Mostly by fractional distillation. Atmospheric air is drawn into a system and filtered to remove impurities and water vapor, then repeatedly cooled in distillation towers until the oxygen liquefies and then is drained out of the tower.
It can also be produced by electrolysis of water, but this method is less than ideal, as it requires as much input energy as it gets back.
To go into deeper detail on u/veespike’s excellent explanation – air is a mixture of gases, roughly 70% Nitrogen, 20% Oxygen, and then some CO2, methane, and other random gases. Just like water melts and freezes at given temperatures, gasses will melt and freeze at given temperatures and each one is unique and different.
So if take a jar of room temperature air and start cooling it down really, really cold, like 200 degrees below zero, at a certain point the CO2 will liquify and fall out of the air like rain drops. I can pour out the liquid CO2 and now I have a a cup of liquid CO2 and jar of air without any CO2 gas. I can keep cooling it further and then the nitrogen will liquify and I can pour that out and have a glass of CO2, a glass of nitrogen, and a jar of whatever’s left in the air.
If I do this right, I can get a glass of liquid oxygen and only liquid oxygen.
I used examples, but Oxygen liquifies at around -294F. Nitrogen is -320F and CO2 is -109F.
So if I cooled air to -108F it’s just cold air. One more degree down and the CO2 ‘rains’ out as liquid CO2. If I kept removing heat the temperature would stay the same for bit but the CO2 would all rain out. Once the CO2 is gone the temperature will fall again until -294F when the Oxygen rains out, and Nitrogen at -320F.
Pressure swing absorption is pretty common for lower grade oxygen. You pump filtered and dry air into a tank filled with a sorbent material. At the increased pressure the nitrogen in the air sticks to the surface of the sorbent stuff in the tank. Then you let the gases our and the nitrogen mostly stays. So you go from 78% nitrogen, 20% oxygen, 2% other stuff to 88-93% oxygen and the rest other stuff with basically no nitrogen. It’s much less energy intensive then cryogenic distillation and is more than good enough for things like medical oxygen.
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