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A Quantum Model of Atoms: Waves and Particles

7. A Quantum Model of Atoms: Waves and Particles. Chapter Outline. 7.1 Light Waves Properties of Waves The Behavior of Waves 7.2 Atomic Spectra 7.3 Particles of Light and Quantum Theory 7.4 The Hydrogen Spectrum and the Bohr Model 7.5 Electron Waves

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A Quantum Model of Atoms: Waves and Particles

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  1. 7 A Quantum Model of Atoms: Waves and Particles

  2. Chapter Outline • 7.1 Light Waves • Properties of Waves • The Behavior of Waves • 7.2 Atomic Spectra • 7.3 Particles of Light and Quantum Theory • 7.4 The Hydrogen Spectrum and the Bohr Model • 7.5 Electron Waves • 7.6 Quantum Numbers and Electron Spin • 7.7 The Sizes and Shapes of Atomic Orbitals • 7.8 The Periodic Table and Filling the Orbitals of Multielectron Atoms • 7.9 Electron Configurations of Ions • 7.10 The Sizes of Atoms and Ions • 7.11 Ionization Energies • 7.12 Electron Affinities

  3. The Electromagnetic Spectrum Continuous range of radiant energy (also called electromagnetic radiation).

  4. Electromagnetic Radiation Perpendicular oscillating fields: Electric Magnetic

  5. Wavelength ():the distance from crest to crest or trough to trough. Frequency (): the number of crests of a wave passing a stationary point of reference per unit of time. Amplitude: the height of the crest or depth of the trough. Wave Properties of Light

  6. Wave Properties (cont.) • Wavelength and frequency are related by: = c • Where  is wavelength (in meters),  is frequency (in hertz, or s-1), and c is the speed of light in vacuum. • c is a constant; in a vacuum c= 2.998× 108m/s.

  7. Behavior of Waves • Refraction: • Bending of light waves when passing between media of different densities (e.g. air to water). • Diffraction: • Bending of light waves as they pass around the edge of an object or through narrow openings. • Interference: • The interaction of waves in which their amplitudes either reinforce (constructive) or cancel (destructive).

  8. Diffraction and Interference Diffraction as light passes through slits.

  9. Diffraction and Interference (cont.) • Propagating waves overlap in/out of phase.

  10. Chapter Outline 7.1 Light Waves 7.2 Atomic Spectra 7.3 Particles of Light and Quantum Theory 7.4 The Hydrogen Spectrum and the Bohr Model 7.5 Electron Waves 7.6 Quantum Numbers and Electron Spin 7.7 The Sizes and Shapes of Atomic Orbitals 7.8 The Periodic Table and Filling the Orbitals of Multielectron Atoms 7.9 Electron Configurations of Ions 7.10 The Sizes of Atoms and Ions 7.11 Ionization Energies 7.12 Electron Affinities

  11. Atomic Spectra a) Fraunhofer lines (dark-line spectra) b) Sodium emission (bright-line spectra)

  12. Atomic Emission Spectra Bright-line emission spectra of other elements.

  13. Absorption Spectra: Dark Lines a) Hydrogen b) Helium c) Neon

  14. Chapter Outline • 7.1 Light Waves • 7.2 Atomic Spectra • 7.3 Particles of Light and Quantum Theory • Quantum Theory • Photoelectric Effect • Wave–Particle Duality • 7.4 The Hydrogen Spectrum and the Bohr Model • 7.5 Electron Waves • 7.6 Quantum Numbers and Electron Spin • 7.7 The Sizes and Shapes of Atomic Orbitals • 7.8 The Periodic Table and Filling the Orbitals of Multielectron Atoms • 7.9 Electron Configurations of Ions • 7.10 The Sizes of Atoms and Ions • 7.11 Ionization Energies • 7.12 Electron Affinities

  15. Particles vs. Waves • Incandescence not consistent with Maxwell equations or wave theory. • Most intense emissions in the infrared region • Little or no emissions in UV

  16. Quantum Theory • Radiant energy is “quantized.” • Having values restricted to whole-number multiples of a specific base value. • Quantum – the smallest discrete quantity of a particular form of energy. • Photon – a quantum of electromagnetic radiation. • Energy of photon: E = h or E = hc/ • h = 6.626 x 10–34J•s (Planck’s constant)

  17. Quantized States • Quantized states: Discrete energy levels (e.g., steps) • Unquantized states: smooth transition between levels (e.g., ramp)

  18. Photoelectric Effect If radiation is below threshold energy, no electrons are released. • Photoelectric effect: • Phenomenon of light striking a metal surface and producing an electric current (flow of electrons).

