can cold air become water in a vacuum cylinder?
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That's a confusing question: water isn't air. So either you mean condensing water vapor that is in the air or liquefying the air. Either can be done if cold enough, but trying to do it in a vacuum works against you.
Yes, water vapor
If you apply a cold enough temperature, then yes it is possible.
Why would the vacuum work against me?
Check out the phase diagram for water or a steam table; As you reduce the pressure, the boiling point goes down.
Let me re-state "cold air" as "a mixture of cold air and water". Is that OK?
The answer to your question will depend upon the Temperature and the value of the pressure in your "vacuum cylinder". If you look at the Phase Diagram in this link, it shows how the phases of water depend on the pressure / temperature combination. There is a "triple Point" for water (around 0°C) at which water can exist as solid, vapour and liquid and move from one to another. The state of the water, as you leave that point in various directions, is shown on the diagram. One simple fact emerges and that is that water can't exist as liquid towards the bottom left corner of the diagram, for any finite value of low pressure, there is a temperature at which water will pass from  vapour to sold and back, without going through a liquid phase (called Sublimation).
The presence of Air, doesn't make any significant difference to the situation.
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