Ch. 1: Atoms: The Quantum World

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. frequency velocity The energy of a photon is conserved. An electron will be ejected when hν > Φ because Ek,electron will benon-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!!!!

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. electron density The Born interpretation • At a node: • Ψ2 = 0 (no electron density) • Ψ passes through 0

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 we get B = 0 the other boundary condition makes 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. values of n Particle in a Box

22. Lsmall Llarge Changing the Box • As L increases: • energies of levels decrease • separations between levels decrease

23. lowest density highest density wavefunction (Ψ) probability density (Ψ2)

24. Ψ passes through 0 Ψ2 = 0 Locating Nodes Number of nodes = n – 1

25. colatitude azimuth radius 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. 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) ml = 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. radial nodes Wavefunction (Ψ) is nonzero at the nucleus (r = 0). For an s-orbital, there is a nonzero probability density (Ψ2) 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. Plot of wavefunction is for yellow lobe along blue arrow axis. 2p-orbital n = 2 l = 1, 0, or -1 no radial nodes 1 nodal plane

42. nodal planes The three p-orbitals The labels “x”, “y”, and “z” do not correspond directly to ml 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 orbitals 2 electrons per orbital Allowed subshells Maximum of 32 electrons for n = 4 shell

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

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

48. Z is the atomic number. Relative Energies of Orbitals in a Multi-electron Atom 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