Your house or apartment has a refrigerator. Very likely it has an air conditioner. The purpose of these devices is to make air that is cooler than its environment even colder. The first does so by blowing hot air out into a warm room, the second by blowing it out to the hot outdoors. You’ve probably felt the hot air exhausted by an air conditioner compressor or coming out from beneath the refrigerator.

At first glance, a refrigerator or air conditioner may seem to violate the second law of thermodynamics. After all, doesn’t the second law forbid heat from being transferred from a colder object to a hotter object? Not quite: The second law says that heat is not spontaneously transferred from a colder to a hotter object. A refrigerator or air conditioner requires electric power to operate. They do cause heat to be transferred from cold to hot, but the transfer is “assisted” rather than spontaneous.

A refrigerator is any closed-cycle device that uses external work $W_{\text{in}}$ to remove heat $Q_{\text{C}}$ from a cold reservoir and exhaust heat $Q_{\text{H}}$ to a hot reservoir. FIGURE 21.10 is the energy-transfer diagram of a refrigerator. The cold reservoir is the air inside the refrigerator or the air inside your house on a summer day. To keep the air cold, in the face of inevitable “heat leaks,” the refrigerator or air conditioner compressor continuously removes heat from the cold reservoir and exhausts heat into the room or outdoors. You can think of a refrigerator as “pumping heat uphill,” much as a water pump lifts water uphill.

FIGURE 21.10 The energy-transfer diagram of a refrigerator.

Because a refrigerator, like a heat engine, is a cyclical device, $\text{Δ}{E}_{\text{th}}=0$. Conservation of energy requires

To move energy from a colder to a hotter reservoir, a refrigerator must exhaust more heat to the outside than it removes from the inside. This has significant implications for whether or not you can cool a room by leaving the refrigerator door open.

The thermal efficiency of a heat engine was defined as “what you get (useful work $W_{\text{out}}$)” versus “what you had to pay (fuel to supply $Q_{\text{H}}$).” By analogy, we define the coefficient of performance K of a refrigerator to be

What you get, in this case, is the removal of heat from the cold reservoir. But you have to pay the electric company for the work needed to run the refrigerator. A better refrigerator will require less work to remove a given amount of heat, thus having a larger coefficient of performance.

A perfect refrigerator would require no work $(W_{\text{in}}=0)$ and would have $K_{\text{perfect}}=\infty$. But if Figure 21.10 had no work input, it would look like Figure 21.2c. That device was forbidden by the second law of thermodynamics because, with no work input, heat would move spontaneously from cold to hot.

We noted in Chapter 20 that the second law of thermodynamics can be stated several different but equivalent ways. We can now give a third informal statement:

Any real refrigerator or air conditioner must use work to move energy from the cold reservoir to the hot reservoir, hence $K<\infty$.