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College Physics for AP® Courses

13.2 Thermal Expansion of Solids and Liquids

College Physics for AP® Courses13.2 Thermal Expansion of Solids and Liquids
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  1. Preface
  2. 1 Introduction: The Nature of Science and Physics
    1. Connection for AP® Courses
    2. 1.1 Physics: An Introduction
    3. 1.2 Physical Quantities and Units
    4. 1.3 Accuracy, Precision, and Significant Figures
    5. 1.4 Approximation
    6. Glossary
    7. Section Summary
    8. Conceptual Questions
    9. Problems & Exercises
  3. 2 Kinematics
    1. Connection for AP® Courses
    2. 2.1 Displacement
    3. 2.2 Vectors, Scalars, and Coordinate Systems
    4. 2.3 Time, Velocity, and Speed
    5. 2.4 Acceleration
    6. 2.5 Motion Equations for Constant Acceleration in One Dimension
    7. 2.6 Problem-Solving Basics for One Dimensional Kinematics
    8. 2.7 Falling Objects
    9. 2.8 Graphical Analysis of One Dimensional Motion
    10. Glossary
    11. Section Summary
    12. Conceptual Questions
    13. Problems & Exercises
    14. Test Prep for AP® Courses
  4. 3 Two-Dimensional Kinematics
    1. Connection for AP® Courses
    2. 3.1 Kinematics in Two Dimensions: An Introduction
    3. 3.2 Vector Addition and Subtraction: Graphical Methods
    4. 3.3 Vector Addition and Subtraction: Analytical Methods
    5. 3.4 Projectile Motion
    6. 3.5 Addition of Velocities
    7. Glossary
    8. Section Summary
    9. Conceptual Questions
    10. Problems & Exercises
    11. Test Prep for AP® Courses
  5. 4 Dynamics: Force and Newton's Laws of Motion
    1. Connection for AP® Courses
    2. 4.1 Development of Force Concept
    3. 4.2 Newton's First Law of Motion: Inertia
    4. 4.3 Newton's Second Law of Motion: Concept of a System
    5. 4.4 Newton's Third Law of Motion: Symmetry in Forces
    6. 4.5 Normal, Tension, and Other Examples of Force
    7. 4.6 Problem-Solving Strategies
    8. 4.7 Further Applications of Newton's Laws of Motion
    9. 4.8 Extended Topic: The Four Basic Forces—An Introduction
    10. Glossary
    11. Section Summary
    12. Conceptual Questions
    13. Problems & Exercises
    14. Test Prep for AP® Courses
  6. 5 Further Applications of Newton's Laws: Friction, Drag, and Elasticity
    1. Connection for AP® Courses
    2. 5.1 Friction
    3. 5.2 Drag Forces
    4. 5.3 Elasticity: Stress and Strain
    5. Glossary
    6. Section Summary
    7. Conceptual Questions
    8. Problems & Exercises
    9. Test Prep for AP® Courses
  7. 6 Gravitation and Uniform Circular Motion
    1. Connection for AP® Courses
    2. 6.1 Rotation Angle and Angular Velocity
    3. 6.2 Centripetal Acceleration
    4. 6.3 Centripetal Force
    5. 6.4 Fictitious Forces and Non-inertial Frames: The Coriolis Force
    6. 6.5 Newton's Universal Law of Gravitation
    7. 6.6 Satellites and Kepler's Laws: An Argument for Simplicity
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
    12. Test Prep for AP® Courses
  8. 7 Work, Energy, and Energy Resources
    1. Connection for AP® Courses
    2. 7.1 Work: The Scientific Definition
    3. 7.2 Kinetic Energy and the Work-Energy Theorem
    4. 7.3 Gravitational Potential Energy
    5. 7.4 Conservative Forces and Potential Energy
    6. 7.5 Nonconservative Forces
    7. 7.6 Conservation of Energy
    8. 7.7 Power
    9. 7.8 Work, Energy, and Power in Humans
    10. 7.9 World Energy Use
    11. Glossary
    12. Section Summary
    13. Conceptual Questions
    14. Problems & Exercises
    15. Test Prep for AP® Courses
  9. 8 Linear Momentum and Collisions
    1. Connection for AP® courses
    2. 8.1 Linear Momentum and Force
    3. 8.2 Impulse
    4. 8.3 Conservation of Momentum
    5. 8.4 Elastic Collisions in One Dimension
    6. 8.5 Inelastic Collisions in One Dimension
    7. 8.6 Collisions of Point Masses in Two Dimensions
    8. 8.7 Introduction to Rocket Propulsion
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  10. 9 Statics and Torque
    1. Connection for AP® Courses
    2. 9.1 The First Condition for Equilibrium
    3. 9.2 The Second Condition for Equilibrium
    4. 9.3 Stability
    5. 9.4 Applications of Statics, Including Problem-Solving Strategies
    6. 9.5 Simple Machines
    7. 9.6 Forces and Torques in Muscles and Joints
