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College Physics

31.6 Binding Energy

College Physics31.6 Binding Energy
  1. Preface
  2. 1 Introduction: The Nature of Science and Physics
    1. Introduction to Science and the Realm of Physics, Physical Quantities, and Units
    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. Introduction to One-Dimensional Kinematics
    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
  4. 3 Two-Dimensional Kinematics
    1. Introduction to Two-Dimensional Kinematics
    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
  5. 4 Dynamics: Force and Newton's Laws of Motion
    1. Introduction to Dynamics: Newton’s Laws of Motion
    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 Forces
    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
  6. 5 Further Applications of Newton's Laws: Friction, Drag, and Elasticity
    1. Introduction: Further Applications of Newton’s Laws
    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
  7. 6 Uniform Circular Motion and Gravitation
    1. Introduction to Uniform Circular Motion and Gravitation
    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
  8. 7 Work, Energy, and Energy Resources
    1. Introduction to Work, Energy, and Energy Resources
    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
  9. 8 Linear Momentum and Collisions
    1. Introduction to Linear Momentum and Collisions
    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
  10. 9 Statics and Torque
    1. Introduction to Statics and Torque
    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
  11. 10 Rotational Motion and Angular Momentum
    1. Introduction to Rotational Motion and Angular Momentum
    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
  12. 11 Fluid Statics
    1. Introduction to Fluid Statics
    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
  13. 12 Fluid Dynamics and Its Biological and Medical Applications
    1. Introduction to Fluid Dynamics and Its Biological and Medical Applications
    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
  14. 13 Temperature, Kinetic Theory, and the Gas Laws
    1. Introduction to Temperature, Kinetic Theory, and the Gas Laws
    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
  15. 14 Heat and Heat Transfer Methods
    1. Introduction to Heat and Heat Transfer Methods
    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
  16. 15 Thermodynamics
    1. Introduction to Thermodynamics
    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
  17. 16 Oscillatory Motion and Waves
    1. Introduction to Oscillatory Motion and Waves
    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
  18. 17 Physics of Hearing
    1. Introduction to the Physics of Hearing
    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
  19. 18 Electric Charge and Electric Field
    1. Introduction to Electric Charge and Electric Field
    2. 18.1 Static Electricity and Charge: Conservation of Charge
    3. 18.2 Conductors and Insulators
    4. 18.3 Coulomb’s Law
    5. 18.4 Electric Field: Concept of a Field Revisited
    6. 18.5 Electric Field Lines: Multiple Charges
    7. 18.6 Electric Forces in Biology
    8. 18.7 Conductors and Electric Fields in Static Equilibrium
    9. 18.8 Applications of Electrostatics
    10. Glossary
    11. Section Summary
    12. Conceptual Questions
    13. Problems & Exercises
  20. 19 Electric Potential and Electric Field
    1. Introduction to Electric Potential and Electric Energy
    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
  21. 20 Electric Current, Resistance, and Ohm's Law
    1. Introduction to Electric Current, Resistance, and Ohm's Law
    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
  22. 21 Circuits and DC Instruments
    1. Introduction to Circuits and DC Instruments
    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
  23. 22 Magnetism
    1. Introduction to Magnetism
    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
  24. 23 Electromagnetic Induction, AC Circuits, and Electrical Technologies
    1. Introduction to Electromagnetic Induction, AC Circuits and Electrical Technologies
    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
  25. 24 Electromagnetic Waves
    1. Introduction to Electromagnetic Waves
    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
  26. 25 Geometric Optics
    1. Introduction to Geometric Optics
    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
  27. 26 Vision and Optical Instruments
    1. Introduction to Vision and Optical Instruments
    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
  28. 27 Wave Optics
    1. Introduction to Wave Optics
    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
  29. 28 Special Relativity
    1. Introduction to Special Relativity
    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
  30. 29 Introduction to Quantum Physics
    1. Introduction to Quantum Physics
    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
  31. 30 Atomic Physics
    1. Introduction to Atomic Physics
    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
  32. 31 Radioactivity and Nuclear Physics
    1. Introduction to Radioactivity and Nuclear Physics
    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
  33. 32 Medical Applications of Nuclear Physics
    1. Introduction to Applications of Nuclear Physics
    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
  34. 33 Particle Physics
    1. Introduction to Particle Physics
    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
  35. 34 Frontiers of Physics
    1. Introduction to Frontiers of Physics
    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. Index

