A firebox is at 750 K, and the ambient temperature is 300 K. The efficiency of a Carnot engine d

Name: Problem 44, Chapter 18: Principles of Physics a Calculus Based Text
Uploaded: 2020-07-29T05:45:11-08:00
Duration: 7 min 9 s
Channel: Sanjeev Kumar

Question

A firebox is at $750 \mathrm{K}$, and the ambient temperature is $300 \mathrm{K}$. The efficiency of a Carnot engine doing $150 \mathrm{J}$ of work as it transports energy between these constanttemperature baths is $60.0 \% .$ The Carnot engine must take in energy $150 \mathrm{J} / 0.600=250 \mathrm{J}$ from the hot reservoir and must put out $100 \mathrm{J}$ of energy by heat into the environment. To follow Carnot's reasoning, suppose some other heat engine S could have an efficiency of $70.0 \%$. (a) Find the energy input and exhaust energy output of engine $\mathrm{S}$ as it does $150 \mathrm{J}$ of work. (b) Let engine $\mathrm{S}$ operate as in part (a) and run the Carnot engine in reverse between the same reservoirs. The output work of engine $\mathrm{S}$ is the input work for the Carnot refrigerator. Find the total energy transferred to or from the firebox and the total energy transferred to or from the environment as both engines operate together. (c) Explain how the results of parts (a) and (b) show that the Clausius statement of the second law of thermodynamics is violated. (d) Find the energy input and work output of engine $\mathrm{S}$ as it puts out exhaust energy of $100 \mathrm{J}$. Let engine $\mathrm{S}$ operate as in part (c) and contribute $150 \mathrm{J}$ of its work output to running the Carnot engine in reverse. Find (e) the total energy the firebox puts out as both engines operate together, (f) the total work output, and (g) the total energy transferred to the environment. (h) Explain how the results show that the Kelvin-Planck statement of the second law is violated. Therefore, our assumption about the efficiency of engine $S$ must be false. (i) Let the engines operate together through one cycle as in part (d). Find the change in entropy of the Universe. (j) Explain how the result of part (i) shows that the entropy statement of the second law is violated.

   A firebox is at $750 \mathrm{K}$, and the ambient temperature is $300 \mathrm{K}$. The efficiency of a Carnot engine doing $150 \mathrm{J}$ of work as it transports energy between these constanttemperature baths is $60.0 \% .$ The Carnot engine must take in energy $150 \mathrm{J} / 0.600=250 \mathrm{J}$ from the hot reservoir and must put out $100 \mathrm{J}$ of energy by heat into the environment. To follow Carnot's reasoning, suppose some other heat engine S could have an efficiency of $70.0 \%$. (a) Find the energy input and exhaust energy output of engine $\mathrm{S}$ as it does $150 \mathrm{J}$ of work. (b) Let engine $\mathrm{S}$ operate as in part (a) and run the Carnot engine in reverse between the same reservoirs. The output work of engine $\mathrm{S}$ is the input work for the Carnot refrigerator. Find the total energy transferred to or from the firebox and the total energy transferred to or from the environment as both engines operate together. (c) Explain how the results of parts (a) and (b) show that the Clausius statement of the second law of thermodynamics is violated. (d) Find the energy input and work output of engine $\mathrm{S}$ as it puts out exhaust energy of $100 \mathrm{J}$. Let engine $\mathrm{S}$ operate as in part
(c) and contribute $150 \mathrm{J}$ of its work output to running the Carnot engine in reverse. Find (e) the total energy the firebox puts out as both engines operate together,
(f) the total work output, and (g) the total energy transferred to the environment. (h) Explain how the results show that the Kelvin-Planck statement of the second law is violated. Therefore, our assumption about the efficiency of engine $S$ must be false. (i) Let the engines operate together through one cycle as in part (d). Find the change in entropy of the Universe. (j) Explain how the result of part (i) shows that the entropy statement of the second law is violated.

