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PHYS274 Spring 2017 • Quiz • Homework #9 due today (Apr. 5) – Chap 41 in its entirety 1 Q31.1 The Pauli Exclusion Principle states that: A) No two electrons can occupy the same quantum state B) Two electrons can occupy the same quantum state but no more than two C) No two electrons can occupy the ground state, but this is ok for excited states D) No two photons can occupy the same quantum state. 2 Q31.1 The Pauli Exclusion Principle states that: A) No two electrons can occupy the same quantum state B) Two electrons can occupy the same quantum state but no more than two C) No two electrons can occupy the ground state, but this is ok for excited states D) No two photons can occupy the same quantum state. 3 Q31.2 Which of the following statements is true? A. The Zeeman effect refers to the splitting of atomic energy levels in the presence of an electric field. B. The Zeeman effect refers to the splitting of atomic energy levels in the presence of a magnetic field. C. The Zeeman effect refers to the splitting of atomic energy levels under applied pressure in one-direction D. The Zeeman effect refers to the splitting of atomic energy levels at high temperature. 4 Q31.2 Which of the following statements is true? A. The Zeeman effect refers to the splitting of atomic energy levels in the presence of an electric field. B. The Zeeman effect refers to the splitting of atomic energy levels in the presence of a magnetic field. C. The Zeeman effect refers to the splitting of atomic energy levels under applied pressure in one-direction D. The Zeeman effect refers to the splitting of atomic energy levels at high temperature. 5 Q31.3 The n=3 state of a hydrogen like atom has an energy of -1.51eV. If a strong magnetic field is applied, how many hyperfine energy levels are observed? A. B. C. D. E. 1 2 3 4 5 6 Magnetic moments and the Zeeman effect • Electron states with nonzero orbital angular momentum (l = 1, 2, 3, …) have a magnetic dipole moment due to the electron motion. Hence these states are affected if the atom is placed in a magnetic field. The result, called the Zeeman effect, is a shift in the energy of states with nonzero ml. This is shown below. 7 Q31.3 The n=3 state of a hydrogen like atom has an energy of -1.51eV. If a strong magnetic field is applied, how many hyperfine energy levels are observed? A. B. C. D. E. 1 2 3 4 5 8 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. 9 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 10 Back to the QM hydrogen atom 11 Back to the QM hydrogen atom In the 1s state, the most likely single place to find the electron is: A) r = 0 B) r = aB C) Why are you confusing us so much? 12 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 px=superposition (addition of m=-1 and m=+1) py=superposition (subtraction of m=-1 and m=+1) (chemistry) 13 Note: Physics vs Chemistry view of orbits 2p wave functions (Physics view) (n=2, l=1) Dumbbell Orbits (chemistry) px m=1 m=-1 m=0 pz py px=superposition (addition of m=-1 and m=+1) py=superposition (subtraction of m=-1 and m=+1) 14 Chemistry: Shells – set of orbitals with similar energy 1s2 2s2, 2p6 (px2, py2, pz2) 3s2, 3p6, 3d10 l n These are the wave functions (orbitals) we just found: n=1, 2, 3 … = Principal Quantum Number En = − E1 / n 2 (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) | L |= l (l + 1) m = ... -1, 0, 1.. = z-component of Angular Momentum (restricted to –l to l) 15 Energy Level Diagram for Hydrogen l=0 (s) n=3 n=2 3s 2s l=1 (p) 3p l=2 (d) 3d 2p In HYDROGEN, energy only depends on n, not l and m. (NOT true for multi-electron atoms!) n=1 1s l=0,m=0 16 Schrodinger’s solution for multi-electron atoms What’s different for these cases? Potential energy (V) changes ! (Now more protons AND other electrons) V (for q1) = kqnucleusq1/rn-1 + kq2q1/r2-1 + kq3q1/r3-1 + …. Need to account for all the interactions among the electrons Must solve for all electrons at once! (use matrices) 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. 17 Implications of QM for chemistry Electron configuration in atoms: How do the electrons fit into the available orbitals? What are energies of orbitals? 3d Total Energy 3p 3s 2p 2s 1s 18 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 3d 3p Total Energy H He Li Be B C N O Oxygen = 1s2 2s2 2p4 3s 2p e e e 2s e e 1s e e e Shell not full reactive Shell full stable 19 Schrodinger predicts wave functions and energies l=1 l=0 4p Energy 4s 3s 2s 1s 3p l=2 3d m=-2,-1,0,1,2 Li Na 2p m=-1,0,1 Why would the behavior of Li be similar to Na? a. because shape of outer most electron is similar. b. because energy of outer most electron is similar. c. both a and b d. some other reason 20 Schrodinger predicts wave functions and energies l=1 l=0 4p Energy 4s 3s 2s 1s 3p l=2 3d m=-2,-1,0,1,2 Li Na 2p m=-1,0,1 Why would the behavior of Li be similar to Na? a. because shape of outer most electron is similar. b. because energy of outer most electron is similar. ✔ c. both a and b d. some other reason 21 PHYS272: Gauss’ Law Electric field is determined by the charge enclosed in the Gaussian sphere 22 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). 23 Ground-state electron configurations 24 Ground-state electron configurations (Zoom in) 25 The Periodic Table and Quantum Mechanics 26 The periodic table and Quantum Mechanics 27 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. 28 What element is this ? What is its electron configuration ? l=0 4p l=1 m=-1,0,1 l=2 m=-2,-1,0,1,2 3d Energy 4s 3s 3p 2p 2s What is the electron configuration for an atom with 20 electrons ? (Write it out !) a. 1s2, 2s2, 2p6, 3s2, 3p4 b. 1s2, 2s2, 2p6, 3s2, 3p6, 3d2 c. 1s2, 2s2, 2p6, 3s2, 3p6, 4s2, 3d6 d. 1s2, 2s2, 2p6, 3s2, 3p6, 4s2 e. none of the above 1s 29 What element is this ? What is its electron configuration ? l=0 4p l=1 m=-1,0,1 l=2 m=-2,-1,0,1,2 3d Energy 4s 3s 3p 2p 2s 1s What is the electron configuration for an atom with 20 electrons ? (Write it out !) a. 1s2, 2s2, 2p6, 3s2, 3p4 b. 1s2, 2s2, 2p6, 3s2, 3p6, 3d2 c. 1s2, 2s2, 2p6, 3s2, 3p6, 4s2, 3d6 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”) 30 More on calcium l=0 l=1 l=2 m=-1,0,1 m=-2,-1,0,1,2 4p 3d N.B. 3d is above 4s level 4th Shell Calcium has 3 complete shells. Energy 4s 3p 3s 3rd Shell Incomplete shell: Chemical behavior & bonding determined by electrons in outer most shell (furthest from the nucleus). 4 2p 2s 2st Shell 2 1 3 1s 1st Shell 31 Mystery of calcium energy levels l=0 l=1 l=2 m=-1,0,1 m=-2,-1,0,1,2 4p 3d N.B. 3d level is above 4s level 4th Shell Energy 4s 3p 3s 3rd Shell 2p 2s 1s 2st Shell 1st Shell 32 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. 33 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 34 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 35 For next time • Moving on to QM of Molecules Read material in advance Concepts require wrestling with material •Homework #9 due tonight (Apr. 5) –Chap 41 in its entirety 36