The more tightly bound a system is, the stronger the forces that hold it together and the greater the energy required to pull it apart. We can therefore learn about nuclear forces by examining how tightly bound the nuclei are. We define the binding energy (BE) of a nucleus to be the energy required to completely disassemble it into separate protons and neutrons. We can determine the BE of a nucleus from its rest mass. The two are connected through Einstein’s famous relationship E=(Δm)c2E=(Δm)c2 size 12{E= \( Δm \) c rSup { size 8{2} } } {}. A bound system has a smaller mass than its separate constituents; the more tightly the nucleons are bound together, the smaller the mass of the nucleus.

Imagine pulling a nuclide apart as illustrated in Figure 31.24. Work done to overcome the nuclear forces holding the nucleus together puts energy into the system. By definition, the energy input equals the binding energy BE. The pieces are at rest when separated, and so the energy put into them increases their total rest mass compared with what it was when they were glued together as a nucleus. That mass increase is thus Δm=BE/c2Δm=BE/c2 size 12{Δm="BE"/c rSup { size 8{2} } } {}. This difference in mass is known as mass defect. It implies that the mass of the nucleus is less than the sum of the masses of its constituent protons and neutrons. A nuclide AXAX size 12{"" lSup { size 8{A} } X} {} has ZZ size 12{Z} {} protons and NN size 12{N} {} neutrons, so that the difference in mass is

Δm=(Zmp+Nmn)mtot.Δm=(Zmp+Nmn)mtot. size 12{Δm= \( ital "Zm" rSub { size 8{p} } + ital "Nm" rSub { size 8{n} } \) - m rSub { size 8{"tot"} } } {}
31.60

Thus,

BE=(Δm)c2=[(Zmp+Nmn)mtot]c2,BE=(Δm)c2=[(Zmp+Nmn)mtot]c2, size 12{"BE"= \( Δm \) c rSup { size 8{2} } = \[ \( ital "Zm" rSub { size 8{p} } + ital "Nm" rSub { size 8{n} } \) - m rSub { size 8{"tot"} } \] c rSup { size 8{2} } } {}
31.61

where mtotmtot size 12{m rSub { size 8{"tot"} } } {} is the mass of the nuclide AXAX, mpmp is the mass of a proton, and mnmn is the mass of a neutron. Traditionally, we deal with the masses of neutral atoms. To get atomic masses into the last equation, we first add ZZ size 12{Z} {} electrons to mtotmtot size 12{m rSub { size 8{"tot"} } } {}, which gives mAXmAX size 12{m left ("" lSup { size 8{A} } X right )} {}, the atomic mass of the nuclide. We then add ZZ size 12{Z} {} electrons to the ZZ size 12{Z} {} protons, which gives Zm1HZm1H size 12{ ital "Zm" left ("" lSup { size 8{1} } H right )} {}, or ZZ size 12{Z} {} times the mass of a hydrogen atom. Thus the binding energy of a nuclide AXAX size 12{"" lSup { size 8{A} } X} {} is

BE = { [ Zm ( 1 H ) + Nm n ] m ( A X )} c 2 . BE = { [ Zm ( 1 H ) + Nm n ] m ( A X )} c 2 .
31.62

The atomic masses can be found in Appendix A, most conveniently expressed in unified atomic mass units u (1u=931.5MeV/c21u=931.5MeV/c2 size 12{1" u"="931" "." 5 "MeV"/c rSup { size 8{2} } } {}). BE is thus calculated from known atomic masses.

The image shows some spherical protons and neutrons pulled out from a nucleus. The work done to pull them apart is binding energy.
Figure 31.24 Work done to pull a nucleus apart into its constituent protons and neutrons increases the mass of the system. The work to disassemble the nucleus equals its binding energy BE. A bound system has less mass than the sum of its parts, especially noticeable in the nuclei, where forces and energies are very large.

Things Great and Small

Nuclear Decay Helps Explain Earth’s Hot Interior

A puzzle created by radioactive dating of rocks is resolved by radioactive heating of Earth’s interior. This intriguing story is another example of how small-scale physics can explain large-scale phenomena.

