Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 4.14.
A heat pump is essentially a backwards air conditioner, in that it extracts an amount of heat from the cold outside air using an amount of electrical energy and dumps an amount of heat into an inside room in order to heat it. We can analyze its performance using the same logic as for a refrigerator. The coefficient of performance in this case is the amount of heat dumped into the room divided by the energy (work) needed to achieve this, so
From conservation of energy
Thus the COP is always greater than 1.
In an ideal refrigerator (e.g. one working on a reversed Carnot cycle) the entropy gained in absorbing is equal to the entropy lost in expelling , so
The upper limit on the COP is therefore
A heat pump is more efficient than a purely electric heater (that is, a heater that generates heat by electrical resistance rather than by transferring heat from outside to inside) since in an electric heater, the work is equal to , while in a heat pump, this work can be used to acquire an additional amount of heat which is expelled into the room along with the electrical work . In other words, with a heat pump you have while with an electric heater, you have only .