Coefficient of performance

The coefficient of performance (COP) is the ratio of the heat a refrigeration device usefully moves to the work it consumes. For a refrigerator, COP = Q_c/W; for a heat pump, COP = Q_h/W. It replaces 'efficiency' because a refrigerator produces no work, so work-out-over-heat-in is meaningless for it. Applying the Carnot heat ratio Q_h/Q_c = T_h/T_c gives the ceilings COP_fridge ≤ T_c/(T_h − T_c) and COP_pump ≤ T_h/(T_h − T_c), attainable only by a reversible cycle.

A COP greater than 1 is routine and is not a violation of anything. A domestic freezer at −18 °C in a 25 °C kitchen has a reversible ceiling near 5.9: six joules of heat moved per joule of electricity. No energy is created — the machine is a pump, not a source, and there is no law requiring that relocating a joule cost as much as the joule itself. The two COPs of a single machine differ by exactly one, COP_pump = COP_fridge + 1, which is the first law read from the warm side.

The ceiling depends only on the two temperatures, and its behaviour at the extremes is instructive. As T_h → T_c the ceiling diverges: moving heat across no gradient is free. As T_c → 0 the fridge ceiling collapses toward zero, so each joule extracted costs more work than the last, without bound. That divergence in cost is the third law of thermodynamics appearing, unbidden, in a formula about kitchen appliances. Second-law efficiency — actual COP divided by the Carnot COP — is the honest way to grade a real device; domestic units score 0.4–0.5.

The concept follows directly from Sadi Carnot's 1824 analysis, though Carnot himself wrote about engines rather than refrigerators. The term and its regulatory use are twentieth-century: COP and its seasonal variants (SEER, SCOP) are now the legal efficiency labels on appliances in most jurisdictions, which makes the coefficient of performance one of the few quantities in this branch to appear on a sticker in a shop.