Refrigerators and their Efficiency (Qualitative)

A refrigerator -- and, viewed from the opposite point of use, a heat pump used for heating -- is, in precise thermodynamic terms, nothing but a heat engine run backward. Instead of letting heat flow naturally from hot to cold in order to extract useful work, a refrigerator has external work $W$ deliberately supplied to it (typically by an electric compressor motor), and uses that work to force heat to flow the "wrong" way: it draws a quantity of heat $Q_2$ out of a cold space -- the inside of the refrigerator cabinet, or the room being cooled by an air conditioner -- and rejects a larger quantity of heat, $Q_1 = Q_2 + W$, to the warmer surroundings outside (the kitchen, or the outdoors). This is exactly, and unavoidably, the physical situation described by the Clausius statement of the second law (Section 11.9): heat never moves spontaneously "uphill," from a colder body to a hotter one, and doing so always demands that external work be supplied to force it.

Why only a qualitative idea is needed here. Unlike a heat engine, whose performance is naturally captured by an efficiency $\eta = W/Q_1$ -- a fraction that is always less than one, telling us what portion of heat input becomes useful work -- a refrigerator's job is the opposite: it uses work to move heat, and a larger, not a smaller, amount of heat moved for a given amount of work supplied is what makes it a better refrigerator, not a worse one. So rather than an "efficiency" in the same sense as an engine, a refrigerator's performance is more naturally judged by how much heat it removes from the cold space for every unit of work it consumes -- and, at this qualitative level, simply understanding this trade-off, and how it changes with the temperatures involved, is the important physical idea.

For reference, this idea can be written compactly as a coefficient of performance, $\beta = Q_2/W$ (the more heat $Q_2$ removed per unit of work $W$ supplied, the higher, and better, this number is). For an ideal, reversible (Carnot) refrigerator working between the same two temperatures $T_1$ and $T_2$ as a Carnot engine, this can be shown to work out to

$\beta_{\text{Carnot}} = \frac{T_2}{T_1 - T_2}$ …