AP Physics 2 Exam Question:

Consider a heat engine operating between a hot reservoir at temperature $T_h$ and a cold reservoir at temperature $T_c$. The engine absorbs $Q_h$ amount of heat from the hot reservoir and rejects $Q_c$ amount of heat to the cold reservoir. The engine also performs work, $W$, during each cycle.

a) Define entropy and explain how entropy changes in a heat engine.

b) Derive the efficiency of a heat engine in terms of the temperatures of the hot and cold reservoirs.

c) A heat engine operates between a hot reservoir at 400 K and a cold reservoir at 300 K. During each cycle, 500 J of heat is absorbed from the hot reservoir and 300 J of heat is rejected to the cold reservoir. Determine the work done by the engine during each cycle. Assume idealized conditions.

Answer:

a) Entropy can be understood as a measure of the randomness or disorder of a system. In the context of a heat engine, entropy is related to the distribution of energy. As the engine absorbs heat from the hot reservoir, the entropy of the hot reservoir decreases while the entropy of the engine increases. Similarly, as the engine rejects heat to the cold reservoir, the entropy of the cold reservoir increases while the entropy of the engine decreases. The total change in entropy of the engine and its surroundings combined must always be greater than or equal to zero, according to the Second Law of Thermodynamics.

b) The efficiency of a heat engine is defined as the ratio of the work done by the engine to the heat absorbed from the hot reservoir:

We can also express the heat rejected to the cold reservoir in terms of the heat absorbed from the hot reservoir and the work done by the engine:

Substituting this into the efficiency equation gives:

c) The efficiency equation can be further simplified using the given temperature values.

Given: $T_h = 400 \, \text{K}$, $T_c = 300 \, \text{K}$, $Q_h = 500 \, \text{J}$, $Q_c = 300 \, \text{J}$

Using the equation $\text{Efficiency} = \frac{W}{Q_h - Q_c}$, we can now substitute the given values:

The efficiency equation cannot be further simplified without knowing the value of work done by the engine.

The work done by the engine during each cycle can be determined using the First Law of Thermodynamics:

Substituting the given values:

Therefore, the work done by the engine during each cycle is 200 J.