Coefficient of performance (COP) of a heat pump operating on the reversed Carnot cycle: choose the correct expression in terms of absolute temperatures.

Introduction / Context:
The reversed Carnot cycle is an idealized benchmark for heat pumps and refrigerators. Understanding the coefficient of performance (COP) formulas helps compare real systems to the theoretical maximum performance.

Given Data / Assumptions:

• Heat pump moves heat from a cold reservoir at Tc (K) to a hot reservoir at Th (K).
• Ideal (reversed Carnot) processes; no irreversibilities.
• Temperatures are absolute (Kelvin).

Concept / Approach:
For a reversed Carnot heat pump, the COP is defined as desired effect (heat delivered to hot space) divided by net work input. The Carnot relation gives a simple temperature-based expression.

Step-by-Step Solution:
For a heat pump: COP_HP = Qh / W.For reversed Carnot: Qh / Qc = Th / Tc and W = Qh - Qc.Therefore COP_HP = Qh / (Qh - Qc) = Th / (Th - Tc).Hence, the correct expression is COP_HP = Th / (Th - Tc).

Verification / Alternative check:
Refrigerator COP_RF = Qc / W = Tc / (Th - Tc); heat-pump COP exceeds refrigerator COP by 1 since COP_HP = COP_RF + 1.

Why Other Options Are Wrong:

• COP_HP = Tc / (Th - Tc) is for a refrigerator, not a heat pump.
• (Th - Tc)/Th and Th/Tc are not COPs for Carnot heat pumps.
• 1/(Th - Tc) lacks units consistency and temperature scaling.

Common Pitfalls:
Using Celsius instead of Kelvin; mixing up heat pump and refrigerator COP expressions.

Final Answer:
COP_HP = Th / (Th - Tc)