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College Physics

30.9 The Pauli Exclusion Principle

College Physics30.9 The Pauli Exclusion Principle
  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

Multiple-Electron Atoms

All atoms except hydrogen are multiple-electron atoms. The physical and chemical properties of elements are directly related to the number of electrons a neutral atom has. The periodic table of the elements groups elements with similar properties into columns. This systematic organization is related to the number of electrons in a neutral atom, called the atomic number, ZZ size 12{n} {}. We shall see in this section that the exclusion principle is key to the underlying explanations, and that it applies far beyond the realm of atomic physics.

In 1925, the Austrian physicist Wolfgang Pauli (see Figure 30.57) proposed the following rule: No two electrons can have the same set of quantum numbers. That is, no two electrons can be in the same state. This statement is known as the Pauli exclusion principle, because it excludes electrons from being in the same state. The Pauli exclusion principle is extremely powerful and very broadly applicable. It applies to any identical particles with half-integral intrinsic spin—that is, having s=1/2, 3/2, ...s=1/2, 3/2, ... size 12{s=1/2,`3/2, "." "." "." "." } {} Thus no two electrons can have the same set of quantum numbers.

Pauli Exclusion Principle

No two electrons can have the same set of quantum numbers. That is, no two electrons can be in the same state.

A black and white portrait of Austrian physicist Wolfgang Pauli.
Figure 30.57 The Austrian physicist Wolfgang Pauli (1900–1958) played a major role in the development of quantum mechanics. He proposed the exclusion principle; hypothesized the existence of an important particle, called the neutrino, before it was directly observed; made fundamental contributions to several areas of theoretical physics; and influenced many students who went on to do important work of their own. (credit: Nobel Foundation, via Wikimedia Commons)

Let us examine how the exclusion principle applies to electrons in atoms. The quantum numbers involved were defined in Quantum Numbers and Rules as n, l,ml, sn, l,ml, s, and msms size 12{m rSub { size 8{s} } } {}. Since ss size 12{s} {} is always 1/21/2 size 12{1/2} {} for electrons, it is redundant to list ss, and so we omit it and specify the state of an electron by a set of four numbers n,l,ml,msn,l,ml,ms. For example, the quantum numbers 2, 1, 0,1/22, 1, 0,1/2 size 12{ left (2,` 1,` 0,` - 1/2 right )} {} completely specify the state of an electron in an atom.

Since no two electrons can have the same set of quantum numbers, there are limits to how many of them can be in the same energy state. Note that nn size 12{n} {} determines the energy state in the absence of a magnetic field. So we first choose nn size 12{n} {}, and then we see how many electrons can be in this energy state or energy level. Consider the n=1n=1 size 12{n=1} {} level, for example. The only value ll size 12{l} {} can have is 0 (see Table 30.1 for a list of possible values once nn size 12{n} {} is known), and thus mlml can only be 0. The spin projection msms can be either +1/2+1/2 or 1/21/2, and so there can be two electrons in the n=1n=1 state. One has quantum numbers 1, 0, 0,+1/21, 0, 0,+1/2, and the other has 1, 0, 0,1/21, 0, 0,1/2. Figure 30.58 illustrates that there can be one or two electrons having n=1n=1 size 12{n=1} {}, but not three.

The figure here shows configuration of electrons. At the top, the key shows two purple balls, which depict electrons. The upward directed arrow on the first ball or electron shows its spin is plus one half, and the downward arrow on the second electron shows the opposite spin that is minus one half. Two other sections show the electronic configurations of electrons for two levels, n equal to one and n equal to two. One section shows the allowed configurations of the electron in the n is equal to one and two levels, and the second section for the configurations which are not allowed. In the allowed section, n is equal to two has three vacant shells and one electron in each of the outer two shells, one with spin up and one with spin down; and n is equal to one configuration has two shells containing one each spin up and spin down electron and the three other shells containing combinations of both spins each. For the not allowed section, n is equal to two have all vacant shells and n is equal to one have unevenly balanced electrons in its shells.
Figure 30.58 The Pauli exclusion principle explains why some configurations of electrons are allowed while others are not. Since electrons cannot have the same set of quantum numbers, a maximum of two can be in the n=1n=1 size 12{n=1} {} level, and a third electron must reside in the higher-energy n=2n=2 size 12{n=2} {} level. If there are two electrons in the n=1n=1 size 12{n=1} {} level, their spins must be in opposite directions. (More precisely, their spin projections must differ.)

