Question

Does a refrigerator that has a higher COP necessarily have a higher second-law efficiency than one with a lower COP? Explain.

Short answer

Answer: No, a refrigerator with a higher COP does not necessarily have a higher second-law efficiency than one with a lower COP. The second-law efficiency depends on the comparison of the actual and ideal COPs.

Step-by-step solution

Understand the Coefficient of Performance (COP)

The Coefficient of Performance (COP) is a measure of a refrigerator's efficiency. It is defined as the ratio of the desired effect (e.g., cooling) to the energy input required to achieve that effect. For refrigerators, the COP is given by the formula: \[ COP = \frac{Q_{L}}{W} \] where \(Q_{L}\) is the amount of heat removed from the low-temperature reservoir, and \(W\) is the work input required.

Understand the Second-law Efficiency

The second-law efficiency is an indicator of how close a device is to its maximum possible efficiency (i.e., its efficiency compared to an ideal device operating under the same conditions). It is the ratio of the actual output to the maximum possible output given the same energy input and is given by the formula: \[ \eta _{II} = \frac{\eta _{actual}}{\eta _{ideal}} \] where \(\eta _{actual}\) is the actual efficiency and \(\eta _{ideal}\) is the ideal efficiency.

Relationship between COP and Second-law Efficiency

To find out if a refrigerator with a higher COP necessarily has a higher second-law efficiency than one with a lower COP, let's examine the relationship between these two efficiencies. The ideal efficiency for a refrigerator is determined by the Carnot cycle, which is given by: \[ COP_{ideal} = \frac{T_L}{T_H - T_L} \] where \(T_L\) is the low-temperature reservoir (COP is highest at lower absolute temperatures), and \(T_H\) is the high-temperature reservoir. As a result, the second-law efficiency can be written as: \[ \eta _{II} = \frac{COP_{actual}}{COP_{ideal}} \]

Conclusion

The relationship between the actual COP and the second-law efficiency depends on the actual and ideal COP values. Therefore, a refrigerator with a higher COP does not necessarily have a higher second-law efficiency than one with a lower COP, as it depends on the comparison of the actual and ideal COPs.