Discussion
If \(Q_{L}\) represents the magnitude of heat transfer from a low temperature reservoir to a cyclic device and \(Q_{H}\) represents the magnitude of heat transfer from a cyclic device to a high temperature reservoir, then for the same \(Q_{L}\) and \(Q_{H}\), the coefficient performance of a refrigerator \((COP_{R})\) and the coefficient performance of a heat pump \((COP_{HP})\) can be related as
Answer: B
The relation is \(COP_{HP} = COP_{R} + 1\).
For the same device, with work input \(W = Q_{H} - Q_{L}\):
\[ \begin{aligned} COP_{R} &= \frac{Q_{L}}{Q_{H} - Q_{L}} \\ COP_{HP} &= \frac{Q_{H}}{Q_{H} - Q_{L}} = \frac{Q_{L} + (Q_{H} - Q_{L})}{Q_{H} - Q_{L}} = COP_{R} + 1 \end{aligned} \]
A heat pump delivers all the heat extracted from the cold side plus the work input, which is why its COP is always one more than the refrigerator's. The other relations do not follow; in particular the product of the two COPs is not 1, and \(COP_{HP}\) can never be less than \(COP_{R}\).
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