Lecture 5.5

1. Lecture 5.5 Quantum Numbers 1.9-1.13 2-Sept Assigned HW See Website Due: Monday 6-Sept

2. Review 1.7-1.9 • Particle in a box • Simplest model to understand quantized energy (sine wave) • Allowable wavelengths are confined to certain values • So Energy is confined to specific values. • Probability Density gives us a 2D map of likely location • Hydrogen atom • Simplest real system. Z = 1 and # electrons = 1 • Attraction between electron and nucleus define a box length – like restriction of electron radius from nucleus • Will restrict the allowable wavelengths!  we predict quantized energy levels • Schrödinger derived Rydberg’s equation from first principles • Similar equation for all one electron systems • Principle Quantum Number (n) defines the energy of one electron atom • Wavefunction allows us to determine the shape of an atomic orbital.

3. Wavefunctions for 1 Electron Systems • 3 variables (r,θ,and Φ) are needed to completely describe position in 3 dimensions • Radial R(r) • r distance from the origin • Angular  Y(θ,Φ) • Φ  orientation around the z axis • Θ  angle from the x axis Shape of orbital

4. Atomic Orbitals Take Shape, n=1 Bohr radius

5. Radial Wavefunctions and n = 2 • When n = 2, there are 2 possible solutions for the radialwavefunction • Describe the difference in energy between these two wavefunctions. • All electrons with the same principle quantum number belong to the same ‘Shell’ • For a 1 electron system, electrons in the same shell have the same energy • The different solutions to R(r) suggest that they have different properties You know all of these terms!

6. Radial Wavefunctions and ‘l’ • Since we have multiple radial wavefunctions, a simple way to differentiate them would be nice • New quantum number: l  orbital angular momentum • Rule for assigning l: • l = 0,1,2,…..,n-1 • There will always be n values of l • Example: List the possible values for l if n = 4.

7. What does l mean? • Gives atomic orbitals their shape in 2D! Node: region of no electron density

8. 2D Orbital Shape Dependence on n • What happens if we vary n, but keep l = 0? • Do all n have l = 0? How many nodes in each case? Remind me, what is a node? General Rule  number of nodes = n

9. 2D Probability and n Most notable difference  radius increases with n

10. n=2, l = 1; P-orbitals How many nodes? Where?

11. n=3, l = 1; P-orbitals How many nodes? Where? Phases?

12. n=3, l = 2; D-orbitals How many nodes? Where? Phases? In this case, two dimensions fail at following the rule: # nodes = n

13. Angular Wavefunctions • Now if we inspect the whole wavefunction, we see for l > 0, there is an angular dependence • This will dictate the orientation in 3D space

14. Y(θ,Φ) and the Magnetic Quantum Number • To make it easy, we define a convention for describing Y(θ,Φ) • Magnetic Quantum Number (ml) • ml = -l l If n = 1, what values can ml have? How about n = 2? -1 0 1

15. Orbital Quantum Numbers Summary • We need 3 Quantum Numbers to completely describe an atomic orbital • n  • l  • ml  • All of this comes from Schrödinger and his famous equation • Ψ predicts l and ml • n is dictated by E

16. Orbital Quantum Numbers Summary Can we find all 3 nodes now?

17. n=3, l = 2; D-orbitals Revisited How many nodes? 3 Where? Phases?

18. Orbitals and 3D Orientation

19. Orbitals and 3D Orientation n = 3, l = 2 ml

20. Now to Describe the Electron • We need 1 more Quantum Number to finish describing the electrons that occupy the atomic orbital • ms spin magnetic quantum number • Two possible values: • This value describes how the electron is rotating • This influences the magnetic properties, as suggested by the name • Right Hand Rule

21. That’s It! • Using the 4 quantum number you know, you can now COMPLETELY describe an electron • n  ENERGY!  ineteger • l  probable distance from the nucleus • ml  3D orientation • ms  spin • List all the possible sets of quantum numbers for an electron in a 1s orbital l= 0  n-1 ml = -ll ms = ±

22. That’s It! • List all the possible sets of quantum number for an electron in a 2p orbital • Describe the difference in energy between these possible sets. 1 n l 1 0 2 ml -1 ms

23. Multi-electron Systems • What happens when we don’t have a hydrogen atom? • How will these differences influence the attractive energy that an electron feels from the nucleus?

