1. Early Quantum Mechanics
Chapter 27

2. Three Major Discoveries
Light is both a wave and a particle
Electron Orbitals are Quantized
The electron (and all matter) is both a wave and a particle
Planck
Einstein
Compton
Bohr
De Broglie
(Later Schrodinger/Heisenberg)

3. Wein’s Law Treat’s light solely as a wave Hot “blackbodies” radiate EM
The hotter the object, the shorter the peak wavelength
Sun (~6000 K) emits in blue and UV
A 3000 K object emits in IR
2.90 X 10-3 mK = lpeakT

5. Wein’s Law: Ex 1
Estimate the temperature of the sun. The Suns light peak at about 500 nm (blue) 500 nm = 500 X 10-9 m T = 2.90 X 10-3 mK lpeak T = 2.90 X 10-3 mK = 6000 K 500 X 10-9 m

6. Wein’s Law: Ex 2
Suppose a star has a surface temperature of about 3500 K. Estimate the wavelength of light produced X 10-3 mK = lpeakT l peak = 2.90 X 10-3 mK T l peak = 2.90 X 10-3 mK = 8.29 X 10-7 m = 830 nm 3500 K

7. Planck’s Quantum Hypothesis
Energy of any atomic or molecular vibration is a whole number
Photon – the light particle
Photons emitted come in “packets”
E = hf
h = X J s (Planck’s constant)

9. Photons: Ex 1
Calculate the energy of a photon of wavelength 600 nm. 600 nm X 1 X 10-9m = 6 X10-7 m 1 nm c = lf f = c/l = (3X108 m/s)/(6 X10-7 m) = 5 X 1014 s-1 E = hf E = (6.626 X J s)(5 X 1014 s-1) = 3.3 X J

10. Photons: Ex 1a
Convert your answer from the previous problem to electron Volts (1 eV = 1.6 X J)

11. Photons: Ex 2
Calculate the energy of a photon of wavelength 450 nm (blue light).

12. Photons: Ex 2a
Convert your answer from the previous problem to electron Volts (1 eV = 1.6 X J)

13. Photons: Ex 3
Estimate the number of photons emitted by a 100 Watt lightbulb per second. Assume each photon has a wavelength of 500 nm. 500 nm X 1 X 10-9m = 5 X10-7 m 1 nm c = lf f = c/l = (3X108 m/s)/(5 X10-7 m) = 6 X 1014 s-1

14. 100 Watts = 100 J/s (we are looking at 1 second) E = nhf n = E/hf n = 100 J (6.626 X J s)(6 X 1014 s-1) n = 2.5 X 1020

15. Photoelectric Effect (Einstein)
When light shines on a metal, electrons are emitted
Can detect a current from the electrons
Used in light meter, scanners, digital cameras (photodiodes rather than tubes)

16. Three Key Points
Below a certain frequency, no electrons are emitted
Greater intensity light produces more electrons
Greater Frequency light produces no more electrons, but the come off with greater speed

17. Low Frequency
High Frequency
Not enough energy to eject electron
Can eject electron
Energy of photon is greater than W (ionization energy

18. More intensity
More photons
More electrons ejected with same KE

19. Greater Frequency No more electrons ejected
Electrons come off with greater speed (KE)

20. hf = KE + W
hf = energy of the photon KE = Maximum KE of the emitted electron W = Work function to eject electron

21. Metal
Work Function (eV) Na
2.46 Al
4.08 Cu
4.70 Zn
4.31 Ag
4.73 Pt
6.35 Pb
4.14 Fe
4.50

22. hf = KE + W: Ex 1
What is the maximum kinetic energy of an electron emitted from a sodium atom whose work function (Wo) is 2.28 eV when illuminated with 410 nm light? 410 nm = 410 X 10-9 m or 4.10 X 10-7 m 2.28 eV X 1.60 X 10-19J = 3.65 X J 1 eV

23. c = lf f = c/l f = c/(4.10 X 10-7 m) = 7.32 X 1014 s-1 hf = KE + W KE = hf – W KE = (6.626 X J s)(7.32 X 1014 s-1) X J KE = 1.20 X J or 0.75 eV

24. hf = KE + W: Ex 2
What is the maximum kinetic energy of an electron emitted from a sodium atom whose work function (Wo) is 2.28 eV when illuminated with 550 nm light? ANS: 2.25 eV

