1. Ch. 1: Atoms: The Quantum World CHEM 4A: General Chemistry with Quantitative Analysis Fall 2009 Instructor: Dr. Orlando E. Raola Santa Rosa Junior College

2. Overview 1.1The nuclear atom 1.2 Characteristics of electromagnetic radiation 1.3 Atomic spectra 1.4 Radiation, quanta, photons 1.5 Wave-particle duality 1.6 Uncertainty principle

4. An electron will be ejected when hν > Φ because E k,electron will be non-zero frequencyvelocity The energy of a photon is conserved.

5. WARNING The following material contains heavy mathematical machinery, including integrals and differential equations. The purpose is to show you how scientist arrived at very important conclusions that will allow you to understand everyday chemistry. You do not have to memorize or even attempt to write down all the numerous mathematical expressions. DO NOT RUN AWAY. THEY ARE PERFECTLY TAME AND BEYOND THIS POINT, EVERYTHING IS DOWNHILL!!!!

7. Constructive interference (peak + peak) Destructive interference (peak + trough) Diffraction Pattern of Electrons

8. Waves show diffraction… Small angle x-ray diffraction on colloidal crystal, from http://www.chem.uu.nl/fcc/www/peopleindex/andrei/andrei.htm

9. Electrons show diffraction… Electron diffraction taken from a crystalline sample, from http://www.matter.org.uk/diffraction/electron/electron_diffraction.htm

10. therefore electrons are waves!

11. ill defined location well defined momentum well defined location ill defined momentum Heisenberg Uncertainty Principle (1927)

12. Heinsenberg’s Uncertainty Principle As a result from the analysis of many experiments and thoughtful theoretical derivations, Heinsenberg (1927) expressed the principle that the momentum and the position of a particle cannot be determined simultaneously with arbitrary precision. In fact the product of the uncertainties in these two variables is always at least as large as Planck constant over 4 .

13. Heisenberg Uncertainty Principle (1927) In its mathematical expression:

14. Example 1.7

15. At a node: Ψ 2 = 0 (no electron density) Ψ passes through 0 electron density The Born interpretation

16. Erwin Schrödinger Features of the equation: Solutions exist for only certain cases. The left side is often written as HΨ. H is known as the “hamiltonian”.

17. The Schrödinger equation

18. The Particle-in-a-box problem For the conditions in the box V(x) = 0 everywhere, energy is only kinetic, and has solutions which gives an expression for E

19. The Particle-in-a-box problem From the boundary conditions the other boundary condition makes we get B = 0 and the expression for E becomes

20. The Particle-in-a-box problem To find the constant A, we apply the normalization condition, since the particle has to be somewhere inside the box: and then and the wavefunction for the particle in a box is

21. Particle in a Box values of n

22. Changing the Box L small L large As L increases: energies of levels decrease separations between levels decrease

23. wavefunction (Ψ) probability density (Ψ 2 ) lowest density highest density

24. Locating Nodes Ψ passes through 0 Ψ 2 = 0 Number of nodes = n – 1

25. radius colatitude azimuth Spherical polar coordinates

26. General formula of wavefunctions for the hydrogen atom For n = 1

27. General formula of wavefunctions for the hydrogen atom For n = 2 and

28. Quantum numbers n: principal quantum number determines the energy indicates the size of the orbital : angular momentum quantum number, relates to the shape of the orbital m : magnetic quantum number, possible orientations of the angular momentum around an arbitrary axis.

29. principal quantum number orbital angular momentum quantum number magnetic quantum number

30. Electron probability in the ground-state H atom.

31. Radial probability distribution

32. Allowable Combinations of Quantum Numbers l = 0, 1, …, (n – 1)m l = l, (l – 1),..., -l

33. No two electrons in the same atom have the same four quantum numbers.

34. Higher probability of finding an electron Lower probability of finding an electron

35. most probable radii The most probable radius increases as n increases.

36. radial nodes boundary surface 90% likelihood of finding electron within

37. radial nodes Wavefunction (Ψ) is nonzero at the nucleus (r = 0). For an s-orbital, there is a nonzero probability density (Ψ 2 ) at the nucleus.

38. n = 1 l = 0 no radial nodes

39. n = 2 l = 0 1 radial node

40. n = 3 l = 0 2 radial nodes

41. 2p-orbital n = 2 l = 1, 0, or -1 no radial nodes 1 nodal plane Plot of wavefunction is for yellow lobe along blue arrow axis.

