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Chapter 8: The Quantum Mechanical Atom

Chapter 8: The Quantum Mechanical Atom. Chemistry: The Molecular Nature of Matter, 6E Jespersen/Brady/Hyslop. Electromagnetic Energy. Electromagnetic Radiation Light energy or wave Travels through space at speed of light in vacuum c = speed of light = 2.9979 × 10 8 m/s

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Chapter 8: The Quantum Mechanical Atom

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  1. Chapter 8: The QuantumMechanical Atom Chemistry: The Molecular Nature of Matter, 6E Jespersen/Brady/Hyslop

  2. Electromagnetic Energy Electromagnetic Radiation • Light energy or wave • Travels through space at speed of light in vacuum • c = speed of light = 2.9979 × 108 m/s • Successive series of these waves or oscillations Waves or Oscillations • Systematic fluctuations in intensities of electrical and magnetic forces • Varies regularly with time • Exhibit wide range of energy

  3. Properties of Waves Wavelength () • Distance between two successive peaks or troughs • Units are in meters, centimeters, nanometers Frequency () • Number of waves per second that pass a given point in space • Units are in Hertz(Hz = cycles/sec = 1/sec = s–1) Related by    = c

  4. Properties of Waves Amplitude • Maximum and minimum height • Intensity of wave, or brightness • Varies with time as travels through space Nodes • Points of zero amplitude • Place where wave goes though axis • Distance between nodes is constant nodes

  5. Learning Check: Converting from Wavelength to Frequency The bright red color in fireworks is due to emission of light when Sr(NO3)2 is heated. If the wavelength is ~650 nm, what is the frequency of this light?  = 4.61 × 1014 s–1 = 4.6 × 1014 Hz

  6. Your Turn! WCBS broadcasts at a frequency of 880 kHz. What is the wavelength of their signal? • 341 m • 293 m • 293 mm • 341 km • 293 mm

  7. Electromagnetic Spectrum • Comprised of all frequencies of light • Divided into regions according to wavelengths of radiation high energy, short waves low energy, long waves

  8. Electromagnetic Spectrum Visible light • Band of wavelengths that human eyes can see • 400 to 700 nm • Make up spectrum of colors White light • Combination of all these colors • Can separate white light into the colors with a prism

  9. Important Experiments in Atomic Theory Late 1800’s: • Matter and energy believed to be distinct • Matter: made up of particles • Energy: light waves Beginning of 1900’s: • Several experiments proved this idea incorrect • Experiments showed that electrons acted like: • Tiny charged particles in some experiments • Waves in other experiments

  10. Particle Theory of Light • Max Planck and Albert Einstein (1905) • Electromagnetic radiation is stream of small packets of energy • Quanta of energy or photons • Each photon travels with velocity = c • Waves with frequency =  • Energy of photon of electromagnetic radiation is proportional to its frequency • Energy of photon E =h • h = Planck’s constant = 6.626 × 10–34 J s

  11. Learning Check What is the frequency, in sec–1, of radiation which has an energy of 3.371 × 10–19 joules per photon? = 5.087 × 1014 s–1

  12. Your Turn! A microwave oven uses radiation with a frequency of 2450 MHz (megahertz, 106 s–1) to warm up food. What is the energy of such photons in joules? • 1.62 × 10–30 J • 3.70 × 1042 J • 3.70 × 1036 J • 1.62 × 1044 J • 1.62 × 10–24 J

  13. PhotoelectricEffect • Shine light on metal surface • Below certain frequency () • Nothing happens • Even with very intense light (high amplitude) • Above certain frequency () • Number of electrons ejected increases as intensity increases • Kinetic energy (KE) of ejected electrons increases as frequency increases • KE = h – BE • h = energy of light shining on surface • BE = binding energy of electron

  14. Means that Energy is Quantized • Can occur only in discrete units of size h • 1 photon = 1 quantum of energy • Energy gained or lost in whole number multiples of h E = nh • If n = NA, then one mole of photons gained or lost E = 6.02 × 1023 h If light is required to start reaction • Must have light above certain frequency to start reaction • Below minimum threshold energy, intensity is NOT important

  15. Learning Check How much energy is contained in one mole of photons, each with frequency 2.00 × 1013? E = 6.02 × 1023h E = (6.02×1023 mol–1)(6.626×10–34 J∙s)(2.00×1013 s–1) E = 7.98 × 103 J/mol

  16. Your Turn! If a mole of photons has an energy of 1.60 × 10–3 J/mol, what is the frequency of each photon? Assume all photons have the same frequency. • 8.03 × 1028 Hz • 2.12 × 10–14 Hz • 3.20 × 1019 Hz • 5.85 × 10–62 Hz • 4.01 × 106 Hz

  17. For Example: Photosynthesis • If you irradiate plants with infrared and microwave radiation • No photosynthesis • Regardless of light intensity • If you irradiate plants with visible light • Photosynthesis occurs • More intense light now means more photosynthesis

  18. Electronic Structure of Atom Clues come from: 1. Study of light absorption • Electron absorbs energy • Moves to higher energy “excited state” 2. Study of light emission • Electron loses photon of light • Drops back down to lower energy “ground state”

  19. Continuous Spectrum • Continuous unbroken spectrum of all colors • i.e., visible light through a prism • Sunlight • Incandescent light bulb • Very hot metal rod

  20. Discontinuous or Line Spectrum • Consider light given off when spark passes through gas under vacuum • Spark (electrical discharge) excites gas molecules or atoms • Spectrum that has only a few discrete lines • Also called atomic spectrum or emission spectrum • Each element has unique emission spectrum

