## Refrigerators and the Second Law of Thermodynamics

#### Introduction

The Second Law of Thermodynamics states that heat will spontaneously always flow from a hot region to a cold region. By itself it never flows the other way, but can be made to do so under the influence of an external agency. The Second Law of Thermodynamics also states that this outside influence must do some work.
In a kitchen refrigerator the inside of a closed box is to be kept cool by removing heat from the inside and depositing it on the outside. Because the heat will not move freely from the cold inside to the hot outside it must be made to do so using an intermediate fluid which absorbs heat on the inside, then carries outside of the box and releases the heat to the air (see figure 1.) This fluid circulates in a pipe which passes in and out of the back of the refrigerator, kept moving by a compressor driven by an electric motor. It is the work done by this compressor (using electrical energy from the household electricity supply) that makes the refrigerator work without violating the Second Law of Thermodynamics.

#### Refrigerators and the First Law

Any refrigerator takes in energy from the region to be cooled (or kept cold) and deposits heat energy into some region outside of the refrigerator, such as your kitchen (see figure 2.) In order to work there has to be some work done by the compressor and its electric motor. Using the First Law of Thermodnamics we can write

 QC - QH = -W

(Note: since work in done one the refrigerator by another device, the compressor, rather than by the refrigerator itself, according to the sign convention which is part of the first law, the work done is negative.)

Suppose that 2.4 MJ of work is used to remove 5.2 MJ of heat from the inside of the refirgerator, then an amount of heat QH = QC + W = 5.2 MJ + 2.4 MJ = 7.6 MJ must be added to the kitchen.

#### Efficiency of a refrigerator

The efficiency of a refrigerator (known as the coefficient of performance, COP) is defined as

 COP = Amount of heat removed from the inside of the refrigerator = QC Work done to operate the refrigerator W

For example, if 20 MJ are removed from the inside of the refrigertor by doing 7.5 MJ of work, then the coefficient of performance is equal to 20/7.5 = 2.67.

#### The Carnot Refrigerator

Just as the Carnot engine is the mathematical limit to the maximum possible efficiency of a heat engine, the Carnot refirgerator is the mathematical limit to the COP (efficiency) of a refrigerator. Although it cannot be built, it does tell us the best performance that we can hope for from a real refrigerator. As with the Carnot engine, the COPc of a Carnot refrigerator depends on only two numbers, the temperature of the region to be kept cold (TC) and the temperature of the region where the heat is transfered (TH). It is equal to

 COPc = TC TH - TC

Suppose we have a Carnot refrigerator used to keep food cold (TC = 3 oC = 276 K) in a kitchn whose temperature is TC = 22 oC = 295 K.

• Its COPc would be 276/(295-276) = 276/19 = 14.5
• If a real refrigerator is only 35 % as good as the Carnot refrigerator, then the real refrigerator would have a coefficient of performance equal to 35 % * 14.5 = 5.08
• If we need to remove 300 kJ of heat (QC) from the refrigerator, then using the definition of the coefficient of performance the amount of work that we need is W = QC / COP = 300 kJ / 5.08 = 59 kJ
• The total heat energy (QH) deposited in the kitchen is equal to QH = QC + W = 300 kJ + 59 kJ = 359 kJ.