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
    12. Test Prep for AP® Courses
  11. 10 Rotational Motion and Angular Momentum
    1. Connection for AP® Courses
    2. 10.1 Angular Acceleration
    3. 10.2 Kinematics of Rotational Motion
    4. 10.3 Dynamics of Rotational Motion: Rotational Inertia
    5. 10.4 Rotational Kinetic Energy: Work and Energy Revisited
    6. 10.5 Angular Momentum and Its Conservation
    7. 10.6 Collisions of Extended Bodies in Two Dimensions
    8. 10.7 Gyroscopic Effects: Vector Aspects of Angular Momentum
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  12. 11 Fluid Statics
    1. Connection for AP® Courses
    2. 11.1 What Is a Fluid?
    3. 11.2 Density
    4. 11.3 Pressure
    5. 11.4 Variation of Pressure with Depth in a Fluid
    6. 11.5 Pascal’s Principle
    7. 11.6 Gauge Pressure, Absolute Pressure, and Pressure Measurement
    8. 11.7 Archimedes’ Principle
    9. 11.8 Cohesion and Adhesion in Liquids: Surface Tension and Capillary Action
    10. 11.9 Pressures in the Body
    11. Glossary
    12. Section Summary
    13. Conceptual Questions
    14. Problems & Exercises
    15. Test Prep for AP® Courses
  13. 12 Fluid Dynamics and Its Biological and Medical Applications
    1. Connection for AP® Courses
    2. 12.1 Flow Rate and Its Relation to Velocity
    3. 12.2 Bernoulli’s Equation
    4. 12.3 The Most General Applications of Bernoulli’s Equation
    5. 12.4 Viscosity and Laminar Flow; Poiseuille’s Law
    6. 12.5 The Onset of Turbulence
    7. 12.6 Motion of an Object in a Viscous Fluid
    8. 12.7 Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  14. 13 Temperature, Kinetic Theory, and the Gas Laws
    1. Connection for AP® Courses
    2. 13.1 Temperature
    3. 13.2 Thermal Expansion of Solids and Liquids
    4. 13.3 The Ideal Gas Law
    5. 13.4 Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature
    6. 13.5 Phase Changes
    7. 13.6 Humidity, Evaporation, and Boiling
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
    12. Test Prep for AP® Courses
  15. 14 Heat and Heat Transfer Methods
    1. Connection for AP® Courses
    2. 14.1 Heat
    3. 14.2 Temperature Change and Heat Capacity
    4. 14.3 Phase Change and Latent Heat
    5. 14.4 Heat Transfer Methods
    6. 14.5 Conduction
    7. 14.6 Convection
    8. 14.7 Radiation
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  16. 15 Thermodynamics
    1. Connection for AP® Courses
    2. 15.1 The First Law of Thermodynamics
    3. 15.2 The First Law of Thermodynamics and Some Simple Processes
    4. 15.3 Introduction to the Second Law of Thermodynamics: Heat Engines and Their Efficiency
    5. 15.4 Carnot’s Perfect Heat Engine: The Second Law of Thermodynamics Restated
    6. 15.5 Applications of Thermodynamics: Heat Pumps and Refrigerators
    7. 15.6 Entropy and the Second Law of Thermodynamics: Disorder and the Unavailability of Energy
    8. 15.7 Statistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  17. 16 Oscillatory Motion and Waves
    1. Connection for AP® Courses
    2. 16.1 Hooke’s Law: Stress and Strain Revisited
    3. 16.2 Period and Frequency in Oscillations
    4. 16.3 Simple Harmonic Motion: A Special Periodic Motion
    5. 16.4 The Simple Pendulum
    6. 16.5 Energy and the Simple Harmonic Oscillator
    7. 16.6 Uniform Circular Motion and Simple Harmonic Motion
    8. 16.7 Damped Harmonic Motion
    9. 16.8 Forced Oscillations and Resonance
    10. 16.9 Waves
    11. 16.10 Superposition and Interference
    12. 16.11 Energy in Waves: Intensity
    13. Glossary
    14. Section Summary
    15. Conceptual Questions
    16. Problems & Exercises
    17. Test Prep for AP® Courses
  18. 17 Physics of Hearing
    1. Connection for AP® Courses
    2. 17.1 Sound
    3. 17.2 Speed of Sound, Frequency, and Wavelength
    4. 17.3 Sound Intensity and Sound Level
    5. 17.4 Doppler Effect and Sonic Booms
    6. 17.5 Sound Interference and Resonance: Standing Waves in Air Columns
    7. 17.6 Hearing
    8. 17.7 Ultrasound
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  19. 18 Electric Charge and Electric Field
    1. Connection for AP® Courses
    2. 18.1 Static Electricity and Charge: Conservation of Charge
    3. 18.2 Conductors and Insulators
    4. 18.3 Conductors and Electric Fields in Static Equilibrium
