CHEM 1A L7: "QM Orbitals" 2/2/2010 © 2004 M. Kubinec 1 Review © 2004 M.Kubinec L08-1 © 2004 M.Kubinec L19-9 Wave FunctionsWave Functions Energy: h 2 8π 2 m - d 2 dx 2 Ψ(x)(x) = E T Ψ(x) E n = h 2 8m n 2 L 2 1 4 9 16 E n Ψ(x)(x) Ψ 2 (x) = E T Ψ(x)(x) nπ L xsin nπ L 2 h 2 8π 2 m A n=1n=1 n=2n=2 n=3n=3 n=4n=4 Orbital Energies nn:: Principal Q. N. n: 1, 2, 3 . . . R 9 3 -R ∞ Ionized ∞ 0 nE -Z 2 © 2004 M.Kubinec L07-2 R ∞ ==2.18 x 102.18 x 10 --1818 J = 3.29 x10 15 Hz N 0 R ∞ = 1312 kJmol --11 Ground 1 -R ∞ 2 4 - ∞ E n = n 2 R ∞ ChemQuiz © 6.4 Which transition in He + has the same wavelength as n = as n = 2 12 1 i H? Ionized 9 3 2 ∞ --RR ∞ --RR ∞ © 2004 M.Kubinec L07-3 C) 4 2B) B) 4 14 1A) 2 1 in Ground 4 1 --RR ∞ CHEM 1A L7: "QM Orbitals" 2/2/2010 © 2004 M. Kubinec 2 Energy of Transitions R ∞ = 2.18 x 10 --1818 J N A R ∞ = 1312 kJmol --11 E n = R ∞ -Z 2 n 2 © 2004 M.Kubinec L08-5 ΔE= ΔE = hR ∞ --ZZ 2 n f n f 2 1 n i n i 2 1 © 2004 M.Kubinec L07-6 © A.Pines,M.Kubinec UCB L9-5 H-Atom, One On -ElectronElect on IonsIons l = 0 (s) y z x 2s Radial NodeNode n = 1 n = 2 y z x 1s + m l = 0 l = 0 (s) m l = 0 + - © A.Pines,M.Kubinec UCB L9-6 n = 2 l = 1 (p) n = 3 l = 0, 1, 2 3s, 3p, 3d 3d Orbitals 2p y 2p z 2p x - y z x + - y z x + - y z x + Angular Node m l = -1, +1 ml = 0 Quantum Numbers ℓ : A l M m t m nn::PrincipalPrincipal n: 1, 2, 3 . . . Q.N. values orbital property Energy Total nodes =n-1 Shape © 2004 M.Kubinec L07-7 ngu ar Mo en u m ℓ : Magnetic ℓ::0 0 (s)(s), 1 , 1 (p)(p),,2 2 (d)(d),,nn--11 m ℓ ::--ℓ, , --((ℓ--1),..,0,..,(1),..,0,..,(ℓ--1), 1), ℓ Orientation Angular nodes = ℓ CHEM 1A L7: "QM Orbitals" 2/2/2010 © 2004 M. Kubinec 3 HH--Atom, OneAtom, One--ElectronElectron IonsIons nn==11 nn==22 m ℓ = 0 ℓ ==0 (0 (ss)) m ℓ = 0 ℓ ==0 (0 (ss)) © 2004 M.Kubinec L07-8 y z x 2s Radial Node y z x 1s + + - (a) The probability distribution for the hydrogen 1s orbital in three1s orbital in three-- dimensional space (b) The probability density of the electron © 2004 M.Kubinec L07-9 Three representations of the hydrogen 1s, 2s, and 3s electron © 2004 M.Kubinec L07-10 CHEM 1A L7: "QM Orbitals" 2/2/2010 © 2004 M. Kubinec 4 2p y 2p z 2p x z z z + Angular Node m ℓ = = --1, +11, +1 m ℓ = 0 nn= 2= 2 ℓ = 1 (= 1 (pp)) © 2004 M.Kubinec L07-11 nn= 3= 3 ℓ = 0, 1, 2 3s, 3p, 3d3s, 3p, Orbitals - y x + - y x + - y x ℓ = angular nodes; n-1 = total nodes Representation of the 2p orbitals © 2004 M.Kubinec L07-12 A cross section of the electron probability distribution for a 3p orbital. © 2004 M.Kubinec L07-13 CHEM 1A L7: "QM Orbitals" 2/2/2010 © 2004 M. Kubinec 5 © 2004 M.Kubinec L07-14 Representation of the 3d orbitals © 2004 M.Kubinec L07-15 Representation of the 3d orbitals © 2004 M.Kubinec L07-16 CHEM 1A L7: "QM Orbitals" 2/2/2010 © 2004 M. Kubinec 6 Representation of the 4f orbitals in terms of their boundary surfaces. © 2004 M.Kubinec L07-17 Which has the most radial nodes? - y z + - Y z + ChemQuiz 7.1 © 2004 M.Kubinec L07-18 A) A) 4f4f B) B) 3d3d C) C) 2s2s x x Lecture Complete © 2004 M.Kubinec L07-20 HNitsche Microsoft PowerPoint - L07S10_QM Orbitals 02-03-10.ppt [Compatibility Mode]