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College Physics for AP® Courses

31.5 Half-Life and Activity

College Physics for AP® Courses31.5 Half-Life and Activity
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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 half-life.
  • Define dating.
  • Calculate the age of old objects by radioactive dating.

The information presented in this section supports the following AP® learning objectives and science practices:

  • 7.C.3.1 The student is able to predict the number of radioactive nuclei remaining in a sample after a certain period of time, and also predict the missing species (alpha, beta, gamma) in a radioactive decay. (S.P. 6.4)

Unstable nuclei decay. However, some nuclides decay faster than others. For example, radium and polonium, discovered by the Curies, decay faster than uranium. This means they have shorter lifetimes, producing a greater rate of decay. In this section we explore half-life and activity, the quantitative terms for lifetime and rate of decay.

Half-Life

Why use a term like half-life rather than lifetime? The answer can be found by examining Figure 31.21, which shows how the number of radioactive nuclei in a sample decreases with time. The time in which half of the original number of nuclei decay is defined as the half-life, t1/2t1/2 size 12{t rSub { size 8{1/2} } } {}. Half of the remaining nuclei decay in the next half-life. Further, half of that amount decays in the following half-life. Therefore, the number of radioactive nuclei decreases from NN size 12{N} {} to N/2N/2 size 12{N/2} {} in one half-life, then to N/4N/4 size 12{N/4} {} in the next, and to N/8N/8 size 12{N/8} {} in the next, and so on. If NN size 12{N} {} is a large number, then many half-lives (not just two) pass before all of the nuclei decay. Nuclear decay is an example of a purely statistical process. A more precise definition of half-life is that each nucleus has a 50% chance of living for a time equal to one half-life t1/2t1/2 size 12{t rSub { size 8{1/2} } } {}. Thus, if NN size 12{N} {} is reasonably large, half of the original nuclei decay in a time of one half-life. If an individual nucleus makes it through that time, it still has a 50% chance of surviving through another half-life. Even if it happens to make it through hundreds of half-lives, it still has a 50% chance of surviving through one more. The probability of decay is the same no matter when you start counting. This is like random coin flipping. The chance of heads is 50%, no matter what has happened before.

The figure shows a radioactive decay graph of number of nuclides in thousands versus time in multiples of half-life. The number of radioactive nuclei decreases exponentially and finally approaches zero after about ten half-lives.
Figure 31.21 Radioactive decay reduces the number of radioactive nuclei over time. In one half-life t1/2t1/2 size 12{t rSub { size 8{1/2} } } {}, the number decreases to half of its original value. Half of what remains decay in the next half-life, and half of those in the next, and so on. This is an exponential decay, as seen in the graph of the number of nuclei present as a function of time.

There is a tremendous range in the half-lives of various nuclides, from as short as 10231023 size 12{"10" rSup { size 8{ - "23"} } } {} s for the most unstable, to more than 10161016 size 12{"10" rSup { size 8{"16"} } } {} y for the least unstable, or about 46 orders of magnitude. Nuclides with the shortest half-lives are those for which the nuclear forces are least attractive, an indication of the extent to which the nuclear force can depend on the particular combination of neutrons and protons. The concept of half-life is applicable to other subatomic particles, as will be discussed in Particle Physics. It is also applicable to the decay of excited states in atoms and nuclei. The following equation gives the quantitative relationship between the original number of nuclei present at time zero (N0N0 size 12{N rSub { size 8{0} } } {}) and the number (NN size 12{N} {}) at a later time tt size 12{t} {}:

N=N0eλt,N=N0eλt, size 12{N=N rSub { size 8{0} } e rSup { size 8{ - λt} } } {}
31.36

where e=2.71828...e=2.71828... size 12{e=2 "." "71828" "." "." "." } {} is the base of the natural logarithm, and λλ size 12{λ} {} is the decay constant for the nuclide. The shorter the half-life, the larger is the value of λλ size 12{λ} {}, and the faster the exponential eλteλt size 12{e rSup { size 8{ - λt} } } {} decreases with time. The relationship between the decay constant λλ size 12{λ} {} and the half-life t1/2t1/2 size 12{t rSub { size 8{1/2} } } {} is

λ= ln(2) t1/2 0.693t1/2.λ= ln(2) t1/2 0.693t1/2. size 12{λ= { {0 "." "693"} over {t rSub { size 8{1/2} } } } } {}
31.37

To see how the number of nuclei declines to half its original value in one half-life, let t=t1/2t=t1/2 size 12{t=t rSub { size 8{1/2} } } {} in the exponential in the equation N=N0eλtN=N0eλt . This gives N=N0 eλt =N0e−0.693 =0.500N0N=N0 eλt =N0e−0.693 =0.500N0. For integral numbers of half-lives, you can just divide the original number by 2 over and over again, rather than using the exponential relationship. For example, if ten half-lives have passed, we divide NN size 12{N} {} by 2 ten times. This reduces it to N/1024N/1024 size 12{ {N} slash {"1024"} } {}. For an arbitrary time, not just a multiple of the half-life, the exponential relationship must be used.

