A Heat Pump

If you heat your home using electric heat, 1000 J of electrical energy can be transformed into 1000 J of heat. An alternate way of heating is to use a heat pump, which extracts heat from a lower-temperature region (outside the house) and transfers it to the higher-temperature region (inside the house). Let's say the work done in the process is 1000 J, and the temperatures are T_H = 17°C = 290 K and T_L = -23 °C = 250 K. What is the maximum amount of heat that can be transferred into the house?

Something less than 1000 J
1000 J
Something more than 1000 J

The best we can do is determined by the Carnot relationship:

T_L

T_H
=
|Q_L|

|Q_H|

Therefore: |Q_L| =
T_L

T_H
|Q_H|

Using this in the energy equation gives:
|Q_H| = |Q_L| + W =
T_L

T_H
|Q_H| + W

|Q_H| * ( 1 -
T_L

T_H
) = W

|Q_H| =
W T_H

T_H - T_L

For our numerical example this gives:
|Q_H| =
1000 * 290

290 - 250
= 1000 * 7.25 = 7250 J

This is why heat pumps are much better than electric heaters. Instead of 1000 J of work going to 1000 J of heat we have 1000 J of work producing over 7000 J of heat.