Final answer:
Refrigerators, air conditioners, and heat pumps are most efficient when there's a small difference between the high and low temperatures in their cycle, as this allows for a higher COP and less energy input required by the compressor.
Step-by-step explanation:
The correct answer is that refrigerators, air conditioners, and heat pumps operate most cost-effectively when there's a small temperature difference between T₁ (the high temperature reservoir) and T₀ (the low temperature reservoir) because their coefficients of performance (COP) are higher. The COP is a measure of the efficiency of a refrigeration cycle or a heat pump and is given by the ratio of heat transferred to or from the room to the work input needed to transfer that heat. A smaller temperature difference allows for less work to be input (lower energy consumption), as the compressor doesn't need to work as hard to raise the temperature of the gas during the cycle.
During the compression phase, when the electrically driven compressor raises the temperature and pressure of the refrigerant gas, less energy is needed if the temperature difference it has to achieve is less. Secondly, when the refrigerant condenses and gives off heat to the room, a smaller temperature differential between the inside of the room and the refrigerant means less heat needs to be removed or added, leading to a more energy-efficient cycle. Therefore, systems are designed to work most efficiently when the temperature difference between the high temperature inside and the low temperature outside is minimized.