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University Physics Volume 2

1.3 Thermal Expansion

University Physics Volume 21.3 Thermal Expansion
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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:
  • Answer qualitative questions about the effects of thermal expansion
  • Solve problems involving thermal expansion, including those involving thermal stress

The expansion of alcohol in a thermometer is one of many commonly encountered examples of thermal expansion, which is the change in size or volume of a given system as its temperature changes. The most visible example is the expansion of hot air. When air is heated, it expands and becomes less dense than the surrounding air, which then exerts an (upward) force on the hot air and makes steam and smoke rise, hot air balloons float, and so forth. The same behavior happens in all liquids and gases, driving natural heat transfer upward in homes, oceans, and weather systems, as we will discuss in an upcoming section. Solids also undergo thermal expansion. Railroad tracks and bridges, for example, have expansion joints to allow them to freely expand and contract with temperature changes, as shown in Figure 1.5.

Photograph a shows an expansion joint as a small gap on a road. Photograph b shows Auckland Harbour Bridge.
Figure 1.5 (a) Thermal expansion joints like these in the (b) Auckland Harbour Bridge in New Zealand allow bridges to change length without buckling. (credit: modification of works by “ŠJů”/Wikimedia Commons)

What is the underlying cause of thermal expansion? As previously mentioned, an increase in temperature means an increase in the kinetic energy of individual atoms. In a solid, unlike in a gas, the molecules are held in place by forces from neighboring molecules; as we saw in Oscillations, the forces can be modeled as in harmonic springs described by the Lennard-Jones potential. Energy in Simple Harmonic Motion shows that such potentials are asymmetrical in that the potential energy increases more steeply when the molecules get closer to each other than when they get farther away. Thus, at a given kinetic energy, the distance moved is greater when neighbors move away from each other than when they move toward each other. The result is that increased kinetic energy (increased temperature) increases the average distance between molecules—the substance expands.

For most substances under ordinary conditions, it is an excellent approximation that there is no preferred direction (that is, the solid is “isotropic”), and an increase in temperature increases the solid’s size by a certain fraction in each dimension. Therefore, if the solid is free to expand or contract, its proportions stay the same; only its overall size changes.

Linear Thermal Expansion

According to experiments, the dependence of thermal expansion on temperature, substance, and original length is summarized in the equation

dLdT=αLdLdT=αL
(1.1)

where ΔLΔL is the change in length L,ΔTL,ΔT is the change in temperature, and αα is the coefficient of linear expansion, a material property that varies slightly with temperature. As αα is nearly constant and also very small, for practical purposes, we use the linear approximation:

ΔL=αLΔT.ΔL=αLΔT.
(1.2)

Table 1.2 lists representative values of the coefficient of linear expansion. As noted earlier, ΔTΔT is the same whether it is expressed in units of degrees Celsius or kelvins; thus, αα may have units of 1/°C1/°C or 1/K with the same value in either case. Approximating αα as a constant is quite accurate for small changes in temperature and sufficient for most practical purposes, even for large changes in temperature. We examine this approximation more closely in the next example.

Material Coefficient of Linear Expansion α(1/°C)α(1/°C) Coefficient of Volume Expansion β(1/°C)β(1/°C)
Solids
Aluminum 25×10−625×10−6 75×10−675×10−6
Brass 19×10−619×10−6 56×10−656×10−6
Copper 17×10−617×10−6 51×10−651×10−6
Gold 14×10−614×10−6 42×10−642×10−6
Iron or steel 12×10−612×10−6 35×10−635×10−6
Invar (nickel-iron alloy) 0.9×10−60.9×10−6 2.7×10−62.7×10−6
Lead 29×10−629×10−6 87×10−687×10−6
Silver 18×10−618×10−6 54×10−654×10−6
Glass (ordinary) 9×10−69×10−6 27×10−627×10−6
Glass (Pyrex®) 3×10−63×10−6 9×10−69×10−6
Quartz 0.4×10−60.4×10−6 1×10−61×10−6
Concrete, brick ~12×10−6~12×10−6 ~36×10−6~36×10−6
Marble (average) 2.5×10−62.5×10−6 7.5×10−67.5×10−6
Liquids
Ether 1650×10−61650×10−6
Ethyl alcohol 1100×10−61100×10−6
Gasoline 950×10−6950×10−6
Glycerin 500×10−6500×10−6
Mercury 180×10−6180×10−6
Water 210×10−6210×10−6
Gases
Air and most other gases at atmospheric pressure 3400×10−63400×10−6
Table 1.2 Thermal Expansion Coefficients

Thermal expansion is exploited in the bimetallic strip (Figure 1.6). This device can be used as a thermometer if the curving strip is attached to a pointer on a scale. It can also be used to automatically close or open a switch at a certain temperature, as in older or analog thermostats.

