Question:

A heat engine operates between temperatures T_h and T_c. The engine absorbs heat Q_h from the hot reservoir at temperature T_h and releases heat Q_c to the cold reservoir at temperature T_c. The engine also performs work W. The efficiency of this heat engine is represented by the Carnot efficiency, given by the equation:

η = 1 - T_c/T_h

1. Define entropy and explain how it relates to the Carnot efficiency of a heat engine.
2. A heat engine operates between a hot reservoir with a temperature of 600 K and a cold reservoir with a temperature of 300 K. If the engine absorbs 20 J of heat from the hot reservoir and its efficiency is 40%, calculate the work done by the engine.

Answer:

1. Entropy is a thermodynamic property that describes the randomness or disorder in a system. In a simplified sense, entropy can be thought of as a measure of how energy is dispersed or spread out within a system. The second law of thermodynamics states that for any process to occur, the total entropy of an isolated system must either increase or remain constant. This means that as energy flows through a heat engine, the total entropy of the system (including the engine and the reservoirs it interacts with) will either increase or remain constant.

The Carnot efficiency of a heat engine can be related to entropy by considering the maximum possible efficiency a heat engine can achieve, which is given by the Carnot efficiency equation:

η = 1 - T_c/T_h

where T_c is the temperature of the cold reservoir and T_h is the temperature of the hot reservoir. This equation shows that the efficiency of a heat engine is dependent on the temperature difference between the hot and cold reservoirs. The greater the temperature difference, the higher the efficiency. Entropy plays a role in this relationship because the increase or conservation of entropy within the system determines the maximum possible efficiency (given by the Carnot efficiency) that can be achieved.

2. Given information:
   • T_h = 600 K (temperature of the hot reservoir)
   • T_c = 300 K (temperature of the cold reservoir)
   • Q_h = 20 J (heat absorbed from the hot reservoir)
   • Efficiency (η) = 40% = 0.40

To calculate the work done by the engine, we can use the equation for efficiency, η = W / Q_h. Rearranging the equation, we get:

W = η * Q_h

Substituting the given values, we have:

W = 0.40 * 20 J

W = 8 J

Therefore, the work done by the engine is 8 J.