The Quantum Mechanical Atom CHAPTER 8 Chemistry: The Molecular Nature of Matter, 6 th edition By Jesperson, Brady, & Hyslop

CHAPTER 8: Quantum Mechanical Atom 2 Learning Objectives  Light as Waves, Wavelength and Frequency  The Photoelectric Effect, Light as Particles and the Relationship between Energy and Frequency  Atomic Emission and Energy Levels  The Bohr Model and its Failures  Electron Diffraction and Electrons as Waves  Quantum Numbers, Shells, Subshells, and Orbitals  Electron Configuration, Noble Gas Configuration and Orbital Diagrams  Aufbau Principle, Hund’s Rule, and Pauli Exclusion Principle, Heisenberg Uncertainty Principle  Valence vs Inner Core Electrons  Nuclear Charge vs Electron Repulsion  Periodic Trends: Atomic Radius, Ionization Energy, and Electron Affinity

Particle-Wave Duality Light Exhibits Interference Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 3 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

Particle-Wave Duality Are Electrons Waves or Particles? Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 4 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

Particle-Wave Duality Standing vs Traveling Waves Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 5 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

Particle-Wave Duality Standing Wave on a Wire Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 6 Integer number (n) of peaks and troughs is required Wavelength is quantized: L is the length of the string

Particle-Wave Duality Standing Wave on a Wire Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 7 Has both wave-like and particle-like properties Energy of moving electron on a wire is E =½ mv 2 Wavelength is related to the quantum number, n, and the wire length:

Particle-Wave Duality Electron on a Wire Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 8 Standing wave Half-wavelength must occur integer number of times along wire’s length de Broglie’s equation relates the mass and speed of the particle to its wavelength m = mass of particle v = velocity of particle

Particle-Wave Duality Electron on a Wire Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 9 Starting with the equation of the standing wave and the de Broglie equation Combining with E = ½mv 2, substituting for v and then λ, we get Combining gives:

Particle-Wave Duality De Broglie & Quantized Energy Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 10 Electron energy quantized – Depends on integer n Energy level spacing changes when positive 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

Particle-Wave Duality Ex: Wavelength of an Electron Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 11 What is the de Broglie wavelength associated with an electron of mass 9.11 × 10 –31 kg traveling at a velocity of 1.0 × 10 7 m/s? = 7.27 × 10 –11 m

Particle-Wave Duality Ex: Wavelength of an Electron Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 12 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 E. 2.3 × 10 –31 m

Particle-Wave Duality Wave Functions Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 13 Schrödinger’s equation – Solutions give wave functions and energy levels of electrons Wave function – Wave that corresponds to electron – Called orbitals for electrons in atoms Amplitude of wave function squared – Can be related to probability of finding electron at that given point Nodes – Regions where electrons will not be found

Quantum Numbers Orbitals Characterized by 3 Quantum #’s Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 14 Quantum Numbers: – Shorthand – Describes characteristics of electron’s position – 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 m ℓ = magnetic quantum number – Divides subshells into individual orbitals

Quantum Numbers Principal Quantum Number (n) Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 15 Allowed values: positive integers from 1 to  – n = 1, 2, 3, 4, 5, …  Determines: – Size of orbital – Total energy of orbital R H hc = 2.18 × 10 –18 J/atom For given atom, – Lower n = Lower (more negative) E = More stable

Quantum Numbers Orbital Angular Momentum ( ℓ ) Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 16 – Allowed values: 0, 1, 2, 3, 4, 5…(n – 1) – Letters: s, p, d, f, g, h Orbital designation number nℓ letter Possible values of ℓ depend on n – n different values of ℓ for given n Determines Shape of orbital

Quantum Numbers Magnetic Quantum Number (m ℓ ) Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 17 Allowed values: from –ℓ to 0 to +ℓ – Ex. when ℓ=2 then m ℓ can be –2, –1, 0, +1, +2 Possible values of m ℓ depend on ℓ – There are 2ℓ+1 different values of m ℓ for given ℓ Determines orientation of orbital in space To designate specific orbital, you need three quantum numbers – n, ℓ, m ℓ

Quantum Numbers n, ℓ, and m ℓ Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 18

Quantum Numbers Multiple Electrons in Orbitals Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 19 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

Quantum Numbers Spin Quantum Number (m s ) Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 20 Arises out of behavior of electron in magnetic field electron acts like a top Spinning charge is like a magnet – Electron behave like tiny magnets Leads to two possible directions of electron spin – Up and down – North and south Possible Values: +½  ½  

Quantum Numbers Pauli Exclusion Principle Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 21 No two electrons in same atom can have same set of all four quantum numbers (n, ℓ, m ℓ, m s ) Can only have two electrons per orbital Two electrons in same orbital must have opposite spin – Electrons are said to be paired

Quantum Numbers Number of Orbitals Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 22

Quantum Numbers Magnetic Properties Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 23 Two electrons in same orbital have different spins – Spins paired—diamagnetic – Sample not attracted to magnetic field – Magnetic effects tend to cancel each other Two electrons in different orbital with same spin – Spins unpaired—paramagnetic – Sample attracted to a magnetic field – Magnetic effects add Measure extent of attraction – Gives number of unpaired spins

Quantum Numbers Ex: Number of Electrons Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 24 What is the maximum number of electrons allowed in a set of 4p orbitals? A. 14 B. 6 C. 0 D. 2 E. 10

Electron Configurations Ground State Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 25 1s1s2s2s2p2p Electron Configurations Distribution of electrons among orbitals of atom 1.List subshells that contain electrons 2.Indicate their electron population 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

Electron Configurations Energy Level Diagram Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 26  How to put electrons into a diagram?  Need some rules

Electron Configurations Aufbau Principle Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 27 Building-up principle Pauli Exclusion Principle Two electrons per orbital Fill following the order suggested by the periodic table Spins must be paired

Electron Configurations Hund’s Rule Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 28 If you have more than one orbital all at the same energy – Put one electron into each orbital with spins parallel (all up) until all are half filled – After orbitals are half full, pair up electrons Why? Repulsion of electrons in same region of space Empirical observation based on magnetic properties

Electron Configurations Orbital Diagram & Electron Configurations: e.g. N, Z = 7 Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 29 Each arrow represents electron 1s 2 2s 2 2p 31s 2 2s 2 2p 3

Electron Configurations Orbital Diagram and Electron Configurations: e.g. V, Z = 23 Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 30 Each arrow represents an electron 1s 2 2s 2 2p 2 3s 2 3p 2 4s 2 3d 31s 2 2s 2 2p 2 3s 2 3p 2 4s 2 3d 3

Electron Configurations Ex: Orbital Diagrams & Electron Configurations Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 31 Give electron configurations and orbital diagrams for Na and As Na Z = 11 As Z = 33 1s 2 2s 2 2p 2 3s 11s 2 2s 2 2p 2 3s 1 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 4p 3

Electron Configurations Ex: Ground State Electron Configurations Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 32 What is the correct ground state electron configuration for Si? A. 1s 2 2s 2 2p 6 3s 2 3p 6 B. 1s 2 2s 2 2p 6 3s 2 3p 4 C. 1s 2 2s 2 2p 6 2d 4 D. 1s 2 2s 2 2p 6 3s 2 3p 2 E. 1s 2 2s 2 2p 6 3s 1 3p 3

Problem Set B