COP versus heat-engine efficiency Is the coefficient of performance (COP) simply the reciprocal of a heat engine’s thermal efficiency?

Introduction / Context:Students often conflate COP of refrigerators/heat pumps with thermal efficiency of heat engines. They are different ratios with different numerators and denominators.

Given Data / Assumptions:

• Heat engine: η = W_out / Q_in (0 < η < 1).
• Refrigerator: COP_R = Q_L / W_in.
• Heat pump: COP_HP = Q_H / W_in.

Concept / Approach:COPs can exceed 1 because the device moves heat using work input. There is no universal reciprocal relationship between COP and engine efficiency; they involve different heat terms at different reservoirs.

Step-by-Step Solution:Write definitions: η, COP_R, COP_HP.For Carnot devices: COP_R,Carnot = T_L / (T_H − T_L), COP_HP,Carnot = T_H / (T_H − T_L).Neither is simply 1/η_Carnot, which equals 1 / (1 − T_L/T_H).

Verification / Alternative check:Numerical example: T_H = 300 K, T_L = 270 K → η_Carnot = 1 − 270/300 = 0.1. Then COP_R,Carnot = 270/30 = 9, while 1/η_Carnot = 10. Not equal.

Why Other Options Are Wrong:

• True / “only” variants: they incorrectly assert a general reciprocal relation.
• “Only at Carnot limits”: not even then, as shown.

Common Pitfalls:Assuming all performance ratios invert each other; forgetting COP depends on temperature lift, not only on efficiency definitions.

Final Answer:False