42. The three p-orbitals nodal planes The labels “x”, “y”, and “z” do not correspond directly to m l values (-1, 0, 1).

43. nodal planes The five d-orbitals n = 3, 4, … l = 2, 1, 0, -1, -2 dark orange (+) light orange (–)

44. The seven f-orbitals n = 4, 5, … l = 3, 2, 1, 0, -1, -2, -3 dark purple (+) light purple (–)

45. Allowed subshells Allowed orbitals 2 electrons per orbital Maximum of 32 electrons for n = 4 shell

46. Silver atoms (with one unpaired electron) Atoms with one type of electron spin Atoms with other type of electron spin Stern and Gerlach Experiment: Electron Spin

47. Spin States of an Electron Spin magnetic quantum number (m s ) has two possible values:

48. Relative Energies of Orbitals in a Multi-electron Atom After Z = 20, 4s orbitals have higher energies than 3d orbitals. Z is the atomic number.

49. Probability maxmima for orbitals within a given shell are close together. A 3s-electron has a greater probability of being found near the nucleus than 3p- and 3d-electrons due to contribution of peaks located closer to the nucleus.

50. Paired spins Parallel spins Lower energy Higher energy

51. Electron Configurations: H and He 1s electron (n, l, m l, m s ) 1, 0, 0, (+½ or –½) 1s electrons (n, l, m l, m s ) 1, 0, 0, +½ 1, 0, 0, –½)

52. Electron Configurations: Li and Be 1s electrons (n, l, m l, m s ) 1, 0, 0, +½ 1, 0, 0, –½ 2s electron * 2, 0, 0, +½ * one possible assignment 1s electrons (n, l, m l, m s ) 1, 0, 0, +½ 1, 0, 0, –½ 2s electrons 2, 0, 0, +½ 2, 0, 0, –½

53. Electron Configurations: B and C 1s electrons (n, l, m l, m s ) 1, 0, 0, +½ 1, 0, 0, –½ 2s electrons 2, 0, 0, +½ 2, 0, 0, –½ 2p electron* 2, 1, +1, +½ * one possible assignment 1s electrons (n, l, m l, m s ) 1, 0, 0, +½ 1, 0, 0, –½ 2s electrons 2, 0, 0, +½ 2, 0, 0, –½ 2p electrons* 2, 1, +1, +½ 2, 1, 0, +½ * one possible assignment

54. subshell being filled Filling order for orbitals maximum number of electrons in subshell

55. The Hydrogen atom: atomic orbitals The potential in a hydrogen atom can be expressed as Schrödinger (1927) found that the exact solutions for his equation give expression for the energy as

56. An atomic orbital is specified by three quantum numbers. n the principal quantum number - a positive integer ℓ the angular momentum quantum number - an integer from 0 to n-1 m ℓ the magnetic moment quantum number - an integer from -ℓ to +ℓ Quantum Numbers and Atomic Orbitals

57. 1.Principal (n = 1, 2, 3,...) - related to size and energy of the orbital. 2.Angular Momentum (ℓ = 0 to n  1) - relates to shape of the orbital. 3.Magnetic (m ℓ = ℓ to  ℓ) - relates to orientation of the orbital in space relative to other orbitals. 4.Electron Spin (m s = + 1 / 2,  1 / 2 ) - relates to the spin states of the electrons. Quantum Numbers

58. Table 7.2 The Hierarchy of Quantum Numbers for Atomic Orbitals Name, Symbol (Property) Allowed ValuesQuantum Numbers Principal, n (size, energy) Angular momentum, ℓ (shape) Magnetic, m ℓ (orientation) Positive integer (1, 2, 3,...) 0 to n-1 - ℓ,…,0,…,+ ℓ 1 0 0 2 01 0 3 0 12 0 0 +1 0+1 0 +2-2

59. Sample Problem 7.5 SOLUTION: PLAN: Determining Quantum Numbers for an Energy Level PROBLEM: What values of the angular momentum ( ℓ ) and magnetic (m ℓ ) quantum numbers are allowed for a principal quantum number (n) of 3? How many orbitals are allowed for n = 3? Follow the rules for allowable quantum numbers found in the text. l values can be integers from 0 to n-1; m ℓ can be integers from - ℓ through 0 to + ℓ. For n = 3, ℓ = 0, 1, 2 For ℓ = 0 m ℓ = 0 For ℓ = 1 m ℓ = -1, 0, or +1 For ℓ = 2 m ℓ = -2, -1, 0, +1, or +2 There are 9 m ℓ values and therefore 9 orbitals with n = 3.

