Question

(a) A Carnot engine operates between a hot reservoir at \(322 \mathrm{~K}\) and a cold reservoir at \(258 \mathrm{~K}\). If it absorbs \(568 \mathrm{~J}\) of heat per cycle at the hot reservoir, how much work per cycle does it deliver? (b) If the same engine, working in reverse, functions as a refrigerator between the same two reservoirs, how much work per cycle must be supplied to transfer \(1230 \mathrm{~J}\) of heat from the cold reservoir?

Short answer

For the first part, the work done by the Carnot engine can be calculated using the formula for efficiency of a Carnot engine and the absorbed heat. The work done by the engine working as a refrigerator for the second part can be found using the formula for Coefficient of Performance and the absorbed heat.

Step-by-step solution

Calculation of work done by the Carnot Engine

The efficiency \(\eta\) of a Carnot engine is given by, \(\eta=1-\frac{T_c}{T_h}\), where \(T_h\) and \(T_c\) are the absolute temperatures of the hot and cold reservoir respectively. In the problem, the engine absorbs \(Q_h=568J\) of heat from the hot reservoir. The work \(W\) delivered by the engine per cycle is given by: \(W=Q_h \cdot \eta\). Plugging in the values: \(T_h=322K\), \(T_c=258K\) and \(Q_h=568J\) into the equation, the work done can be calculated.

Calculation of work done by the Carnot Refrigerator

When operating as a fridge, the Carnot cycle runs in reverse. The Coefficient of Performance (COP) for a refrigerator is given by \(COP_r=\frac{T_c}{T_h-T_c}\). When the Carnot engine works as a fridge, the work \(W_r\) required per cycle is given by, \(W_r=\frac{Q_c}{COP_r}\), where \(Q_c\) is the heat absorbed from the cold reservoir. The heat absorbed from the cold reservoir is given to be \(Q_c = 1230J\). Therefore, we can calculate the work done by plugging the values \(T_h=322K\), \(T_c=258K\) and \(Q_c=1230J\) into the formula.

Key concepts

Thermodynamics

Thermodynamics is a branch of physics focused on understanding how heat is transferred and transformed into different energy forms. At the heart of this field are the laws of thermodynamics which govern the principles of heat energy and its interaction with matter. The first law, often referred to as the law of energy conservation, asserts that energy cannot be created or destroyed, only transformed. The second law introduces the concept of entropy, suggesting that energy systems tend to move towards disorder and that in any energy transfer, some energy is lost as heat to the surroundings. Understanding these principles is key when analyzing systems like heat engines, which convert thermal energy into work.

In the context of the given exercise, thermodynamics helps us describe the operation of a Carnot engine, identifying the relationships between the heat absorbed, the work done, and the temperatures of the hot and cold reservoirs. By applying thermodynamics principles, one can deduce the amount of work a heat engine can deliver or the work needed to function as a refrigerator.

Heat Engines

Heat engines are devices that convert heat energy into mechanical work, often through the expansion of gases and fluids. The fundamental operation of these engines is to absorb heat from a high temperature source (the 'hot reservoir'), convert some of this energy to work, and release the remaining heat to a lower temperature sink (the 'cold reservoir').

A crucial aspect is the efficiency of these engines, which reflects the ratio of the work done to the heat energy absorbed from the hot reservoir. The efficiency depends on the temperatures of the two reservoirs but will always be less than 100% due to the second law of thermodynamics. In our exercise example, when the Carnot engine absorbs 568 J of heat, we apply these concepts to calculate how much work is converted from the absorbed heat.

Carnot Cycle

The Carnot Cycle represents an idealized model for heat engine performance, consisting of two isothermal processes (at constant temperature) and two adiabatic processes (no heat exchange). During the isothermal expansion, the engine does work while absorbing heat from the hot reservoir. In the adiabatic expansion and compression phases, the engine's temperature changes without heat exchange. Finally, during isothermal compression, the engine expels heat to the cold reservoir.

The Carnot Cycle is reversible, which means it can operate as a refrigerator or heat pump when reversed. The efficiency of a Carnot engine only depends on the temperatures of the hot and cold reservoirs and represents the maximum possible efficiency that any heat engine operating between these temperatures could achieve. Given the temperatures provided in the problem, we can determine the exact efficiency of the Carnot Cycle to calculate the work output.

Efficiency of Heat Engines

The efficiency of a heat engine is defined as the ratio of work output to heat input, represented by the formula \(\eta = \frac{W}{Q_h}\), where \(W\) is the work produced and \(Q_h\) is the heat absorbed from the hot reservoir. For a Carnot engine, the efficiency depends on the temperatures of the hot and cold reservoirs and is expressed as \(\eta = 1 - \frac{T_c}{T_h}\).

This equation is pivotal in solving part (a) of our exercise. By substituting in the given temperature values, we obtain the engine's efficiency and then calculate the work produced by multiplying this efficiency by the heat input. For refrigerators and heat pumps, we talk about the Coefficient of Performance (COP) instead of efficiency, which measures how effectively a refrigerator transfers heat from the cold reservoir relative to the work input. We use similar principles to address part (b) of the exercise, which concerns the engine in its role as a refrigerator.