A Carnot refrigerator is operated between two heat reservoirs at temperatures of 320 K and 270 K. (a) If in each cycle the refrigerator receives 415 J of heat energy from the reservoir at 270 K, how many joules of heat energy does it deliver to the reservoir at 320 K?

Verified step by step guidance

1. Identify the temperatures of the two heat reservoirs. The cold reservoir is at 270 K and the hot reservoir is at 320 K.
2. Calculate the coefficient of performance (COP) for the Carnot refrigerator using the formula: \( COP = \frac{T_{cold}}{T_{hot} - T_{cold}} \), where \( T_{cold} \) and \( T_{hot} \) are the temperatures of the cold and hot reservoirs, respectively.
3. Given that the refrigerator receives 415 J of heat energy from the cold reservoir, use the COP to calculate the work input required by the refrigerator. The formula is: \( W = \frac{Q_{cold}}{COP} \), where \( Q_{cold} \) is the heat extracted from the cold reservoir.
4. Calculate the total heat delivered to the hot reservoir using the energy conservation principle. The formula is: \( Q_{hot} = Q_{cold} + W \), where \( Q_{hot} \) is the heat delivered to the hot reservoir.
5. Summarize the process: The refrigerator extracts heat from the cold reservoir, work is done on the system to transfer this heat, and then the total heat (initial heat plus work done) is delivered to the hot reservoir.

Key Concepts

Carnot Cycle

The Carnot cycle is a theoretical thermodynamic cycle that provides the maximum possible efficiency for a heat engine or refrigerator operating between two heat reservoirs. It consists of four reversible processes: two isothermal (constant temperature) and two adiabatic (no heat exchange). Understanding this cycle is crucial for analyzing the performance of real-world refrigerators and heat engines.

Coefficient of Performance (COP)

The Coefficient of Performance (COP) is a measure of the efficiency of a refrigerator or heat pump, defined as the ratio of the heat removed from the cold reservoir to the work input. For a Carnot refrigerator, the COP can be calculated using the temperatures of the hot and cold reservoirs, indicating how effectively the refrigerator transfers heat. A higher COP signifies a more efficient refrigerator.

Heat Transfer

Heat transfer refers to the movement of thermal energy from one object or substance to another due to a temperature difference. In the context of the Carnot refrigerator, it involves the absorption of heat from the cold reservoir and the rejection of heat to the hot reservoir. Understanding the principles of heat transfer is essential for calculating the energy dynamics in thermodynamic systems.