Radioactive dating plays a role in determining the approximate age of the Earth. The oldest rocks on Earth solidified about 3.5×1093.5×109 size 12{3 "." 5 times "10" rSup { size 8{9} } } {} years ago—a number determined by uranium-238 dating. These rocks could only have solidified once the surface of the Earth had cooled sufficiently. The temperature of the Earth at formation can be estimated based on gravitational potential energy of the assemblage of pieces being converted to thermal energy. Using heat transfer concepts discussed in Thermodynamics it is then possible to calculate how long it would take for the surface to cool to rock-formation temperatures. The result is about 109109 size 12{"10" rSup { size 8{9} } } {} years. The first rocks formed have been solid for 3.5×1093.5×109 size 12{3 "." 5 times "10" rSup { size 8{9} } } {} years, so that the age of the Earth is approximately 4.5×1094.5×109 size 12{4 "." 5 times "10" rSup { size 8{9} } } {} years. There is a large body of other types of evidence (both Earth-bound and solar system characteristics are used) that supports this age. The puzzle is that, given its age and initial temperature, the center of the Earth should be much cooler than it is today (see Figure 31.25).

The figure shows that the center of the Earth cools by three heat transfer methods. Convection heat transfer in the center region, then conduction heat transfer moves thermal energy to the surface, and finally radiation heat transfer from the surface to space.
Figure 31.25 The center of the Earth cools by well-known heat transfer methods. Convection in the liquid regions and conduction move thermal energy to the surface, where it radiates into cold, dark space. Given the age of the Earth and its initial temperature, it should have cooled to a lower temperature by now. The blowup shows that nuclear decay releases energy in the Earth’s interior. This energy has slowed the cooling process and is responsible for the interior still being molten.

We know from seismic waves produced by earthquakes that parts of the interior of the Earth are liquid. Shear or transverse waves cannot travel through a liquid and are not transmitted through the Earth’s core. Yet compression or longitudinal waves can pass through a liquid and do go through the core. From this information, the temperature of the interior can be estimated. As noticed, the interior should have cooled more from its initial temperature in the 4.5×1094.5×109 size 12{4 "." 5 times "10" rSup { size 8{9} } } {} years since its formation. In fact, it should have taken no more than about 109109 size 12{4 "." 5 times "10" rSup { size 8{9} } } {} years to cool to its present temperature. What is keeping it hot? The answer seems to be radioactive decay of primordial elements that were part of the material that formed the Earth (see the blowup in Figure 31.25).

Nuclides such as 238U238U and 40K40K have half-lives similar to or longer than the age of the Earth, and their decay still contributes energy to the interior. Some of the primordial radioactive nuclides have unstable decay products that also release energy— 238U238U size 12{"" lSup { size 8{"238"} } U} {} has a long decay chain of these. Further, there were more of these primordial radioactive nuclides early in the life of the Earth, and thus the activity and energy contributed were greater then (perhaps by an order of magnitude). The amount of power created by these decays per cubic meter is very small. However, since a huge volume of material lies deep below the surface, this relatively small amount of energy cannot escape quickly. The power produced near the surface has much less distance to go to escape and has a negligible effect on surface temperatures.

A final effect of this trapped radiation merits mention. Alpha decay produces helium nuclei, which form helium atoms when they are stopped and capture electrons. Most of the helium on Earth is obtained from wells and is produced in this manner. Any helium in the atmosphere will escape in geologically short times because of its high thermal velocity.

What patterns and insights are gained from an examination of the binding energy of various nuclides? First, we find that BE is approximately proportional to the number of nucleons AA in any nucleus. About twice as much energy is needed to pull apart a nucleus like 24Mg24Mg compared with pulling apart 12C12C, for example. To help us look at other effects, we divide BE by AA size 12{A} {} and consider the binding energy per nucleon, BE/ABE/A size 12{ {"BE"} slash {A} } {}. The graph of BE/ABE/A size 12{ {"BE"} slash {A} } {} in Figure 31.26 reveals some very interesting aspects of nuclei. We see that the binding energy per nucleon averages about 8 MeV, but is lower for both the lightest and heaviest nuclei. This overall trend, in which nuclei with AA size 12{A} {} equal to about 60 have the greatest BE/ABE/A size 12{ {"BE"} slash {A} } {} and are thus the most tightly bound, is due to the combined characteristics of the attractive nuclear forces and the repulsive Coulomb force. It is especially important to note two things—the strong nuclear force is about 100 times stronger than the Coulomb force, and the nuclear forces are shorter in range compared to the Coulomb force. So, for low-mass nuclei, the nuclear attraction dominates and each added nucleon forms bonds with all others, causing progressively heavier nuclei to have progressively greater values of BE/ABE/A size 12{ {"BE"} slash {A} } {}. This continues up to A60A60 size 12{A approx "60"} {}, roughly corresponding to the mass number of iron. Beyond that, new nucleons added to a nucleus will be too far from some others to feel their nuclear attraction. Added protons, however, feel the repulsion of all other protons, since the Coulomb force is longer in range. Coulomb repulsion grows for progressively heavier nuclei, but nuclear attraction remains about the same, and so BE/ABE/A size 12{ {"BE"} slash {A} } {} becomes smaller. This is why stable nuclei heavier than A40A40 size 12{A approx "40"} {} have more neutrons than protons. Coulomb repulsion is reduced by having more neutrons to keep the protons farther apart (see Figure 31.27).

