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University Physics Volume 2

4.5 The Carnot Cycle

University Physics Volume 24.5 The Carnot Cycle
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  1. Preface
  2. Unit 1. Thermodynamics
    1. 1 Temperature and Heat
      1. Introduction
      2. 1.1 Temperature and Thermal Equilibrium
      3. 1.2 Thermometers and Temperature Scales
      4. 1.3 Thermal Expansion
      5. 1.4 Heat Transfer, Specific Heat, and Calorimetry
      6. 1.5 Phase Changes
      7. 1.6 Mechanisms of Heat Transfer
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 2 The Kinetic Theory of Gases
      1. Introduction
      2. 2.1 Molecular Model of an Ideal Gas
      3. 2.2 Pressure, Temperature, and RMS Speed
      4. 2.3 Heat Capacity and Equipartition of Energy
      5. 2.4 Distribution of Molecular Speeds
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 3 The First Law of Thermodynamics
      1. Introduction
      2. 3.1 Thermodynamic Systems
      3. 3.2 Work, Heat, and Internal Energy
      4. 3.3 First Law of Thermodynamics
      5. 3.4 Thermodynamic Processes
      6. 3.5 Heat Capacities of an Ideal Gas
      7. 3.6 Adiabatic Processes for an Ideal Gas
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 4 The Second Law of Thermodynamics
      1. Introduction
      2. 4.1 Reversible and Irreversible Processes
      3. 4.2 Heat Engines
      4. 4.3 Refrigerators and Heat Pumps
      5. 4.4 Statements of the Second Law of Thermodynamics
      6. 4.5 The Carnot Cycle
      7. 4.6 Entropy
      8. 4.7 Entropy on a Microscopic Scale
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  3. Unit 2. Electricity and Magnetism
    1. 5 Electric Charges and Fields
      1. Introduction
      2. 5.1 Electric Charge
      3. 5.2 Conductors, Insulators, and Charging by Induction
      4. 5.3 Coulomb's Law
      5. 5.4 Electric Field
      6. 5.5 Calculating Electric Fields of Charge Distributions
      7. 5.6 Electric Field Lines
      8. 5.7 Electric Dipoles
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    2. 6 Gauss's Law
      1. Introduction
      2. 6.1 Electric Flux
      3. 6.2 Explaining Gauss’s Law
      4. 6.3 Applying Gauss’s Law
      5. 6.4 Conductors in Electrostatic Equilibrium
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 7 Electric Potential
      1. Introduction
      2. 7.1 Electric Potential Energy
      3. 7.2 Electric Potential and Potential Difference
      4. 7.3 Calculations of Electric Potential
      5. 7.4 Determining Field from Potential
      6. 7.5 Equipotential Surfaces and Conductors
      7. 7.6 Applications of Electrostatics
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 8 Capacitance
      1. Introduction
      2. 8.1 Capacitors and Capacitance
      3. 8.2 Capacitors in Series and in Parallel
      4. 8.3 Energy Stored in a Capacitor
      5. 8.4 Capacitor with a Dielectric
      6. 8.5 Molecular Model of a Dielectric
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    5. 9 Current and Resistance
      1. Introduction
      2. 9.1 Electrical Current
      3. 9.2 Model of Conduction in Metals
      4. 9.3 Resistivity and Resistance
      5. 9.4 Ohm's Law
      6. 9.5 Electrical Energy and Power
      7. 9.6 Superconductors
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    6. 10 Direct-Current Circuits
      1. Introduction
      2. 10.1 Electromotive Force
      3. 10.2 Resistors in Series and Parallel
      4. 10.3 Kirchhoff's Rules
      5. 10.4 Electrical Measuring Instruments
      6. 10.5 RC Circuits
      7. 10.6 Household Wiring and Electrical Safety
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    7. 11 Magnetic Forces and Fields
      1. Introduction
      2. 11.1 Magnetism and Its Historical Discoveries
      3. 11.2 Magnetic Fields and Lines
      4. 11.3 Motion of a Charged Particle in a Magnetic Field
      5. 11.4 Magnetic Force on a Current-Carrying Conductor
      6. 11.5 Force and Torque on a Current Loop
      7. 11.6 The Hall Effect
      8. 11.7 Applications of Magnetic Forces and Fields
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    8. 12 Sources of Magnetic Fields
      1. Introduction
      2. 12.1 The Biot-Savart Law
      3. 12.2 Magnetic Field Due to a Thin Straight Wire
      4. 12.3 Magnetic Force between Two Parallel Currents
      5. 12.4 Magnetic Field of a Current Loop
      6. 12.5 Ampère’s Law
      7. 12.6 Solenoids and Toroids
      8. 12.7 Magnetism in Matter
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    9. 13 Electromagnetic Induction
      1. Introduction
      2. 13.1 Faraday’s Law
      3. 13.2 Lenz's Law
      4. 13.3 Motional Emf
      5. 13.4 Induced Electric Fields
      6. 13.5 Eddy Currents
      7. 13.6 Electric Generators and Back Emf
      8. 13.7 Applications of Electromagnetic Induction
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    10. 14 Inductance
      1. Introduction
      2. 14.1 Mutual Inductance
      3. 14.2 Self-Inductance and Inductors
      4. 14.3 Energy in a Magnetic Field
      5. 14.4 RL Circuits
      6. 14.5 Oscillations in an LC Circuit
      7. 14.6 RLC Series Circuits
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    11. 15 Alternating-Current Circuits
      1. Introduction
      2. 15.1 AC Sources
      3. 15.2 Simple AC Circuits
      4. 15.3 RLC Series Circuits with AC
      5. 15.4 Power in an AC Circuit
      6. 15.5 Resonance in an AC Circuit
      7. 15.6 Transformers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    12. 16 Electromagnetic Waves
      1. Introduction
      2. 16.1 Maxwell’s Equations and Electromagnetic Waves
      3. 16.2 Plane Electromagnetic Waves
      4. 16.3 Energy Carried by Electromagnetic Waves
      5. 16.4 Momentum and Radiation Pressure
      6. 16.5 The Electromagnetic Spectrum
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  4. A | Units
  5. B | Conversion Factors
  6. C | Fundamental Constants
  7. D | Astronomical Data
  8. E | Mathematical Formulas
  9. F | Chemistry
  10. G | The Greek Alphabet
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
    14. Chapter 14
    15. Chapter 15
    16. Chapter 16
  12. Index

