How does a diving bell NOT get flooded with water, and does it work with other liquids?

Question

If the air in an upside-down container (such as a diving bell) weighs much less than water, then why does it not get compressed down until it is the same mass per volume, when submerged even to a shallow depth? For example, as I understand it (though admittedly I am not an expert), a diving bell filled with air at 1 atmosphere (1 bar), when dragged down to a certain depth in the ocean, can be, say, several hundred times denser than it originally was. But air that is 300 times as dense is still far LESS heavy per volume than water. Yet, if kept at the same spot, it stays that way and doesn't get more compressed, even though it is lighter, correct?

What other forces make that happen?

Also, does the afore described process ALSO happen with other liquids i.e. would a diving bell work in, say, liquid hydrogen or liquid helium (possibly with a gas other than air, one that has a much lower freezing point)?

Answer 1

The diving bell works because the pressure of a gas, approximately $ P = nRT / V$, can equal the pressure of a liquid, $P = \rho g h$, with no need to be at the same density. If we set them equal and hold n constant (we don't pump more air into the bell) we can solve for V, the volume of air in the bell.

$V = \frac{nRT}{\rho g h}$

A diving bell wouldn't work in an ultra high pressure fluid like liquid helium because the air can't compress enough to equal the pressure. The bell would immediately fill with flash-vaporized helium, which would quickly revert to liquid as pressure equalized, with some trace mixture of liquid oxygen, nitrogen, etc, from the air. When this process happens in water, it's called cavitation, and it can occur when fast propellers cause water pressure behind the blade to drop below the boiling pressure of water at that temperature.