Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 4.13.
An air conditioner is an example of a refrigerator in which the cold reservoir is the room to be cooled and the hot reservoir is the outside atmosphere. On a hot day, the rate at which heat leaks into an air conditioned room from the outside is roughly proportional to the temperature difference between the outside and inside. In that case, the work done to remove an amount of heat in time is
where is the heat expelled to the outside and is a constant.
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 work required to maintain a temperature of is therefore
Thus lowering the inside temperature by a small amount can have a large effect on the work required to maintain this temperature, and thus on the cost of running the air conditioner. For example, suppose the outside temperature is and the inside temperature is . If we wish to lower the inside temperature by only one degree, the extra work required is
We need to use 27% more power to achieve a single degree more cooling. This is one reason why it is much more economical to bear with a slightly higher indoor temperature on a hot day.