Learning Objectives

  • Describe the Carnot cycle with the roles of all four processes involved
  • Outline the Carnot principle and its implications
  • Demonstrate the equivalence of the Carnot principle and the second law of thermodynamics

In the early 1820s, Sadi Carnot (1786−1832), a French engineer, became interested in improving the efficiencies of practical heat engines. In 1824, his studies led him to propose a hypothetical working cycle with the highest possible efficiency between the same two reservoirs, known now as the Carnot cycle. An engine operating in this cycle is called a Carnot engine. The Carnot cycle is of special importance for a variety of reasons. At a practical level, this cycle represents a reversible model for the steam power plant and the refrigerator or heat pump. Yet, it is also very important theoretically, for it plays a major role in the development of another important statement of the second law of thermodynamics. Finally, because only two reservoirs are involved in its operation, it can be used along with the second law of thermodynamics to define an absolute temperature scale that is truly independent of any substance used for temperature measurement.

With an ideal gas as the working substance, the steps of the Carnot cycle, as represented by Figure 4.11, are as follows.

  1. Isothermal expansion. The gas is placed in thermal contact with a heat reservoir at a temperature Th.Th. The gas absorbs heat QhQh from the heat reservoir and is allowed to expand isothermally, doing work W1.W1. Because the internal energy EintEint of an ideal gas is a function of the temperature only, the change of the internal energy is zero, that is, ΔEint=0ΔEint=0 during this isothermal expansion. With the first law of thermodynamics, ΔEint=QW,ΔEint=QW, we find that the heat absorbed by the gas is
    Qh=W1=nRThlnVNVM.Qh=W1=nRThlnVNVM.
    The figure shows four steps of Carnot cycle, namely isothermal expansion, adiabatic expansion, isothermal compression and adiabatic compression.
    Figure 4.11 The four processes of the Carnot cycle. The working substance is assumed to be an ideal gas whose thermodynamic path MNOP is represented in Figure 4.12.
    The first part of the figure shows a graph corresponding to four steps of Carnot cycle. The x-axis is V and y-axis is p. The second part shows a downward arrow Q subscript h at T subscript h which splits into a downward arrow Q subscript c at T subscript c and a right arrow W.
    Figure 4.12 The total work done by the gas in the Carnot cycle is shown and given by the area enclosed by the loop MNOPM.
  2. Adiabatic expansion. The gas is thermally isolated and allowed to expand further, doing work W2.W2. Because this expansion is adiabatic, the temperature of the gas falls—in this case, from ThtoTc.ThtoTc. From pVγ=constantpVγ=constant and the equation of state for an ideal gas, pV=nRTpV=nRT, we have
    TVγ1=constant,TVγ1=constant,

