Section 25.6 Second Law Bootcamp
Subsection 25.6.1 Heat Engine
Problem 25.6.1. Efficiency of a Steam Engine.
Follow the link: Checkpoint 25.2.2.
Problem 25.6.2. Work Produced and Efficiecy of a Thermal Engine.
Follow the link: Checkpoint 25.2.3.
Subsection 25.6.2 Carnot Engine
Problem 25.6.3. The Efficiency of a Carnot Engine.
Follow the link: Checkpoint 25.3.2.
Problem 25.6.4. Heat Flow in a Carnot Engine.
Follow the link: Checkpoint 25.3.3.
Problem 25.6.5. Power of a Carnot Engine.
Follow the link: Checkpoint 25.3.4.
Problem 25.6.6. Carnot Cycle with Monatomic Ideal Gas.
Follow the link: Checkpoint 25.3.5.
Problem 25.6.7. Numerical Study of Carnot Cycle.
Follow the link: Checkpoint 25.3.6.
Subsection 25.6.3 Carnot Refrigerator
Problem 25.6.8. Freezing Water in a Refrigerator.
Follow the link: Checkpoint 25.4.2.
Problem 25.6.9. Performance of a Kitchen Refrigerator.
Follow the link: Checkpoint 25.4.3.
Problem 25.6.10. Work Required to Cool a Bucket of Water.
Follow the link: Checkpoint 25.4.4.
Problem 25.6.11. Power to Maintain Temperature in a Carnot Refrigerator.
Follow the link: Checkpoint 25.4.5.
Subsection 25.6.4 Real Engines
Problem 25.6.12. The Real Efficiency of a Power Plant.
Follow the link: Checkpoint 25.5.1.
Problem 25.6.13. Efficiency of a Coal Power Plant.
Follow the link: Checkpoint 25.5.2.
Subsection 25.6.5 Miscellaneous
Problem 25.6.14. Carnot Engine Providing Energy to Run an Electric Motor.
An engine working in a Carnot cycle between two heat baths of temperatures \(600\text{ K}\) and \(273\text{ K}\) runs an electric motor that uses \(60\text{ W}\) of power. If each cycle is completed in 10 seconds, how much heat does the engine absorb from the higher-temperature bath in each cycle?
Use \(W = P\Delta t\) to find the work the engine should produce.
\(1100\ \text{J}\text{.}\)
The efficiency
The work output in each cycle is
Therefore
Problem 25.6.15. Using a Carnot Engine to Run a Carnot Refrigerator.
A Carnot cycle working between \(100^{\circ}\text{C}\) and \(30^{\circ}\text{C}\) is used to drive a refrigerator between \(-10^{\circ}\text{C}\) and \(30^{\circ}\text{C}\text{.}\) (a) How much energy must the Carnot engine produce per second so that the refrigerator is able to discard 10 Joules of energy per second? (b) How much heat does the Carnot engine take in each second from its hot bath source to produce the work required to run the refrigerator at \(10\text{ J/s}\) rate?
First find the work required to discard 10 J.
\(8.1\text{ J}\)
(a) The definition of the coefficient of performance of the refrigerator gives the work required to extract \(Q_C\) from the refrigerator.
The coefficient of performance \(\beta\) for a Carnot refrigerator can be computed from the temperatures inside and outside.
Therefore we need the engine to supply the following energy per second.
(b) Since we need \(1.52\text{ J}\) of energy from the engine we need
Thus, in the end we need to spend \(8.1\text{J}\) of thermal energy from the ultimate source in the engine/refrigerator combined system to take out \(10\text{ J}\) of energy from inside the refrigerator assuming the temperature of the bath at \(30^{\circ}\text{C}\) does not change.
Problem 25.6.16. Carnot Engine Driving a Carnot Refrigerator.
The work output of a Carnot engine operating between temperatures \(T_1\) and \(T_2\) with \(T_1>T_2\) is used to drive a refrigerator between temperatures \(T_3\) and \(T_4\) where \(T_3>T_4\text{.}\) Find the ratio of heat taken from thermal baths \(T_1\) and \(T_4 \) in terms of the four temperatures.
Work in one cycle of engine will be the work used by the refrigerator.
\(\left(T_3/T_4 -1 \right)/\left(1-T_2/T_1 \right)\text{.}\)
In this problem, we have a Carnot engine and a Carnot refrigrator. Let us denote the quantities of refrigerator by a prime and those of the engine without a prime. We wish the work produced by the engine to be equal to the work used by the refrigerator.
Dividing both sides by $Q_H Q'_C$ gives
Carnot cycles have
Therefore,