B Does gas expanding in a container cause displacement of the container?

[bobdavis]

If particles are trapped in one half of a massless container by a barrier, and the barrier is removed so that the particles are allowed to expand to fill the whole container, it seems the center of mass would be displaced to the center of the container, whereas before it was located at the center of the half of the container where the particles were originally contained. But the center of mass can't be displaced in the rest frame of the center of mass due to conservation of momentum, so it seems the expansion of the particles into the other half of the container must be compensated by a displacement of the whole container such that the center of mass remains constant. So an observer outside the container but originally at rest with the container should see the container move in the direction opposite to the expansion.

Do I have a correct understanding? If so how is the displacement explained in terms of the collision interactions? How does an observer comoving with the container explain the displacement of the center of mass of the particle system?

EDIT: container doesn't need to be massless, I was trying to simplify the problem but maybe it's simpler if mass of container is included so that particle/container collisions can be dealt with with standard collision theory

 

[hmmm27]

Not entirely sure what the question is.

how is the displacement explained in terms of the collision interactions?

Stuff on one side continues to hit that side, imparting momentum, while stuff on the other side - that used to be in the middle of the container - has to wait until it travels to the other end of the container before doing same.

How does an observer comoving with the container explain the displacement of the center of mass of the particle system?

"So, all of a sudden the CoM moved", I would imagine.

 

[BvU]

Hi,

I must say you are bringing together things that are usually looked at separately. But as a thought experiment it's a nice concoction.

But the center of mass can't be displaced in the rest frame of the center of mass

To me that sounds like a tautology.

Let me transmogrificate your thought experiment to the following: we have a sturdy container with a sturdy barrier in the middle:

Fill the right half with water and put the whole thing on wheels so it can move freely left and right.
Suppose the whole thing is 1 x 1 x 2 m so the center of mass is at 0.5 m to the right of the barrier

What happens when the barrier is removed instantaneously ?
My guess is that if the sloshing has come to an end, the center of mass is still in the same place (no horizontal external forces have worked on the system). So the container has moved to the right over a distance of 0.5 m.

(If the sturdy walls and the barrier represent a considerable mass compared to the 1000 kg of water, some allowance has to be made and it will be a little less than 0.5 m)​

Would that be intrinsically different from your scenario ?

$\$

 

[A.T.]

How does an observer co-moving with the container explain the displacement of the center of mass of the particle system?

It's a non-inertial frame, so there is no expectation of momentum conservation. If you want to apply Newtons 2nd Law to the particle system in that frame, you have to introduce inertial forces.

 

[bobdavis]

Not entirely sure what the question is.

Stuff on one side continues to hit that side, imparting momentum, while stuff on the other side - that used to be in the middle of the container - has to wait until it travels to the other end of the container before doing same.

"So, all of a sudden the CoM moved", I would imagine.

Regarding the collision theory, I think that makes sense informally

Regarding the observer comoving with the container I guess I mean how does that observer account for the force that causes the movement

Would that be intrinsically different from your scenario ?

No looks like the same scenario to me

It's a non-inertial frame, so there is no expectation of momentum conservation. If you want to apply Newtons 2nd Law to the particle system in that frame, you have to introduce inertial forces.

I see, thanks, I didn't account for whether the frame was inertial or non-inertial. So in particular it's non-inertial because it *does* undergo the displacement and this requires an acceleration

 

[bobdavis]

A couple more questions: would the container be expected to oscillate around the center of mass in a predictable way? i.e. if the removal of the barrier causes the container to undergo a large acceleration in the direction opposite to the expansion, will the particles colliding with the far wall of the empty side of the container cause another large acceleration in the direction of the expansion? If so would the oscillations be expected to continue indefinitely or would they be expected to die out? If not would the container be expected to simply move its center to the center of mass and stop?

 

[jbriggs444]

A couple more questions: would the container be expected to oscillate around the center of mass in a predictable way? i.e. if the removal of the barrier causes the container to undergo a large acceleration in the direction opposite to the expansion, will the particles colliding with the far wall of the empty side of the container cause another large acceleration in the direction of the expansion? If so would the oscillations be expected to continue indefinitely or would they be expected to die out? If not would the container be expected to simply move its center to the center of mass and stop?

I would expect a damped oscillation as the contents "slosh" back and forth before damping to a new thermal equilibrium.

 

[russ_watters]

Regarding the observer comoving with the container I guess I mean how does that observer account for the force that causes the movement

The observer doesn't need to know what's inside the box as long as the COM stays inside the box.

 

[vanhees71]

It depends on what's in the other half of the box. Consider the following three cases, using quantum theory of an ideal (Bose or Fermi) gas and discuss the equilibrium entropy before and after taking out the dividing wall:

(a) the other half of the box is empty (vacuum)
(b) the other half of the box is filled with a gas at the same temperature and pressure consisting of different kinds of atoms or molecules
(c) as (b) but with the gas consisting of the very same kind of atoms or molecules

Finally discuss the classical limit!

 

[Chestermiller]

If particles are trapped in one half of a massless container by a barrier, and the barrier is removed so that the particles are allowed to expand to fill the whole container, it seems the center of mass would be displaced to the center of the container, whereas before it was located at the center of the half of the container where the particles were originally contained. But the center of mass can't be displaced in the rest frame of the center of mass due to conservation of momentum, so it seems the expansion of the particles into the other half of the container must be compensated by a displacement of the whole container such that the center of mass remains constant. So an observer outside the container but originally at rest with the container should see the container move in the direction opposite to the expansion.

Do I have a correct understanding?

Yes. Assuming there is vacuum initially in the right half of the container, when the barrier vanishes, the force exerted by the gas on the left wall of the container must also vanish. This is because the container is massless, and thus must always have no net force acting on it. In order for this force on the left wall to vanish, the force per unit area on the left face must drop from the initial pressure to zero. This happens because the face immediately begins moving to the left, causing a viscous tensile stress to develop within the gas in close proximity to the wall. This is consistent with the 3D tensorially correct version of Newton's law of viscosity applied to the gas in the region. So, in the end, the center of mass of the gas will not move, but the container will be displaced to the left, such that the center of mass of the gas will finally be at the center of the container.

Some of the answers have referred to "sloshing" of the gas within the container during the process. This is correct, but, in the end, this sloshing will be damped by viscous stresses in the gas. So the viscous behavior of the gas plays a large role in establishing the final equilibrium configuration of this system.

 

[bobdavis]

Thank you all for the responses. So is the damping of the oscillation entirely due to the viscous stresses? i.e. without viscous stresses would the oscillations continue indefinitely?

 

[Chestermiller]

Thank you all for the responses. So is the damping of the oscillation entirely due to the viscous stresses? i.e. without viscous stresses would the oscillations continue indefinitely?

In my judgment, yes.