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My question is, what (if anything) will happen to the temperature? Will it change or remain constant? I'm wondering because for an ideal gas there is the relation P*V=n*R*T. We already said we keep n constant and R is of course constant. If T is also constant, then follows another question:

If the pressure is increased enough, the gas will eventually change to liquid form. But is the temperature still the same? So if we start at for example room temperature (20 degrees C) and decrease the volume until the gas liquifies, does that mean the boiling temperature of the gas will increase significantly (actually becoming equal to the starting temperature), since it liquifies without changing temperature?