Radioactive dating is a clever use of naturally occurring radioactivity. Its most famous application is carbon-14 dating. Carbon-14 has a half-life of 5730 years and is produced in a nuclear reaction induced when solar neutrinos strike 14N14N size 12{"" lSup { size 8{"14"} } N} {} in the atmosphere. Radioactive carbon has the same chemistry as stable carbon, and so it mixes into the ecosphere, where it is consumed and becomes part of every living organism. Carbon-14 has an abundance of 1.3 parts per trillion of normal carbon. Thus, if you know the number of carbon nuclei in an object (perhaps determined by mass and Avogadro’s number), you multiply that number by 1.3×10121.3×1012 to find the number of 14C14C nuclei in the object. When an organism dies, carbon exchange with the environment ceases, and 14C14C is not replenished as it decays. By comparing the abundance of 14C14C in an artifact, such as mummy wrappings, with the normal abundance in living tissue, it is possible to determine the artifact’s age (or time since death). Carbon-14 dating can be used for biological tissues as old as 50 or 60 thousand years, but is most accurate for younger samples, since the abundance of 14C14C nuclei in them is greater. Very old biological materials contain no 14C14C at all. There are instances in which the date of an artifact can be determined by other means, such as historical knowledge or tree-ring counting. These cross-references have confirmed the validity of carbon-14 dating and permitted us to calibrate the technique as well. Carbon-14 dating revolutionized parts of archaeology and is of such importance that it earned the 1960 Nobel Prize in chemistry for its developer, the American chemist Willard Libby (1908–1980).

One of the most famous cases of carbon-14 dating involves the Shroud of Turin, a long piece of fabric purported to be the burial shroud of Jesus (see Figure 31.22). This relic was first displayed in Turin in 1354 and was denounced as a fraud at that time by a French bishop. Its remarkable negative imprint of an apparently crucified body resembles the then-accepted image of Jesus, and so the shroud was never disregarded completely and remained controversial over the centuries. Carbon-14 dating was not performed on the shroud until 1988, when the process had been refined to the point where only a small amount of material needed to be destroyed. Samples were tested at three independent laboratories, each being given four pieces of cloth, with only one unidentified piece from the shroud, to avoid prejudice. All three laboratories found samples of the shroud contain 92% of the 14C14C size 12{"" lSup { size 8{"14"} } C} {} found in living tissues, allowing the shroud to be dated (see Example 31.4).

The figure shows two images of Jesus. Left image is very faint and hardly visible but the right image shows a much clearer picture.
Figure 31.22 Part of the Shroud of Turin, which shows a remarkable negative imprint likeness of Jesus complete with evidence of crucifixion wounds. The shroud first surfaced in the 14th century and was only recently carbon-14 dated. It has not been determined how the image was placed on the material. (credit: Butko, Wikimedia Commons)

Example 31.4 How Old Is the Shroud of Turin?

Calculate the age of the Shroud of Turin given that the amount of 14C14C size 12{"" lSup { size 8{"14"} } C} {} found in it is 92% of that in living tissue.

Strategy

Knowing that 92% of the 14C14C remains means that N/N0=0.92N/N0=0.92 size 12{N/N rSub { size 8{0} } =0 "." "92"} {}. Therefore, the equation N=N0eλtN=N0eλt size 12{N=N rSub { size 8{0} } e rSup { size 8{ - λt} } } {} can be used to find λtλt size 12{λt} {}. We also know that the half-life of 14C14C is 5730 y, and so once λtλt size 12{λt} {} is known, we can use the equation λ=0.693t1/2λ=0.693t1/2 size 12{λ= { {0 "." "693"} over {t rSub { size 8{1/2} } } } } {} to find λλ size 12{λ} {} and then find tt size 12{t} {} as requested. Here, we postulate that the decrease in 14C14C is solely due to nuclear decay.

