Lecture 7 CHEM 101 Fall 2024.pdf
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Principles of Chemistry CHEM 101 Lecture 7 1 Rn,l (r) Wave function for 1s orbital of H 1s (r) 𝑟 R1,0 (r) = 𝝍𝟏𝒔 (𝑟) = 𝑁1𝑠 exp...
Principles of Chemistry CHEM 101 Lecture 7 1 Rn,l (r) Wave function for 1s orbital of H 1s (r) 𝑟 R1,0 (r) = 𝝍𝟏𝒔 (𝑟) = 𝑁1𝑠 exp − 𝑎0 (r) = N e – r/a0 R1,0 (r) = 1s 1s 1s (r) = N1s e – 0/a0 r=0 1s 2s (r) = N e – 0 1s 1s 1s (r) = N1s 1 0 a0 2a0 3a0 r 1s (r) = N1s e – 0/a0 a r = a0 1s (r) = N1s e – 1 1s (r) = N1s 0.368 1s (r) = N1s 0.135 r = 2 a0 1s (r) = N1s 0.00674 r = 5 a0 1s (r) = N1s 0.0497 r = 3 a0 1s (r) = N1s 0.000045 r = 10 a0 2 Rn,l (r) 1s (r) = N1s e – r/a0 Wave function for 2s orbital of H R2,0 (r) = 2s (r) = N2s (2 – r/a0 ) e -r/2a0 2s (r) = N2s (2 – 3a0/a0 ) e – 3a0/2a0 r = 3 a0 (r) = N (2 – 3) e-1.5 2s 2s r = 10 a0 r=0 2s (r) = N2s (2 – 0/a0 ) e – 0/2a0 2s (r) = N2s (– 1) 0.223 = – N2s 0.223 2s (r) = – N2s 0.054 2s (r) = N2s (2 – 0 ) e-0 2s (r) = N2s (2 – 4a0/a0 ) e – 4a0/2a0 r = 4 a0 2s (r) = N2s (2) 1 = N2s 2 (r) = N (2 – 4) e-2 2s 2s 2s (r) = N2s (2 – a0/a0 ) e – a0/2a0 r = a0 2s (r) = N2s (– 2) 0.135 = – N2s 0.27 2s (r) = N (2 – 1 ) e-0.5 2s (r) = N2s (2 – 5a0/a0 ) e – 5a0/2a0 Radial node 2s 2s r = 5 a0 2s (r) = N2s (1 ) 0.606 = N2s 0.606 (r) = N (2 – 5) e-2.5 r = 2a0 2s 2s 2s (r) = N2s (2 – 2a0/a0 ) e – 2a0/2a0 r = 2 a0 2s (r) = N2s (– 3) 0.082 = – N2s 0.246 0 r (r) = N (2 – 2 ) e-1 2s 2s 2s (r) = N2s (2 – 8a0/a0 ) e – 8a0/2a0 r = 8 a0 2s (r) = N2s (0 ) 0.369 = 0 2s (r) = N2s (2 – 8) e-4 r = 4a0 2s (r) = N2s (– 6) 0.0183 = – N2s 0.101 3 Rn,l (r) Wave function for 2s orbital of H Wave function for 1s orbital of H 1s (r) -r/2a0 R2,0 (r) = 2s (r) = N2s (2 – r/a0 ) e 2s 2s (r) = N2s 2 r=0 2s (r) = N2s 0.606 r = a0 2s (r) = 0 r = 2 a0 (r) 2s (r) = – N2s 0.22 r = 3 a0 2s (r) = – N2s 0.27 r = 4 a0 2s (r) = – N2s 0.246 r = 5 a0 Radial Node 0 a0 2a0 3a0 r 2s (r) = – N2s 0.101 r = 8 a0 r = 2a0 2s (r) = – N2s 0.054 r = 10 a0 0 r r = 4a0 1s 2s 4 Rn,l (r) Wave function for 3s orbital of H 3s = N3s {27 –18( r/a0 )+2 ( r/a0 )2} e-r/3a0 3s (r) Radial Node Radial Node r 5 Rn,l (r) Radial part of the wavefunctions 1s 2s 3s 0 r 0 r 0 r No radial node 1 radial node 2 radial node 6 Rn,l (r) 2s (r) = N2s (2 – r/a0 ) e-r/2a0 2p = N2p (10) e-5 r = 10 a0 Wave function for 2p orbital of H 2p = N2p (10) 0.007 = N2p0.07 R2,1 (r) = 2p = N2p (r/a0 ) e -r/2a0 2p = N2p (4) e-2 r = 4 a0 2p = N2p (4) 0.135 = N2p0.54 2p = N2p (11) e-5.5 r = 11 a0 2p = N2p (0/a0 ) e -0/2a0 r=0 2p = N2p (11) 0.004 = N2p0.04 2p = N2p (5) e-2.5 r = 5 a0 2p = N2p (0) 1 = 0 2p = N2p (12) e-6 r = 12 a0 2p = N2p (5) 0.082 = N2p0.41 2p = N2p (a0/a0 ) e-a0/2a0 r = a0 2p = N2p (12) 0.002 = N2p0.02 2p = N2p (6) e-3 r = 6 a0 2p = N2p (1) e-0.5 2p = 0 r=0 2p = N2p (6) 0.05 = N2p0.3 2p = N2p0.606 r = a0 2p = N2p0.756 2p = N2p (1) 0.606 = N2p0.606 2p = N2p (7) e-3.5 r = 7 a0 2p = N2p0.669 r = 2a0 r = 3a0 r = 2 a0 2p = N2p (7) 0.03 = N2p0.21 2p = N2p0.54 r = 4a0 2p = N2p (2a0/a0) e-1 2p = N2p0.41 r = 5a0 2p = N2p (2) 0.378 = N2p0.756 2p = N2p (8) e-4 r = 8 a0 2p = N2p0.3 r = 6a0 2p = N2p0.21 r = 7a0 2p = N2p (8) 0.018 = N2p0.14 2p = N2p0.14 r = 8a0 2p = N2p (3a0/a0) e-1.5 r = 3 a0 2p = N2p0.09 r = 9a0 2p = N2p (9) e-4.5 r = 9 a0 2p = N2p0.07 r = 10a0 2p = N2p (3) 0.223 = N2p0.669 2p = N2p0.04 r = 11a0 2p = N2p (9) 0.011 = N2p0.09 2p = N2p0.02 r = 12a0 Rn,l (r) Wave function for 2p orbital of H R2,1 (r) = 2p = N2p (r/a0 ) e -r/2a0 2p = 0 r=0 2p = N2p0.606 r = a0 (r) 2p 2p = N2p0.756 r = 2a0 2p = N2p0.669 r = 3a0 2p = N2p0.54 r = 4a0 2p = N2p0.41 r = 5a0 2p = N2p0.3 r = 6a0 2p = N2p0.21 r = 7a0 r 2p = N2p0.14 r = 8a0 2p = N2p0.09 r = 9a0 2p = N2p0.07 r = 10a0 Angular Node, nodal plane 2p = N2p0.04 r = 11a0 2p = N2p0.02 r = 12a0 8 Rn,l (r) (r) Wave function for 3p orbital of H 2p R3,1 (r) = 3p = N3p {6(r/a0)– (r/a0)2 } e-r/3a0 r 3p Angular Node, nodal plane (r) r Angular Node, nodal plane Radial Node 9 Rn,l (r) Radial part of the wavefunctions (r) 1s (r) 2s (r) 3s 0 r 0 r 0 r (r) 2p (r) 3p 0 r 0 r 10 Probability of finding the electron in a thin shell of radius r and thickness r. r Area of a sphere of radius r = 4 r2 Volume of a shell of radius r and thickness r = 4 r2 r = V Probability of electron in volume V = [1s(r)]2 4 r2 r Radial distribution function (RDF) P1s (r) = [1s(r)]2 V RDF P1s (r) = [1s(r)]2 4 r2 If r is kept constant, then RDF 1. [1𝑠 (𝑟)]2 depends only on two terms: 2. 𝑟2 Radial distribution Functions (RDF) of ns orbitals 1s (r) = [R1s (r)] 2s (r) = [R2s (r)] P1s (r) = [R1s (r)]2 r2 P2s (r) = [R2s (r)]2 r2 1s 2s rmp = a0 P2s (r) RDF r a0 2a0 3a0 r [1s (r)]2 Radial node r2 r2 r2 r a0 2a0 3a0 r 12 Radial distribution Functions (RDF) of ns orbitals 3s (r) = [R3s (r)] P3s (r) = [R3s (r)]2 r2 P2s (r) P3s (r) 3s r r Radial node 2 Radial nodes [1s (r)]2 P1s (r) = [1s (r)]2 r2 RDF r2 r2 r2 r r a0 2a0 3a0 r 13 a0 2a0 3a0 Radial distribution Functions (RDF) of 1s, 2s, and 3s orbitals rmp = a0 1s rmp = ~5a0 P(r) rmp= ~11a0 2s 3s r 14 Radial distribution Functions (RDF) of 2p and 3p orbitals 2p (r) = [R2p (r)] 3p (r) = [R2p (r)] P2p (r) = [R2p (r)]2 r2 P3p (r) = [R2p (r)]2 r2 2p 3p P3p (r) P2p (r) r r nodes node 15 Radial distribution Functions (RDF) of 1s, 2s and 2p orbitals rmp = a0 1s P(r) rmp = ~4a0 2p 2s rmp = ~5a0 r 16
