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University Physics Volume 2

3.6 Adiabatic Processes for an Ideal Gas

University Physics Volume 23.6 Adiabatic Processes for an Ideal Gas
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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

By the end of this section, you will be able to:
  • Define adiabatic expansion of an ideal gas
  • Demonstrate the qualitative difference between adiabatic and isothermal expansions

When an ideal gas is compressed adiabatically (Q=0),(Q=0), work is done on it and its temperature increases; in an adiabatic expansion, the gas does work and its temperature drops. Adiabatic compressions actually occur in the cylinders of a car, where the compressions of the gas-air mixture take place so quickly that there is no time for the mixture to exchange heat with its environment. Nevertheless, because work is done on the mixture during the compression, its temperature does rise significantly. In fact, the temperature increases can be so large that the mixture can explode without the addition of a spark. Such explosions, since they are not timed, make a car run poorly—it usually “knocks.” Because ignition temperature rises with the octane of gasoline, one way to overcome this problem is to use a higher-octane gasoline.

Another interesting adiabatic process is the free expansion of a gas. Figure 3.13 shows a gas confined by a membrane to one side of a two-compartment, thermally insulated container. When the membrane is punctured, gas rushes into the empty side of the container, thereby expanding freely. Because the gas expands “against a vacuum” (p=0)(p=0), it does no work, and because the vessel is thermally insulated, the expansion is adiabatic. With Q=0Q=0 and W=0W=0 in the first law, ΔEint=0,ΔEint=0, so Einti=EintfEinti=Eintf for the free expansion.

The figure on the left is an illustration of the initial equilibrium state of a container with a partition in the middle dividing it into two chambers.  The outer walls are insulated. The chamber on the left is full of gas, indicated by blue shading and many small dots representing the gas molecules. The right chamber is empty. The figure on the right is an illustration of the final equilibrium state of the container. The partition has a hole in it. The entire container, on both sides of the partition, is full of gas, indicated by blue shading and many small dots representing the gas molecules. The dots in the second, final equilibrium state, illustration are less dense than in the first, initial state illustration.
Figure 3.13 The gas in the left chamber expands freely into the right chamber when the membrane is punctured.

If the gas is ideal, the internal energy depends only on the temperature. Therefore, when an ideal gas expands freely, its temperature does not change.

A quasi-static, adiabatic expansion of an ideal gas is represented in Figure 3.14, which shows an insulated cylinder that contains 1 mol of an ideal gas. The gas is made to expand quasi-statically by removing one grain of sand at a time from the top of the piston. When the gas expands by dV, the change in its temperature is dT. The work done by the gas in the expansion is dW=pdV;dQ=0dW=pdV;dQ=0 because the cylinder is insulated; and the change in the internal energy of the gas is, from Equation 3.9, dEint=CVndT.dEint=CVndT. Therefore, from the first law,

CVndT=0pdV=pdV,CVndT=0pdV=pdV,

so

dT=pdVCVn.dT=pdVCVn.
The figure is an illustration of a container. The walls and bottom are filled with a thick layer of insulation. The chamber of the container is closed from above by a piston. Inside the chamber is a gas. There is a pile of sand on top of the piston, and a hand with tweezers is removing grains from the pile.
Figure 3.14 When sand is removed from the piston one grain at a time, the gas expands adiabatically and quasi-statically in the insulated vessel.

Also, for 1 mol of an ideal gas,

d(pV)=d(RnT),d(pV)=d(RnT),

so

pdV+Vdp=RndTpdV+Vdp=RndT

and

dT=pdV+VdpRn.dT=pdV+VdpRn.

We now have two equations for dT. Upon equating them, we find that

CVnVdp+(CVn+Rn)pdV=0.CVnVdp+(CVn+Rn)pdV=0.

Now, we divide this equation by npV and use Cp=CV+RCp=CV+R. We are then left with

CVdpp+CpdVV=0,CVdpp+CpdVV=0,

which becomes

dpp+γdVV=0,dpp+γdVV=0,

where we define γγ as the ratio of the molar heat capacities:

γ=CpCV.γ=CpCV.
(3.11)

Thus,

dpp+γdVV=0dpp+γdVV=0

and

lnp+γlnV=constant.lnp+γlnV=constant.

