less than the dry lapse rate, it will take a long time, compared to convective time scales, for the temperature in the tube to equilibrate. But it will equilibrate; the time for the thermal gradient to be equalized is not infinite.

Exactly, nonconvecting air is an extremely bad conductor of heat. Air bubbles are the isolating ingredient of most insulation materials. I agree that your opinion that air still conducts some heat is very helpful!

 

[DrStupid]

Because the volume of the gas is constrained by the tube. In the atmosphere, the volume of the air is unconstrained. More precisely, the volume of a given parcel of air in the atmosphere is only constrained by the weight of the air above it. In a tube of gas, the volume is also constrained by the walls of the tube.

That does not explain why the gas cannot expand. Due to the hydrostatic equilibrium there is a vertical pressure gradient. In the result convection leads to a change of pressure. What prevents the raising gas from adiabatic expansion when the pressure decreases?

 

[PeterDonis]

What prevents the raising gas from adiabatic expansion when the pressure decreases?

If the rising gas is expanding, some other gas in the tube must be contracting, because the total volume of gas in the tube is constrained to be constant. In adiabatic expansion, such as in the atmosphere, the expansion of a rising parcel of air does not force any other parcel of air to contract, because the total volume of air in the atmosphere is not constrained.

 

[DrStupid]

If the rising gas is expanding, some other gas in the tube must be contracting, because the total volume of gas in the tube is constrained to be constant.

Yes, of course. But that does not mean that the gas cannot expand.

In adiabatic expansion, such as in the atmosphere, the expansion of a rising parcel of air does not force any other parcel of air to contract, because the total volume of air in the atmosphere is not constrained.

Raising air doesn't leave a vacuum behind. It is replaced by other air which is replaced itself and so on. In the result the total volume is almost not affected by convection. For every volume which raises and expands there is a similar volume which descends and contracts.

 

[OrangeDog]

People are being overly pretentious in this thread for some reason. The answer is true that the kinetic energy should theoretically increase between the top and bottom of the tube. Maybe if the tube is really, really long, maybe the size of a sky scraper, we would see a difference. However, there are a few caveats:

1: When speaking of molecules and thermodynamic quantities we speak of averages over ensembles rather than of an individual molecule. So, say your tube is 1 meter wide by 100 meters high, yeah you could calculate the energy lost due to potential and gain in kinetics, temperature, etc is xxx, but do you know how many molecules are in a unit 1 meter disk? You can calculate this and estimate the energy gained per molecule and youll find it is quite small.

2: Molecules collide with each other. These collisions are another way to transfer energy. You can calculate approximately how many collisions per second via kinetic theory. You'll see that in addition to potential and kinetic, we have vibrational, rotational, electrical, magnetic, etc modes of energy to consider. We can't account for this energy distribution exactly except for special cases.

So, the short answer is that yes, the measured kinetic energy should theoretically be higher, but ON AVERAGE, from an ensemble perspective the change in energies is negligible.​

 

[PeterDonis]

The answer is true that the kinetic energy should theoretically increase between the top and bottom of the tube.

The kinetic energy density will be higher at the bottom than at the top, yes. But kinetic energy density is not the same thing as temperature. (If you go back and look at my post #41, you will see both of these points discussed.) The OP's question was about temperature.

but do you know how many molecules are in a unit 1 meter disk?

By the density. If you know the type of molecules (i.e., the chemical constitution of the gas), and you know their mass density, then you know how many molecules there are per unit volume.

Molecules collide with each other. These collisions are another way to transfer energy.

Yes, and this his how heat transfer via conduction occurs in a gas confined in a tube which has a temperature gradient. This will happen whether or not the tube is in a gravity field.

 

[PeterDonis]

the total volume is almost not affected by convection.

Almost? Or exactly? Only "exactly" will do for the case of a confined tube. But in the atmosphere, it is at best "almost". That makes a difference.

 

[DrStupid]

Almost? Or exactly?

In the tube? Of course.
In the atmosphere? Of course not.

What's your point?

 

[Khashishi]

It's amazing that nobody actually answered the question yet. The short answer is that the pressure gradient cancels out the gravity resulting in no average acceleration of a parcel of gas.

The OP is thinking that gravity is accelerating particles downward so the kinetic energy of particles should be larger at the bottom than the top. This would be true for non-interacting particles, but for a gas with pressure, the pressure gradient is accelerating particles upward exactly canceling the downward acceleration by gravity.

Temperature and pressure only exist as average statistical quantities, but if you prefer to zoom in and look at the molecules, you will see that there are more molecules hitting your molecule from below than from above, so the molecules aren't (on average) accelerating downward under gravity. And spoiler alert: the density and pressure decrease exponentially as you go up,
$P = P_0 e^{-mgh/kT}$
as given by Boltzmann distribution or by Bernoulli's equation.

 

[jartsa]

The OP is thinking that gravity is accelerating particles downward so the kinetic energy of particles should be larger at the bottom than the top. This would be true for non-interacting particles, but for a gas with pressure, the pressure gradient is accelerating particles upward exactly canceling the downward acceleration by gravity.