  19. Photoelectric Effect • Explained by quantum theory: • Photons of sufficient energy (h) dislodge e– from metal surface. • Work function () – amount of energy needed to dislodge an electron from the surface of a metal: = h0 • Where  = work function; 0 = threshold frequency. • Kinetic energy of ejected electrons: KEelectron = h– 

  20. Wave–Particle Duality • Wavelike behavior of radiant energy: • Refraction • Diffraction/interference • Particle-like behavior of radiant energy: • Photoelectric effect • Quantized packets of energy

  21. Chapter Outline • 7.1 Light Waves • 7.2 Atomic Spectra • 7.3 Particles of Light and Quantum Theory • 7.4 The Hydrogen Spectrum and the Bohr Model • Hydrogen Emission Spectum • Bohr Model of Hydrogen • 7.5 Electron Waves • 7.6 Quantum Numbers and Electron Spin • 7.7 The Sizes and Shapes of Atomic Orbitals • 7.8 The Periodic Table and Filling the Orbitals of Multielectron Atoms • 7.9 Electron Configurations of Ions • 7.10 The Sizes of Atoms and Ions • 7.11 Ionization Energies • 7.12 Electron Affinities

  22. The Hydrogen Spectrum • Johann Balmer (1825–1898) • Fit frequencies (ν) of four brightest lines in the Hydrogen emission spectrum: (n > 2) • Johannes Rydberg (1854–1919) • Revised Balmer’s equation by changing frequency to wavenumber (1/): (n1 < n2)

  23. Practice: The Rydberg Equation - Solve: (1/λ) = (1.0097 × 10–2 nm–1) (1/22 – 1/42) = 1.893 × 10–3 nm–1 λ = (1/1.893 × 10–3) = 528 nm What is the wavelength of the line in the visible spectrum corresponding to n1 = 2 and n2 = 4?

  24. Practice: The Rydberg Equation - Collect and Organize: We are given values for n1 and n2, and are looking for the wavelength (λ). What is the wavelength of the line in the visible spectrum corresponding to n1 = 2 and n2 = 4?

  25. Practice: The Rydberg Equation - Think About It: The answer (528 nm) corresponds to a wavelength in the visible region, and so is consistent with expectations. What is the wavelength of the line in the visible spectrum corresponding to n1 = 2 and n2 = 4?

  26. The Bohr Model • Electrons in H atoms: • occupy discrete energy levels and exist only in the available energy levels. • may move between energy levels by either absorbing or emitting energy. • have energy levels designated by specific value for n. • Difference between energy levels (ΔE):

  27. Electronic States Energy level – an allowed state that an electron can occupy in an atom. Ground state – lowest-energy level available to an electron in an atom (n = 1). Excited state – any energy state above the ground state. Electron transition – movement of an electron between energy levels.

  28. Show electron transitions (i.e., movement of electrons between energy levels)  = absorption of energy  = emission of energy Energy-Level Diagrams

  29. The Bohr Model (cont.) • Strengths: • Accurately predicts energy needed to remove an electron from an atom (ionization) • Allowed scientists to begin using quantum theory to explain matter at atomic level • Limitations: • Does not account for spectra of multielectron atoms • Movement of electrons in atoms is less clearly defined than Bohr allowed

  30. Chapter Outline • 7.1 Light Waves • 7.2 Atomic Spectra • 7.3 Particles of Light and Quantum Theory • 7.4 The Hydrogen Spectrum and the Bohr Model • 7.5 Electron Waves • De Broglie Wavelengths • Heisenberg Uncertainty Principle • 7.6 Quantum Numbers and Electron Spin • 7.7 The Sizes and Shapes of Atomic Orbitals • 7.8 The Periodic Table and Filling the Orbitals of Multielectron Atoms • 7.9 Electron Configurations of Ions • 7.10 The Sizes of Atoms and Ions • 7.11 Ionization Energies • 7.12 Electron Affinities

  31. Particles or Waves? • De Broglie (1892–1987) • If electromagnetic radiation behaves as a particle, could a particle in motion, such as an electron, behave as a wave? • De Broglie’s equation: • = hc/E = hc/mc2 = h/mc (or = h/mv) •  = de Broglie wavelength, m = mass of electron (in kg), v = velocity (in m/s), and h = Planck’s constant.

  32. Linear Waves • Nodes – regions of standing waves that experience no displacement. • Wavelength,λ= 2L • Harmonic frequencies: • L = nλ/2 • For n = 2½: no nodes, so no standing wave.

  33. Electrons as Waves • Electrons behave like circular waves oscillating around nucleus. • No defined stationary ends • Stable circular waves of circumference = n a) stable b) not stable

  34. The Uncertainty Principle • Wave theory vs. particle theory: • Each yield different answers for location of electron around nucleus. • Impossible to predict both location and velocity accurately. • Heisenberg Uncertainty Principle: • We cannot determine both the position and the momentum of an electron in an atom at the same time.

