A Carnot engine extracts \(890 \mathrm{J}\) from a \(550 \mathrm{K}\) reservoir during each cycle and rejects \(470 \mathrm{J}\) to a cooler reservoir. It operates at 22 cycles per second. Find (a) the work done during each cycle, (b) its efficiency, (c) the temperature of the cool reservoir, and (d) its mechanical power output.

Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy and physical properties of matter. The laws of thermodynamics govern the principles of heat transfer and the conversion of energy from one form to another. At the heart of these laws is the concept of energy conservation and the tendency for systems to evolve toward a state of equilibrium.

A critical aspect of thermodynamics is the study of thermal engines, like the Carnot engine, which are designed to convert heat energy into mechanical work. These engines operate between two thermal reservoirs—hot and cold—and perform work by transferring heat from the hot reservoir to the cold one. The theoretical model of a Carnot engine sets an upper limit on the efficiency that any engine operating between two temperatures can achieve.

The efficiency of a Carnot cycle is a measure of how well the engine converts heat into work. It is defined as the ratio of the work output to the heat input from the hot reservoir. Due to its reversible and thus idealized nature, the Carnot cycle is the most efficient cycle possible for a heat engine.

The formula for the efficiency (\(η\)) of a Carnot engine is \(η = 1 - \frac{T_c}{T_h}\), where \(T_h\) is the absolute temperature of the hot reservoir and \(T_c\) is the absolute temperature of the cool reservoir. In practical problems, efficiency can also be calculated by dividing the work done per cycle (\(W\)) by the heat input (\(Q_h\)): \(η = \frac{W}{Q_h}\). However, it's essential to remember that no real engine can achieve the Carnot efficiency due to irreversibilities and practical constraints.

Heat reservoirs are key components of thermal engines, acting as sources and sinks of thermal energy. A hot reservoir provides the thermal energy needed to be converted into mechanical work, while a cold reservoir serves as the recipient for the engine's waste heat.

In the context of the Carnot engine, the effectiveness of energy transfer and the engine's performance are significantly influenced by the temperatures of these reservoirs. The primary characteristic of an idealized heat reservoir is that its temperature remains constant despite heat transfer. This ideal assumption simplifies calculations and helps in understanding the theoretical bounds of engine efficiency.

The mechanical power output of a heat engine is the rate at which work is performed by the engine. It is an essential metric because it directly relates to the amount of usable energy the engine can generate over time.

For a Carnot engine, the mechanical power output (\(P\)) can be calculated by multiplying the work done per cycle (\(W\)) by the number of cycles that occur each second (frequency, \(f\)): \(P = W \times f\). In the given exercise, the engine's power output is 9.24 kW, which signifies how much work the engine can do in a given second. Understanding how to calculate power output is crucial because it helps in designing systems that require a certain amount of work within a specified time frame, such as electricity generators.