25. hf = KE + W: Ex 3
What is the maximum wavelength of light (cutoff wavelength) that will eject an electron from an Aluminum sample? Aluminum’s work function (Wo) is 4.08 eV? 4.08 eV X 1.60 X 10-19J = 6.53 X J 1 eV

26. hf = KE + W hf = 0 + W (looking for bare minimum) c = lf f = c/l hf = W hc = W l l = hc = (6.626 X J s)(3.0 X 108 m/s) W 6.53 X J l = 3.04 X 10-7 m = 304 nm (UV)

27. Photon/Matter Interactions
Electron excitation (photon disappears)
Ionization/photoelectric effect (photon disappears)
Scattering by nucleus or electron
Pair production (photon disappears)

28. Electron Excitation
Photon is absorbed (disappears)
Electron jumps to an excited state
Ionization/Photoelectric Effect
Photon is absorbed (disappears)
Electron is propelled out of the atom

29. Scattering
Photon collides with a nucleus or electron
Photon loses some energy
Speed does not change, but the wavelength increases
Pair Production
Photon closely approaches a nucleus
Photon disappears
An electron and positron are created.

30. 3. Scattering: Compton Effect
Electrons and nuclie can scatter photons
Scattered photon is at a lower frequency than incident photon
Some of the energy is transferred to the electron or nucleus

31. l’ = l + h (1 – cos q) moc l’ = wavelength of scattered photon l = wavelength of incident photon mo = rest mass of particle q = angle of incidence

32. Compton Effect: Ex 1
X-rays of wavelength nm are scatterd from a block of carbon. What will be the wavelength of the X-rays scattered at 0o? l’ = l + h (1 – cos q) moc l’ = 140 X10-9m + (6.626 X J s)(1 – cos 0) (9.11 X kg)(3 X 108m/s) l’ = 140 X10-9m + 0 l’ = 140 nm

33. Compton Effect: Ex 2
What will be the wavelength of the X-rays scattered at 90o? l’ = l + h (1 – cos q) moc l’ = 140 X10-9m + (6.626 X J s)(1 – cos 90) (9.11 X kg)(3 X 108m/s) l’ = 140 X10-9m X m l’ = 142 nm

34. Compton Effect: Ex 3
What will be the wavelength of the X-rays scattered at 180o? l’ = l + h (1 – cos q) moc l’ = 140 X10-9m + (6.626 X J s)(1 – cos 180) (9.11 X kg)(3 X 108m/s) l’ = 140 X10-9m X m l’ = 145 nm (straight back)

35. 4. Pair Production Photon Disappears e- and e+ are produced
They have opposite direction (law of conservation of momentum)
When e- and e+ collide they annihilate each other  a new photon appears

37. Principle of Complimentarity
Neils Bohr
Any experiment can only observe light’s wave or particle properties, not both
Different “faces” that light shows
Wave
Particle
Prism
Blackbody Radiation
Photoelectric effect

38. The Discovery of the Electron (Thomson)
Cathode Ray Tube
Charged particles produced (affected by magnetic field)

39. Concluded that atom must have positive and negative parts
Electron – negative part of the atom
Only knew the e/m ratio
Plum Pudding Model

40. Charge and Mass of the Electron (Millikan)
Oil drop experiment
Determines charge on electron (uses electric field to counteract gravity)
Quantized
e = X C
m = 9.11 X kg

41. The Nucleus (Rutherford)
Gold Foil Experiment
Discovers nucleus (disproves Plum Pudding Model)
Planetary Model

43. Wave Nature of Matter Everything has both wave and particle properties
DeBroglie Wavelength
E2 = p2c2 + m2c4 (consider a photon)
E2 = p2c2 (photon has no mass)
E = pc
E = hf
hf = pc
c = lf

44. hf = plf p = mv (for a particle) hf = mvlf h = mvl l = h mv Everything has a wavelength
Diffraction pattern of electrons scattered off aluminum foil

45. DeBroglie Wavelength: Ex 1
Calculate the wavelength of a baseball of mass 0.20 kg moving at 15 m/s l = h mv l = (6.626 X J s) (0.20 kg)(15 m/s) l = 2.2 X m

46. DeBroglie Wavelength: Ex 2
Calculate the wavelength of an electron moving at 2.2 X 106 m/s l = h mv l = (6.626 X J s) (9.11 X kg)(2.2 X 106 m/s) l = 3.3 X m or 0.33 nm