Principles of Physics a Calculus Based Text

Raymond A. Serway,… 5th Edition

Chapter 18, Problem 44

Instant Answer

Solved by Expert Sanjeev Kumar

07/29/2020

Step 1

In this case, $T_C = 300 \, K$ and $T_H = 750 \, K$, so the efficiency of the Carnot engine is $\eta = 1 - \frac{300}{750} = 0.6$ or $60\%$. Show more…

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A firebox is at $750 \mathrm{K}$, and the ambient temperature is $300 \mathrm{K}$. The efficiency of a Carnot engine doing $150 \mathrm{J}$ of work as it transports energy between these constanttemperature baths is $60.0 \% .$ The Carnot engine must take in energy $150 \mathrm{J} / 0.600=250 \mathrm{J}$ from the hot reservoir and must put out $100 \mathrm{J}$ of energy by heat into the environment. To follow Carnot's reasoning, suppose some other heat engine S could have an efficiency of $70.0 \%$. (a) Find the energy input and exhaust energy output of engine $\mathrm{S}$ as it does $150 \mathrm{J}$ of work. (b) Let engine $\mathrm{S}$ operate as in part (a) and run the Carnot engine in reverse between the same reservoirs. The output work of engine $\mathrm{S}$ is the input work for the Carnot refrigerator. Find the total energy transferred to or from the firebox and the total energy transferred to or from the environment as both engines operate together. (c) Explain how the results of parts (a) and (b) show that the Clausius statement of the second law of thermodynamics is violated. (d) Find the energy input and work output of engine $\mathrm{S}$ as it puts out exhaust energy of $100 \mathrm{J}$. Let engine $\mathrm{S}$ operate as in part (c) and contribute $150 \mathrm{J}$ of its work output to running the Carnot engine in reverse. Find (e) the total energy the firebox puts out as both engines operate together, (f) the total work output, and (g) the total energy transferred to the environment. (h) Explain how the results show that the Kelvin-Planck statement of the second law is violated. Therefore, our assumption about the efficiency of engine $S$ must be false. (i) Let the engines operate together through one cycle as in part (d). Find the change in entropy of the Universe. (j) Explain how the result of part (i) shows that the entropy statement of the second law is violated.

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Key Concepts

Carnot Engine

A Carnot engine is an idealized heat engine that operates on a reversible Carnot cycle between two thermal reservoirs. It serves as a theoretical model that establishes the maximum possible efficiency any heat engine can achieve when operating between two fixed temperatures, thus setting the benchmark for real engine performance.

Efficiency of Heat Engines

The efficiency of a heat engine is defined as the ratio of the work output to the heat input from the hot reservoir. This concept underscores the limitations imposed by the second law of thermodynamics, as even the ideal Carnot engine does not convert all the absorbed heat into work due to the inevitable expulsion of some energy as waste heat.

Clausius Statement of the Second Law

The Clausius statement asserts that heat cannot spontaneously flow from a colder body to a hotter body without external work. This principle implies that any process claiming to transport thermal energy in the reverse direction of natural heat flow—i.e., from cold to hot—without an external work input, is thermodynamically impossible.

Kelvin-Planck Statement of the Second Law

The Kelvin-Planck statement declares that it is impossible for any engine operating in a cyclic process to convert all the absorbed heat entirely into work, with no other effect. This principle emphasizes that some waste heat must always be expelled to a lower temperature reservoir, reaffirming the inherent limitations of energy conversion in heat engines.

Entropy and the Second Law

Entropy is a measure of disorder or randomness in a system, and the second law of thermodynamics states that the total entropy of an isolated system can never decrease over time. In the context of cyclic processes, while individual engine components may show changes in entropy, the overall entropy change of the universe must be zero for reversible processes or positive for irreversible ones, ensuring that spontaneous processes lead to increased disorder.

Reversible Processes

Reversible processes are idealized processes that occur infinitely slowly, ensuring that the system remains in thermodynamic equilibrium at each step. They are key to understanding the benchmark performance of devices like the Carnot engine, as they represent the maximum possible efficiency and involve no generation of entropy, making them essential for the formulation of the theoretical limits dictated by the second law.

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