  21. Line Spectrum

  22. Atomic Spectra • Atomic line spectra are rather complicated • Line spectrum of hydrogen is simplest • Single electron • First success in explaining quantized line spectra • First studied extensively • J.J. Balmer • Found empirical equation to fit lines in visible region of spectrum • J. Rydberg • More general equation explains all emission lines in H atom spectrum (infrared, visible, and UV)

  23. Rydberg Equation • Can be used to calculate all spectral lines of hydrogen • The values for n correspond to allowed energy levels for atom

  24. Learning Check: Using Rydberg Equation Consider the Balmer series where n1 = 2 Calculate  (in nm) for the transition from n2 = 6 down to n1 = 2. = 24,373 cm–1  = 410.3 nm Violet line in spectrum

  25. Learning Check A photon undergoes a transition from nhigher down to n = 2 and the emitted light has a wavelength of 650.5 nm? n2 = 3

  26. Your Turn! What is the wavelength of light (in nm) that is emitted when an excited electron in the hydrogen atom falls from n = 5 to n = 3? • 1.28 × 103 nm • 1.462 × 104 nm • 7.80 × 102 nm • 7.80 × 10–4 nm • 3.65 × 10–7 nm

  27. Significance of Atomic Spectra • Atomic line spectra tells us • When excited atom loses energy • Only fixed amounts of energy can be lost • Only certain energy photons are emitted • Electron restricted to certain fixed energy levels in atoms • Energy of electron is quantized • Simple extension of Planck's Theory • Any theory of atomic structure must account for • Atomic spectra • Quantization of energy levels in atom

  28. What Does Quantized Mean? • Energy is quantized if only certain discrete values are allowed • Presence of discontinuities makes atomic emission quantized Potential Energy of Rabbit

  29. Bohr Model of Atom • First theoretical model of atom to successfully account for Rydberg equation • Quantization of energy in hydrogen atom • Correctly explained atomic line spectra • Proposed that electrons moved around nucleus like planets move around sun • Move in fixed paths or orbits • Each orbit has fixed energy

  30. Energy for Bohr Model of H • Equation for energy of electron in H atom • Ultimately b relates to RH by b = RHhc • OR • Where b = RHhc = 2.1788 × 10–18 J/atom • Allowed values of n = 1, 2, 3, 4, … • n = quantum number • Used to identify orbit

  31. Energy Level Diagram for H Atom • Absorption of photon • Electron raised to higher energy level • Emission of photon • Electron falls to lower energy level • Energy levels are quantized • Every time an electron drops from one energy level to a lower energy level • Same frequency photon is emitted • Yields line spectra

  32. Hydrogen Balmer Series: nfinal = 2

  33. Bohr Model of Hydrogen Atom • n = 1 First Bohr orbit • Most stable energy state equals the ground state which is the lowest energy state • Electron remains in lowest energy state unless disturbed How to change the energy of the atom? • Add energy in the form of light: E = h • Electron raised to higher n orbit n = 2, 3, 4, …  • Higher n orbits = excited states = less stable • So electron quickly drops to lower energy orbit and emits photon of energy equal to E between levels E = Eh – Elh = higher l = lower

  34. Bohr’s Model Fails • Theory could not explain spectra of multi-electron atoms • Theory doesn’t explain collapsing atom paradox • If electron doesn’t move, atom collapses • Positive nucleus should easily capture electron • Vibrating charge should radiate and lose energy

  35. Your Turn! In Bohr's atomic theory, when an electron moves from one energy level to another energy level more distant from the nucleus, • energy is emitted • energy is absorbed • no change in energy occurs • light is emitted • none of these

  36. 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

  37. Diffraction and Electrons • Light • Exhibits interference • Has particle-like nature • Electrons • Known to be particles • Also demonstrate interference

  38. 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

  39. Standing Wave on a Wire • Integer number (n) of peaks and troughs is required • Wavelength is quantized: • L is the length of the string

  40. How Do We Describe an Electron? • Has both wave-like and particle-like properties • Energy of moving electron on a wire is E =½mv2 • Wavelength is related to the quantum number, n, and the wire length:

  41. Electron on Wire—Theories 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

  42. Electron on Wire—Theories Starting with the equation of the standing wave and the de Broglie equation Combining with E = ½mv2, substituting for v and then λ, we get Combining gives:

  43. de Broglie Explains 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

  44. Learning Check: Calculate Wavelength for 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?  = 7.27 × 10–11m

  45. Your Turn! Calculate the de Broglie wavelength of a baseball with a mass of 0.10 kg and traveling at a velocity of 35 m/s. • 1.9 × 10–35m • 6.6 × 10–33m • 1.9 × 10–34m • 2.3 × 10–33m • 2.3 × 10–31m

  46. 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

  47. Orbitals Characterized by Three Quantum Numbers: 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

  48. n = Principal Quantum Number • Allowed values: positive integers from 1 to  • n = 1, 2, 3, 4, 5, …  • Determines: • Size of orbital • Total energy of orbital • RHhc = 2.18 × 10–18 J/atom • For given atom, • Lower n = Lower (more negative) E = More stable

  49. ℓ = Orbital Angular Momentum Quantum Number • Allowed values: 0, 1, 2, 3, 4, 5…(n – 1) • Letters: s, p, d, f, g, h Orbital designation numbernℓletter • Possible values of ℓdepend on n • n different values of ℓ for given n • Determines • Shape of orbital

  50. mℓ = Magnetic Quantum Number • Allowed values: from –ℓ to 0 to +ℓ • Ex. when ℓ=2 thenmℓ 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

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