The figure shows a graph of binding energy per nucleon versus atomic mass for different elements. From the graph it can be observed that elements with atomic mass near sixty have greater binding energy per nucleon.
Figure 31.26 A graph of average binding energy per nucleon, BE/ABE/A size 12{ {"BE"} slash {A} } {}, for stable nuclei. The most tightly bound nuclei are those with AA size 12{A} {} near 60, where the attractive nuclear force has its greatest effect. At higher AA size 12{A} {} s, the Coulomb repulsion progressively reduces the binding energy per nucleon, because the nuclear force is short ranged. The spikes on the curve are very tightly bound nuclides and indicate shell closures.
The image shows a bunch of spherical nucleons inside a nucleus. A circular dashed path is shown which depicts the range of nuclear force and the nucleons inside that range feel nuclear force directly.
Figure 31.27 The nuclear force is attractive and stronger than the Coulomb force, but it is short ranged. In low-mass nuclei, each nucleon feels the nuclear attraction of all others. In larger nuclei, the range of the nuclear force, shown for a single nucleon, is smaller than the size of the nucleus, but the Coulomb repulsion from all protons reaches all others. If the nucleus is large enough, the Coulomb repulsion can add to overcome the nuclear attraction.

There are some noticeable spikes on the BE/ABE/A size 12{ {"BE"} slash {A} } {} graph, which represent particularly tightly bound nuclei. These spikes reveal further details of nuclear forces, such as confirming that closed-shell nuclei (those with magic numbers of protons or neutrons or both) are more tightly bound. The spikes also indicate that some nuclei with even numbers for ZZ size 12{Z} {} and NN size 12{N} {}, and with Z=NZ=N size 12{Z=N} {}, are exceptionally tightly bound. This finding can be correlated with some of the cosmic abundances of the elements. The most common elements in the universe, as determined by observations of atomic spectra from outer space, are hydrogen, followed by 4He4He, with much smaller amounts of 12C12C and other elements. It should be noted that the heavier elements are created in supernova explosions, while the lighter ones are produced by nuclear fusion during the normal life cycles of stars, as will be discussed in subsequent chapters. The most common elements have the most tightly bound nuclei. It is also no accident that one of the most tightly bound light nuclei is 4He4He, emitted in αα decay.

Example 31.7 What Is BE/ABE/A size 12{ {"BE"} slash {A} } {} for an Alpha Particle?

Calculate the binding energy per nucleon of 4He4He size 12{"" lSup { size 8{4} } "He"} {}, the αα size 12{α} {} particle.

Strategy

To find BE/ABE/A, we first find BE using the Equation BE={[Zm(1H)+Nmn]m(AX)}c2BE={[Zm(1H)+Nmn]m(AX)}c2 and then divide by AA. This is straightforward once we have looked up the appropriate atomic masses in Appendix A.

