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Run a Carnot Engine Backwards: COP = Tc / (Th - Tc)

A refrigerator is a heat engine run in reverse: work is put in, heat is taken from the cold space and a larger amount is rejected to the warm surroundings. Its figure of merit is the coefficient of performance (COP), not the efficiency.

NSEJS · Thermal Physics · 6 linked questions · 100% coverage

Know this first

Refrigerators and Coefficient of Performance is the central idea in this question pattern. Identify the condition in the stem before applying the method below.

Study deeper: NCERT Science · Thermal Physics

COP and the heat balance

For a refrigerator,

COP = Qc / W, where Qc is the heat removed from the cold space and W is the work supplied

For a reversible refrigerator,

COP = Tc / (Th - Tc) with temperatures in kelvin, and the heat rejected is Qh = Qc + W

A refrigerator that moves 360 J per second from -3 °C (270 K) to 27 °C (300 K) has

COP = 270/30 = 9, so the power consumed is 360/9 = 40 W

Moving 180 J per second from 0 °C to 30 °C has

COP = 273/30, about 9.1, so the power is about 19.8 W

Finding temperatures from the COP

Invert the formula when the COP is given: with COP = 5 and a freezer at -20 °C (253 K), Th - 253 = 253/5 = 50.6 K, so the surroundings are at about 303.6 K, roughly 30.6 °C. For a refrigerator working between 4 °C (277 K) and 30 °C (303 K), COP = 277/26, about 10.7, so removing 300 calories per second requires about 28 calories per second of work.

A Carnot engine used as a refrigerator

A Carnot engine of efficiency η run backwards has

COP = (1 - η)/η

An engine of efficiency 1/20 used as a refrigerator therefore has COP = (19/20)/(1/20) = 19, so 20 J of work removes 19 x 20 = 380 J from the cold reservoir.

Check with the heat balance:

Qh = Qc + W = 400 J, and Qc/Qh = 380/400 = 19/20 matches the engine

OlympIQ exam lens

✓COP = Qc / W; for a reversible refrigerator COP = Tc / (Th - Tc), kelvin only
✓Heat rejected to the surroundings = Qc + W
✓Power = heat removed per second divided by COP
✓Invert COP = Tc/(Th - Tc) to find Th from a given COP
✓An engine with efficiency η, run backwards, has COP = (1 - η)/η

OlympIQ trap

Do not call COP an efficiency or expect it to be below 1. A refrigerator's COP is usually well above 1 because it moves more heat than the work it consumes; the smaller the temperature gap, the larger the COP.

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