When you warm up your room which has a fixed volume, do you necessarily increase the internal energy of the air enclosed by it?
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Unless you live in a sealed room where no air can escape, the pressure better be constant. This means as the air warms, it's pushed out of the room. If you now use the Ideal Gas Law, you will realize that since p and V of the room has not changed, the value for NRT remains constant, and as a result the internal energy of the air which is proportional to NRT must also be constant.