    5. 18.4 Coulomb’s Law
    6. 18.5 Electric Field: Concept of a Field Revisited
    7. 18.6 Electric Field Lines: Multiple Charges
    8. 18.7 Electric Forces in Biology
    9. 18.8 Applications of Electrostatics
    10. Glossary
    11. Section Summary
    12. Conceptual Questions
    13. Problems & Exercises
    14. Test Prep for AP® Courses
  20. 19 Electric Potential and Electric Field
    1. Connection for AP® Courses
    2. 19.1 Electric Potential Energy: Potential Difference
    3. 19.2 Electric Potential in a Uniform Electric Field
    4. 19.3 Electrical Potential Due to a Point Charge
    5. 19.4 Equipotential Lines
    6. 19.5 Capacitors and Dielectrics
    7. 19.6 Capacitors in Series and Parallel
    8. 19.7 Energy Stored in Capacitors
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  21. 20 Electric Current, Resistance, and Ohm's Law
    1. Connection for AP® Courses
    2. 20.1 Current
    3. 20.2 Ohm’s Law: Resistance and Simple Circuits
    4. 20.3 Resistance and Resistivity
    5. 20.4 Electric Power and Energy
    6. 20.5 Alternating Current versus Direct Current
    7. 20.6 Electric Hazards and the Human Body
    8. 20.7 Nerve Conduction–Electrocardiograms
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  22. 21 Circuits, Bioelectricity, and DC Instruments
    1. Connection for AP® Courses
    2. 21.1 Resistors in Series and Parallel
    3. 21.2 Electromotive Force: Terminal Voltage
    4. 21.3 Kirchhoff’s Rules
    5. 21.4 DC Voltmeters and Ammeters
    6. 21.5 Null Measurements
    7. 21.6 DC Circuits Containing Resistors and Capacitors
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
    12. Test Prep for AP® Courses
  23. 22 Magnetism
    1. Connection for AP® Courses
    2. 22.1 Magnets
    3. 22.2 Ferromagnets and Electromagnets
    4. 22.3 Magnetic Fields and Magnetic Field Lines
    5. 22.4 Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field
    6. 22.5 Force on a Moving Charge in a Magnetic Field: Examples and Applications
    7. 22.6 The Hall Effect
    8. 22.7 Magnetic Force on a Current-Carrying Conductor
    9. 22.8 Torque on a Current Loop: Motors and Meters
    10. 22.9 Magnetic Fields Produced by Currents: Ampere’s Law
    11. 22.10 Magnetic Force between Two Parallel Conductors
    12. 22.11 More Applications of Magnetism
    13. Glossary
    14. Section Summary
    15. Conceptual Questions
    16. Problems & Exercises
    17. Test Prep for AP® Courses
  24. 23 Electromagnetic Induction, AC Circuits, and Electrical Technologies
    1. Connection for AP® Courses
    2. 23.1 Induced Emf and Magnetic Flux
    3. 23.2 Faraday’s Law of Induction: Lenz’s Law
    4. 23.3 Motional Emf
    5. 23.4 Eddy Currents and Magnetic Damping
    6. 23.5 Electric Generators
    7. 23.6 Back Emf
    8. 23.7 Transformers
    9. 23.8 Electrical Safety: Systems and Devices
    10. 23.9 Inductance
    11. 23.10 RL Circuits
    12. 23.11 Reactance, Inductive and Capacitive
    13. 23.12 RLC Series AC Circuits
    14. Glossary
    15. Section Summary
    16. Conceptual Questions
    17. Problems & Exercises
    18. Test Prep for AP® Courses
  25. 24 Electromagnetic Waves
    1. Connection for AP® Courses
    2. 24.1 Maxwell’s Equations: Electromagnetic Waves Predicted and Observed
    3. 24.2 Production of Electromagnetic Waves
    4. 24.3 The Electromagnetic Spectrum
    5. 24.4 Energy in Electromagnetic Waves
    6. Glossary
    7. Section Summary
    8. Conceptual Questions
    9. Problems & Exercises
    10. Test Prep for AP® Courses
  26. 25 Geometric Optics
    1. Connection for AP® Courses
    2. 25.1 The Ray Aspect of Light
    3. 25.2 The Law of Reflection
    4. 25.3 The Law of Refraction
    5. 25.4 Total Internal Reflection
    6. 25.5 Dispersion: The Rainbow and Prisms
    7. 25.6 Image Formation by Lenses
    8. 25.7 Image Formation by Mirrors
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  27. 26 Vision and Optical Instruments
    1. Connection for AP® Courses
    2. 26.1 Physics of the Eye
    3. 26.2 Vision Correction
    4. 26.3 Color and Color Vision
    5. 26.4 Microscopes
    6. 26.5 Telescopes
    7. 26.6 Aberrations
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
    12. Test Prep for AP® Courses
  28. 27 Wave Optics
    1. Connection for AP® Courses
