Hello everybody. I have trouble of proving the theoretical maximum of heat pump Coefficent of Performance. The thing I'm trying to calculate is for heat pump pumping heat from colder reservoir to the hotter reservoir: COP=Qh/W Qh - heat supplied to the hot reservoir(output) W - mechanical work consumed by the pump(input) The formulas in the wikipedia seems to be wrong and I want to derive the correct formulas from the thermodynamic laws. With current Wikipedia formula it is normal to get COP >2. Search wikipedia:Coefficent of Performance. As I calculate the heat balance of the basic air to air heat pump, any COP values above 2 will violate the second law of the thermodynamics, and therefore make perpetual motion machine possible. So is what is the correct formula and what is the way to prove it? I argued a lot with my friend about that, when he wanted to buy the heat pump to heat his office. Eventually he still bought the pump because I could not provide the calculation and he decided to believe what the sales man told him (COP=3 or something). But now he do not seem very happy with the electricity bill. Thanks in advance.