I kind of agree, but the same laws of thermodynamics should apply to gas of interacting particles and gas of non-interacting particles.

Dense gas:
There are random temperature fluctuations in the gas, and warm parcels of gas tend to float upwards while cool parcels of gas tend to sink downwards. This effect cancels the effect of randomly moving parcels of gas heating up when moving downwards and cooling down when moving upwards. And there's an uniform temperature in the tube.

Non-dense gas, just a few molecules in a tube:
A molecule that happens to lose most of its speed when colliding with the upper part of the tube starts to move downwards because of gravity, while a particle that loses most of its speed when colliding with the bottom part of the tube stays near the bottom. This effect cancels the effect of randomly moving molecule speeding up when moving downwards and slowing down when moving upwards. And there's an uniform temperature in the tube.Ignore the following part, if you think gravitational redshift is off topics.

Photon gas:
Any photon moving distance d down experiences a x percent energy increase. Nothing at all cancels this effect, so the temperature in the photon gas tube is almost uniform but not exactly uniform.

As the laws of thermodynamics should be same for dense gas, non-dense gas, and photon gas, I should change "uniform temperature" to "almost uniform temperature" in the dense gas and non-dense gas cases, but I won't, because this may be off topics.

 

[DrStupid]

I should change "uniform temperature" to "almost uniform temperature"

Better change it in "uniform as seen from a certain position". Every observer will see the same temperature in any direction but this temperature is not identical for observers at different height.

 

[PeterDonis]

Every observer will see the same temperature in any direction but this temperature is not identical for observers at different height.

Are you referring to the effect of gravitational time dilation? Or are you claiming that there is a temperature gradient because of the gravitational potential? The latter claim, as has already been discussed, is not correct; more precisely, if there is a temperature gradient, thermal conduction will transfer heat so as to remove it and make the temperature uniform throughout the tube.

I mention gravitational time dilation because, in principle, an observer at the top of the tube has a slightly higher "rate of time flow" than an observer at the bottom of the tube, and this will, in principle, affect the temperature they measure. Both observers will measure the entire tube to have the same temperature, but it will be a slightly different temperature for the two observers. But this effect is extremely small for, say, the field of the Earth--about one part in $10^{16}$ for a one meter high tube at the Earth's surface. So we can ignore it here.

 

[DrStupid]

Are you referring to the effect of gravitational time dilation?

Yes.

But this effect is extremely small for, say, the field of the Earth

That has already been mentioned.

 

[jartsa]

Better change it in "uniform as seen from a certain position". Every observer will see the same temperature in any direction but this temperature is not identical for observers at different height.

I wonder how many readers would get anything about that?

On the other hand, blueshift of falling photons is nothing very odd. Let's consider blueshift of "falling" photons in an accelerating spacecraft :

A spaceship with uniform temperature turns rocket motors on, radiative heat starts flowing from the "ceiling" to the "floor", clearly there's a heat pump working there, right? After some time the heat flow stops, because a dynamic equilibrium is reached. The heat pump is still on, but heat is not flowing, and the heat pump is doing no work as it's keeping a constant temperature difference between the ceiling and the floor. People on board may think it's gravity that's causing the temperature difference.

 

[Zachary Smith]

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.

Wouldn't the molecules moving up be converting Kinetic energy to potential energy as they slow?

 

[Greenarrow]

What about gaining or losing KE due to collisions? I don't have an answer to the OP's question, but my 2 cents suggestion is that his question deals with the microscopic behavior of the gas and he is wondering about the macroscopic effect of temperature as a result. Isn't the net energy change within a gas balance out? or, it violates the law of conservation of energy. The gas inside the tube is a system in itself and is defined by the pressure, volume, temperature and number of moles, which define the state of the gas at an instant. Any temperature changes within the gas has to derive the energy from within itself and cancel out.

 

[Khashishi]

You can fool yourself with the conservation of energy if you don't consider the boundaries. Consider adiabatic expansion of a gas. The temperature of the gas decreases as it expands. Where does the energy go? It goes into work done on the boundaries of the container as you increase the volume of the container. At a microscopic level, some molecules slow down as they bounce off of the wall of the container as the wall moves away from the center of the chamber. Now, the walls don't have to be real walls; you can just consider an imaginary bubble around a region of gas, as long as the gas is collisional within the bubble. The pressure is essentially the force per unit area on the surface of an imaginary bubble due to collisions.

To tie it back to the column of air, you need to consider the net force on a particle. The gravitational force and the pressure force.

 

[Let'sthink]

If we consider the gas in a tube and gravity together as a system, then its internal energy cannot change unless heat is supplied or work is done on this system. The work done by gravity is internal work. I think the answer is pressure at the bottom is more and density is also more but the temperature remains same throughout. One more thing is to be noted that temperature is a macroscopic variable which has meaning for a group of molecules and an individual molecule is not supposed to have temperature, pressure or density.