24. Multi-electron Energies • Energy contributions to multiple electron systems • Attraction between the nucleus and the 1st electron • Attraction between the nucleus and the 2nd electron • Repulsion between 1st electron and 2nd electron r1 r2 Why is there a factor of Z in the terms that include the nucleus? Why are attractive energies (-) and repulsions (+)? r3

25. Multi-electron Energies - Shielding • Consider Li. • Z = ___ • As we fill in the electrons, not all have the same energy! • 1st electron feels NO repulsion because there are no other electrons • 2nd electron feels a repulsion from the 1st electron • 3rd electron feels repulsion from both of these electron. +3 +3 +3

26. Shielding and Effective Nuclear Charge • This decreased stabilization (increased energy) resulting from adding additional electrons is termed shielding. • Adding electrons around an atom’s nucleus minimizes the potential stabilizing energy that the next electron can feel. • Effective Nuclear Charge (Zeff) Other electrons r

27. Shielding and Shell Energy • We saw in the hydrogen atom that as n increases, the energy increases or decreases? E n = n = n =

28. Shielding and Shell Energy • We saw in the hydrogen atom that as n increases, the energy increases or decreases? E • Same is true for multiple electron atoms n = n = n =

29. Shielding and Orbital Energies • Electron shielding results in a shift in the stabilizing energy associated with orbitals within the same shell • Consider n = 2. • Which orbital do you think should have the lowest energy? • Why?

30. Shielding and Orbital Energies Within a given shell, atomic orbital energies follow this trend s < p < d < f E Note that each of these orbitals have a unique set of quantum numbers. f d degenerate Electrons occupy the orbitals according to lowest energy first. How many electrons can each orbital hold? These orbitals are all the same energy p s Which orbital is this? n = 4, l = 0, ml = 0

31. Shielding and Orbital Energies • General scheme  increasing energy of atomic orbitals The 4s orbital is occupied before the 3d because it has a lower energy. Why might this be? Hund’s rule Add Z electrons to the orbitals in the order shown here. Never more than 2 electrons per orbital. If multiple isoenergeticorbitals are available, add electrons with parallel spins to EACH orbital before adding a 2nd electron to any one orbital.

32. Shielding and Orbital Energies • This scheme can also be seen in the periodic table: Why are the blocks named the way that they are? What do you think the numbers below each block mean?

33. Filling in Atomic Orbitals Let’s fill in the electrons of Helium How many electrons? Shorthand: 1s2 E Electrons occupy the orbitals according to lowest energy first. 2s Helium Ground State 1s Orbitals can hold 2 electrons!

34. Shielding and Orbital Energies Let’s fill in the electrons of Helium How many electrons? E Shorthand? 1s12s1 2s Helium Excited State 1s

35. Shielding and Orbital Energies Beryllium How many electrons? C Shorthand: 1s22s2 E 2p 2s 1s

36. Shielding and Orbital Energies C Shorthand: 1s22s12p1 What would the excited state of Be look like? E What happens when it relaxes……? 2p ENERGY 2s 1s

37. Shielding and Orbital Energies C Iron How many electrons? Shorthand: [Ar]4s23d6 E We’re too lazy to write out ALL the orbitals! Lazy chemists approach: Pick an atom that represents the last full shell (this well be a noble gas). Put it in brackets Count the number of electrons in the valence shell. Fill in the valence electrons as we’ve done in the previous examples. 3d 4s [Ar]

38. Sample Problems Write the ground state electron configuration for Bismuth. 1st Excited State. Now Promethium (Z = 61)

39. Periodic Trends – Atomic Radius • Atomic Radius is dictated by the number of shells and the charge of the nucleus vs. total charge of electrons. • Shell – The atomic radius increases with the shell – so increases and we go DOWN the periodic table • Example: The atomic radius of K > Na • Within the same shell, we’re adding protons and electrons • Shielding is still ~the same, so increasing Z decreases radius • Example: The atomic radius of Cl < S < P General Trend: decreases from left to right and increases down

40. Periodic Trends – Atomic Radius Decreasing radius

41. Periodic Trends – Ionic Radius • What influence will adding an electron have to the radius of an atom? • Have we added any protons? • So we have NO additional attraction  radius increases • What happens if we remove an electron? • Remove repulsive force from electron-electron interactions • Remaining electrons feel more (+)  radius decreases

42. Periodic Trends – Ionic Radius Same # electrons, so why different size?

43. Periodic Trends – Ionization Energy • The amount of energy to remove an electron from an atom. • Directly related to the energy of the electron • 1st ionization energy < 2nd ionization energy, etc. 0 E • Easiest to remember that this value is inversely related to atomic radius • When the radius is small, the electron is bound tightly, so high ionization energy 2s 1s Compare the ionization energy of F to As

44. Periodic Trends – Ionization Energy N > O 0 E 2s There is a special stabilization energy when degenerate energy orbitals are completely full and half full. 1s

45. Shielding and Orbital Energies Within a given shell, atomic orbital energies follow this trend s < p < d < f E Note that each of these orbitals have a unique set of quantum numbers. f d degenerate Electrons occupy the orbitals according to lowest energy first. How many electrons can each orbital hold? These orbitals are all the same energy p s Which orbital is this? n = 4, l = 0, ml = 0