    2. 27.1 The Wave Aspect of Light: Interference
    3. 27.2 Huygens's Principle: Diffraction
    4. 27.3 Young’s Double Slit Experiment
    5. 27.4 Multiple Slit Diffraction
    6. 27.5 Single Slit Diffraction
    7. 27.6 Limits of Resolution: The Rayleigh Criterion
    8. 27.7 Thin Film Interference
    9. 27.8 Polarization
    10. 27.9 *Extended Topic* Microscopy Enhanced by the Wave Characteristics of Light
    11. Glossary
    12. Section Summary
    13. Conceptual Questions
    14. Problems & Exercises
    15. Test Prep for AP® Courses
  29. 28 Special Relativity
    1. Connection for AP® Courses
    2. 28.1 Einstein’s Postulates
    3. 28.2 Simultaneity And Time Dilation
    4. 28.3 Length Contraction
    5. 28.4 Relativistic Addition of Velocities
    6. 28.5 Relativistic Momentum
    7. 28.6 Relativistic Energy
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
    12. Test Prep for AP® Courses
  30. 29 Introduction to Quantum Physics
    1. Connection for AP® Courses
    2. 29.1 Quantization of Energy
    3. 29.2 The Photoelectric Effect
    4. 29.3 Photon Energies and the Electromagnetic Spectrum
    5. 29.4 Photon Momentum
    6. 29.5 The Particle-Wave Duality
    7. 29.6 The Wave Nature of Matter
    8. 29.7 Probability: The Heisenberg Uncertainty Principle
    9. 29.8 The Particle-Wave Duality Reviewed
    10. Glossary
    11. Section Summary
    12. Conceptual Questions
    13. Problems & Exercises
    14. Test Prep for AP® Courses
  31. 30 Atomic Physics
    1. Connection for AP® Courses
    2. 30.1 Discovery of the Atom
    3. 30.2 Discovery of the Parts of the Atom: Electrons and Nuclei
    4. 30.3 Bohr’s Theory of the Hydrogen Atom
    5. 30.4 X Rays: Atomic Origins and Applications
    6. 30.5 Applications of Atomic Excitations and De-Excitations
    7. 30.6 The Wave Nature of Matter Causes Quantization
    8. 30.7 Patterns in Spectra Reveal More Quantization
    9. 30.8 Quantum Numbers and Rules
    10. 30.9 The Pauli Exclusion Principle
    11. Glossary
    12. Section Summary
    13. Conceptual Questions
    14. Problems & Exercises
    15. Test Prep for AP® Courses
  32. 31 Radioactivity and Nuclear Physics
    1. Connection for AP® Courses
    2. 31.1 Nuclear Radioactivity
    3. 31.2 Radiation Detection and Detectors
    4. 31.3 Substructure of the Nucleus
    5. 31.4 Nuclear Decay and Conservation Laws
    6. 31.5 Half-Life and Activity
    7. 31.6 Binding Energy
    8. 31.7 Tunneling
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  33. 32 Medical Applications of Nuclear Physics
    1. Connection for AP® Courses
    2. 32.1 Medical Imaging and Diagnostics
    3. 32.2 Biological Effects of Ionizing Radiation
    4. 32.3 Therapeutic Uses of Ionizing Radiation
    5. 32.4 Food Irradiation
    6. 32.5 Fusion
    7. 32.6 Fission
    8. 32.7 Nuclear Weapons
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
    13. Test Prep for AP® Courses
  34. 33 Particle Physics
    1. Connection for AP® Courses
    2. 33.1 The Yukawa Particle and the Heisenberg Uncertainty Principle Revisited
    3. 33.2 The Four Basic Forces
    4. 33.3 Accelerators Create Matter from Energy
    5. 33.4 Particles, Patterns, and Conservation Laws
    6. 33.5 Quarks: Is That All There Is?
    7. 33.6 GUTs: The Unification of Forces
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
    12. Test Prep for AP® Courses
  35. 34 Frontiers of Physics
    1. Connection for AP® Courses
    2. 34.1 Cosmology and Particle Physics
    3. 34.2 General Relativity and Quantum Gravity
    4. 34.3 Superstrings
    5. 34.4 Dark Matter and Closure
    6. 34.5 Complexity and Chaos
    7. 34.6 High-Temperature Superconductors
    8. 34.7 Some Questions We Know to Ask
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  36. A | Atomic Masses
  37. B | Selected Radioactive Isotopes
  38. C | Useful Information
  39. D | Glossary of Key Symbols and Notation
  40. 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
    17. Chapter 17
    18. Chapter 18
    19. Chapter 19
    20. Chapter 20
    21. Chapter 21
    22. Chapter 22
    23. Chapter 23
    24. Chapter 24
    25. Chapter 25
    26. Chapter 26
    27. Chapter 27
    28. Chapter 28
    29. Chapter 29
    30. Chapter 30
    31. Chapter 31
    32. Chapter 32
    33. Chapter 33
    34. Chapter 34
  41. Index