Figure a shows two vertical strips attached to each other. It is labeled T0. Figure b shows the same two strips bent towards the right. It is labeled T greater than T0.
Figure 1.6 The curvature of a bimetallic strip depends on temperature. (a) The strip is straight at the starting temperature, where its two components have the same length. (b) At a higher temperature, this strip bends to the right, because the metal on the left has expanded more than the metal on the right. At a lower temperature, the strip would bend to the left.

Example 1.2

Calculating Linear Thermal Expansion The main span of San Francisco’s Golden Gate Bridge is 1275 m long at its coldest. The bridge is exposed to temperatures ranging from 15°C15°C to 40°C40°C. What is its change in length between these temperatures? Assume that the bridge is made entirely of steel.

Strategy Use the equation for linear thermal expansion ΔL=αLΔTΔL=αLΔT to calculate the change in length, ΔLΔL. Use the coefficient of linear expansion αα for steel from Table 1.2, and note that the change in temperature ΔTΔT is 55°C.55°C.

Solution Substitute all of the known values into the equation to solve for ΔLΔL:

ΔL=αLΔT=(12×10−6°C)(1275m)(55°C)=0.84m.ΔL=αLΔT=(12×10−6°C)(1275m)(55°C)=0.84m.

Significance Although not large compared with the length of the bridge, this change in length is observable. It is generally spread over many expansion joints so that the expansion at each joint is small.

Thermal Expansion in Two and Three Dimensions

Unconstrained objects expand in all dimensions, as illustrated in Figure 1.7. That is, their areas and volumes, as well as their lengths, increase with temperature. Because the proportions stay the same, holes and container volumes also get larger with temperature. If you cut a hole in a metal plate, the remaining material will expand exactly as it would if the piece you removed were still in place. The piece would get bigger, so the hole must get bigger too.

Thermal Expansion in Two Dimensions

For small temperature changes, the change in area ΔAΔA is given by

ΔA=2αAΔTΔA=2αAΔT
(1.3)

where ΔAΔA is the change in area A,ΔTA,ΔT is the change in temperature, and αα is the coefficient of linear expansion, which varies slightly with temperature. (The derivation of this equation is analogous to that of the more important equation for three dimensions, below.)

Figure shows a circle inside a square. The circle is outlined by another, slightly bigger circle. The bigger circle is a dashed outline. Similarly, the square is outlined by a bigger, dashed square. Figure b is similar to figure a except that the inner circle is cut out of the square. Figure c is a cuboid surrounded by a bigger, dashed cuboid.
Figure 1.7 In general, objects expand in all directions as temperature increases. In these drawings, the original boundaries of the objects are shown with solid lines, and the expanded boundaries with dashed lines. (a) Area increases because both length and width increase. The area of a circular plug also increases. (b) If the plug is removed, the hole it leaves becomes larger with increasing temperature, just as if the expanding plug were still in place. (c) Volume also increases, because all three dimensions increase.

Thermal Expansion in Three Dimensions

The relationship between volume and temperature dVdTdVdT is given by dVdT=βVΔTdVdT=βVΔT, where ββ is the coefficient of volume expansion. As you can show in Exercise 1.60, β=3αβ=3α. This equation is usually written as

ΔV=βVΔT.ΔV=βVΔT.
(1.4)

Note that the values of ββ in Table 1.2 are equal to 3α3α except for rounding.

Volume expansion is defined for liquids, but linear and area expansion are not, as a liquid’s changes in linear dimensions and area depend on the shape of its container. Thus, Table 1.2 shows liquids’ values of ββ but not αα.

In general, objects expand with increasing temperature. Water is the most important exception to this rule. Water does expand with increasing temperature (its density decreases) at temperatures greater than 4°C(40°F)4°C(40°F). However, it is densest at +4°C+4°C and expands with decreasing temperature between +4°C+4°C and 0°C0°C (40°Fto32°F40°Fto32°F), as shown in Figure 1.8. A striking effect of this phenomenon is the freezing of water in a pond. When water near the surface cools down to 4°C,4°C, it is denser than the remaining water and thus sinks to the bottom. This “turnover” leaves a layer of warmer water near the surface, which is then cooled. However, if the temperature in the surface layer drops below 4°C4°C, that water is less dense than the water below, and thus stays near the top. As a result, the pond surface can freeze over. The layer of ice insulates the liquid water below it from low air temperatures. Fish and other aquatic life can survive in 4°C4°C water beneath ice, due to this unusual characteristic of water.

Figure shows a graph of density of fresh water in grams per cubic centimeter versus temperature in degree Celsius. The graph starts at 0.99985 at 0 degrees and rises to a maximum y value of just under 1 at 4 degrees Celsius. It then curves down to 0.99950 at 12 degrees Celsius.
Figure 1.8 This curve shows the density of water as a function of temperature. Note that the thermal expansion at low temperatures is very small. The maximum density at 4°C4°C is only 0.0075%0.0075% greater than the density at 2°C2°C, and 0.012%0.012% greater than that at 0°C0°C. The decrease of density below 4°C4°C occurs because the liquid water approachs the solid crystal form of ice, which contains more empty space than the liquid.

