Lecture 7 9/11/2024 KE → p → λ → where standing waves are allowed principal quantum # (n), n = 1, 2, 3, 4... angular quantum # (l), l = 0, 1, 2... (n-1) magnetic quantum # (m_l), m_l = -l, ...(l-1), 0, ...(l-1), l orbitals: 3D standing waves telling where an e- can be found n-1 = total nodes. l = angular nodes. l = 0 -> s orbital, l = 1 -> p, l = 2 -> d, l = 3 -> f orbital Y_1,0,0 = 1s e- Y_2,0,0 = 2s Y_2,1,0 = 2pz Y_2,1,1 = 2px Y_2,1,-1 = 2py Y_3,0,0 = 3s (2 - Zr/a_0) e- exp. decay no nodes (2-Zr/2a_0) most likely r (Zr/2a_0) 99.9% probability (Zr/a_0) 99.9% probability (2-Zr/a_0) (2-Zr/2a_0) (Zr/a_0) (2-Zr/2a_0) (2-Zr/a_0) (2-Zr/3a_0) (27-18Zr/a_0 + 2Z^2r^2/a_0^2) as you get farther from the nucleus, the chances of finding the e- decrease; mathematical solution w/o physical meaning probability of finding the e-: steeper than Y as going to less chance since the e- is attracted to the nucleus Y_2 multiplied by 4πr^2; very small chance of finding e- at nucleus (3D(2πr -> 2D)) radial distribution function a_0 = 0.53 Å "houses" farther away get a weaker signal; attraction but more "houses" are farther away; larger circumference encompasses more electrons no house @ radio tower" nucleus maximum chance of finding the e- most probable place depends on nuclear charge (Z) 2s orbital (Ψ_2,0,0) radial node Ψ = (2 - Zr/a_0)e^(-Zr/2a_0) at r = 2a_0, term = 0 no probability O at nucleus, 4πr^2 then at 0, bc node (radial), then max probability, and exp. decay n=3 3s, 3p, 3d n=2 2s, 2p n=1 1s energy depends on n. 2px: (Zr/a_0)e^(-Zr/2a_0) Sine cos ∅ 2py: (Zr/a_0)e^(-Zr/2a_0) Sine sin ∅ 2pz: (Zr/a_0)e^(-Zr/2a_0) cos ∅ (at 90° ∅ = 0), radial node only goes to 0@nucleus; 2/a_0 radial node must be a certain value (90°), but r and ∅ can be anything (plane cutting thru) plane if l = 1 three different allowed solutions for any p orbital (px, py, pz); different orientations by m_l l = 2 -> d orbital (2 angular nodes) dxy dz^2 dxz dyz dx²-y² 2-2, -1, 0, 1, 2 l = 2, 0, 1, 2 2 angular nodes (2-2r/a_0)e^(-Zr/2a_0) (Zr/a_0) (2-2r/a_0) (Zr/a_0) (2-2r/a_0) (Zr/a_0) (2-2r/a_0) (Zr/a_0)