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Don't lock the kids in a jar! :D

But if not how are they going to learn?

Putting a hole in the lid and covering it with micropore tape or polyfill would allow gas exchange without allowing microorganisms to escape or infect the jar.

Would the gas exchange membrane also facilitate evaporation of the water?

In my experience, no. If it does, it's negligible.

It would be neat to aerate it without breaking the closed system. Maybe a magnetic connection through the glass. Motor on the outside and prop on the inside... Does add complexity though.

Probably almost every wet-bench lab has a magnetic stir bar, as in [0].

[0] https://www.youtube.com/watch?v=N3BoXEWCF2E

But that would just mix in air from the top of the jar?

Oh I guess I don't understand what it means "to aerate" then.

If you're not changing the air, then it will already equalize very quickly without your help.