Shells and Subshells

Because of the Pauli exclusion principle, only hydrogen and helium can have all of their electrons in the n=1n=1 size 12{n=1} {} state. Lithium (see the periodic table) has three electrons, and so one must be in the n=2n=2 size 12{n=2} {} level. This leads to the concept of shells and shell filling. As we progress up in the number of electrons, we go from hydrogen to helium, lithium, beryllium, boron, and so on, and we see that there are limits to the number of electrons for each value of nn size 12{n} {}. Higher values of the shell nn size 12{n} {} correspond to higher energies, and they can allow more electrons because of the various combinations of l,mll,ml size 12{l, m rSub { size 8{l} } } {}, and msms size 12{m rSub { size 8{s} } } {} that are possible. Each value of the principal quantum number nn size 12{n} {} thus corresponds to an atomic shell into which a limited number of electrons can go. Shells and the number of electrons in them determine the physical and chemical properties of atoms, since it is the outermost electrons that interact most with anything outside the atom.

The probability clouds of electrons with the lowest value of ll size 12{l} {} are closest to the nucleus and, thus, more tightly bound. Thus when shells fill, they start with l=0l=0 size 12{l=0} {}, progress to l=1l=1 size 12{l=1} {}, and so on. Each value of ll size 12{l} {} thus corresponds to a subshell.

The table given below lists symbols traditionally used to denote shells and subshells.

Shell Subshell
n n size 12{n} {} l l size 12{l} {} Symbol
1 0 s s size 12{s} {}
2 1 p p size 12{p} {}
3 2 d d size 12{d} {}
4 3 f f size 12{f} {}
5 4 g g size 12{g} {}
5 h h size 12{h} {}
62 i i size 12{i} {}
Table 30.2 Shell and Subshell Symbols

To denote shells and subshells, we write nlnl size 12{ ital "nl"} {} with a number for nn size 12{n} {} and a letter for ll size 12{l} {}. For example, an electron in the n=1n=1 size 12{n=1} {} state must have l=0l=0 size 12{l=1} {}, and it is denoted as a 1s1s size 12{1s} {} electron. Two electrons in the n=1n=1 size 12{n=1} {} state is denoted as 1s21s2 size 12{1s rSup { size 8{2} } } {}. Another example is an electron in the n=2n=2 size 12{n=2} {} state with l=1l=1 size 12{l=1} {}, written as 2p2p size 12{2p} {}. The case of three electrons with these quantum numbers is written 2p32p3 size 12{2p rSup { size 8{3} } } {}. This notation, called spectroscopic notation, is generalized as shown in Figure 30.59.

Diagram illustrating the components of the expression 2 times p to the third power, where 2 is the pricncipal quantum number n, p is the angular momentum quantum number, represented by a script letter l, and the exponent 3 is the number of electrons.
Figure 30.59

Counting the number of possible combinations of quantum numbers allowed by the exclusion principle, we can determine how many electrons it takes to fill each subshell and shell.

Example 30.4 How Many Electrons Can Be in This Shell?

List all the possible sets of quantum numbers for the n=2n=2 size 12{n=2} {} shell, and determine the number of electrons that can be in the shell and each of its subshells.

Strategy

Given n=2n=2 size 12{n=2} {} for the shell, the rules for quantum numbers limit ll size 12{l} {} to be 0 or 1. The shell therefore has two subshells, labeled 2s2s size 12{2s} {} and 2p2p size 12{2p} {}. Since the lowest ll size 12{l} {} subshell fills first, we start with the 2s2s size 12{2s} {} subshell possibilities and then proceed with the 2p2p size 12{2p} {} subshell.

Solution

It is convenient to list the possible quantum numbers in a table, as shown below.

Image contains a table listing all possible quantum numbers for the n equals 2 shell. The table shows that there are a total of two electrons in the 2 s subshell and six electrons in the 2 p subshell, for a total of eight electrons in the shell.
Figure 30.60

Discussion

It is laborious to make a table like this every time we want to know how many electrons can be in a shell or subshell. There exist general rules that are easy to apply, as we shall now see.