Solution

The binding energy for a nucleus is given by the equation

BE={[Zm(1H)+Nmn]m(AX)}c2.BE={[Zm(1H)+Nmn]m(AX)}c2.
31.63

For 4He4He size 12{"" lSup { size 8{4} } "He"} {}, we have Z=N=2Z=N=2 size 12{Z=N=2} {}; thus,

BE={[2m(1H)+2mn]m(4He)}c2.BE={[2m(1H)+2mn]m(4He)}c2.
31.64

Appendix A gives these masses as m(4He)=4.002602 um(4He)=4.002602 u, m(1H)=1.007825 um(1H)=1.007825 u, and mn=1.008665 umn=1.008665 u size 12{m rSub { size 8{n} } =0 "." "008665"`" u"} {}. Thus,

BE=(0.030378 u)c2.BE=(0.030378 u)c2. size 12{"BE"= \( 0 "." "030378 u" \) c rSup { size 8{2} } } {}
31.65

Noting that 1 u=931.5 MeV/c21 u=931.5 MeV/c2 size 12{"1u"="931" "." "5 MeV/"c rSup { size 8{2} } } {}, we find

BE=(0.030378)(931.5 MeV/c2)c2=28.3 MeV.BE=(0.030378)(931.5 MeV/c2)c2=28.3 MeV. size 12{"BE"= \( 0 "." "030378" \) \( "931" "." "5 MeV/"c rSup { size 8{2} } \) c rSup { size 8{2} } ="28" "." 3" MeV"} {}
31.66

Since A=4A=4 size 12{A=4} {}, we see that BE/ABE/A size 12{ {"BE"} slash {A} } {} is this number divided by 4, or

BE/A=7.07 MeV/nucleon.BE/A=7.07 MeV/nucleon. size 12{"BE"/A=7 "." "07"" MeV/nucleon"} {}
31.67

Discussion

This is a large binding energy per nucleon compared with those for other low-mass nuclei, which have BE/A3 MeV/nucleonBE/A3 MeV/nucleon. This indicates that 4He4He is tightly bound compared with its neighbors on the chart of the nuclides. You can see the spike representing this value of BE/ABE/A for 4He4He on the graph in Figure 31.26. This is why 4He4He is stable. Since 4He4He is tightly bound, it has less mass than other A=4A=4 nuclei and, therefore, cannot spontaneously decay into them. The large binding energy also helps to explain why some nuclei undergo αα decay. Smaller mass in the decay products can mean energy release, and such decays can be spontaneous. Further, it can happen that two protons and two neutrons in a nucleus can randomly find themselves together, experience the exceptionally large nuclear force that binds this combination, and act as a 4He4He unit within the nucleus, at least for a while. In some cases, the 4He4He escapes, and αα decay has then taken place.

There is more to be learned from nuclear binding energies. The general trend in BE/ABE/A size 12{"BE"/A} {} is fundamental to energy production in stars, and to fusion and fission energy sources on Earth, for example. This is one of the applications of nuclear physics covered in Medical Applications of Nuclear Physics. The abundance of elements on Earth, in stars, and in the universe as a whole is related to the binding energy of nuclei and has implications for the continued expansion of the universe.

Problem-Solving Strategies

For Reaction And Binding Energies and Activity Calculations in Nuclear Physics

  1. Identify exactly what needs to be determined in the problem (identify the unknowns). This will allow you to decide whether the energy of a decay or nuclear reaction is involved, for example, or whether the problem is primarily concerned with activity (rate of decay).
  2. Make a list of what is given or can be inferred from the problem as stated (identify the knowns).
  3. For reaction and binding-energy problems, we use atomic rather than nuclear masses. Since the masses of neutral atoms are used, you must count the number of electrons involved. If these do not balance (such as in β+β+ size 12{β rSup { size 8{+{}} } } {} decay), then an energy adjustment of 0.511 MeV per electron must be made. Also note that atomic masses may not be given in a problem; they can be found in tables.
  4. For problems involving activity, the relationship of activity to half-life, and the number of nuclei given in the equation R=0.693Nt1/2R=0.693Nt1/2 size 12{R= { {1 "." "693"N} over {t rSub { size 8{1/2} } } } } {} can be very useful. Owing to the fact that number of nuclei is involved, you will also need to be familiar with moles and Avogadro’s number.
  5. Perform the desired calculation; keep careful track of plus and minus signs as well as powers of 10.
  6. Check the answer to see if it is reasonable: Does it make sense? Compare your results with worked examples and other information in the text. (Heeding the advice in Step 5 will also help you to be certain of your result.) You must understand the problem conceptually to be able to determine whether the numerical result is reasonable.
PhET Explorations: Nuclear Fission

Start a chain reaction, or introduce non-radioactive isotopes to prevent one. Control energy production in a nuclear reactor!

Figure 31.28
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