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_c = -23 °C = 250 K. What is the maximum amount of heat that can be transferred into the house?

1. Something less than 1000 J
2. 1000 J
3. Something more than 1000 J

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

Tc Th

=

|Qc| |Qh|

Therefore:     |Qc|

=

Tc Th

|Qh|

Using this in the energy equation gives:

|Qh| = |Qc| + W

=

Tc Th

|Qh| + W

|Qh| *

(

1 -

Tc Th

)

= W

|Qh|

=

W Th Th - Tc

For our numerical example this gives:

|Qh|

=

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.