Kinetic energy and momentum of an expanding gas

Let's take the simplest case of an ideal gas undergoing adiabatic expansion against a vacuum of finite volume, without any heat and work transfer into or out of the system.

The gas moves in bulk to fill the space, while its internal energy and temperature drop.

Is part or all of this ΔU converted to kinetic energy? When equilibrium is reached and the gas becomes homogeneous in temperature and density once more, does this kinetic energy revert back into internal energy? If it does, will this process cause the temperature to increase?

Or, does the gas expand evenly and gradually 'brakes' down consuming kinetic energy as it approaches the final volume, and eventually naturally stop by itself the very moment the total volume is reached?

How is momentum conserved in either case?

• Are you talking about a reversible expansion or a Joule expansion? If it's a Joule expansion the internal energy doesn't change, while if it's a reversible expansion the system has to do work on its surroundings. Oct 2 '15 at 8:35

1) These sorts of things are actually being studied nowadays with trapped atomic gases. These gases are released from optical traps and expand into what is essentially a perfect vacuum. If you want, they can be recaptured, although a typical trap does not have sharp walls'', but harmonic confinement.

2) Yes, if you expand into a vacuum then you just convert internal energy into (collective) kinetic energy $\frac{1}{2}\rho u^2$, where $\rho$ is the mass density and $u$ is the fluid velocity. If you expand forever then temperature will go to zero, and all energy is kinetic.

3) If you recapture then the result will depend somewhat on the nature of the gas and the walls.

i) Ideal gas, soft walls (harmonic confinement). The systems rebounds from the wall and starts to undergo periodic oscillations. The energy remains kinetic. Since the gas is ideal, the motion is undamped.

ii) Real gas, soft walls. Same thing, but collective oscillations are damped by viscosity, Systems settles down, energy goes back to internal (that means for ideal gas equation of state $T$ goes back to initial value), entropy has increased.

iii) Ideal gas, sharp walls. Rebound creates shock waves. This is a complicated process, since you will get a messy systems of interacting shocks. Even ideal shocks create entropy, so this systems may settle down as in ii).

iv) Real gas, sharp wall. As in iii), but this definitely settles down as in ii).

In the case of a wall separating a vacuum and a gas disappearing inside a rigidly held container. The energy inside the container would remain constant so the temperature would remain the same (for an ideal gas). The gas would be pushing on the wall that did not disappear while the gas was accelerating, transferring momentum to it. Then when the gas hits the wall on the other side there would be another momentum transfer.

In the case where the wall moved over slowly, the gas would be doing work on the wall and therefore losing energy to it. In this case the temperature would decrease.

• So in the first case there is no difference between internal energy and collective kinetic energy since the gas is ideal? Oct 3 '15 at 0:40