A Carnot engine operates between temperatures of 5℃ and 500℃. The output is used to run a Carnot refrigerator operating between -5℃ and 25℃. How many joules of heat energy does the refrigerator exhaust into the room for each joule of heat energy used by the heat engine?

Verified step by step guidance

1. Convert all temperatures from Celsius to Kelvin for both the Carnot engine and the refrigerator. Use the formula: Temperature in Kelvin = Temperature in Celsius + 273.15.
2. Calculate the efficiency of the Carnot engine using the formula: \(\eta_{engine} = 1 - \frac{T_{cold, engine}}{T_{hot, engine}}\), where \(T_{cold, engine}\) and \(T_{hot, engine}\) are the cold and hot reservoir temperatures in Kelvin, respectively.
3. Calculate the coefficient of performance (COP) of the Carnot refrigerator using the formula: \(COP_{refrigerator} = \frac{T_{cold, refrigerator}}{T_{hot, refrigerator} - T_{cold, refrigerator}}\), where \(T_{cold, refrigerator}\) and \(T_{hot, refrigerator}\) are the cold and hot reservoir temperatures in Kelvin, respectively.
4. Determine the amount of heat energy the refrigerator uses by multiplying the output of the Carnot engine by the efficiency of the engine. This gives the work input to the refrigerator.
5. Calculate the heat exhausted into the room by the refrigerator using the formula: \(Q_{exhausted} = COP_{refrigerator} \times W_{input}\), where \(W_{input}\) is the work input to the refrigerator.

Key Concepts

Carnot Engine

A Carnot engine is an idealized heat engine that operates on the Carnot cycle, which is the most efficient cycle possible between two temperature reservoirs. It converts heat energy from a hot reservoir into work while rejecting some heat to a cold reservoir. The efficiency of a Carnot engine depends solely on the temperatures of the hot and cold reservoirs, given by the formula η = 1 - (T_c/T_h), where T_c and T_h are the absolute temperatures of the cold and hot reservoirs, respectively.

Carnot Refrigerator

A Carnot refrigerator is an idealized refrigeration cycle that operates between two temperature reservoirs, transferring heat from a cold space to a hot space. Its efficiency is defined by the coefficient of performance (COP), which is the ratio of heat removed from the cold reservoir to the work input. The COP for a Carnot refrigerator is given by COP = T_c / (T_h - T_c), where T_c is the absolute temperature of the cold reservoir and T_h is that of the hot reservoir.

Heat Transfer and Energy Conservation

Heat transfer refers to the movement of thermal energy from one object or system to another due to a temperature difference. In the context of thermodynamic cycles, the principle of energy conservation states that energy cannot be created or destroyed, only transformed. This means that the heat energy input to the Carnot engine will be equal to the work done by the engine plus the heat rejected, which is crucial for calculating the heat exhausted by the refrigerator based on the energy used by the engine.