Learning Objectives

By the end of this section, you will be able to:

  • Define and describe thermal expansion.
  • Calculate the linear expansion of an object given its initial length, change in temperature, and coefficient of linear expansion.
  • Calculate the volume expansion of an object given its initial volume, change in temperature, and coefficient of volume expansion.
  • Calculate thermal stress on an object given its original volume, temperature change, volume change, and bulk modulus.
Figure 13.10 Thermal expansion joints like these in the Auckland Harbour Bridge in New Zealand allow bridges to change length without buckling. (credit: Ingolfson, Wikimedia Commons)

The expansion of alcohol in a thermometer is one of many commonly encountered examples of thermal expansion, the change in size or volume of a given mass with temperature. Hot air rises because its volume increases, which causes the hot air’s density to be smaller than the density of surrounding air, causing a buoyant (upward) force on the hot air. The same happens in all liquids and gases, driving natural heat transfer upwards in homes, oceans, and weather systems. 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.

What are the basic properties of thermal expansion? First, thermal expansion is clearly related to temperature change. The greater the temperature change, the more a bimetallic strip will bend. Second, it depends on the material. In a thermometer, for example, the expansion of alcohol is much greater than the expansion of the glass containing it.

What is the underlying cause of thermal expansion? As is discussed in Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature, an increase in temperature implies an increase in the kinetic energy of the individual atoms. In a solid, unlike in a gas, the atoms or molecules are closely packed together, but their kinetic energy (in the form of small, rapid vibrations) pushes neighboring atoms or molecules apart from each other. This neighbor-to-neighbor pushing results in a slightly greater distance, on average, between neighbors, and adds up to a larger size for the whole body. For most substances under ordinary conditions, there is no preferred direction, and an increase in temperature will increase the solid’s size by a certain fraction in each dimension.

Linear Thermal Expansion—Thermal Expansion in One Dimension

The change in length ΔLΔL size 12{ΔL} {} is proportional to length LL size 12{L} {}. The dependence of thermal expansion on temperature, substance, and length is summarized in the equation

ΔL=αLΔT,ΔL=αLΔT, size 12{ΔL=αLΔT} {}
13.7

where ΔLΔL size 12{ΔL} {} is the change in length LL size 12{L} {}, ΔTΔT size 12{ΔT} {} is the change in temperature, and αα size 12{α} {} is the coefficient of linear expansion, which varies slightly with temperature.

Table 13.2 lists representative values of the coefficient of linear expansion, which may have units of 1/ºC1/ºC size 12{1/°C} {} or 1/K. Because the size of a kelvin and a degree Celsius are the same, both αα size 12{α} {} and ΔTΔT size 12{ΔT} {} can be expressed in units of kelvins or degrees Celsius. The equation ΔL=αLΔT ΔL=αLΔT size 12{ΔL=αLΔT} {} is accurate for small changes in temperature and can be used for large changes in temperature if an average value of αα size 12{α} {} is used.

Material Coefficient of linear expansion α(1/ºC)α(1/ºC) size 12{α \( 1/°C \) } {} Coefficient of volume expansion β(1/ºC)β(1/ºC) size 12{β \( 1/°C \) } {}
Solids
Aluminum 25 × 10 6 25 × 10 6 size 12{"25" times "10" rSup { size 8{–6} } } {} 75 × 10 6 75 × 10 6 size 12{"75"´"10" rSup { size 8{ +- 6} } } {}
Brass 19 × 10 6 19 × 10 6 size 12{"19" times "10" rSup { size 8{–6} } } {} 56 × 10 6 56 × 10 6 size 12{"56"´"10" rSup { size 8{ +- 6} } } {}
Copper 17 × 10 6 17 × 10 6 size 12{"17" times "10" rSup { size 8{–6} } } {} 51 × 10 6 51 × 10 6 size 12{"51" times "10" rSup { size 8{–6} } } {}
Gold 14 × 10 6 14 × 10 6 size 12{"14" times "10" rSup { size 8{–6} } } {} 42 × 10 6 42 × 10 6 size 12{"42" times "10" rSup { size 8{–6} } } {}
Iron or Steel 12 × 10 6 12 × 10 6 size 12{"12" times "10" rSup { size 8{–6} } } {} 35 × 10 6 35 × 10 6 size 12{"35" times "10" rSup { size 8{–6} } } {}
Invar (Nickel-iron alloy) 0 . 9 × 10 6 0 . 9 × 10 6 size 12{0 "." 9 times "10" rSup { size 8{–6} } } {} 2 . 7 × 10 6 2 . 7 × 10 6 size 12{2 "." 7 times "10" rSup { size 8{–6} } } {}
Lead 29 × 10 6 29 × 10 6 size 12{"29" times "10" rSup { size 8{–6} } } {} 87 × 10 6 87 × 10 6 size 12{"87" times "10" rSup { size 8{–6} } } {}
Silver 18 × 10 6 18 × 10 6 size 12{"18" times "10" rSup { size 8{–6} } } {} 54 × 10 6 54 × 10 6 size 12{"54" times "10" rSup { size 8{–6} } } {}
Glass (ordinary) 9 × 10 6 9 × 10 6 size 12{9 times "10" rSup { size 8{–6} } } {} 27 × 10 6 27 × 10 6 size 12{"27" times "10" rSup { size 8{–6} } } {}
Glass (Pyrex®) 3 × 10 6 3 × 10 6 size 12{3 times "10" rSup { size 8{–6} } } {} 9 × 10 6 9 × 10 6 size 12{9 times "10" rSup { size 8{–6} } } {}
Quartz 0 . 4 × 10 6 0 . 4 × 10 6 size 12{0 "." 4´"10" rSup { size 8{ +- 6} } } {} 1 × 10 6 1 × 10 6 size 12{1 times "10" rSup { size 8{–6} } } {}
Concrete, Brick ~ 12 × 10 6 ~ 12 × 10 6 size 12{ "~" "12"´"10" rSup { size 8{ +- 6} } } {} ~ 36 × 10 6 ~ 36 × 10 6 size 12{ "~" "36" times "10" rSup { size 8{–6} } } {}
Marble (average) 2 . 5 × 10 6 2 . 5 × 10 6 size 12{2 "." 5´"10" rSup { size 8{ +- 6} } } {} 7 . 5 × 10 6 7 . 5 × 10 6 size 12{7 "." 5 times "10" rSup { size 8{–6} } } {}
Liquids
Ether 1650 × 10 6 1650 × 10 6 size 12{"1650" times "10" rSup { size 8{–6} } } {}
Ethyl alcohol 1100 × 10 6 1100 × 10 6 size 12{"1100" times "10" rSup { size 8{–6} } } {}
Petrol 950 × 10 6 950 × 10 6 size 12{"950" times "10" rSup { size 8{–6} } } {}
Glycerin 500 × 10 6 500 × 10 6 size 12{"500" times "10" rSup { size 8{–6} } } {}
Mercury 180 × 10 6 180 × 10 6 size 12{"180" times "10" rSup { size 8{–6} } } {}
Water 210 × 10 6 210 × 10 6 size 12{"210" times "10" rSup { size 8{–6} } } {}
Gases
Air and most other gases at atmospheric pressure 3400 × 10 6 3400 × 10 6 size 12{"3400" times "10" rSup { size 8{–6} } } {}
Table 13.2 Thermal Expansion Coefficients at 20ºC20ºC size 12{"20"°C} {}1