    so that
    ThVNγ1=TcVOγ1.ThVNγ1=TcVOγ1.
  3. Isothermal compression. The gas is placed in thermal contact with a cold reservoir at temperature TcTc and compressed isothermally. During this process, work W3W3 is done on the gas and it gives up heat QcQc to the cold reservoir. The reasoning used in step 1 now yields
    Qc=nRTclnVOVP,Qc=nRTclnVOVP,

    where QcQc is the heat dumped to the cold reservoir by the gas.
  4. Adiabatic compression. The gas is thermally isolated and returned to its initial state by compression. In this process, work W4W4 is done on the gas. Because the compression is adiabatic, the temperature of the gas rises—from TctoThTctoTh in this particular case. The reasoning of step 2 now gives
    TcVPγ1=ThVMγ1.TcVPγ1=ThVMγ1.

    The total work done by the gas in the Carnot cycle is given by
    W=W1+W2W3W4.W=W1+W2W3W4.

This work is equal to the area enclosed by the loop shown in the pV diagram of Figure 4.12. Because the initial and final states of the system are the same, the change of the internal energy of the gas in the cycle must be zero, that is, ΔEint=0ΔEint=0. The first law of thermodynamics then gives

W=QΔEint=(QhQc)0,W=QΔEint=(QhQc)0,

and

W=QhQc.W=QhQc.

To find the efficiency of this engine, we first divide QcbyQh:QcbyQh:

QcQh=TcThlnVO/VPlnVN/VM.QcQh=TcThlnVO/VPlnVN/VM.

When the adiabatic constant from step 2 is divided by that of step 4, we find

VOVP=VNVM.VOVP=VNVM.

Substituting this into the equation for Qc/Qh,Qc/Qh, we obtain

QcQh=TcTh.QcQh=TcTh.

Finally, with Equation 4.2, we find that the efficiency of this ideal gas Carnot engine is given by

e=1TcTh.e=1TcTh.
(4.5)

An engine does not necessarily have to follow a Carnot engine cycle. All engines, however, have the same net effect, namely the absorption of heat from a hot reservoir, the production of work, and the discarding of heat to a cold reservoir. This leads us to ask: Do all reversible cycles operating between the same two reservoirs have the same efficiency? The answer to this question comes from the second law of thermodynamics discussed earlier: All reversible engine cycles produce exactly the same efficiency. Also, as you might expect, all real engines operating between two reservoirs are less efficient than reversible engines operating between the same two reservoirs. This too is a consequence of the second law of thermodynamics shown earlier.

The cycle of an ideal gas Carnot refrigerator is represented by the pV diagram of Figure 4.13. It is a Carnot engine operating in reverse. The refrigerator extracts heat QcQc from a cold-temperature reservoir at TcTc when the ideal gas expands isothermally. The gas is then compressed adiabatically until its temperature reaches Th,Th, after which an isothermal compression of the gas results in heat QhQh being discarded to a high-temperature reservoir at Th.Th. Finally, the cycle is completed by an adiabatic expansion of the gas, causing its temperature to drop to Tc.Tc.

The first part of the figure shows a graph for one cycle of the Carnot refrigerator. The x-axis is V and y-axis is p. The second part shows an upward arrow Q subscript c at T subscript c which becomes arrow Q subscript h at T subscript h after arrow W is added from the left.
Figure 4.13 The work done on the gas in one cycle of the Carnot refrigerator is shown and given by the area enclosed by the loop MPONM.

The work done on the ideal gas is equal to the area enclosed by the path of the pV diagram. From the first law, this work is given by

W=QhQc.W=QhQc.

An analysis just like the analysis done for the Carnot engine gives

QcTc=QhTh.QcTc=QhTh.

When combined with Equation 4.3, this yields

KR=TcThTcKR=TcThTc
(4.6)

for the coefficient of performance of the ideal-gas Carnot refrigerator. Similarly, we can work out the coefficient of performance for a Carnot heat pump as

KP=QhQhQc=ThThTc.KP=QhQhQc=ThThTc.
(4.7)

We have just found equations representing the efficiency of a Carnot engine and the coefficient of performance of a Carnot refrigerator or a Carnot heat pump, assuming an ideal gas for the working substance in both devices. However, these equations are more general than their derivations imply. We will soon show that they are both valid no matter what the working substance is.

