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The Quantum Mechanical Atom CHAPTER 8 Chemistry: The Molecular Nature of Matter, 6th edition By Jesperson, Brady, & Hyslop CHAPTER 8: Quantum Mechanical Atom Learning Objectives q Light as Waves, Wavelength and Frequency q The Photoelectric Effect, Light as Particles and the Relationship between Energy and Frequency q Atomic Emission and Energy Levels q The Bohr Model and its Failures q Electron Diffraction and Electrons as Waves q Quantum Numbers, Shells, Subshells, and Orbitals q Electron Configuration, Noble Gas Configuration and Orbital Diagrams q Aufbau Principle, Hund’s Rule, and Pauli Exclusion Principle, Heisenberg Uncertainty Principle q Valence vs Inner Core Electrons q Nuclear Charge vs Electron Repulsion q Periodic Trends: Atomic Radius, Ionization Energy, and Electron Affinity 2 Particle-Wave Duality Light Exhibits Interference Constructive interference – Waves in-phase lead to greater amplitude – They add together Destructive interference – Waves out-of-phase lead to lower amplitude – They cancel out Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 3 Particle-Wave Duality Are Electrons Waves or Particles? Light behaves like both a particle and a wave: – Exhibits interference – Has particle-like nature When studying behavior of electrons: – Known to be particles – Also demonstrate interference Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 4 Particle-Wave Duality Standing vs Traveling Waves Traveling wave – Produced by wind on surfaces of lakes and oceans Standing wave – Produced when guitar string is plucked – Center of string vibrates – Ends remain fixed Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 5 Particle-Wave Duality Standing Wave on a Wire • Integer number (n) of peaks and troughs is required • Wavelength is quanMzed: • L is the length of the string 2L != n Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 6 Particle-Wave Duality Standing Wave on a Wire • Has both wave-‐like and parMcle-‐like properMes • Energy of moving electron on a wire is E =½ mv 2 • Wavelength is related to the quantum number, n, and the wire length: 2L != n Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 7 Particle-Wave Duality Electron on a Wire Standing wave • Half-‐wavelength must occur integer number of Mmes along wire s length 2L != n de Broglie s equa3on relates the mass and speed of the parMcle to its wavelength h h ! = v = mv !m • m = mass of parMcle • v = velocity of parMcle Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 8 Particle-Wave Duality Electron on a Wire StarMng with the equaMon of the standing wave and the de Broglie equaMon 2L v= h != n !m Combining with E = ½mv 2, subsMtuMng for v and then λ, we get ! $ " h# " h# # # # !E = m # # = # & " h n h # !m # m! # !E = # &= # # # m #L / n & $mL# " % Combining gives: ( 2 ) 2 nh E = 8mL2 Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 9 Particle-Wave Duality De Broglie & Quantized Energy • Electron energy quanMzed – Depends on integer n • Energy level spacing changes when posiMve charge in nucleus changes – Line spectra different for each element • Lowest energy allowed is for n =1 • Energy cannot be zero, hence atom cannot collapse Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 10 Particle-Wave Duality Ex: Wavelength of an Electron What is the de Broglie wavelength associated with an electron of mass 9.11 × 10 –31 kg traveling at a velocity of 1.0 × 107 m/s? 6.626 × 10 J s 1 kg m /s λ= × (1.0 × 10 m/s)(9.11 × 10 kg) 1J −34 7 2 2 −31 λ = 7.27 × 10–11 m Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 11 Particle-Wave Duality Ex: Wavelength of an Electron Calculate the de Broglie wavelength of a baseball with a mass of 0.10 kg and traveling at a velocity of 35 m/s. A. 1.9 × 10–35 m B. 6.6 × 10–33 m # !.!"! ! #$"%& '() & # #(-.(+" ,)" & C. 1.9 × 10–34 m (( ! %% (( ! = %% D. 2.3 × 10–33 m %*(+,) ! $.#$(-. ' $ #(' $ ' –31 E. 2.3 × 10 m Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 12 Particle-Wave Duality Wave Functions Schrödinger s equa3on – SoluMons give wave funcMons and energy levels of electrons Wave func3on – Wave that corresponds to electron – Called orbitals for electrons in atoms Amplitude of wave funcMon squared – Can be related to probability of finding electron at that given point Nodes – Regions where electrons will not be found Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 13 Quantum Numbers Orbitals Characterized by 3 Quantum #’s Quantum Numbers: – Shorthand – Describes characterisMcs of electron s posiMon – Predicts its behavior n = principal quantum number – All orbitals with same n are in same shell ℓ = secondary quantum number – Divides shells into smaller groups called subshells mℓ = magne3c quantum number – Divides subshells into individual orbitals Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 14 Quantum Numbers Principal Quantum Number (n) • Allowed values: posiMve integers from 1 to ∞ – n = 1, 2, 3, 4, 5, … ∞ • Determines: – Size of orbital E =! Z 2RH hc n2 – Total energy of