Solution

Solving the equation N=N0eλtN=N0eλt size 12{N=N rSub { size 8{0} } e rSup { size 8{ - λt} } } {} for N/N0N/N0 size 12{N/N rSub { size 8{0} } } {} gives

NN0=eλt.NN0=eλt. size 12{ { {N} over {N rSub { size 8{0} } } } =e rSup { size 8{-λt} } } {}
31.38

Thus,

0.92=eλt.0.92=eλt. size 12{0 "." "92"=e rSup { size 8{ - λt} } } {}
31.39

Taking the natural logarithm of both sides of the equation yields

ln0.92=–λtln0.92=–λt size 12{"ln "0 "." "92""=-"λt} {}
31.40

so that

0.0834=λt.0.0834=λt. size 12{ - 0 "." "0834"= - λt} {}
31.41

Rearranging to isolate tt size 12{t} {} gives

t=0.0834λ.t=0.0834λ. size 12{t= { {0 "." "0834"} over {λ} } } {}
31.42

Now, the equation λ=0.693t1/2λ=0.693t1/2 size 12{λ= { {0 "." "693"} over {t rSub { size 8{1/2} } } } } {} can be used to find λλ size 12{λ} {} for 14C14C size 12{"" lSup { size 8{"14"} } C} {}. Solving for λλ size 12{λ} {} and substituting the known half-life gives

λ=0.693t1/2=0.6935730 y.λ=0.693t1/2=0.6935730 y. size 12{λ= { {0 "." "693"} over {t rSub { size 8{1/2} } } } = { {0 "." "693"} over {"5730"" y"} } } {}
31.43

We enter this value into the previous equation to find tt size 12{t} {}:

t=0.08340.6935730 y=690 y.t=0.08340.6935730 y=690 y. size 12{t= { {0 "." "0834"} over { { {0 "." "693"} over {"5730"" y"} } } } ="690"" y"} {}
31.44

Discussion

This dates the material in the shroud to 1988–690 = a.d. 1300. Our calculation is only accurate to two digits, so that the year is rounded to 1300. The values obtained at the three independent laboratories gave a weighted average date of a.d. 1320±601320±60 size 12{"1320" +- "60"} {}. The uncertainty is typical of carbon-14 dating and is due to the small amount of 14C14C size 12{"" lSup { size 8{"14"} } C} {} in living tissues, the amount of material available, and experimental uncertainties (reduced by having three independent measurements). It is meaningful that the date of the shroud is consistent with the first record of its existence and inconsistent with the period in which Jesus lived.

There are other forms of radioactive dating. Rocks, for example, can sometimes be dated based on the decay of 238U238U. The decay series for 238U238U ends with 206Pb206Pb, so that the ratio of these nuclides in a rock is an indication of how long it has been since the rock solidified. The original composition of the rock, such as the absence of lead, must be known with some confidence. However, as with carbon-14 dating, the technique can be verified by a consistent body of knowledge. Since 238U238U has a half-life of 4.5×1094.5×109 y, it is useful for dating only very old materials, showing, for example, that the oldest rocks on Earth solidified about 3.5×1093.5×109 size 12{3 "." 5 times "10" rSup { size 8{9} } } {} years ago.

Activity, the Rate of Decay

What do we mean when we say a source is highly radioactive? Generally, this means the number of decays per unit time is very high. We define activity RR size 12{R} {} to be the rate of decay expressed in decays per unit time. In equation form, this is

R=ΔNΔtR=ΔNΔt size 12{R= { {ΔN} over {Δt} } } {}
31.45

where ΔNΔN size 12{ΔN} {} is the number of decays that occur in time ΔtΔt size 12{Δt} {}. The SI unit for activity is one decay per second and is given the name becquerel (Bq) in honor of the discoverer of radioactivity. That is,

1Bq=1 decay/s.1Bq=1 decay/s. size 12{1" Bq"="1 decay/s"} {}
31.46

Activity RR size 12{R} {} is often expressed in other units, such as decays per minute or decays per year. One of the most common units for activity is the curie (Ci), defined to be the activity of 1 g of 226Ra226Ra, in honor of Marie Curie’s work with radium. The definition of curie is

1 Ci=3.70×1010 Bq, 1 Ci=3.70×1010 Bq, size 12{1" Ci"=3 "." "70" times "10" rSup { size 8{"10"} } " Bq"} {}
31.47

or 3.70×10103.70×1010 size 12{3 "." "70" times "10" rSup { size 8{"10"} } } {} decays per second. A curie is a large unit of activity, while a becquerel is a relatively small unit. 1 MBq=100 microcuries (μCi)1 MBq=100 microcuries (μCi) size 12{"1 MBq"="100 microcuries " \( μ"Ci" \) } {}. In countries like Australia and New Zealand that adhere more to SI units, most radioactive sources, such as those used in medical diagnostics or in physics laboratories, are labeled in Bq or megabecquerel (MBq).