Example 1.3

Calculating Thermal Expansion Suppose your 60.0-L (15.9-gal(15.9-gal-gal) steel gasoline tank is full of gas that is cool because it has just been pumped from an underground reservoir. Now, both the tank and the gasoline have a temperature of 15.0°C.15.0°C. How much gasoline has spilled by the time they warm to 35.0°C35.0°C?

Strategy The tank and gasoline increase in volume, but the gasoline increases more, so the amount spilled is the difference in their volume changes. We can use the equation for volume expansion to calculate the change in volume of the gasoline and of the tank. (The gasoline tank can be treated as solid steel.)

Solution

  1. Use the equation for volume expansion to calculate the increase in volume of the steel tank:
    ΔVs=βsVsΔT.ΔVs=βsVsΔT.
  2. The increase in volume of the gasoline is given by this equation:
    ΔVgas=βgasVgasΔT.ΔVgas=βgasVgasΔT.
  3. Find the difference in volume to determine the amount spilled as
    Vspill=ΔVgasΔVs.Vspill=ΔVgasΔVs.

Alternatively, we can combine these three equations into a single equation. (Note that the original volumes are equal.)

Vspill=(βgasβs)VΔT=[(95035)×10−6/°C](60.0L)(20.0°C)=1.10L.Vspill=(βgasβs)VΔT=[(95035)×10−6/°C](60.0L)(20.0°C)=1.10L.

Significance This amount is significant, particularly for a 60.0-L tank. The effect is so striking because the gasoline and steel expand quickly. The rate of change in thermal properties is discussed later in this chapter.

If you try to cap the tank tightly to prevent overflow, you will find that it leaks anyway, either around the cap or by bursting the tank. Tightly constricting the expanding gas is equivalent to compressing it, and both liquids and solids resist compression with extremely large forces. To avoid rupturing rigid containers, these containers have air gaps, which allow them to expand and contract without stressing them.

Check Your Understanding 1.1

Does a given reading on a gasoline gauge indicate more gasoline in cold weather or in hot weather, or does the temperature not matter?

Thermal Stress

If you change the temperature of an object while preventing it from expanding or contracting, the object is subjected to stress that is compressive if the object would expand in the absence of constraint and tensile if it would contract. This stress resulting from temperature changes is known as thermal stress. It can be quite large and can cause damage.

To avoid this stress, engineers may design components so they can expand and contract freely. For instance, in highways, gaps are deliberately left between blocks to prevent thermal stress from developing. When no gaps can be left, engineers must consider thermal stress in their designs. Thus, the reinforcing rods in concrete are made of steel because steel’s coefficient of linear expansion is nearly equal to that of concrete.

To calculate the thermal stress in a rod whose ends are both fixed rigidly, we can think of the stress as developing in two steps. First, let the ends be free to expand (or contract) and find the expansion (or contraction). Second, find the stress necessary to compress (or extend) the rod to its original length by the methods you studied in Static Equilibrium and Elasticity on static equilibrium and elasticity. In other words, the ΔLΔL of the thermal expansion equals the ΔLΔL of the elastic distortion (except that the signs are opposite).

Example 1.4

Calculating Thermal Stress Concrete blocks are laid out next to each other on a highway without any space between them, so they cannot expand. The construction crew did the work on a winter day when the temperature was 5°C5°C. Find the stress in the blocks on a hot summer day when the temperature is 38°C38°C. The compressive Young’s modulus of concrete is Y=20×109N/m2Y=20×109N/m2.

Strategy According to the chapter on static equilibrium and elasticity, the stress F/A is given by

FA=YΔLL0,FA=YΔLL0,

where Y is the Young’s modulus of the material—concrete, in this case. In thermal expansion, ΔL=αL0ΔT.ΔL=αL0ΔT. We combine these two equations by noting that the two ΔLsΔLs are equal, as stated above. Because we are not given L0L0 or A, we can obtain a numerical answer only if they both cancel out.

Solution We substitute the thermal-expansion equation into the elasticity equation to get

FA=YαL0ΔTL0=YαΔT,FA=YαL0ΔTL0=YαΔT,

and as we hoped, L0L0 has canceled and A appears only in F/A, the notation for the quantity we are calculating.

Now we need only insert the numbers:

FA=(20×109N/m2)(12×10−6/°C)(38°C5°C)=7.9×106N/m2.FA=(20×109N/m2)(12×10−6/°C)(38°C5°C)=7.9×106N/m2.

Significance The ultimate compressive strength of concrete is 20×106N/m2,20×106N/m2, so the blocks are unlikely to break. However, the ultimate shear strength of concrete is only 2×106N/m2,2×106N/m2, so some might chip off.

Check Your Understanding 1.2

Two objects A and B have the same dimensions and are constrained identically. A is made of a material with a higher thermal expansion coefficient than B. If the objects are heated identically, will A feel a greater stress than B?

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