L08S07_QuantumHAtom 02-02-07

# L08S07_QuantumHAtom 02-02-07 - CHEM 1A L8"Quantum...

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Unformatted text preview: CHEM 1A L8: "Quantum H-Atom" Review Schrdinger equation: h2 d2 Quantum Mechanics - h d (x) + V(x) (x) = ET (x) V(x) 82m dx2 Kinetic Potential = 0 Total (x) = A sin(kx) + B cos(kx) sin(k cos(k (0) = 0 0 0 x x 2004 M.Kubinec bead on a string L L B=0 n k= L (L) = 0 L19-5 Probability/Normalization: L Probability Energy: Wave Functions (x) 2(x) En 16 A2 sin2 0 n x dx = 1 L L - h2 d2 (x) = ET (x) 82m dx2 2 n n A sin L x 82m L n=4 A2 (sin(kx)cos(kx) + x) = 1 A= 2 L 0 0 L L h2 2(x) dx = 1 = ET (x) n=3 9 (x) = 2004 M.Kubinec 2 L sin n x L particle must be somewhere L19-8 0 En = h2n2 8mL2 n=2 n=1 4 2004 M.Kubinec L08-1 1 2004 M.Kubinec L19-9 Wave Functions Orbitals, Quantum Numbers, Energies Orbitals, H-Atom, One-Electron Ions Onen=1 l = 0 (s) (s ml = 0 n=2 l = 0 (s) (s ml = 0 Radial Node z er (r,, ) ||2 : Prob( e-) (r, Prob( (r, 2 y n: 1, 2, 3 . . . l: 0,1, 2, n-1 nml: -l, -(l-1),..,0,..,(l-1), l 1),..,0,..,(l z 1s + L9-3 x -Z2 R? En = n2 n=2 ml = -1, +1 z 2s + - y x y L9-5 R? = Rydberg's Constant = 2.18 x 10-18 J x A.Pines,M.Kubinec UCB A.Pines,M.Kubinec UCB l = 1 (p) (p ml = 0 Energy Levels z ? Ionized -R? -R? 4 9 l=0 ml=-1 l=1 0 1 l=2 -2 -1 0 1 2 z - 2px Angular Node z 2py + + y x - y x + - 2pz 3 3s 2s 3p 3p 3p 2p 2p 2p 3d (5 orbitals) y 2 x n = 3 l = 0, 1, 2 A.Pines,M.Kubinec UCB 3s, 3p, 3d Orbitals L9-6 1 Ground -R? 1s Notation: nlml L08-2 L9-9 2004 M.Kubinec A.Pines,M.Kubinec UCB Coordinates and quantum numbers Wave Functions e- z (r,, ) (r, (r, Three spherical polar coordinates r y x || 2 : Probability ( e-) n, , m Three quantum numbers 2004 M.Kubinec L08-3 2004 M. Kubinec 1 CHEM 1A L8: "Quantum H-Atom" Quantum Numbers Q.N. n : Principal values orbital property Energy Shape n: 1, 2, 3 . . . Total nodes =n-1 : Angular Momentum : 0 (s), 1 (p), 2 (d), n-1 (s), (p), (d), Angular nodes = m : Magnetic m: -, -(-1),..,0,..,(-1), 1),..,0,..,( 2004 M.Kubinec Orientation L08-4 Orbital Energies n : Principal Q. N. -Z 2 R En = n2 = 2.18 x 10-18 J = 3.29 x1015 Hz N0R = 1312 kJmol-1 R 2004 M.Kubinec n 3 2 n: 1, 2, 3 . . . Ionized E 0 -R -R 4 9 1 Ground -R L08-5 ChemQuiz 8.1 H To which ion does the E-Level diagram `X' correspond ? 2 1 4 3 2 X -R 4 -R 1) H + 2) He + 3) Li 2+ 1 2004 M.Kubinec L08-6 2004 M. Kubinec 2 CHEM 1A L8: "Quantum H-Atom" ChemQuiz 8.1 Solution H 2 1 E1 = 12 12 R En = -Z 2 R n2 4 3 2 E2 = X -R 4 -R -Z 2 22 R Z(X) = 2 1 1) H + 2) He + 3) Li 2+ L08-7 2004 M.Kubinec ChemQuiz 8.2 Which transition in + He has the same wavelength as n = 2 1 in H? 3 2 Ionized -R -R 4 9 1) 2 1 2004 M.Kubinec 2) 4 1 3) 4 2 1 Ground -R L08-8 ChemQuiz 8.2 Solution -R 4 2 H Ionized 4 3 2 -R -4 R 9 4 -R 1 -R 1) 2 1 2004 M.Kubinec 2) 4 1 3) 4 2 1 Ground - 4 R L08-9 2004 M. Kubinec 3 CHEM 1A L8: "Quantum H-Atom" H-Atom, One-Electron Ions n=1 = 0 (s) (s m = 0 n=2 = 0 (s) (s m = 0 Radial Node z 1s + z 2s + - y x y L08-10 x 2004 M.Kubinec (a) The probability distribution for the hydrogen 1s orbital in threethreedimensional space (b) The probability density of the electron 2004 M.Kubinec L08-11 Three representations of the hydrogen 1s, 2s, and 3s electron 2004 M.Kubinec L08-12 2004 M. Kubinec 4 CHEM 1A L8: "Quantum H-Atom" m = -1, +1 n=2 = 1 (p) z 2py + m = 0 z - 2px Angular Node z y x + 2pz + y x - y x n = 3 = 0, 1, 2 2004 M.Kubinec 3s, 3p, 3d Orbitals L08-13 = angular nodes; n-1 = total nodes Representation of the 2p orbitals 2004 M.Kubinec L08-14 A cross section of the electron probability distribution for a 3p orbital. 2004 M.Kubinec L08-15 2004 M. Kubinec 5 CHEM 1A L8: "Quantum H-Atom" 2004 M.Kubinec L08-16 Representation of the 3d orbitals 2004 M.Kubinec L08-17 Representation of the 3d orbitals 2004 M.Kubinec L08-18 2004 M. Kubinec 6 CHEM 1A L8: "Quantum H-Atom" Representation of the 4f orbitals in terms of their boundary surfaces. 2004 M.Kubinec L08-19 ChemQuiz 8.3 For which orbital is the plot shown and 2 vs. ? 0 2 1) 2py 2004 M.Kubinec 2) 2pz 3) 2px L08-20 ChemQuiz 8.3 Solution z + y =/2 = 0 /2 x 1) 2py 2004 M.Kubinec 2 2) 2pz 3) 2px L08-21 2004 M. Kubinec 7 CHEM 1A L8: "Quantum H-Atom" ChemQuiz 8.4 Which has the most radial nodes? y x x z y + - - + z 1) 4f 2004 M.Kubinec 2) 3d 3) 2s L08-22 ChemQuiz 8.4 Solution Total nodes are n-1 n = number of angular nodes f=3 d=2 s=0 1) 4f 2004 M.Kubinec x - + y z 2) 3d 3) 2s L08-23 Lecture Complete 2004 M.Kubinec L08-24 2004 M. Kubinec 8 ...
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