Intuitively, you would expect the activity of a source to depend on two things: the amount of the radioactive substance present, and its half-life. The greater the number of radioactive nuclei present in the sample, the more will decay per unit of time. The shorter the half-life, the more decays per unit time, for a given number of nuclei. So activity RR size 12{R} {} should be proportional to the number of radioactive nuclei, NN size 12{N} {}, and inversely proportional to their half-life, t1/2t1/2 size 12{t rSub { size 8{1/2} } } {}. In fact, your intuition is correct. It can be shown that the activity of a source is

R=0.693Nt1/2R=0.693Nt1/2 size 12{R= { {0 "." "693"N} over {t rSub { size 8{1/2} } } } } {}
31.48

where NN size 12{N} {} is the number of radioactive nuclei present, having half-life t1/2t1/2 size 12{t rSub { size 8{1/2} } } {}. This relationship is useful in a variety of calculations, as the next two examples illustrate.

Example 31.5 How Great Is the 14C14C size 12{"" lSup { size 8{"14"} } C} {} Activity in Living Tissue?

Calculate the activity due to 14C14C size 12{"" lSup { size 8{"14"} } C} {} in 1.00 kg of carbon found in a living organism. Express the activity in units of Bq and Ci.

Strategy

To find the activity RR size 12{R} {} using the equation R=0.693Nt1/2R=0.693Nt1/2 size 12{R= { {0 "." "693"N} over {t rSub { size 8{1/2} } } } } {}, we must know NN size 12{N} {} and t1/2t1/2 size 12{t rSub { size 8{1/2} } } {}. The half-life of 14C14C size 12{"" lSup { size 8{"14"} } C} {} can be found in Appendix B, and was stated above as 5730 y. To find NN size 12{N} {}, we first find the number of 12C12C size 12{"" lSup { size 8{"12"} } C} {} nuclei in 1.00 kg of carbon using the concept of a mole. As indicated, we then multiply by 1.3×10121.3×1012 size 12{1 "." 3×"10" rSup { size 8{ +- "12"} } } {} (the abundance of 14C14C size 12{"" lSup { size 8{"14"} } C} {} in a carbon sample from a living organism) to get the number of 14C14C size 12{"" lSup { size 8{"14"} } C} {} nuclei in a living organism.

Solution

One mole of carbon has a mass of 12.0 g, since it is nearly pure 12C12C size 12{"" lSup { size 8{"12"} } C} {}. (A mole has a mass in grams equal in magnitude to AA size 12{A} {} found in the periodic table.) Thus the number of carbon nuclei in a kilogram is

N(12C)= 6.02 × 1023 mol–1 12.0 g/mol ×(1000 g) =5.02×1025. N(12C)= 6.02 × 1023 mol–1 12.0 g/mol ×(1000 g) =5.02×1025.
31.49

So the number of 14C14C size 12{"" lSup { size 8{"14"} } C} {} nuclei in 1 kg of carbon is

N(14C)= (5.02×1025 )(1.3×10−12)=6.52×1013.N(14C)= (5.02×1025 )(1.3×10−12)=6.52×1013. size 12{N \( rSup { size 8{"14"} } C \) = \( 5 "." "02" times "10" rSup { size 8{"25"} } \) \( 1 "." 3 times "10" rSup { size 8{ - "12"} } \) =6 "." "52" times "10" rSup { size 8{"13"} } } {}
31.50

Now the activity RR size 12{R} {} is found using the equation R=0.693Nt1/2R=0.693Nt1/2 size 12{R= { {0 "." "693"N} over {t rSub { size 8{1/2} } } } } {}.

