Question

The thermal efficiency of a reversible heat engine operating between two given thermal reservoirs is $0.4$. The device is used either as a refrigerator or as a heat pump between the same reservoirs. Then the coefficient of performance as a refrigerator $(\text{COP})_\text{R}$ and the coefficient of performance as a heat pump $(\text{COP})_\text{HP}$ are:

• A. $(\text{COP})_\text{R} = (\text{COP})_\text{HP} = 0.6$
• B. $(\text{COP})_\text{R} = 2.5; (\text{COP})_\text{HP} = 1.5$
• C. $(\text{COP})_\text{R} = 1.5; (\text{COP})_\text{HP} = 2.5$
• D. $(\text{COP})_\text{R} = (\text{COP})_\text{HP} = 2.5$

Hint:

For any cyclic device operating between the same reservoirs, the heat pump $COP$ is always exactly $1$ greater than the refrigerator $COP$: $COP_{\text{HP}} - COP_{\text{R}} = 1$.

Answer 1

The Correct Option is C

Concept: For reversible cyclic devices operating between the same two thermal reservoirs at temperatures $T_{\text{H}}$ and $T_{\text{L}}$, there are direct algebraic relationships connecting the engine efficiency ($\eta$), the refrigerator coefficient of performance ($COP_{\text{R}}$), and the heat pump coefficient of performance ($COP_{\text{HP}}$): 1) The $COP$ of a reversible heat pump is the reciprocal of the thermal efficiency of the corresponding heat engine: \[ COP_{\text{HP}} = \frac{1}{\eta} \] 2) The relationship between a heat pump and a refrigerator operating between the same limits is: \[ COP_{\text{HP}} = COP_{\text{R}} + 1 \implies COP_{\text{R}} = COP_{\text{HP}} - 1 \]

Step 1: Calculate the Coefficient of Performance of the Heat Pump $(\text{COP})_\text{HP}$.
We are given the thermal efficiency $\eta = 0.4$. Applying the reciprocal relationship: \[ COP_{\text{HP}} = \frac{1}{0.4} = \frac{10}{4} = 2.5 \]

Step 2: Calculate the Coefficient of Performance of the Refrigerator $(\text{COP})_\text{R}$.
Using the relationship between the two coefficients: \[ COP_{\text{R}} = COP_{\text{HP}} - 1 = 2.5 - 1 = 1.5 \] Thus, we find $(\text{COP})_\text{R} = 1.5$ and $(\text{COP})_\text{HP} = 2.5$. This matches Option C.