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I think you'll find that when you compress the gas you'll need a chamber or gas bottle to contain it. This along will be heavy. At work I have large bottles of Hydrogen, which is a lighter gas, these bottles are 40 litres in capacity (5 feet high) and have up to 200bar pressure. The bottle full weighs well over 60 Kg's

I've never seen these bottles float away lol

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RE:

Can I compress helium??

If i compressed a lot of helium into an air compressor, would the compressor start to float into the air?

If so why don't we use this as a more space effecient blimp?

The helium is only compressed while it's in the tank, so that more gas can fit in it. Once it comes out of the tank, it will most definitely make a balloon float.

Helium can be compressed and because the idea of density of an object is to be as big of a volume and as little of a mass as possible since density = mass/volume, so if the blimp is more dense then air it wont float.

You change phase at some point to a liquid, and then the volume is different.

See what the density of liquid helium is, and see if it is greater then or less then the density of air.