Every chemical has a relationship between heat and pressure. Some of them, when you compress them get hot in proportion to the pressure. Then when you expand them they get cold in relation to the drop in pressure.
So you find a chemical that does this very efficiently (refrigerant), compress it so it gets really hot. Run that through a radiator so that it comes down to ambient temperature. Then expand it, and it will come out colder than it started because you exhausted all that heat while it was compressed. Run air over that and you have air conditioning.
If you want to heat the room, you just do that in reverse. You run air over the compressed refrigerant to heat up the room. Then you expand it and run it through a radiator so it warms up to ambient. (The need to warm it up is why heat pumps lose efficiency at low outside temps.)
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