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Chapter 5: Problem 37

Prove that a cyclic device that violates the KelvinPlanck statement of the second law also violates the Clausius statement of the second law.

Short Answer

Expert verified
A device violating Kelvin-Planck implies heat transfer from cold to hot without work, contradicting Clausius, hence violating both statements.

Step by step solution

01

Understand the Kelvin-Planck Statement

The Kelvin-Planck statement of the second law of thermodynamics states that it is impossible for a cyclic process to convert heat completely into work without any other effect. Essentially, no heat engine can have 100% efficiency, as some energy will always be lost as waste heat to the surroundings.
02

Understand the Clausius Statement

The Clausius statement of the second law of thermodynamics states that it is impossible for a process to have the sole result of transferring heat from a cooler body to a hotter body without an external work input. This means heat does not naturally flow from cold to hot spontaneously.
03

Assume Violation of Kelvin-Planck Statement

Assume there is a device that violates the Kelvin-Planck statement by converting all absorbed heat into work without expelling any heat to a cooler reservoir. This implies 100% efficiency in a cyclic process.
04

Connect the Violation to the Clausius Statement

Use the hypothetical device from Step 3 to drive a heat pump that moves heat from a cold reservoir to a hot reservoir without work input. Since the device produces net work, it could sustain a process where heat only moves from cold to hot without external energy, violating the Clausius statement.
05

Conclude the Infeasibility

Since the hypothetical scenario allows the violation of the Clausius statement due to the assumed violation of the Kelvin-Planck statement, this demonstrates that the Kelvin-Planck violation leads to a logical contradiction with the second law of thermodynamics as expressed by the Clausius statement.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kelvin-Planck Statement
The Kelvin-Planck statement is a fundamental principle in the second law of thermodynamics. It asserts that it is impossible for any heat engine operating in a cycle to convert all the heat it absorbs from a hot reservoir entirely into work. In simpler terms, no engine can be 100% efficient. There will always be some waste energy given off as heat to a sink. This means that every time a heat engine produces work, it must also transfer some energy to a colder reservoir. This energy, often in the form of waste heat, is inevitable and highlights the natural energy dispersion that is central to the second law of thermodynamics.
Clausius Statement
The Clausius statement is another crucial aspect of the second law of thermodynamics. It proposes that it is impossible for heat to spontaneously flow from a colder body to a hotter one without any external work being done. Essentially, in the natural order of things, heat will only flow from hot to cold. To reverse this process, such as in a refrigerator or a heat pump, external energy must be supplied to "pump" the heat against its natural direction of flow. This statement underlies the everyday functionality of refrigerators, air conditioners, and heat pumps, which all rely on external work to move heat from cooler to warmer spaces.
Thermodynamic Cycles
Thermodynamic cycles are at the heart of many practical applications of the second law of thermodynamics. They describe a series of thermodynamic processes that return a system to its initial state, often to produce work. An example of such a cycle is the Carnot cycle, used to model the most efficient heat engine possible between two temperatures. Each cycle comprises processes such as isothermal expansion and compression, adiabatic expansion and compression, among others. The concept of cycles helps us understand why it's impossible to achieve 100% conversion of heat to work, emphasizing the Kelvin-Planck statement.
Heat Engines
A heat engine is a device that converts thermal energy into mechanical work by exploiting temperature differences between two reservoirs, commonly referred to as a heat source and a heat sink. The basic operation of a heat engine includes three main steps:
  • Absorption of heat from the heat source.
  • Conversion of part of this heat into work output.
  • Rejection of residual heat to the heat sink.
The efficiency of a heat engine is determined by the portion of absorbed heat that is converted into work. According to the Kelvin-Planck statement, no heat engine can achieve 100% efficiency due to inevitable waste heat. This enforced waste highlights the inescapability of energy dispersion, illustrating the limitations imposed by the second law of thermodynamics.

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Most popular questions from this chapter

After you have driven a car on a trip and it is back home, the car's engine has cooled down and thus is back to the state in which it started. What happened to all the energy released in the burning of gasoline? What happened to all the work the engine gave out?

A lawnmower tractor engine produces 18 hp using \(40 \mathrm{Btu} / \mathrm{s}\) of heat transfer from burning fuel. Find the thermal efficiency and the rate of heat transfer rejected to the ambient.

A \(10-\mathrm{m}^{3}\) tank of air at \(500 \mathrm{kPa}, 600 \mathrm{~K}\) acts as the high-temperature reservoir for a Carnot heat engine that rejects heat at \(300 \mathrm{~K}\). A temperature difference of \(25^{\circ} \mathrm{C}\) between the air tank and the Carnot cycle high temperature is needed to transfer the heat. The heat engine runs until the air temperature has dropped to \(400 \mathrm{~K}\) and then stops. Assume constant specific heat for air and find how much work is given out by the heat engine.

A car engine takes atmospheric air in at \(20^{\circ} \mathrm{C},\) no fuel, and exhausts the air at \(-20^{\circ} \mathrm{C}\), producing work in the process. What do the first and second laws say about that?

A temperature of about \(0.01 \mathrm{~K}\) can be achieved by magnetic cooling. In this process a strong magnetic field is imposed on a paramagnetic salt, maintained at \(1 \mathrm{~K}\) by transfer of energy to liquid helium boiling at low pressure. The salt is then thermally isolated from the helium, the magnetic field is removed, and the salt temperature drops. Assume that \(1 \mathrm{~mJ}\) is removed at an average temperature of \(0.1 \mathrm{~K}\) to the helium by a Carnot cycle heat pump. Find the work input to the heat pump and the COP with an ambient at \(300 \mathrm{~K}\).
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