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How do you separate Hydrogen and Oxygen gases in industrial electrolysis of water process?

In electrolysis, oxygen and hydrogen gas are formed at dissimilar electrodes (oxygen at the anode and hydrogen at the cathode). As these electrodes do not have to be in close proximity, the hydrogen and oxygen will bubble upwards into separate collection vessels.

If you required separating hydrogen and oxygen once mixed, the simplest way I can think of would be to cool the mixture to ~60K. At this point the oxygen would condense and leave hydrogen gas.

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