Still a few iclickers to be mapped to student names Please see my after class or email me your information if your clicker is not mapped. Remember homework assignments are due on Wednesday ! I will add one last assignment on particle physics and cosmology. -Start reading Chap 42

Quantum states and the Pauli exclusion principle The allowed quantum numbers for an atomic electron (see the Table below) are n ≥ 1; 0 ≤ l ≤ n – 1; –l ≤ ml ≤ l; and ms = ±1/2. The Pauli exclusion principle states that if an atom has more than one electron, no two electrons can have the same set of quantum numbers. Warning: I do not know much about chemistry but I do know a little about QM.

Implications of Pauli principle for the 1S state. n=1, l=0, sz1=1/2, sz2=1/2 ✔ n=1, l=0, sz1=1/2, sz2=-1/2 Imagine the “Pauli’s” are point-like electrons

Back to the QM hydrogen atom

Back to the QM hydrogen atom University of Colorado at Boulder clicker question In the 1s state, the most likely single place to find the electron is: r = 0 B) r = aB C) Why are you confusing us so much?

Shapes of hydrogen wave functions: l=1, called p-orbitals: angular dependence (n=2) l=1, m=0: pz = dumbbell shaped. l=1, m=-1: bagel shaped around z-axis (traveling wave) l=1, m=+1 Superposition applies: Dumbbells (chemistry) px=superposition (addition of m=-1 and m=+1) py=superposition (subtraction of m=-1 and m=+1)

Question from last class: Physics vs Chemistry view of orbits: 2p wave functions (Physics view) (n=2, l=1) Dumbbell Orbits (chemistry) px py pz m=1 m=-1 m=0 px=superposition (addition of m=-1 and m=+1) py=superposition (subtraction of m=-1 and m=+1)

Chemistry: Shells – set of orbitals with similar energy 1s2 2s2, 2p6 (px2, py2, pz2) 3s2, 3p6, 3d10 n l These are the wave functions (orbitals) we just found: n=1, 2, 3 … = Principal Quantum Number (for Hydrogen, same as Bohr) l=s, p, d, f … = Angular Momentum Quantum Number =0, 1, 2, 3 (restricted to 0, 1, 2 … n-1) m = ... -1, 0, 1.. = z-component of Angular Momentum (restricted to –l to l)

In HYDROGEN, energy only depends on n, not l and m. Energy Level Diagram for Hydrogen l=0 (s) l=1 (p) l=2 (d) n=3 3s 3p 3d n=2 2s 2p In HYDROGEN, energy only depends on n, not l and m. (NOT true for multi-electron atoms!) A: 4 B: 54 C: 5 D: 27 E: 9 170 responses n=1 l=0,m=0 1s

Schrodinger’s solution for multi-electron atoms What’s different for these cases? Potential energy (V) changes ! (Now more protons AND other electrons) Need to account for all the interactions among the electrons Must solve for all electrons at once! (use matrices) V (for q1) = kqnucleusq1/rn-1 + kq2q1/r2-1 + kq3q1/r3-1 + …. Is very difficult to solve … huge computer programs! Solutions change: - wave functions change - energy of wave functions affected by Z (# of protons) We use the central field approximation. Can use modified single electron solutions.

Implications of QM for chemistry Electron configuration in atoms: How do the electrons fit into the available orbitals? What are energies of orbitals? 3d 3p 3s 2p Total Energy 2s 1s

Implications of QM for chemistry Electron configuration in atoms: How do the electrons fit into the available orbitals? What are energies of orbitals? Filling orbitals … lowest to highest energy, 2 e’s per orbital H Oxygen = 1s2 2s2 2p4 He 3d Li 3p Be 3s B e e e e C 2p Total Energy e e N 2s O Shell not full reactive Shell full stable e e 1s

Schrodinger predicts wave functions and energies l=0 l=1 l=2 4p 3d 4s m=-2,-1,0,1,2 Li 3p Energy 3s Na 2p 2s m=-1,0,1 Why would the behavior of Li be similar to Na? because shape of outer most electron is similar. because energy of outer most electron is similar. c. both a and b d. some other reason 1s

Schrodinger predicts wave functions and energies l=0 l=1 l=2 4p 3d 4s m=-2,-1,0,1,2 Li 3p Energy 3s Na 2p 2s m=-1,0,1 Why would the behavior of Li be similar to Na? because shape of outer most electron is similar. because energy of outer most electron is similar. c. both a and b d. some other reason 1s ✔