The number of electrons that can be in a subshell depends entirely on the value of ll size 12{l} {}. Once ll size 12{l} {} is known, there are a fixed number of values of mlml size 12{m rSub { size 8{l} } } {}, each of which can have two values for msms size 12{m rSub { size 8{s} } } {} First, since mlml size 12{m rSub { size 8{l} } } {} goes from ll size 12{ - l} {} to l in steps of 1, there are 2l+12l+1 size 12{2l+1} {} possibilities. This number is multiplied by 2, since each electron can be spin up or spin down. Thus the maximum number of electrons that can be in a subshell is 22l+122l+1 size 12{2 left (2l+1 right )} {}.

For example, the 2s2s size 12{2s} {} subshell in Example 30.4 has a maximum of 2 electrons in it, since 22l+1=20+1=222l+1=20+1=2 size 12{2 left (2l+1 right )=2 left (0+1 right )=2} {} for this subshell. Similarly, the 2p2p size 12{2p} {} subshell has a maximum of 6 electrons, since 22l+1=22+1=622l+1=22+1=6 size 12{2 left (2l+1 right )=2 left (2+1 right )=6} {}. For a shell, the maximum number is the sum of what can fit in the subshells. Some algebra shows that the maximum number of electrons that can be in a shell is 2n22n2 size 12{2n rSup { size 8{2} } } {}.

For example, for the first shell n=1n=1 size 12{n=1} {}, and so 2n2=22n2=2 size 12{2n rSup { size 8{2} } =2} {}. We have already seen that only two electrons can be in the n=1n=1 size 12{n=1} {} shell. Similarly, for the second shell, n=2n=2 size 12{n=2} {}, and so 2n2=82n2=8 size 12{2n rSup { size 8{2} } =8} {}. As found in Example 30.4, the total number of electrons in the n=2n=2 size 12{n=2} {} shell is 8.

Example 30.5 Subshells and Totals for n=3n=3 size 12{n=3} {}

How many subshells are in the n=3n=3 size 12{n=3} {} shell? Identify each subshell, calculate the maximum number of electrons that will fit into each, and verify that the total is 2n22n2 size 12{2n rSup { size 8{2} } } {}.

Strategy

Subshells are determined by the value of ll size 12{l} {}; thus, we first determine which values of ll size 12{ ital "ls"} {} are allowed, and then we apply the equation “maximum number of electrons that can be in a subshell =22l+1=22l+1 size 12{2 left (2l+1 right )} {}” to find the number of electrons in each subshell.

Solution

Since n=3n=3 size 12{n=3} {}, we know that l l can be 0, 10, 1, or 22; thus, there are three possible subshells. In standard notation, they are labeled the 3s3s, 3p3p, and 3d3d size 12{3d} {} subshells. We have already seen that 2 electrons can be in an ss state, and 6 in a pp size 12{p} {} state, but let us use the equation “maximum number of electrons that can be in a subshell = 22l+122l+1 size 12{2 left (2l+1 right )} {}” to calculate the maximum number in each:

3 s has l = 0 ; thus, 2 2l + 1 = 2 0 + 1 = 2 3 p has l = 1; thus, 2 2l + 1 = 2 2 + 1 = 6 3 d has l = 2; thus, 2 2l + 1 = 2 4 + 1 = 10 Total = 18 ( in the n = 3 shell ) 3 s has l = 0 ; thus, 2 2l + 1 = 2 0 + 1 = 2 3 p has l = 1; thus, 2 2l + 1 = 2 2 + 1 = 6 3 d has l = 2; thus, 2 2l + 1 = 2 4 + 1 = 10 Total = 18 ( in the n = 3 shell )
30.55

The equation “maximum number of electrons that can be in a shell = 2n22n2 size 12{2n rSup { size 8{2} } } {}” gives the maximum number in the n=3n=3 size 12{n=3} {} shell to be

Maximum number of electrons=2n2=232=29=18.Maximum number of electrons=2n2=232=29=18.
30.56

Discussion

The total number of electrons in the three possible subshells is thus the same as the formula 2n22n2 size 12{2n rSup { size 8{2} } } {}. In standard (spectroscopic) notation, a filled n=3n=3 size 12{n=3} {} shell is denoted as 3s23p63d103s23p63d10 size 12{3s rSup { size 8{2} } 3p rSup { size 8{6} } 3d rSup { size 8{"10"} } } {}. Shells do not fill in a simple manner. Before the n=3n=3 size 12{n=3} {} shell is completely filled, for example, we begin to find electrons in the n=4n=4 size 12{n=4} {} shell.