Entering known values gives

R=0.693(6.52×1013)5730 y=7.89×109y–1,R=0.693(6.52×1013)5730 y=7.89×109y–1, size 12{R= { {0 "." "693" \( 6 "." "52"´"10" rSup { size 8{"13"} } \) } over {"5730"" y"} } =7 "." "89"´"10" rSup { size 8{9} } /y} {}
31.51

or 7.89×1097.89×109 size 12{7 "." "89" times "10" rSup { size 8{9} } } {} decays per year. To convert this to the unit Bq, we simply convert years to seconds. Thus,

R = (7.89 × 10 9 y –1 ) 1.00 y 3. 16× 107 s=250 Bq, R = (7.89 × 10 9 y –1 ) 1.00 y 3. 16× 107 s=250 Bq, size 12{R=7 "." "89"´"10" rSup { size 8{9} } /y cdot { {1 "." "00"" y"} over {3 "." "16"´"10" rSup { size 8{7} } " s"} } ="250"" Bq"} {}
31.52

or 250 decays per second. To express RR size 12{R} {} in curies, we use the definition of a curie,

R=250 Bq3.7×1010 Bq/Ci=6.76×109 Ci.R=250 Bq3.7×1010 Bq/Ci=6.76×109 Ci. size 12{R= { {"250"" Bq"} over {3 "." 7´"10" rSup { size 8{"10"} } " Bq/Ci"} } =6 "." "75"´"10" rSup { size 8{-9} } " Ci"} {}
31.53

Thus,

R=6.76nCi.R=6.76nCi. size 12{R=6 "." "75" "nCi"} {}
31.54

Discussion

Our own bodies contain kilograms of carbon, and it is intriguing to think there are hundreds of 14C14C size 12{"" lSup { size 8{"14"} } C} {} decays per second taking place in us. Carbon-14 and other naturally occurring radioactive substances in our bodies contribute to the background radiation we receive. The small number of decays per second found for a kilogram of carbon in this example gives you some idea of how difficult it is to detect 14C14C size 12{"" lSup { size 8{"14"} } C} {} in a small sample of material. If there are 250 decays per second in a kilogram, then there are 0.25 decays per second in a gram of carbon in living tissue. To observe this, you must be able to distinguish decays from other forms of radiation, in order to reduce background noise. This becomes more difficult with an old tissue sample, since it contains less 14C14C size 12{"" lSup { size 8{"14"} } C} {}, and for samples more than 50 thousand years old, it is impossible.

Human-made (or artificial) radioactivity has been produced for decades and has many uses. Some of these include medical therapy for cancer, medical imaging and diagnostics, and food preservation by irradiation. Many applications as well as the biological effects of radiation are explored in Medical Applications of Nuclear Physics, but it is clear that radiation is hazardous. A number of tragic examples of this exist, one of the most disastrous being the meltdown and fire at the Chernobyl reactor complex in the Ukraine (see Figure 31.23). Several radioactive isotopes were released in huge quantities, contaminating many thousands of square kilometers and directly affecting hundreds of thousands of people. The most significant releases were of 131I131I, 90Sr90Sr, 137Cs137Cs, 239Pu239Pu, 238U238U, and 235U235U. Estimates are that the total amount of radiation released was about 100 million curies.

Human and Medical Applications

A person holding a hand held radiation detector near the Chernobyl reactor.
Figure 31.23 The Chernobyl reactor. More than 100 people died soon after its meltdown, and there will be thousands of deaths from radiation-induced cancer in the future. While the accident was due to a series of human errors, the cleanup efforts were heroic. Most of the immediate fatalities were firefighters and reactor personnel. (credit: Elena Filatova)

Example 31.6 What Mass of 137Cs137Cs Escaped Chernobyl?

It is estimated that the Chernobyl disaster released 6.0 MCi of 137Cs137Cs into the environment. Calculate the mass of 137Cs137Cs released.

Strategy

We can calculate the mass released using Avogadro’s number and the concept of a mole if we can first find the number of nuclei NN size 12{N} {} released. Since the activity RR size 12{R} {} is given, and the half-life of 137Cs137Cs size 12{"" lSup { size 8{"137"} } "Cs"} {} is found in Appendix B to be 30.2 y, we can use the equation R=0.693Nt1/2R=0.693Nt1/2 size 12{R= { {0 "." "693"N} over {t rSub { size 8{1/2} } } } } {} to find NN size 12{N} {}.