Example 13.3 Calculating Linear Thermal Expansion: The Golden Gate Bridge

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 15ºC15ºC size 12{–"15"°C} {} to 40ºC40ºC size 12{"40"°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 size 12{ΔL=αL`ΔT} {} to calculate the change in length , ΔLΔL size 12{ΔL} {}. Use the coefficient of linear expansion, αα size 12{α} {}, for steel from Table 13.2, and note that the change in temperature, ΔTΔT size 12{ΔT} {}, is 55ºC55ºC size 12{"55"°C} {}.

Solution

Plug all of the known values into the equation to solve for ΔLΔL size 12{ΔL} {}.

ΔL=αLΔT=12×106ºC1275 m55ºC=0.84 m.ΔL=αLΔT=12×106ºC1275 m55ºC=0.84 m. size 12{ΔL=αLΔT= left ( { {"12" times "10" rSup { size 8{ - 6} } } over {°C} } right ) left ("1275 m" right ) left ("55"°C right )=0 "." "84 m"} {}
13.8

Discussion

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

Objects expand in all dimensions, as illustrated in Figure 13.11. That is, their areas and volumes, as well as their lengths, increase with temperature. Holes 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 plug was still in place. The plug would get bigger, and so the hole must get bigger too. (Think of the ring of neighboring atoms or molecules on the wall of the hole as pushing each other farther apart as temperature increases. Obviously, the ring of neighbors must get slightly larger, so the hole gets slightly larger).

Thermal Expansion in Two Dimensions

For small temperature changes, the change in area ΔAΔA size 12{ΔA} {} is given by

ΔA=2αAΔT,ΔA=2αAΔT, size 12{ΔA=2αAΔT} {}
13.9

where ΔAΔA size 12{ΔA} {} is the change in area AA size 12{A} {}, ΔTΔT size 12{ΔT} {} is the change in temperature, and αα size 12{α} {} is the coefficient of linear expansion, which varies slightly with temperature.

Part a shows the outline of a flat metal plate before and after expansion. After expansion, it has the same shape and ratio of dimensions as before, but it takes up a greater area. Part b shows the outline of a flat metal plate with a hole in it, before and after expansion. The hole expands. Part c shows the outline of a rectangular box before and after expansion. After expansion, the box has the same proportions as before expansion, but it has a greater volume.
Figure 13.11 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 change in volume ΔVΔV size 12{ΔV} {} is very nearly ΔV=3αVΔTΔV=3αVΔT size 12{ΔV=3αVΔT} {}. This equation is usually written as

ΔV=βVΔT,ΔV=βVΔT, size 12{ΔV=βVΔT} {}
13.10

where ββ size 12{β} {} is the coefficient of volume expansion and ββ size 12{β approx 3α} {}. Note that the values of ββ size 12{β} {} in Table 13.2 are almost exactly equal to size 12{3α} {}.

In general, objects will expand with increasing temperature. Water is the most important exception to this rule. Water expands with increasing temperature (its density decreases) when it is at temperatures greater than 4ºC (40ºF)4ºC (40ºF) size 12{4°"C " \( "40"°F \) } {}. However, it expands with decreasing temperature when it is between +4ºC+4ºC size 12{+4°C} {} and 0ºC0ºC size 12{0°C} {}(40ºF(40ºF size 12{ \( "40"°F} {} to 32ºF)32ºF) size 12{"32"°F \) } {}. Water is densest at +4ºC+4ºC size 12{+4°C} {}. (See Figure 13.12.) Perhaps the most striking effect of this phenomenon is the freezing of water in a pond. When water near the surface cools down to 4ºC4ºC size 12{4°C} {} it is denser than the remaining water and thus will sink to the bottom. This “turnover” results in a layer of warmer water near the surface, which is then cooled. Eventually the pond has a uniform temperature of 4ºC4ºC size 12{4°C} {}. If the temperature in the surface layer drops below 4ºC4ºC size 12{4°C} {}, the water is less dense than the water below, and thus stays near the top. As a result, the pond surface can completely freeze over. The ice on top of liquid water provides an insulating layer from winter’s harsh exterior air temperatures. Fish and other aquatic life can survive in 4ºC4ºC size 12{4°C} {} water beneath ice, due to this unusual characteristic of water. It also produces circulation of water in the pond that is necessary for a healthy ecosystem of the body of water.