Shell Filling and the Periodic Table

Table 30.3 shows electron configurations for the first 20 elements in the periodic table, starting with hydrogen and its single electron and ending with calcium. The Pauli exclusion principle determines the maximum number of electrons allowed in each shell and subshell. But the order in which the shells and subshells are filled is complicated because of the large numbers of interactions between electrons.

Element Number of electrons (Z) Ground state configuration
H 1 1 s 1 1 s 1 size 12{1s rSup { size 8{1} } } {}
He 2 1 s 2 1 s 2 size 12{1s rSup { size 8{2} } } {}
Li 3 1 s 2 1 s 2 size 12{1s rSup { size 8{2} } } {} 2 s 1 2 s 1 size 12{2s rSup { size 8{1} } } {}
Be 4 " 2 s 2 2 s 2 size 12{2s rSup { size 8{2} } } {}
B 5 " 2 s 2 2 s 2 size 12{2s rSup { size 8{2} } } {} 2 p 1 2 p 1 size 12{2p rSup { size 8{1} } } {}
C 6 " 2 s 2 2 s 2 size 12{2s rSup { size 8{2} } } {} 2 p 2 2 p 2 size 12{2p rSup { size 8{2} } } {}
N 7 " 2 s 2 2 s 2 size 12{2s rSup { size 8{2} } } {} 2 p 3 2 p 3 size 12{2p rSup { size 8{3} } } {}
O 8 " 2 s 2 2 s 2 size 12{2s rSup { size 8{2} } } {} 2 p 4 2 p 4 size 12{2p rSup { size 8{4} } } {}
F 9 " 2 s 2 2 s 2 size 12{2s rSup { size 8{2} } } {} 2 p 5 2 p 5 size 12{2p rSup { size 8{5} } } {}
Ne 10 " 2 s 2 2 s 2 size 12{2s rSup { size 8{2} } } {} 2 p 6 2 p 6 size 12{2p rSup { size 8{6} } } {}
Na 11 " 2 s 2 2 s 2 size 12{2s rSup { size 8{2} } } {} 2 p 6 2 p 6 size 12{2p rSup { size 8{6} } } {} 3 s 1 3 s 1 size 12{3s rSup { size 8{1} } } {}
Mg 12 " " " 3 s 2 3 s 2 size 12{3s rSup { size 8{2} } } {}
Al 13 " " " 3 s 2 3 s 2 size 12{3s rSup { size 8{2} } } {} 3 p 1 3 p 1 size 12{3p rSup { size 8{1} } } {}
Si 14 " " " 3 s 2 3 s 2 size 12{3s rSup { size 8{2} } } {} 3 p 2 3 p 2 size 12{3p rSup { size 8{2} } } {}
P 15 " " " 3 s 2 3 s 2 size 12{3s rSup { size 8{2} } } {} 3 p 3 3 p 3 size 12{3p rSup { size 8{3} } } {}
S 16 " " " 3 s 2 3 s 2 size 12{3s rSup { size 8{2} } } {} 3 p 4 3 p 4 size 12{3p rSup { size 8{4} } } {}
Cl 17 " " " 3 s 2 3 s 2 size 12{3s rSup { size 8{2} } } {} 3 p 5 3 p 5 size 12{3p rSup { size 8{5} } } {}
Ar 18 " " " 3 s 2 3 s 2 size 12{3s rSup { size 8{2} } } {} 3 p 6 3 p 6 size 12{3p rSup { size 8{6} } } {}
K 19 " " " 3 s 2 3 s 2 size 12{3s rSup { size 8{2} } } {} 3 p 6 3 p 6 size 12{3p rSup { size 8{6} } } {} 4 s 1 4 s 1 size 12{4s rSup { size 8{1} } } {}
Ca 20 " " " " " 4 s 2 4 s 2 size 12{4s rSup { size 8{2} } } {}
Table 30.3 Electron Configurations of Elements Hydrogen Through Calcium

