For a Carnot refrigerator, what is COP_R?

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Multiple Choice

For a Carnot refrigerator, what is COP_R?

Explanation:
The main idea is how much heat you can move from the cold space per unit of work you put in, for a Carnot refrigerator operating between the hot reservoir at T_hot and the cold reservoir at T_cold. COP_R is defined as the heat removed from the cold space divided by the work input: COP_R = Q_c / W. For a Carnot refrigerator, the heat interactions follow Q_h / Q_c = T_hot / T_cold. So Q_h = Q_c (T_hot / T_cold). The work input is W = Q_h − Q_c = Q_c (T_hot / T_cold − 1) = Q_c (T_hot − T_cold) / T_cold. Therefore COP_R = Q_c / W = Q_c / [Q_c (T_hot − T_cold) / T_cold] = T_cold / (T_hot − T_cold). This shows COP_R increases as the temperature difference shrinks, and temperatures should be in absolute units (Kelvin) for correct results. The expression T_hot / (T_hot − T_cold) corresponds to the COP of a Carnot heat pump (COP_HP = Q_h / W), not the refrigerator. For a refrigerator, the correct form uses the cold temperature in the numerator: COP_R = T_cold / (T_hot − T_cold). For example, with T_hot = 320 K and T_cold = 300 K, COP_R = 300 / (320 − 300) = 300 / 20 = 15.

The main idea is how much heat you can move from the cold space per unit of work you put in, for a Carnot refrigerator operating between the hot reservoir at T_hot and the cold reservoir at T_cold. COP_R is defined as the heat removed from the cold space divided by the work input: COP_R = Q_c / W.

For a Carnot refrigerator, the heat interactions follow Q_h / Q_c = T_hot / T_cold. So Q_h = Q_c (T_hot / T_cold). The work input is W = Q_h − Q_c = Q_c (T_hot / T_cold − 1) = Q_c (T_hot − T_cold) / T_cold. Therefore COP_R = Q_c / W = Q_c / [Q_c (T_hot − T_cold) / T_cold] = T_cold / (T_hot − T_cold).

This shows COP_R increases as the temperature difference shrinks, and temperatures should be in absolute units (Kelvin) for correct results. The expression T_hot / (T_hot − T_cold) corresponds to the COP of a Carnot heat pump (COP_HP = Q_h / W), not the refrigerator. For a refrigerator, the correct form uses the cold temperature in the numerator: COP_R = T_cold / (T_hot − T_cold).

For example, with T_hot = 320 K and T_cold = 300 K, COP_R = 300 / (320 − 300) = 300 / 20 = 15.

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