A graph of density of freshwater in grams per cubic centimeter versus temperature in degrees Celsius. The line is convex up. At zero degrees C, the density is just under zero point nine nine nine five grams per cubic centimeter. The density then increases at a decreasing rate until it hits a peak of about zero point nine nine nine nine seven grams per cubic centimeter at about four degrees C. Above four degrees C, the density decreases with increasing temperature.
Figure 13.12 The density of water as a function of temperature. Note that the thermal expansion is actually very small. The maximum density at +4ºC+4ºC size 12{+4°C} {} is only 0.0075% greater than the density at 2ºC2ºC size 12{2°C} {}, and 0.012% greater than that at 0ºC0ºC size 12{0°C} {}.

Making Connections: Real-World Connections—Filling the Tank

Differences in the thermal expansion of materials can lead to interesting effects at the gas station. One example is the dripping of gasoline from a freshly filled tank on a hot day. Gasoline starts out at the temperature of the ground under the gas station, which is cooler than the air temperature above. The gasoline cools the steel tank when it is filled. Both gasoline and steel tank expand as they warm to air temperature, but gasoline expands much more than steel, and so it may overflow.

This difference in expansion can also cause problems when interpreting the gasoline gauge. The actual amount (mass) of gasoline left in the tank when the gauge hits “empty” is a lot less in the summer than in the winter. The gasoline has the same volume as it does in the winter when the “add fuel” light goes on, but because the gasoline has expanded, there is less mass. If you are used to getting another 40 miles on “empty” in the winter, beware—you will probably run out much more quickly in the summer.

Fuel gauge pointing to empty.
Figure 13.13 Because the gas expands more than the gas tank with increasing temperature, you can’t drive as many miles on “empty” in the summer as you can in the winter. (credit: Hector Alejandro, Flickr)

Example 13.4 Calculating Thermal Expansion: Gas vs. Gas Tank

Suppose your 60.0-L (15.9-gal) steel gasoline tank is full of gas, so both the tank and the gasoline have a temperature of 15.0ºC15.0ºC size 12{"15" "." 0°C} {}. How much gasoline has spilled by the time they warm to 35.0ºC35.0ºC size 12{"35" "." 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. (The gasoline tank can be treated as solid steel.) We can use the equation for volume expansion to calculate the change in volume of the gasoline and of the tank.

Solution

1. Use the equation for volume expansion to calculate the increase in volume of the steel tank:

ΔVs=βsVsΔT.ΔVs=βsVsΔT. size 12{ΔV rSub { size 8{s} } =β rSub { size 8{s} } V rSub { size 8{s} } ΔT} {}
13.11

2. The increase in volume of the gasoline is given by this equation:

ΔVgas=βgasVgasΔT.ΔVgas=βgasVgasΔT. size 12{ΔV rSub { size 8{"gas"} } =β rSub { size 8{"gas"} } V rSub { size 8{"gas"} } ΔT} {}
13.12

3. Find the difference in volume to determine the amount spilled as

Vspill=ΔVgasΔVs.Vspill=ΔVgasΔVs. size 12{V rSub { size 8{"spill"} } =ΔV rSub { size 8{"gas"} } - ΔV rSub { size 8{s} } } {}
13.13

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

Vspill = βgasβsVΔT = 950 35 × 10 6 / º C 60.0L20.0ºC = 1.10L. Vspill = βgasβsVΔT = 950 35 × 10 6 / º C 60.0L20.0ºC = 1.10L.
13.14

Discussion

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 in Heat and Heat Transfer Methods.

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 being compressed with extremely large forces. To avoid rupturing rigid containers, these containers have air gaps, which allow them to expand and contract without stressing them.

Thermal Stress

Thermal stress is created by thermal expansion or contraction (see Elasticity: Stress and Strain for a discussion of stress and strain). Thermal stress can be destructive, such as when expanding gasoline ruptures a tank. It can also be useful, for example, when two parts are joined together by heating one in manufacturing, then slipping it over the other and allowing the combination to cool. Thermal stress can explain many phenomena, such as the weathering of rocks and pavement by the expansion of ice when it freezes.

Example 13.5 Calculating Thermal Stress: Gas Pressure

What pressure would be created in the gasoline tank considered in Example 13.4, if the gasoline increases in temperature from 15.0ºC15.0ºC size 12{"15" "." 0°C} {} to 35.0ºC35.0ºC size 12{"35" "." 0°C} {} without being allowed to expand? Assume that the bulk modulus BB size 12{B} {} for gasoline is 1.00×109 N/m21.00×109 N/m2 size 12{1 "." "00" times "10" rSup { size 8{9} } " N/m" rSup { size 8{2} } } {}. (For more on bulk modulus, see Elasticity: Stress and Strain.)

Strategy

To solve this problem, we must use the following equation, which relates a change in volume ΔVΔV size 12{DV} {} to pressure:

ΔV = 1 B F A V 0 , ΔV = 1 B F A V 0 , size 12{ΔV= { {1} over {B} } { {F} over {A} } V rSub { size 8{0} } ,} {}
13.15

where F/AF/A size 12{F/A} {} is pressure, V0V0 size 12{V rSub { size 8{0} } } {} is the original volume, and BB size 12{B} {} is the bulk modulus of the material involved. We will use the amount spilled in Example 13.4 as the change in volume, ΔVΔV size 12{ΔV} {}.

