Ch. 1: Atoms: The Quantum World

Transcription

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

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4 The energy of a photon is conserved. E photon = E kinetic, electron + Work Function of metal hν = 1 2 m e v2 + Φ frequency velocity An electron will be ejected when hν > Φ because E k,electron will be non-zero

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!!!!

6 1 2 m e v 2 = hν Φ y = mx + b

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

8 Waves show diffraction Small angle x-ray diffraction on colloidal crystal, from

9 Electrons show diffraction Electron diffraction taken from a crystalline sample, from

10 therefore electrons are waves! λ = h mv = h p

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

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 4apple. Δp Δx 2π

13 Heisenberg Uncertainty Principle (1927) In its mathematical expression: Δp Δx 1 2

14 Example 1.7 mδv Δx = 2 Δx = = 2mΔv J s kg m s 1 = kg m 2 s 2 s kg m s -1 = m

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

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 2m d 2 ψ dx 2 +V(x)ψ = Eψ Hψ = Eψ

18 The Particle-in-a-box problem For the conditions in the box V(x) = 0 everywhere, energy is only kinetic, and d 2 ψ 2m dx = Eψ 2 has solutions ψ (x) = Asinkx + B cos kx which gives an expression for E E = k 2 h 2 8π 2 m

19 The Particle-in-a-box problem From the boundary conditions ψ (0) = 0 we get B = 0 the other boundary condition ψ (L) = 0 makes k = nπ L and the expression for E becomes E = k 2 h 2 8mL 2

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: nπ x L 0 L 0 ψ (x) 2 dx = A 2 sin 2 and then A = ψ n = 2 L L dx = 1 and the wavefunction for the particle in a box is 2 2 nπ x L sin L n = 1,2,3...

21 Particle in a Box ø n ( x) n x = sin n L L = 1, 2,... values of n

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

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 Spherical polar coordinates colatitude azimuth radius

26 General formula of wavefunctions for the hydrogen atom ψ (r,θ,ϕ) = R(r)Y(θ,ϕ) For n = 1 ψ (r,θ,ϕ) = 2e a r a 0 1 2π 1 2 r a 0 = e ( 3 πa ) a 0 = 4πε 0 2 m e e 2

27 General formula of wavefunctions for the hydrogen atom ψ (r,θ,ϕ) = R(r)Y(θ,ϕ) For n = 2 and E 2 = 1 4 hr ψ (r,θ,ϕ) = a r e r 2a 0 3 4π 1 2 sinθ cosφ = πa 0 r e r 2a 0 sinθ cosφ

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 magnetic quantum number principal quantum number orbital angular momentum 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 boundary surface 90% likelihood of finding electron within radial nodes

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

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

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

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

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

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 Z is the atomic number. After Z = 20, 4s orbitals have higher energies than 3d orbitals.

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 Lower energy Parallel spins 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 Filling order for orbitals subshell being filled maximum number of electrons in subshell

55 The Hydrogen atom: atomic orbitals The potential in a hydrogen atom can be expressed as V(x) = e2 4πε 0 r Schrödinger (1927) found that the exact solutions for his equation give expression for the energy as E = hr n 2 R = m e e 4 8h 3 ε 0 2 n = 1,2,3...

56 Quantum Numbers and Atomic Orbitals 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

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

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

59 Sample Problem 7.5 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? PLAN: 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. SOLUTION: 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 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 PLAN: Combine the n value and l designation to name the sublevel. SOLUTION: Knowing l, we can find m l and the number of orbitals. n l sublevel name possible m l values # of orbitals (a) 3 2 3d -2, -1, 0, 1, 2 5 (b) 2 0 2s 0 1 (c) 5 1 5p -1, 0, 1 3 (d) 4 3 4f -3, -2, -1, 0, 1, 2, 3 7

61 1s 2s 3s

62 The 2p orbitals.

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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

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 = 0 and 1 Therefore, in n = 2 levell there are 2 types of orbitals 2 sublevels For = 0 m = 0 this is a s sublevel For = 1 m = -1, 0, +1 this is a p sublevel with 3 orbitals

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? = 0, 1, 2 and so there are 3 sublevels in level n=3. For = 0, m = 0 apple s sublevel with single orbital For = 1, m = -1, 0, +1 apple p sublevel with 3 orbitals For = 2, m = -2, -1, 0, +1, +2 apple d sublevel with 5 orbitals

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

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

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89 2 s orbital

90 Summary of Quantum Numbers of Electrons in Atoms Name Symbol Permitted Values Property principal n positive integers(1,2,3, ) orbital energy (size) angular l integers from 0 to n-1 orbital shape (The l values momentum 0, 1, 2, and 3 correspond to s, p, d, and f orbitals, respectively.) magnetic m l integers from -l to 0 to +l orbital orientation spin m s +1/2 or -1/2 direction 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

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96 One of the seven possible 4f orbitals.

97 Schematic representation of the energy levels of the hydrogen atom