Solution

Solving the equation R=0.693Nt1/2R=0.693Nt1/2 size 12{R= { {0 "." "693"N} over {t rSub { size 8{1/2} } } } } {} for NN size 12{N} {} gives

N=Rt1/20.693.N=Rt1/20.693. size 12{N= { { ital "Rt""" lSub { size 8{1/2} } } over {0 "." "693"} } } {}
31.55

Entering the given values yields

N=(6.0 MCi)(30.2 y)0.693.N=(6.0 MCi)(30.2 y)0.693. size 12{N= { { \( 6 "." 0" MCi" \) \( "30" "." 2" y" \) } over {0 "." "693"} } } {}
31.56

Converting curies to becquerels and years to seconds, we get

N = ( 6 . 0 × 10 6 Ci ) ( 3 . 7 × 10 10 Bq/Ci ) ( 30.2 y ) ( 3 . 16 × 10 7 s/y ) 0.693 = 3 . 1 × 10 26 . N = ( 6 . 0 × 10 6 Ci ) ( 3 . 7 × 10 10 Bq/Ci ) ( 30.2 y ) ( 3 . 16 × 10 7 s/y ) 0.693 = 3 . 1 × 10 26 . alignl { stack { size 12{N= { { \( 6 "." 0´"10" rSup { size 8{6} } " Ci" \) \( 3 "." 7´"10" rSup { size 8{"10"} } " Bq/Ci" \) \( "30" "." 2" y" \) \( 3 "." "16"´"10" rSup { size 8{7} } " s/y" \) } over {0 "." "693"} } } {} # " "=3 "." 1´"10" rSup { size 8{"26"} } "." {} } } {}
31.57

One mole of a nuclide AXAX size 12{"" lSup { size 8{A} } X} {} has a mass of AA size 12{A} {} grams, so that one mole of 137Cs137Cs size 12{"" lSup { size 8{"137"} } "Cs"} {} has a mass of 137 g. A mole has 6.02 ×10236.02 ×1023 size 12{6 "." "02 " times "10" rSup { size 8{"23"} } } {} nuclei. Thus the mass of 137Cs137Cs size 12{"" lSup { size 8{"137"} } "Cs"} {} released was

m = 137 g 6.02 × 10 23 ( 3 . 1 × 10 26 ) = 70 × 10 3 g = 70 kg . m = 137 g 6.02 × 10 23 ( 3 . 1 × 10 26 ) = 70 × 10 3 g = 70 kg . alignl { stack { size 12{m= left ( { {"137"" g"} over {6 "." "02 "´"10" rSup { size 8{"23"} } } } right ) \( 3 "." 1´"10" rSup { size 8{"26"} } \) ="70"´"10" rSup { size 8{3} } " g"} {} # " "="70 kg" "." {} } } {}
31.58

Discussion

While 70 kg of material may not be a very large mass compared to the amount of fuel in a power plant, it is extremely radioactive, since it only has a 30-year half-life. Six megacuries (6.0 MCi) is an extraordinary amount of activity but is only a fraction of what is produced in nuclear reactors. Similar amounts of the other isotopes were also released at Chernobyl. Although the chances of such a disaster may have seemed small, the consequences were extremely severe, requiring greater caution than was used. More will be said about safe reactor design in the next chapter, but it should be noted that Western reactors have a fundamentally safer design.

Activity RR size 12{R} {} decreases in time, going to half its original value in one half-life, then to one-fourth its original value in the next half-life, and so on. Since R=0.693Nt1/2R=0.693Nt1/2 size 12{R= { {0 "." "693"N} over {t rSub { size 8{1/2} } } } } {}, the activity decreases as the number of radioactive nuclei decreases. The equation for RR size 12{R} {} as a function of time is found by combining the equations N=N0eλtN=N0eλt size 12{N=N rSub { size 8{0} } e rSup { size 8{ - λt} } } {} and R=0.693Nt1/2R=0.693Nt1/2 size 12{R= { {0 "." "693"N} over {t rSub { size 8{1/2} } } } } {}, yielding

R=R0eλt,R=R0eλt, size 12{R=R rSub { size 8{0} } e rSup { size 8{ - λt} } } {}
31.59

where R0R0 size 12{R rSub { size 8{0} } } {} is the activity at t=0t=0 size 12{t=0} {}. This equation shows exponential decay of radioactive nuclei. For example, if a source originally has a 1.00-mCi activity, it declines to 0.500 mCi in one half-life, to 0.250 mCi in two half-lives, to 0.125 mCi in three half-lives, and so on. For times other than whole half-lives, the equation R=R0eλtR=R0eλt size 12{R=R rSub { size 8{0} } e rSup { size 8{ - λt} } } {} must be used to find RR size 12{R} {}.

PhET Explorations: Alpha Decay

Watch alpha particles escape from a polonium nucleus, causing radioactive alpha decay. See how random decay times relate to the half life. Click to open media in new browser.

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