Solution

1. Rearrange the equation for calculating pressure:

P = F A = ΔV V 0 B . P = F A = ΔV V 0 B . size 12{P= { {F} over {A} } = { {ΔV} over {V rSub { size 8{0} } } } B "." } {}
13.16

2. Insert the known values. The bulk modulus for gasoline is B=1.00×109 N/m2B=1.00×109 N/m2 size 12{B=1 "." "00"´"10" rSup { size 8{9} } " N/m" rSup { size 8{2} } } {}. In the previous example, the change in volume ΔV=1.10 LΔV=1.10 L size 12{DV=1 "." "10"" L"} {} is the amount that would spill. Here, V0=60.0 LV0=60.0 L size 12{V rSub { size 8{0} } ="60" "." 0" L"} {} is the original volume of the gasoline. Substituting these values into the equation, we obtain

P = 1 . 10 L 60 . 0 L 1 . 00 × 10 9 Pa = 1 . 83 × 10 7 Pa . P = 1 . 10 L 60 . 0 L 1 . 00 × 10 9 Pa = 1 . 83 × 10 7 Pa . size 12{P= { {1 "." "10 L"} over {"60" "." "0 L"} } left (1 "." "00" times "10" rSup { size 8{9} } " Pa" right )=1 "." "83" times "10" rSup { size 8{7} } " Pa" "." } {}
13.17

Discussion

This pressure is about 2500lb/in22500lb/in2 size 12{"2500"" lb/in" rSup { size 8{2} } } {}, much more than a gasoline tank can handle.

Forces and pressures created by thermal stress are typically as great as that in the example above. Railroad tracks and roadways can buckle on hot days if they lack sufficient expansion joints. (See Figure 13.14.) Power lines sag more in the summer than in the winter, and will snap in cold weather if there is insufficient slack. Cracks open and close in plaster walls as a house warms and cools. Glass cooking pans will crack if cooled rapidly or unevenly, because of differential contraction and the stresses it creates. (Pyrex® is less susceptible because of its small coefficient of thermal expansion.) Nuclear reactor pressure vessels are threatened by overly rapid cooling, and although none have failed, several have been cooled faster than considered desirable. Biological cells are ruptured when foods are frozen, detracting from their taste. Repeated thawing and freezing accentuate the damage. Even the oceans can be affected. A significant portion of the rise in sea level that is resulting from global warming is due to the thermal expansion of sea water.

A cracked asphalt road with a pothole.
Figure 13.14 Thermal stress contributes to the formation of potholes. (credit: Editor5807, Wikimedia Commons)

Metal is regularly used in the human body for hip and knee implants. Most implants need to be replaced over time because, among other things, metal does not bond with bone. Researchers are trying to find better metal coatings that would allow metal-to-bone bonding. One challenge is to find a coating that has an expansion coefficient similar to that of metal. If the expansion coefficients are too different, the thermal stresses during the manufacturing process lead to cracks at the coating-metal interface.

Another example of thermal stress is found in the mouth. Dental fillings can expand differently from tooth enamel. It can give pain when eating ice cream or having a hot drink. Cracks might occur in the filling. Metal fillings (gold, silver, etc.) are being replaced by composite fillings (porcelain), which have smaller coefficients of expansion, and are closer to those of teeth.

Check Your Understanding

Two blocks, A and B, are made of the same material. Block A has dimensions l×w×h=L×2L×Ll×w×h=L×2L×L size 12{l times w times h=L times 2L times L} {} and Block B has dimensions 2L×2L×2L2L×2L×2L size 12{2L times 2L times 2L} {}. If the temperature changes, what is (a) the change in the volume of the two blocks, (b) the change in the cross-sectional area l×wl×w size 12{l times w} {}, and (c) the change in the height hh size 12{h} {} of the two blocks?

There are two rectangular blocks. Block A has its dimensions labeled length equals L, width equals two times L, height equals L. Block B has its dimensions labeled length, width, and height all equal to two times L.
Figure 13.15

Solution

(a) The change in volume is proportional to the original volume. Block A has a volume of L×2L×L=2L3.L×2L×L=2L3. size 12{L´2L´L=2L rSup { size 8{3} } "." } {}. Block B has a volume of 2L×2L×2L=8L3,2L×2L×2L=8L3, size 12{2L´2L´2L=8L rSup { size 8{3} } ,} {} which is 4 times that of Block A. Thus the change in volume of Block B should be 4 times the change in volume of Block A.

(b) The change in area is proportional to the area. The cross-sectional area of Block A is L×2L=2L2,L×2L=2L2, size 12{L´2L=2L rSup { size 8{2} } ,} {} while that of Block B is 2L×2L=4L2.2L×2L=4L2. size 12{2L´2L=4L rSup { size 8{2} } "." } {} Because cross-sectional area of Block B is twice that of Block A, the change in the cross-sectional area of Block B is twice that of Block A.

(c) The change in height is proportional to the original height. Because the original height of Block B is twice that of A, the change in the height of Block B is twice that of Block A.

Footnotes

  • 1 Values for liquids and gases are approximate.
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