60. Sample Problem 7.6 SOLUTION: PLAN: Determining Sublevel Names and Orbital Quantum Numbers PROBLEM:Give the name, magnetic quantum numbers, and number of orbitals for each sublevel with the following quantum numbers: (a) n = 3, ℓ = 2(b) n = 2 ℓ = 0(c) n = 5, ℓ = 1(d) n = 4, ℓ = 3 Combine the n value and ℓ designation to name the sublevel. Knowing ℓ, we can find m ℓ and the number of orbitals. n ℓ sublevel namepossible m ℓ values# of orbitals (a) (b) (c) (d) 3 2 5 4 2 0 1 3 3d 2s 5p 4f -2, -1, 0, 1, 2 0 -1, 0, 1 -3, -2, -1, 0, 1, 2, 3 5 1 3 7

61. 1s2s3s

62. The 2p orbitals.

66. Representation of the 1s, 2s and 3s orbitals in the hydrogen atom

67. Representation of the 2p orbitals of the hydrogen atom

68. Representation of the 3d orbitals

69. Representation of the 4f orbitals

70. Pauli Exclusion Principle In a given atom, no two electrons can have the same set of four quantum numbers (n, ℓ, m ℓ, m s ). Therefore, an orbital can hold only two electrons, and they must have opposite spins.

71. Types of Atomic Orbitals

72. Levels and sublevels When n = 1, then ℓ = 0 and m ℓ = 0 Therefore, in n = 1, there is 1 type of sublevel and that sublevel has a single orbital ℓ (m ℓ has a single value  1 orbital) This sublevel is labeled s (“ess”) Each level has 1 orbital labeled s, and it is SPHERICAL in shape.

73. s orbital are spherical Dot picture of electron cloud in 1s orbital. Surface density 4πr 2  versus distance Surface of 90% probability sphere

74. 1s orbital

75. 2s orbitals

76. 3s orbital

77. p orbitals When n = 2, then ℓ = 0 and 1 Therefore, in n = 2 levell there are 2 types of orbitals — 2 sublevels For ℓ = 0m ℓ = 0 this is a s sublevel For ℓ = 1 m ℓ = -1, 0, +1 this is a p sublevel with 3 orbitals When l = 1, there is a PLANAR NODE through the nucleus

78. p Orbitals The three p orbitals lie 90 o apart in space

79. 2p x Orbital 3p x Orbital

80. d Orbitals When n = 3, what are the values of ℓ? ℓ = 0, 1, 2 and so there are 3 sublevels in level n=3. For ℓ = 0, m ℓ = 0  s sublevel with single orbital For ℓ = 1, m ℓ = -1, 0, +1  p sublevel with 3 orbitals For ℓ = 2, m ℓ = -2, -1, 0, +1, +2  d sublevel with 5 orbitals

81. d Orbitals s orbitals have no planar node (ℓ = 0) and so are spherical. p orbitals have ℓ = 1, and have 1 planar node, and so are “dumbbell” shaped. This means d orbitals (with ℓ = 2) have 2 planar nodes

82. 3d xy Orbital

83. 3d xz Orbital

84. 3d yz Orbital

85. 3d x 2 - y 2 Orbital

86. 3d z 2 Orbital

87. f — Orbitals One of 7 possible f orbitals. All have 3 planar surfaces. Can you find the 3 surfaces here?

88. f — Orbitals

89. Spherical Nodes Orbitals also have spherical nodesOrbitals also have spherical nodes Number of spherical nodes = n - l - 1Number of spherical nodes = n - l - 1 For a 2s orbital: No. of nodes = 2 - 0 - 1 = 1For a 2s orbital: No. of nodes = 2 - 0 - 1 = 1 2 s orbital

90. Summary of Quantum Numbers of Electrons in Atoms NameSymbolPermitted ValuesProperty principalnpositive integers(1,2,3,…)orbital energy (size) angular momentum ℓ integers from 0 to n-1 orbital shape (The ℓ values 0, 1, 2, and 3 correspond to s, p, d, and f orbitals, respectively.) magnetic mℓmℓ integers from - ℓ to 0 to + ℓ orbital orientation spin msms +1/2 or -1/2direction of e - spin

91. Experimental observation of the spin of the electron (Stern and Gerlach, 1920)

92. A comparison of the radial probability distributions of the 2s and 2p orbitals

93. The radial probability distribution for an electron in a 3s orbital. The radial probability distribution for the 3s, 3p, and 3d orbitals.

94. The 3d orbitals

96. One of the seven possible 4f orbitals.

97. Schematic representation of the energy levels of the hydrogen atom