PHYS272: Gauss’ Law Electric field is determined by the charge enclosed in the Gaussian sphere

A multielectron atom and “screening” The figure on the right is a sketch of a lithium atom, which has 3 electrons. The allowed electron states are naturally arranged in shells of different size centered on the nucleus. The n = 1 states make up the K shell, the n = 2 states make up the L shell, and so on. Due to the Pauli exclusion principle, the 1s subshell of the K shell (n = 1, l = 0, ml = 0) can accommodate only two electrons (one with ms = + 1/2, one with ms = –1/2). Hence the third electron goes into the 2s subshell of the L shell (n = 2, l = 0, ml = 0).

Ground-state electron configurations

Ground-state electron configurations (Zoom in)

The Periodic Table and Quantum Mechanics

The periodic table and Quantum Mechanics

The Periodic Table and Quantum Mechanics The outer electrons (“valence electrons”) determine the chemical properties of elements e.g. Na 1s22s22p63s (outer electron loosely bound valence +1); Cl 1s22s22p63s23p5 (missing one electron valence -1) Individually dangerous elements but you can eat NaCl. e.g. He, Ne, Ar are noble gases; all electron states are filled. Real bumper sticker sold by the American Chemical Society.

What element is this ? What is its electron configuration ? 4p 3d 4s What is the electron configuration for an atom with 20 electrons ? (Write it out !) 3p 3s Energy a. 1s2, 2s2, 2p6, 3s2, 3p4 b. 1s2, 2s2, 2p6, 3s2, 3p6, 3d2 2p c. 1s2, 2s2, 2p6, 3s2, 3p6, 4s2, 3d6 2s d. 1s2, 2s2, 2p6, 3s2, 3p6, 4s2 e. none of the above 1s

What element is this ? What is its electron configuration ? 4p 3d 4s What is the electron configuration for an atom with 20 electrons ? (Write it out !) 3p 3s Energy a. 1s2, 2s2, 2p6, 3s2, 3p4 b. 1s2, 2s2, 2p6, 3s2, 3p6, 3d2 2p c. 1s2, 2s2, 2p6, 3s2, 3p6, 4s2, 3d6 2s d. 1s2, 2s2, 2p6, 3s2, 3p6, 4s2 e. none of the above Answer is d! Calcium: Fills lowest energy levels first. Which orbitals are occupied determines chemical behavior (bonding, reactivity, “alkaline earth”) 1s

More on calcium N.B. 3d is above 4s level 4th Shell 3rd Shell 4 3 3 Calcium has 3 complete shells. Incomplete shell: Chemical behavior & bonding determined by electrons in outer most shell (furthest from the nucleus). 4p 4th Shell 3d 4s 3p 3rd Shell 3s Energy 3 4 3 2p 2 2st Shell 2s 1 1st Shell 1s

Mystery of calcium energy levels N.B. 3d level is above 4s level l=1 m=-1,0,1 l=0 l=2 m=-2,-1,0,1,2 4p 4th Shell 3d 4s 3p 3rd Shell 3s Energy 2p 2st Shell 2s 1st Shell 1s

Screening in multi-electron atoms An atom of atomic number Z has a nucleus of charge +Ze and Z electrons of charge –e each. Electrons in outer shells “see” a nucleus of charge +Zeffe, where Zeff < Z, because the nuclear charge is partially “screened” by electrons in the inner shells. This equation works when one electron is screened from the nucleus by other electrons. Warning: Eqn 41.43 is wrong except for hydrogen. Screening is important.

Clicker question on charge screening Potassium has 19 electrons. It is relatively easy to remove one electron but substantially more difficult to then remove a second electron. Why is this? A. The second electron feels a stronger attraction to the other electrons than did the first electron that was removed. B. When the first electron is removed, the other electrons readjust their orbits so that they are closer to the nucleus. C. The first electron to be removed was screened from more of the charge on the nucleus than is the second electron. D. all of the above E. none of the above Answer: C

Clicker question on charge screening Potassium has 19 electrons. It is relatively easy to remove one electron but substantially more difficult to then remove a second electron. Why is this? A. The second electron feels a stronger attraction to the other electrons than did the first electron that was removed. B. When the first electron is removed, the other electrons readjust their orbits so that they are closer to the nucleus. C. The first electron to be removed was screened from more of the charge on the nucleus than is the second electron. D. all of the above E. none of the above