  35. Chapter Outline 7.1 Light Waves 7.2 Atomic Spectra 7.3 Particles of Light and Quantum Theory 7.4 The Hydrogen Spectrum and the Bohr Model 7.5 Electron Waves 7.6 Quantum Numbers and Electron Spin 7.7 The Sizes and Shapes of Atomic Orbitals 7.8 The Periodic Table and Filling the Orbitals of Multielectron Atoms 7.9 Electron Configurations of Ions 7.10 The Sizes of Atoms and Ions 7.11 Ionization Energies 7.12 Electron Affinities

  36. Electron Wave Equations • Erwin Schrödinger (1925) • Developed mathematical equations to describe behavior of electron waves; became the basis of quantum mechanics. • Wave functions (ᴪ) • ᴪ describes the motion of electron waves as they vary with location and time. • ᴪ2 defines an orbital, or region of high probability for locating an electron.

  37. Wave Numbers and Quantum Equations • Mathematical solutions to wave equations (2) identified by unique combination of three integers • Quantum numbers: • Principle quantum number (n) indicates shell and relative size of orbital(s). n = 1, 2, 3,… • Angular momentum quantum number (ℓ)defines shape of orbital and subshell. ℓ = 0 →(n– 1) • Magnetic quantum number (mℓ) defines orientation of orbital around nucleus. mℓ = –ℓ → +ℓ

  38. Quantum Numbers (cont.)

  39. Quantum Number Relationships

  40. Practice: Electron Orbitals - Collect and Organize: We know the value n = 4, and need to identify the letter designations of the subshells, and the total number of orbitals. For the n = 4 level or shell: • What are the letter designations of all the subshells? • What is the total number of orbitals?

  41. Practice: Electron Orbitals • Analyze: If n = 4, we know the possible range of values for the other quantum numbers and can determine the identities of the subshells and the total number of orbitals from the relationships provided: ℓ = 0  (n – 1), and mℓ = (–ℓ +ℓ). For the n = 4 level or shell: • What are the letter designations of all the subshells? • What is the total number of orbitals?

  42. Practice: Electron Orbitals - Solve: For n = 4, there are 4 possible subshells: - ℓ = 0 for s orbital (1 possible orbital) - ℓ = 1 for porbitals (3 possible) - ℓ = 2 for dorbitals (5 possible) - ℓ = 3 for forbitals (7 possible) - Total number of orbitals = 16 For the n = 4 level or shell: • What are the letter designations of all the subshells? • What is the total number of orbitals?

  43. Practice: Electron Orbitals - Think About It: The values obtained are consistent with the quantum number relationships in Figure 7.18. For the n = 4 level or shell: • What are the letter designations of all the subshells? • What is the total number of orbitals?

  44. Electron Spin • Not all spectral features explained by wave equations: • Appearance of “doublets” in atoms with a single electron in outermost shell • Electron Spin • Up or down

  45. Spin Magnetic Quantum Number • Spinning of electron creates magnetic field oriented either “up” or “down.” • Fourth quantum number (ms) describes electron spin; ms = +½ or –½. • Pauli Exclusion Principle: • No two electrons in an atom can have the same set of four quantum numbers.

  46. Practice: Quantum Numbers - Collect and Organize: We are given a set of quantum numbers and must determine which combinations are allowed. Which of the following combinations of quantum number are allowed? • n = 1, ℓ = 1, mℓ = 0 • n = 3, ℓ = 0, mℓ = 0 • n = 1, ℓ = 0, mℓ = –1 • n = 2, ℓ = 1, mℓ = 2

  47. Practice: Quantum Numbers • Analyze: From the relationships between the quantum numbers and possible values provided in Table 7.1, we can determine what sets of quantum numbers are allowed and which are not. Which of the following combinations of quantum number are allowed? • n = 1, ℓ = 1, mℓ = 0 • n = 3, ℓ = 0, mℓ = 0 • n = 1, ℓ = 0, mℓ = –1 • n = 2, ℓ = 1, mℓ = 2

  48. Practice: Quantum Numbers - Solve: Set 1 – acceptable Set 2 – acceptable Set 3 – not acceptable. For ℓ = 0, the only possible value for mℓ = 0. Set 4 – not acceptable. For ℓ = 1, the possible values for mℓ = –1 +1. Which of the following combinations of quantum number are allowed? • n = 1, ℓ = 1, mℓ = 0 • n = 3, ℓ = 0, mℓ = 0 • n = 1, ℓ = 0, mℓ = –1 • n = 2, ℓ = 1, mℓ = 2

  49. Chapter Outline • 7.1 Light Waves • 7.2 Atomic Spectra • 7.3 Particles of Light and Quantum Theory • 7.4 The Hydrogen Spectrum and the Bohr Model • 7.5 Electron Waves • 7.6 Quantum Numbers and Electron Spin • 7.7 The Sizes and Shapes of Atomic Orbitals • s, p, and dorbitals • 7.8 The Periodic Table and Filling the Orbitals of Multielectron Atoms • 7.9 Electron Configurations of Ions • 7.10 The Sizes of Atoms and Ions • 7.11 Ionization Energies • 7.12 Electron Affinities

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