Some of you are probably aware of What If, xkcd's blog about interesting physics problems. One episode, Glass Half Empty, concerns itself with what would happen if a glass of water is half water, half vacuum.
One scenario he looks at is water in bottom, vacuum on top, with air rushing in from the top end. He says that after a few hundred ns, the air has filled the vacuum completely. I would very much like to know how he arrives at that estimate!
What I thought about so far:
He cites a max speed of the air molecules of ~1000m/s, but this can't be the only mechanism, cause after 500ns the fastest molecules would've only travelled about 0.5mm.
I figured that the air rushing in would have to happen via a shock, and that the situation described is essentially a shock tube with air as driver and vacuum on the driven side (neglecting the vacuum/water interface for convenience).
However, I could not find a formula which would let me calculate the shock speed (in m/s) for the given configuration. Specifically, I'm not sure the "standard" handbook formulae even apply here, because of vacuum as "driven gas". Can anybody shed light here?
Conditions: Let's say it's dry air at 300K and 1013mbar, and perfect vacuum (or very low pressure if that's necessary). I'm not interested in what happens after the shock arrives at the end of the tube, so no need to open that can of worms.
Thanks for any insight!