AP Physics 2 Exam Question

A Carnot heat engine operates with a reservoir at a high temperature of 600 K and a reservoir at a low temperature of 300 K. The engine operates between these two temperatures with an efficiency of 50%.

a) Calculate the entropy change of the hot reservoir, cold reservoir, and the overall entropy change in the Carnot heat engine.

b) The heat engine absorbs 10,000 J of heat from the hot reservoir. Determine the amount of work done by the engine and the amount of heat rejected to the cold reservoir.

Solution

a) The entropy change of the hot reservoir can be calculated using the equation ∆S_hot = Q_hot / T_hot, where ∆S_hot is the entropy change of the hot reservoir, Q_hot is the amount of heat gained by the hot reservoir, and T_hot is the temperature of the hot reservoir.

Given that the hot reservoir absorbs heat Q_hot = 10,000 J and T_hot = 600 K, we can calculate the entropy change of the hot reservoir:

∆S_hot = 10,000 J / 600 K = 16.67 J/K

The entropy change of the cold reservoir can be calculated similarly using the equation ∆S_cold = -Q_cold / T_cold, where ∆S_cold is the entropy change of the cold reservoir, Q_cold is the amount of heat lost by the cold reservoir, and T_cold is the temperature of the cold reservoir.

Since the Carnot heat engine is operating at an efficiency of 50%, we can use the equation η = 1 - (T_cold / T_hot) to find T_cold:

0.5 = 1 - (T_cold / 600 K)

T_cold = 0.5 * 600 K = 300 K

Now, using Q_hot = Q_cold + W, where W is the work done by the engine, we can find Q_cold:

Q_cold = Q_hot - W = 10,000 J - W

Substituting the heat engine efficiency equation and the known values, we have:

0.5 * 10,000 J = 10,000 J - W

W = 10,000 J * 0.5 = 5,000 J

Therefore, the work done by the engine is 5,000 J.

To calculate the overall entropy change in the Carnot heat engine, we can use the equation ∆S_engine = ∆S_hot + ∆S_cold:

∆S_engine = ∆S_hot + ∆S_cold = 16.67 J/K + (-16.67 J/K) = 0 J/K

Therefore, the overall entropy change in the Carnot heat engine is 0 J/K.

b) The amount of heat rejected to the cold reservoir can be found using the equation Q_cold = 10,000 J - W:

Q_cold = 10,000 J - 5,000 J = 5,000 J

Therefore, the amount of heat rejected to the cold reservoir is 5,000 J.

The amount of work done by the engine is already calculated in part a) and is equal to 5,000 J.

Thus, the amount of work done by the engine is 5,000 J and the amount of heat rejected to the cold reservoir is 5,000 J.