Carnot summarized his study of the Carnot engine and Carnot cycle into what is now known as Carnot’s principle:

Carnot’s Principle

No engine working between two reservoirs at constant temperatures can have a greater efficiency than a reversible engine.

This principle can be viewed as another statement of the second law of thermodynamics and can be shown to be equivalent to the Kelvin statement and the Clausius statement.

Example 4.2

The Carnot Engine A Carnot engine has an efficiency of 0.60 and the temperature of its cold reservoir is 300 K. (a) What is the temperature of the hot reservoir? (b) If the engine does 300 J of work per cycle, how much heat is removed from the high-temperature reservoir per cycle? (c) How much heat is exhausted to the low-temperature reservoir per cycle?

Strategy From the temperature dependence of the thermal efficiency of the Carnot engine, we can find the temperature of the hot reservoir. Then, from the definition of the efficiency, we can find the heat removed when the work done by the engine is given. Finally, energy conservation will lead to how much heat must be dumped to the cold reservoir.

Solution

  1. From e=1Tc/The=1Tc/Th we have
    0.60=1300KTh,0.60=1300KTh,

    so that the temperature of the hot reservoir is
    Th=300K10.60=750K.Th=300K10.60=750K.
  2. By definition, the efficiency of the engine is e=W/Qe=W/Q, so that the heat removed from the high-temperature reservoir per cycle is
    Qh=We=300J0.60=500J.Qh=We=300J0.60=500J.
  3. From the first law, the heat exhausted to the low-temperature reservoir per cycle by the engine is
    Qc=QhW=500J300J=200J.Qc=QhW=500J300J=200J.

Significance A Carnot engine has the maximum possible efficiency of converting heat into work between two reservoirs, but this does not necessarily mean it is 100%100% efficient. As the difference in temperatures of the hot and cold reservoir increases, the efficiency of a Carnot engine increases.

Example 4.3

A Carnot Heat Pump Imagine a Carnot heat pump operates between an outside temperature of 0°C0°C and an inside temperature of 20.0°C20.0°C. What is the work needed if the heat delivered to the inside of the house is 30.0 kJ?

Strategy Because the heat pump is assumed to be a Carnot pump, its performance coefficient is given by KP=Qh/W=Th/(ThTc).KP=Qh/W=Th/(ThTc). Thus, we can find the work W from the heat delivered Qh.Qh.

Solution The work needed is obtained from

W=Qh/KP=Qh(ThTc)/Th=30kJ×(293K273K)/293K=2kJ.W=Qh/KP=Qh(ThTc)/Th=30kJ×(293K273K)/293K=2kJ.

Significance We note that this work depends not only on the heat delivered to the house but also on the temperatures outside and inside. The dependence on the temperature outside makes them impractical to use in areas where the temperature is much colder outside than room temperature.

In terms of energy costs, the heat pump is a very economical means for heating buildings (Figure 4.14). Contrast this method with turning electrical energy directly into heat with resistive heating elements. In this case, one unit of electrical energy furnishes at most only one unit of heat. Unfortunately, heat pumps have problems that do limit their usefulness. They are quite expensive to purchase compared to resistive heating elements, and, as the performance coefficient for a Carnot heat pump shows, they become less effective as the outside temperature decreases. In fact, below about –10°C–10°C, the heat they furnish is less than the energy used to operate them.

The photo shows a heat pump.
Figure 4.14 A photograph of a heat pump (large box) located outside a house. This heat pump is located in a warm climate area, like the southern United States, since it would be far too inefficient located in the northern half of the United States. (credit: modification of work by Peter Stevens)
Check Your Understanding 4.3

A Carnot engine operates between reservoirs at 400°C400°C and 30°C30°C. (a) What is the efficiency of the engine? (b) If the engine does 5.0 J of work per cycle, how much heat per cycle does it absorb from the high-temperature reservoir? (c) How much heat per cycle does it exhaust to the cold-temperature reservoir? (d) What temperatures at the cold reservoir would give the minimum and maximum efficiency?

Check Your Understanding 4.4

A Carnot refrigerator operates between two heat reservoirs whose temperatures are 0°C0°C and 25°C25°C. (a) What is the coefficient of performance of the refrigerator? (b) If 200 J of work are done on the working substance per cycle, how much heat per cycle is extracted from the cold reservoir? (c) How much heat per cycle is discarded to the hot reservoir?

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