COP (sometimes CP), of a heat pump is the ratio of the change in heat at the "output" (the heat reservoir of interest) to the supplied work:
where • is the change in heat at the heat reservoir of interest, and • is the work consumed by the heat pump. (Note: COP has no units, therefore in this equation, heat and work must be expressed in the same units.) The COP for heating and cooling are thus different, because the heat reservoir of interest is different. When one is interested in how well a machine cools, the COP is the ratio of the heat removed from the cold reservoir to input work. However, for heating, the COP is the ratio of the heat removed from the cold reservoir plus the heat added to the hot reservoir by the input work to input work:
where • is the heat moved from the cold reservoir (to the hot reservoir). Contents [hide] • 1 Derivation • 2 Example • 3 Conditions of use • 4 See also • 5 External links • 6 References
 Derivation According to the first law of thermodynamics, in a reversible system we can show that Qhot = Qcold + W and W = Qhot − Qcold, where Qhot is the heat given off by the hot heat reservoir and Qcold is the heat taken in by the cold heat reservoir. Therefore, by substituting for W,
For a heat pump operating at maximum theoretical efficiency (i.e. Carnot efficiency), it can be shown that and , where Thot and Tcold are the temperatures of the hot and cold heat reservoirs respectively. Hence, at maximum theoretical efficiency,
It can also be shown that COPcooling = COPheating − 1. Note that these equations must use the absolute temperature, such as the Kelvin scale. COPheating applies to heat pumps and COPcooling applies to air conditioners or refrigerators. For heat engines, see Efficiency. Values for actual systems will always be less than these theoretical maximums.