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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 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 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 Matter, 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 Matter, 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 Matter, 6E 5 Particle-Wave Duality Standing Wave on a Wire • Integer number (n) of peaks and troughs is required • Wavelength is quantized: • L is the length of the string l= Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 2L n 6 Particle-Wave Duality Standing Wave on a Wire • 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: l= 2L n Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 7 Particle-Wave Duality Electron on a Wire Standing wave • Half-wavelength must occur integer number of times along wire’s length l= 2L n de Broglie’s equation relates the mass and speed of the particle to its wavelength l= h mv v= h lm • m = mass of particle • v = velocity of particle Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 8 Particle-Wave Duality Electron on a Wire Starting with the equation of the standing wave and the de Broglie equation l= 2L n v= h lm Combining with E = ½mv 2, substituting for v and then λ, we get 1 h2 1 h2 E = m 2 2= 2 lm 2 ml 2 Combining gives: n 2h 2 E = 2 8mL æ 1ç h2 E = ç 2 ç m 2L / n è Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E ( ) ö ÷ n 2h 2 = ÷ 2 2 8 mL ÷ ø 9 Particle-Wave Duality De Broglie & Quantized Energy • 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 Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 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 (1.0 10 m/s)(9.11 10 kg) 1J 34 2 2 31 7 = 7.27 × 10–11 m Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 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 æ 6.626 ´ 10-34 J s ö æ 1 kg m2 /s2 ö C. 1.9 × 10–34 m ÷÷ ´ çç ÷÷ l = çç D. 2.3 × 10–33 m 35 m/s ´ 0.10 kg ø è 1J è ø –31 E. 2.3 × 10 m Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 12 Particle-Wave Duality Wave Functions 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 Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 13 Quantum Numbers Orbitals Characterized by 3 Quantum #’s 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 mℓ = magnetic quantum number – Divides subshells into individual orbitals Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 14 Quantum Numbers Principal Quantum Number (n) • Allowed values: positive integers from 1 to – n = 1, 2, 3, 4, 5, … • Determines: – Size of orbital E =- – Total energy of orbital Z 2RH hc n2 • RHhc = 2.18 × 10–18 J/atom • For given atom, – Lower n = Lower (more negative) E = More stable Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 15 Quantum Numbers Orbital Angular Momentum (ℓ) – 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 Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 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 orientation of orbital in space • To designate specific orbital, you need three quantum numbers – n, ℓ, mℓ Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 17 Quantum Numbers n, ℓ, and mℓ Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 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 Matter, 6E 19 Quantum Numbers Spin Quantum Number (ms) • 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 Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 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 Matter, 6E 21 Quantum Numbers Number of Orbitals Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 22 Quantum Numbers Magnetic Properties • 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 Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 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 Matter, 6E 24 Electron Configurations Ground State 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 1s 2s 2p Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 6E 25 Electron Configurations Energy Level Diagram 4f 6s 5p 4d 5s 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 Matter, 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 Matter, 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) 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 Jesperson, Brady, Hyslop. Chemistry: The Molecular Nature of Matter, 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 Matter, 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 Matter, 6E 30 Electron Configurations Ex: Orbital Diagrams & Electron Configurations Give electron configurations and orbital diagrams for Na and As 6s 5p 4d 5s 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 Matter, 6E 31 Electron Configurations Ex: Ground State Electron Configurations What is the correct ground state electron configuration 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 Matter, 6E 32 Problem Set B