47. DeBroglie Wavelength: Ex 3
Calculate the wavelength of an electron that has been accelerated through a potential difference of 100 V V = PE (PE =KE) q V = 1 mv2 2 q v2 = 2qV/m

48. v = (2qV/m)1/2 v=[(2)(1. 602 X 10-19 C)(100V)/(9
v = (2qV/m)1/2 v=[(2)(1.602 X C)(100V)/(9.11 X kg)]1/2 v = 5.9 X 106 m/s l = h mv l = (6.626 X J s) (9.11 X kg)(5.9 X 106 m/s) l = 3.3 X m or 0.33 nm

49. Electron Microscope Electron’s wavelength is smaller than light
Magnetic focusing

50. DNA

51. Line Spectra Discharge tube
Low density gas (acts like isolated atoms)
Run a high voltage through it
Light is emitted
Light is emitted only at certain (discrete) wavelengths
Gases absorb light at the same frequency that they emit

53. Hydrogen
Helium
Solar absorption spectrum

54. Explaining the Lines

55. Lyman Series 1 = R 1 - 1 l 12 n2 Balmer Series l 22 n2 Paschen Series l 32 n2

56. Bohr Model: Hydrogen

57. Electrons orbit in ground state (without radiating energy)
Classically, electrons should radiate energy since that are accelerating because they are changing directions
Jumps to excited state by absorbing a photon
Returns to ground state by emitted a photon

58. Bohr Model

60. hf = Ee - Eg

62. Bohr’s Equation
Worls only for H and other 1-electron atoms (He+, Li2+, Be3+, etc…)
Energy of ionized atom is set at 0
Orbital energies are below zero

63. En = -13. 6 eV n2 En = Energy of an orbital -13
En = eV n2 En = Energy of an orbital eV = Orbital of hydrogen closest to nucleus n = Number of the orbital (1 eV = 1.60 X J)

64. Bohr’s Equation: Ex 1
Calculate the energy of the first three orbitals of hydrogen En = eV n2 E1 = eV 12

65. E2 = -13.6 eV 22 E2 = -3.4 eV E3 = -13.6 eV 32 E3 = -1.51 eV

66. Bohr’s Equation: Ex 2
What wavelength of light is emitted if a hydrogen electron drops from the n=2 to the n=1 orbit? E1 = eV E2 = -3.4 eV DE = (-3.4eV eV) = 10.2 eV DE = (10.2 eV)(1.60 X 10-19J/eV) = 1.63 X 10-18J DE = hf f = c/l

67. DE = hc/l
= hc DE
= (6.626 X J s)(3.0 X 108 m/s)
(1.63 X 10-18J)
l = 1.22 X 10-7 m = 122 nm (UV)

68. Bohr’s Equation: Ex 3
Calculate the wavelength of light emitted if a hydrogen electron drops from the n=6 to the n=2 orbit? (ANS: 410 nm (violet))

69. Bohr’s Equation: Ex 4
Calculate the wavelength of light that must be absorbed to exite a hydrogen electron from the n=1 to the n=3 orbit? (ANS: 103 nm (UV))

70. Bohr Model: Other Atoms
En = (Z2)(-13.6 eV) n2 Z = Atomic number of the element (H=1, He=2, etc)

71. Other Atoms: Ex 1
Calculate the ionization energy of He+. This is the energy needed to move an electron from n=1 to zero. E1 = (22)(-13.6 eV) 12 E1 = eV

72. Other Atoms: Ex 2
What wavelength of light would be required to ionize He+
E1 = eV  X J
E = hf
E = hc/l
= hc = (6.626 X J s)(3.0 X 108 m/s)
E (8.70 X J)
l = 22.8 nm

73. Other Atoms: Ex 3
Calculate E1 for a Li2+ ion. E1 = (32)(-13.6 eV) 12 E1 = eV

74. Other Atoms: Ex 4
What wavelength of light would be emitted from a n=3 to n=1 transition in He+ ? (ANS: l = 34.2 nm)

75. Wave/Particle Duality
DeBroglie
Each e- is actually a standing wave
Only certain wavelengths produce resonance

76. Forbidden Zone

77. Circumference = 2pr 2prn = nl l = h mv mvrn = nh 2p