Examining the above table, you can see that as the number of electrons in an atom increases from 1 in hydrogen to 2 in helium and so on, the lowest-energy shell gets filled first—that is, the n=1n=1 size 12{n=1} {} shell fills first, and then the n=2n=2 size 12{n=2} {} shell begins to fill. Within a shell, the subshells fill starting with the lowest ll size 12{l} {}, or with the ss size 12{s} {} subshell, then the pp size 12{p} {}, and so on, usually until all subshells are filled. The first exception to this occurs for potassium, where the 4s4s size 12{4s} {} subshell begins to fill before any electrons go into the 3d3d size 12{3d} {} subshell. The next exception is not shown in Table 30.3; it occurs for rubidium, where the 5s5s size 12{5s} {} subshell starts to fill before the 4d4d size 12{4d} {} subshell. The reason for these exceptions is that l=0l=0 size 12{l=0} {} electrons have probability clouds that penetrate closer to the nucleus and, thus, are more tightly bound (lower in energy).

Figure 30.61 shows the periodic table of the elements, through element 118. Of special interest are elements in the main groups, namely, those in the columns numbered 1, 2, 13, 14, 15, 16, 17, and 18.

This picture shows the periodic table of the elements.
Figure 30.61 Periodic table of the elements (credit: National Institute of Standards and Technology, U.S. Department of Commerce)

The number of electrons in the outermost subshell determines the atom’s chemical properties, since it is these electrons that are farthest from the nucleus and thus interact most with other atoms. If the outermost subshell can accept or give up an electron easily, then the atom will be highly reactive chemically. Each group in the periodic table is characterized by its outermost electron configuration. Perhaps the most familiar is Group 18 (Group VIII), the noble gases (helium, neon, argon, etc.). These gases are all characterized by a filled outer subshell that is particularly stable. This means that they have large ionization energies and do not readily give up an electron. Furthermore, if they were to accept an extra electron, it would be in a significantly higher level and thus loosely bound. Chemical reactions often involve sharing electrons. Noble gases can be forced into unstable chemical compounds only under high pressure and temperature.

Group 17 (Group VII) contains the halogens, such as fluorine, chlorine, iodine and bromine, each of which has one less electron than a neighboring noble gas. Each halogen has 5 pp size 12{p} {} electrons (a p5p5 size 12{p} {} configuration), while the pp size 12{p} {} subshell can hold 6 electrons. This means the halogens have one vacancy in their outermost subshell. They thus readily accept an extra electron (it becomes tightly bound, closing the shell as in noble gases) and are highly reactive chemically. The halogens are also likely to form singly negative ions, such as C1C1 size 12{C1 rSup { size 8{ - {}} } } {}, fitting an extra electron into the vacancy in the outer subshell. In contrast, alkali metals, such as sodium and potassium, all have a single ss size 12{s} {} electron in their outermost subshell (an s 1 s 1 size 12{s rSup { size 8{1} } } {} configuration) and are members of Group 1 (Group I). These elements easily give up their extra electron and are thus highly reactive chemically. As you might expect, they also tend to form singly positive ions, such as Na+Na+ size 12{"Na" rSup { size 8{+{}} } } {}, by losing their loosely bound outermost electron. They are metals (conductors), because the loosely bound outer electron can move freely.

Of course, other groups are also of interest. Carbon, silicon, and germanium, for example, have similar chemistries and are in Group 4 (Group IV). Carbon, in particular, is extraordinary in its ability to form many types of bonds and to be part of long chains, such as inorganic molecules. The large group of what are called transitional elements is characterized by the filling of the dd size 12{d} {} subshells and crossing of energy levels. Heavier groups, such as the lanthanide series, are more complex—their shells do not fill in simple order. But the groups recognized by chemists such as Mendeleev have an explanation in the substructure of atoms.

PhET Explorations: Stern-Gerlach Experiment

Build an atom out of protons, neutrons, and electrons, and see how the element, charge, and mass change. Then play a game to test your ideas!

Figure 30.62

Footnotes

  • 2 It is unusual to deal with subshells having l l greater than 6, but when encountered, they continue to be labeled in alphabetical order.
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