orbital • RHhc = 2.18 × 10–18 J/atom • For given atom, – Lower n = Lower (more negaMve) E = More stable Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 15 Quantum Numbers Orbital Angular Momentum (ℓ) – Allowed values: 0, 1, 2, 3, 4, 5…(n – 1) – Le>ers: s, p, d, f, g, h Orbital designa3on number nℓ le>er • Possible values of ℓ depend on n – n different values of ℓ for given n • Determines • Shape of orbital Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 16 Quantum Numbers Magnetic Quantum Number (mℓ) • Allowed values: from –ℓ to 0 to +ℓ – Ex. when ℓ=2 then mℓ can be • –2, –1, 0, +1, +2 • Possible values of mℓ depend on ℓ – There are 2ℓ+1 different values of mℓ for given ℓ • Determines orientaMon of orbital in space • To designate specific orbital, you need three quantum numbers – n, ℓ, mℓ Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 17 Quantum Numbers n, ℓ, and mℓ Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 18 Quantum Numbers Multiple Electrons in Orbitals Orbital Designation § Based on first two quantum numbers § Number for n and letter for ℓ § How many electrons can go in each orbital? § Two electrons § Need another quantum number Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 19 Quantum Numbers Spin Quantum Number (ms) • Arises out of behavior of electron in magneMc field • electron acts like a top • Spinning charge is like a magnet – Electron behave like Mny magnets • Leads to two possible direcMons of electron spin – Up and down – North and south Possible Values: +½ ↑ Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E -½ ↓ 20 Quantum Numbers Pauli Exclusion Principle • No two electrons in same atom can have same set of all four quantum numbers (n, ℓ, mℓ , ms) Can only have two electrons per orbital • Two electrons in same orbital must have opposite spin – Electrons are said to be paired Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 21 Quantum Numbers Number of Orbitals Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 22 Quantum Numbers Magnetic Properties • Two electrons in same orbital have different spins – Spins paired—diamagne3c – Sample not a>racted to magneMc field – MagneMc effects tend to cancel each other • Two electrons in different orbital with same spin – Spins unpaired—paramagne3c – Sample a>racted to a magneMc field – MagneMc effects add • Measure extent of a>racMon – Gives number of unpaired spins Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 23 Quantum Numbers Ex: Number of Electrons What is the maximum number of electrons allowed in a set of 4p orbitals? A. 14 B. 6 C. 0 D. 2 E. 10 Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 24 Electron Configurations Ground State Electron Configura3ons • DistribuMon of electrons among orbitals of atom 1. List subshells that contain electrons 2. Indicate their electron populaMon with superscript e.g. N is 1s 2 2s 2 2p 3 Orbital Diagrams • Way to represent electrons in orbitals 1. Represent each orbital with circle (or line) 2. Use arrows to indicate spin of each electron e.g. N is 1s 2s 2p Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 25 Electron Configurations 6s 5s Energy Level Diagram 4f 5p 4d 4p 3d 4s Energy 3p 3s 2p 2s § How to put electrons into a diagram? § Need some rules 1s Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 26 Electron Configurations Aufbau Principle • Building-‐up principle Pauli Exclusion Principle • Two electrons per orbital • Fill following the order suggested by the periodic table • Spins must be paired Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 27 Electron Configurations Hund’s Rule • If you have more than one orbital all at the same energy – Put one electron into each orbital with spins parallel (all up) unMl all are half filled – Aler orbitals are half full, pair up electrons Why? • Repulsion of electrons in same region of space • Empirical observaMon based on magneMc properMes Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 28 Electron Configurations Orbital Diagram & Electron Configurations: e.g. N, Z = 7 4p 3d 4s Energy 3p 3s 2p 2s Each arrow represents electron 1s 2 2s 2 2p 3 1s Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 29 Electron Configurations Orbital Diagram and Electron Configurations: e.g. V, Z = 23 4p 3d 4s Energy 3p 3s 2p 2s Each arrow represents an electron 1s 2 2s 2 2p 2 3s 2 3p 2 4s 2 3d 3 1s Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 30 Electron Configurations Ex: Orbital Diagrams & Electron Configurations Give electron configurations and orbital diagrams for Na and As 6s 5s 5p 4d 4p 3d 4s Energy 3p 3s 2p 2s Na Z = 11 1s 2 2s 2 2p 2 3s 1 As Z = 33 1s 1s 22s 22p 63s 23p 64s 23d 104p 3 Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 31 Electron Configurations Ex: Ground State Electron Configurations What is the correct ground state electron configuraMon for Si? A. 1s 22s 22p 63s 23p 6 B. 1s 22s 22p 63s 23p 4 C. 1s 22s 22p 62d 4 D. 1s 22s 22p 63s 23p 2 E. 1s 22s 22p 63s 13p 3 Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Ma>er, 6E 32 Problem Set B