Lecture 5

1. Lecture 5 The Hydrogen Atom and Quantum Numbers 1.8-1.11 30-Aug Assigned HW 1.37, 1.38, 1.40, 1.42, 1.48a and c, 1.51  1.56, 1.58, 1.60 Due: Monday 6-Sept

2. Review 1.5-1.7 • Photoelectric effect describes the process that occurs when a beam of light is aimed at a solid metal surface • If E > φ, an electron is ejected • Electromagnetic radiation have characteristic properties of particles and waves • 1937 Nobel Prize  diffraction patterns that resulted in this discovery • Louis de Broglie • All particles have wavelike properties • Heisenberg Uncertainty Principle • The position of a particle and it’s momentum (therefore energy) cannot be simultaneously known • Wavefunctions  ψ mathematical description of the position of electrons • Born interpretation  ψ2  probability density

3. Particle in a Box • Imagine a particle confined to a box with a length of L. • Wavelengths are restricted to those with nodes at 0 and L. A simple sine wave describes this system! Why can’t n = 0?

4. Particle in a Box • How do we find the energy of the particle? Allowed wavelengths are:

5. Quantization of Energy • We’ve established that for a particle in a 2D box. • Only certain wavelengths fit into the box, n is restricted to integers. • What does this tell us about the levels of energy? • Think about the Photoelectric Effect. Does this make sense?

6. Changing the Box Length • If we confine the particle to a smaller or larger space. What influence will it have on the energy levels? • Qualitatively  • Quantitavely  5 nm box vs. 500 nm box. Calculate n = 1 and n = 2.

7. Changing the Box Length • If we confine the particle to a smaller or larger space. What influence will it have on the energy levels?

8. What do the Energy Levels Mean?

9. Sample Problems • Use the particle-in-a-box model to calculate the wavelength of the third quanta of a box with a length of 100pm.

10. Sample Problems • Calculate the probability density for the particle in a box model. Much easier than it looks, isn’t it?

11. Sample Problems • Derive an equation that allows the difference between two energy levels to be determined.

12. Simplest Real Model…Hydrogen • Confining an electron to an atom is very similar to confining it to a box. • Why? • As with particle-in-a-box, these restrictions limit the wavelengths that are allowed to fit within this new ‘box’ • What influence will this have on the allowable energy levels of an electron? • Schrödinger again…. • Do you hate him yet? He still had an awesome bowtie. Winthrop’s own Edwin Schrödinger look-a-like

13. Schrödinger and the Hydrogen Atom • For each energy level, we use to determine: • Wavefunction(ψ) • Energy (En) And by ‘we’, I mean they…you don’t have to do this (but you know you want to)

14. Schrödinger and the Hydrogen Atom • Starting point…… • Final product….. • Almost all constants! • h • me • e4 • ε0 n=3 n=2 n=1 Rydberg’s constant Remember this?

15. Energy of Single Electron Systems • A very similar expression can be derived for other single electron systems. • He+ • Li2+ • C5+ The ‘+’ means we’ve removed electrons from the atom and generated a net charge of the indicated value. Calculate # electrons: Atomic Number – Charge Li2+# electrons = Z – 2 = 3 - 2 = 1 electron Only for single electron systems! Verify this equation works for agrees with hydrogen specific equation

16. Sample Problem Calculate the ionization energy of an electron at the 2nd quantum of Li2+. Make sure Li2+ has only one electron Solve the equation

17. Principal Quantum Number • We define n as the principle quantum number. • Describes the energy level that an electron can hang out in • n = 1  ground state • Lowest energy level possible for the given atom • As an electron climbs the energy ‘ladder’ • Does n increase or decrease? • Does the energy increase or decrease? • What happens when n = ∞? • Energy =

18. Wavefunctions for 1 Electron Systems • Now the we can find the energy, we need to find the wavefunctions (ψ). • What will we learn if we know the wavefunctions? • Atomic orbitals! • Directly from probability density (ψ2) Do you remember what this means?

19. 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

20. Wavefunctions and Probability Density, n=1 • This looks more complicated than it is…. n=1 independent of θ and Φ r is the only variable, so…. constant Let’s make this more useful probability density

21. Atomic Orbitals Take Shape • Let’s make this easier to look at: • What does a graph of y vs. x look like? y x

22. Atomic Orbitals Take Shape Bohr radius

23. Radial Distribution Function – P(r) This differs only subtly from ψ2(r) – the difference is due to the volume dependence of ψ2(r) There is effectively no probability of finding an electron more than 5a0 from the nucleus (n=1) for a hydrogen atom Why is P(r) = 0 at the center of the atom? Math – Qualititive –

24. 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!

25. 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.

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

27. 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

28. 2D Probability and n Most notable difference  radius increases with n

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

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

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

32. 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

33. 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

34. 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

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

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

37. Orbitals and 3D Orientation

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

39. 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

40. 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 = ±

41. 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. n l ml ms