The Quantum Mechanical Model of the Atom

1. The Quantum Mechanical Model of the Atom Chapter 7

2. The nature of light • Light is electromagnetic radiation: a wave of oscillating electric & magnetic fields.

3. Wave properties • A wave has wavelength, frequency, and amplitude

4. Wave properties • Wavelength =  (lambda), in m • Frequency =  (nu), in cycles/sec or s-1 • Wavelength & frequency related by wave speed: • Speed of light c = 3.00 x 108 m s-1

5. Example1 • Calculate the frequency of light with  = 589 nm

6. Example 1 • Calculate the frequency of light with  = 589 nm •  = c c = 3.00 x 108 m s-1 • Must convert nm to m so units agree (n = 10-9)

7. Example 2 • Calculate the wavelength in nm of light with  = 9.83 x 1014 s-1

8. Electromagnetic spectrum = all wavelengths of electromagnetic radiation

9. Photoelectric effect • Light shining on metal surface can cause metal to emit e- (measured as electric current)

10. Photoelectric effect • Classical theory • more e- emitted if light either brighter (amplitude) or more energetic (shorter ) • e– still emitted in dim light if given enough time for e- to gather enough energy to escape

11. Photoelectric effect • Observations do not match classical predictions! • A threshold frequency exists for e- emission: no e- emitted below that  regardless of brightness • Above threshold , e- emitted even with dim light • No lag time for e- emission in high , dim light

12. Albert Einstein explainsphotoelectric effect • Light comes in packets or particles called photons • Amount of energy in a photon related to its frequency • h = 6.626 x 10-34 J s

13. Example 3 • What is the energy of a photon with wavelength 242.4 nm?

14. Example 4 • A Cl2 molecule has bond energy = 243 kJ/mol. Calculate the minimum photon frequency required to dissociate a Cl2 molecule.

15. Example 4

16. Example 4 • What color is this photon? • A photon with wavelength 492 nm is blue

17. Example 4 • A Cl2 molecule has bond energy = 243 kJ/mol. Calculate minimum photon frequency to dissociate a Cl2 molecule. What color is this photon? • A photon with wavelength 492 nm is blue

18. Emission spectra • When atom absorbs energy it may re-emit the energy as light

19. Emission spectra • White light spectrum is continuous • Atomic emission spectrum is discontinuous • Each substance has a unique line pattern

20. Hydrogen emission spectrum • Visible lines at • 410 nm (far violet) • 434 nm (violet) • 486 nm (blue-green) • 656 nm (red)

21. Emission spectra • Classical theories could not explain • Why atomic emission spectra were not continuous • Why electron doesn’t continuously emit energy as it spirals into the nucleus

22. Bohr model • Niels Bohr’s model to explain atomic spectra • electron = particle in circular orbit around nucleus • Only certain orbits (called stationary states) can exist rn = orbit radius, n = positive integer, a0 = 53 pm • Electron in stationary state has constant energy RH = 2.179 x 10–18 J • Bohr model is quantized

23. Bohr model • e– can pass only from one allowed orbit to another • When making a transition, a fixed quantum of energy is involved

25. Bohr model • Calculate the wavelength of light emitted when the hydrogen electron jumps from n=4 to n=2 • Photon energy is an absolute amount of energy • Electron absorbs photon, ∆Eelectron is + • Electron emits photon, ∆Eelectron is –

26. Bohr model • Calculate the wavelength of light emitted when the hydrogen electron jumps from n=4 to n=2 • 486 nm corresponds to the blue-green line

27. Example • What wavelength of light will cause the H electron to jump from n=1 to n=3? To what region of the electromagnetic spectrum does this photon belong?

28. Example • What wavelength of light will cause the H electron to pass from n=1 to n=3?

29. Example • What wavelength of light will cause the H electron to pass from n=1 to n=3? • The atom must absorb an ultraviolet photon with  = 103 nm

30. Light has both particle & wave behaviors • Wave nature shown by diffraction

31. Light has both particle & wave behaviors • Particle nature shown by photoelectric effect

32. Electrons also have wave properties • Individual electrons exhibit diffraction, like waves • How can e– be both particle & wave?

33. Complementarity • Without laser, single e– produces diffraction pattern (wave-like) • With laser, single e– makes a flash behind one slit or the other, indicating which slit it went through –– and diffraction pattern is gone (particle-like) • We can never simultaneously see the interference pattern and know which slit the e– goes through

34. Complementarity • Complementary properties exclude each other • If you know which slit the e– passes through (particle), you lose the diffraction pattern (wave) • If you see interference (wave), you lose information about which slit the e– passes through (particle) • Heisenberg uncertainty principle sets limit on what we can know

35. Indeterminancy • Classical outcome is predictable from starting conditions • Quantum-mechanical outcome not predictable but we can describe probability region

36. Electrons & probability • Schrodinger applied wave mechanics to electrons • Equation (wave function, ) describe e– energy • Equation requires 3 integers (quantum numbers) • Plot of 2 gives a probability distribution map of e– location = orbital • Schrodinger wave functions successfully predict energies and spectra for all atoms

37. Quantum numbers • Principal quantum number, n • Determines size and overall energy of orbital • Positive integer 1, 2, 3 . . . • Corresponds to Bohr energy levels

38. Quantum numbers • Angular momentum quantum number, l • Determines shape of orbital • Positive integer 0, 1, 2 . . . (n–1) • Corresponds to sublevels

39. Quantum numbers • Magnetic quantum number, ml • Determines number of orbitals in a sublevel and orientation of each orbital in xyz space • integers –l . . . 0 . . . +l

41. Shapes of orbitals • s orbital (l = 0, ml = 0) • p orbitals (l = 1, ml = –1, 0, +1)

42. Shapes of orbitals • d orbitals (l = 2, ml = –2, –1, 0, +1, +2)

43. What type of orbital is designated by each set of quantum numbers? • n = 5, l = 1, ml = 0 • n = 4, l = 2, ml = –2 • n = 2, l = 0, ml = 0 • Write a set of quantum numbers for each orbital • 4s • 3d • 5p

44. What type of orbital is designated by each set of quantum numbers? • n = 5, l = 1, ml = 0 5p • n = 4, l = 2, ml = –2 4d • n = 2, l = 0, ml = 0 2s • Write a set of quantum numbers for each orbital • 4s n = 4, l = 0, ml = 0 • 3d n = 3, l = 2, ml = –2, –1, 0, +1, or +2 • 5p n = 5, l = 1, ml = –1, 0, or +1

46. Electron configurations • Aufbau principle: e– takes lowest available energy • Hund’s rule: if there are 2 or more orbitals of equal energy (degenerate orbitals), e– will occupy all orbitals singly before pairing • Pauli principle: • Adds a 4th quantum number, ms (spin) • No two e– in an atom can have the same set of 4 quantum numbers ⇒ 2 e– per orbital