Derivation of Reverse Carnot Refrigerator COP:

• Definition of COP: The Coefficient of Performance of a refrigerator is the ratio of heat absorbed from the cold refrigerated space ($Q_1$) to the net work input ($W$):
  $$\text{COP} = \frac{Q_1}{W} = \frac{Q_1}{Q_2 - Q_1}$$
• For a reversible Carnot cycle operating between absolute temperatures $T_1$ (evaporator/low) and $T_2$ (condenser/high):
  $$\frac{Q_1}{Q_2} = \frac{T_1}{T_2}$$
  Substituting temperatures yields:
  $$\text{COP}_{\text{ref}} = \frac{T_1}{T_2 - T_1}$$
  (Note: $T_2 / (T_2 - T_1)$ is the COP of a heat pump, whereas $(T_2 - T_1) / T_2$ is the thermal efficiency of a Carnot heat engine).