A Carnot refrigerator is operated between two heat reservoirs at temperatures of 320 K and 270 K. (b) If the refrigerator completes 165 cycles each minute, what power input is required to operate it?

Verified step by step guidance

1. Calculate the coefficient of performance (COP) for the Carnot refrigerator using the formula: \( COP = \frac{T_{low}}{T_{high} - T_{low}} \), where \( T_{low} \) is the lower temperature reservoir (270 K) and \( T_{high} \) is the higher temperature reservoir (320 K).
2. Determine the amount of heat extracted from the cold reservoir per cycle. This can be calculated using the formula: \( Q_{cold} = COP \times W \), where \( W \) is the work input per cycle.
3. Calculate the total heat extracted from the cold reservoir per minute by multiplying the heat extracted per cycle by the number of cycles per minute (165 cycles).
4. Convert the total heat extracted per minute to power (since power is energy per unit time) by dividing the total heat extracted per minute by 60 seconds.
5. The power input required to operate the refrigerator is equal to the power calculated in the previous step, as the work done per unit time by the refrigerator is equivalent to the power input.

Key Concepts

Carnot Refrigerator

A Carnot refrigerator is an idealized refrigeration cycle that operates between two heat reservoirs, utilizing the principles of thermodynamics. It is defined by its efficiency, which depends on the temperatures of the hot and cold reservoirs. The efficiency is maximized when the temperature difference is minimized, making it a benchmark for real refrigerators.

Coefficient of Performance (COP)

The Coefficient of Performance (COP) is a measure of the efficiency of a refrigerator, defined as the ratio of heat removed from the cold reservoir to the work input required to operate the refrigerator. For a Carnot refrigerator, the COP can be calculated using the formula COP = T_c / (T_h - T_c), where T_c and T_h are the absolute temperatures of the cold and hot reservoirs, respectively.

Power Input

Power input refers to the rate at which work is done or energy is supplied to the refrigerator to maintain its operation. It can be calculated by multiplying the work done per cycle by the number of cycles per second. In the context of the Carnot refrigerator, the power input is essential for determining how much energy is needed to transfer heat from the cold reservoir to the hot reservoir.