Finally, using ln(Ax)=xlnAandlnAB=lnA+lnBln(Ax)=xlnAandlnAB=lnA+lnB, we can write this in the form

pVγ=constant.pVγ=constant.
(3.12)

This equation is the condition that must be obeyed by an ideal gas in a quasi-static adiabatic process. For example, if an ideal gas makes a quasi-static adiabatic transition from a state with pressure and volume p1p1 and V1V1 to a state with p2p2 and V2,V2, then it must be true that p1V1γ=p2V2γ.p1V1γ=p2V2γ.

The adiabatic condition of Equation 3.12 can be written in terms of other pairs of thermodynamic variables by combining it with the ideal gas law. In doing this, we find that

p1γTγ=constantp1γTγ=constant
(3.13)

and

TVγ1=constant.TVγ1=constant.
(3.14)

A reversible adiabatic expansion of an ideal gas is represented on the pV diagram of Figure 3.15. The slope of the curve at any point is

dpdV=ddV(constantVγ)=γpV.dpdV=ddV(constantVγ)=γpV.
The figure is a plot of pressure, p on the vertical axis as a function of volume, V on the horizontal axis. Two curves are plotted. Both are monotonically decreasing and concave up.  One is slightly higher and has a greater curvature. This curve is labeled  “isothermal.” The second curve is below the isothermal curve and has  a slightly smaller curvature. This curve is labeled “adiabatic.”
Figure 3.15 Quasi-static adiabatic and isothermal expansions of an ideal gas.

The dashed curve shown on this pV diagram represents an isothermal expansion where T (and therefore pV) is constant. The slope of this curve is useful when we consider the second law of thermodynamics in the next chapter. This slope is

dpdV=ddVnRTV=pV.dpdV=ddVnRTV=pV.

Because γ>1,γ>1, the isothermal curve is not as steep as that for the adiabatic expansion.

Example 3.7

Compression of an Ideal Gas in an Automobile Engine Gasoline vapor is injected into the cylinder of an automobile engine when the piston is in its expanded position. The temperature, pressure, and volume of the resulting gas-air mixture are 20°C20°C, 1.00×105N/m2,1.00×105N/m2, and 240cm3240cm3, respectively. The mixture is then compressed adiabatically to a volume of 40cm340cm3. Note that in the actual operation of an automobile engine, the compression is not quasi-static, although we are making that assumption here. (a) What are the pressure and temperature of the mixture after the compression? (b) How much work is done by the mixture during the compression?

Strategy Because we are modeling the process as a quasi-static adiabatic compression of an ideal gas, we have pVγ=constantpVγ=constant and pV=nRTpV=nRT. The work needed can then be evaluated with W=V1V2pdVW=V1V2pdV.

Solution

  1. For an adiabatic compression we have
    p2=p1(V1V2)γ,p2=p1(V1V2)γ,

    so after the compression, the pressure of the mixture is
    p2=(1.00×105N/m2)(240×10−6m340×10−6m3)1.40=1.23×106N/m2.p2=(1.00×105N/m2)(240×10−6m340×10−6m3)1.40=1.23×106N/m2.

    From the ideal gas law, the temperature of the mixture after the compression is
    T2=(p2V2p1V1)T1=(1.23×106N/m2)(40×10−6m3)(1.00×105N/m2)(240×10−6m3)·293K=600K=328°C.T2=(p2V2p1V1)T1=(1.23×106N/m2)(40×10−6m3)(1.00×105N/m2)(240×10−6m3)·293K=600K=328°C.
  2. The work done by the mixture during the compression is
    W=V1V2pdV.W=V1V2pdV.

    With the adiabatic condition of Equation 3.12, we may write p as K/Vγ,K/Vγ, where K=p1V1γ=p2V2γ.K=p1V1γ=p2V2γ. The work is therefore
    W=V1V2KVγdV=K1γ(1V2γ11V1γ1)=11γ(p2V2γV2γ1p1V1γV1γ1)=11γ(p2V2p1V1)=111.40[(1.23×106N/m2)(40×10−6m3)(1.00×105N/m2)(240×10−6m3)]=−63J.W=V1V2KVγdV=K1γ(1V2γ11V1γ1)=11γ(p2V2γV2γ1p1V1γV1γ1)=11γ(p2V2p1V1)=111.40[(1.23×106N/m2)(40×10−6m3)(1.00×105N/m2)(240×10−6m3)]=−63J.

Significance The negative sign on the work done indicates that the piston does work on the gas-air mixture. The engine would not work if the gas-air mixture did work on the piston.

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