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Suppose I have a sealed glass cylindrical vessel of radius r and length l that is evacuated. I smash one end.

How long will it take for air to (a) reach the end of the cylinder (b) fill the vacuum and equalise the pressure? Will the air molecules move at the speed of sound?

It seems to me that the speed of sound as a wave is not necessarily the same as the mean velocity of the air molecules themselves. Can someone enlighten me? Thanks.

Note

I'm not certain how well-defined my question is. Suggestions for improvement would be appreciated.

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The molecules in an ideal gas move slightly faster than the speed of sound. Air at atmospheric temperature and pressure is very similar to an ideal gas.

The difference in speed is not because "the collisions between the gas molecules slow down the sound wave." If that was correct, the speed of sound would depend on the pressure of the gas, and it does not. The difference is simply because the molecules are moving in random directions (and the direction changes with every collision) and so are not moving in the same direction that the sound wave propagates.

In air at room temperature, the average molecule speed is about 1.5 times the speed of sound, i.e. about 500 m/s compared with the speed of sound of about 340 m/s.

Assuming the "first" molecules will reach the far end of the tube travelling at the speed of sound in the tube is a reasonable approximation.

The time to "fill the vacuum and equalize the pressure" is not very well defined, since the air entering the tube will overshoot atmospheric pressure and create pressure oscillations depending on the length of the tube. The number of cycles for the oscillations to decay to a "steady" pressure will depend on the rate at which the excess energy is be radiated back into the outside air as "sound."

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