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The Bohr Model; Wave Mechanics and Orbitals

The Bohr Model; Wave Mechanics and Orbitals. Bohr’s Quantum Model of the Atom. Attempt to explain H line emission spectrum Why lines ? Why the particular pattern of lines? Emission lines suggest quantized E states…. nucleus. ( ). E n = -2.18 x 10 -18 J. 1. n 2.

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The Bohr Model; Wave Mechanics and Orbitals

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  1. The Bohr Model; Wave Mechanics and Orbitals

  2. Bohr’s Quantum Model of the Atom • Attempt to explain H line emission spectrum • Why lines? • Why the particular pattern of lines? • Emission lines suggest quantized E states…

  3. nucleus ( ) En = -2.18 x 10-18 J 1 n2 Bohr’s Model of the H Atom • e- occupies only certain quantized energy states • e- orbits the nucleus in a fixed radius circular path • Ee- in the nth state • depends on Coulombic attraction of nucleus(+) and e-(-) • always negative n = 1,2,3,…

  4. n=4 n=3 excited states nucleus n=2 E n=1 n=1 ground state n=2 n=3 n=4 E Levels are spaced increasingly closer together as n First Four e- Energy Levels in Bohr Model

  5. n = 1,2,3,… ( ) En = -2.18 x 10-18 J 1 n2 Energy of H atom e- in n=1 state? • In J/atom: • En=1 = -2.18 x 10-18 J/(12) = -2.18 x 10-18 J/atom • In J/mole: • En=1 = -2.18 x 10-18 J/atom(6.02 x 1023 atoms/mol)(1kJ/1000J) = -1310kJ/mol

  6. n=4 -1.36 x 10-19 J/atom n=3 -2.42 x 10-19 J/atom n=2 -5.45 x 10-19 J/atom E n=1 -2.18 x 10-18 J/atom n=1 n=2 n=3 n=4 First Four e- Energy Levels in Bohr Model the more - , the lower the En

  7. n=4 -1.36 x 10-19 J/atom n=3 -2.42 x 10-19 J/atom n=2 -5.45 x 10-19 J/atom E n=1 -2.18 x 10-18 J/atom n=1 n=2 n=3 n=4 What is DE for e- transition from n=4 to n=1? (Problem 1) DE = En=1 - En=4 = -2.18 x 10-18J/atom - (-1.36 x 10-19J/atom) = -2.04 x 10-18J/atom

  8. What is l of photon released when e- moves from n=4 to n=1? (Problem 1) • Ephoton = |DE| = hc/l 2.04 x 10-18J/atom = (6.63 x 10-34 J•s/photon)(3.00 x 108 m/s)/ l • l = 9.75 x 10-8 m or 97.5 nm A line at 97.5 nm (UV region) is observed in H emission spectrum.

  9. Bohr Model Explains H Emission Spectrum DEn calculated by Bohr’s eqn predicts all l’s (lines). Quantum theory explains the behavior of e- in H. But, the model fails when applied to any multielectron atom or ion.

  10. Wave Mechanics Quantum, Part II

  11. Wave Mechanics • Incorporates Planck’s quantum theory • But very different from Bohr Model • Important ideas • Wave-particle duality • Heisenberg’s uncertainty principle

  12. e- or light wave wave split into pattern slit Wave-Particle Duality • e- can have both particle and wave properties • Particle: e- has mass • Wave: e- can be diffracted like light waves

  13. Wave-Particle Duality • Mathematical expression (deBroglie) • Any particle has a l but wavelike properties are observed only for very small mass particles l =h/mu u = velocitym = mass

  14. Heisenberg’s Uncertainty Principle • Cannot simultaneously measure position (x) and momentum (p) of a small particle • Dx .Dp > h/4p • Dx =uncertainty in position • Dp =uncertainty in momentum p = mu, so p a E

  15. Heisenberg’s Uncertainty Principle Dx .Dp > h/4p • As Dp  0, Dx becomes large • In other words, • If E (or p) of e- is specified, there is large uncertainty in its position • Unlike Bohr Model

  16. Wave Mechanics(Schrodinger) • Wave mechanics = deBroglie + Heisenberg + wave eqns from physics • Leads to series of solutions (wavefunctions, Y) describing allowed En of the e- • Yn corresponds to specific En • Defines shape/volume (orbital) where e- with En is likely to be • (Yn )2gives probability of finding e- in a particular space

  17. Ways to Represent Orbitals (1s) Where 90% of the e- density is found for the 1s orbital (Y1s )2 probability density falls off rapidly as distance from nucleus increases

  18. Quantum Numbers • Q# = conditions under which Yncan be solved • Bohr Model uses a single Q# (n) to describe an orbit • Wave mechanics uses three Q# (n, l, ml) to describe an orbital

  19. Three Q#s Act As Orbital ‘Zip Code’ • n = e- shell (principal E level) • l = e- subshell or orbital type (shape) • ml = particular orbital within the subshell (orientation)

  20. l = 1 (p orbitals) Orbital Shapes l = 0 (s orbitals) these have different ml values

  21. Orbital Shapes l = 2 (d orbitals) these have different ml values

  22. l = 0 l = 1 l = 2 n=1 n=2 n=3 Energy of orbitals in a 1 e- atom orbital 3s 3p 3d 2s 2p E Three quantum numbers (n, l, ml) fully describe each orbital. The mldistinguishes orbitals of the sametype. 1s

  23. Spin Quantum Number,ms • In any sample of atoms, some e- interact one way with magnetic field and others interact another way. • Behavior explained by assuming e- is a spinning charge

  24. Spin Quantum Number,ms ms = +1/2 ms = -1/2 Each orbital (described by n, l, ml) can contain a maximum of two e-, each with a different spin. Each e- is described by four quantum numbers (n, l, ml , ms).

  25. Energy of orbitals in a 1 e- atom orbital 3s 3p 3d 2s 2p E 1s

  26. Filling Order of Orbitals in Multielectron Atoms

  27. p block s block d block n 1 2 3 4 5 6 7 6 7 f block The Quantum Periodic Table l = 1 l = 0 l = 2 l = 3

  28. More About Orbitals and Quantum Numbers

  29. n=1 n=2 n=3 n = principal Q# • n = 1,2,3,… • Two or more e- may have same n value • e- are in the same shell • n =1: e- in 1stshell; n =2: e- in 2ndshell; ... • Defines orbital E and diameter

  30. l = angular momentum or azimuthal Q# • l = 0, 1, 2, 3, … (n-1) • Defines orbital shape • # possible values determines how many orbital types (subshells) are present • Values of l are usually coded l = 0: s orbital l = 1: p orbital l = 2: d orbital l = 3: f orbital A subshell l = 1 is a ‘p subshell’ An orbital in that subshell is a ‘p orbital.’

  31. ml = magnetic Q# • ml = +l to -l • Describes orbital orientation • # possible ml values for a particular l tells how many orbitals of type l are in that subshell If l = 2 then ml = +2, +1, 0, -1, -2 So there are five orbitals in the d (l=2) subshell

  32. Problem:What orbitals are present in n=1 level? In the n=2 level? • If n = 1 • l = 0 (one orbital type, s orbital) • ml = 0 (one orbital of this type) • Orbital labeled 1s • If n = 2 • l = 0 or 1 (two orbital types, s and p) • for l = 0, ml = 0 (one s orbital) • for l = 1, ml = -1, 0, +1 (three p orbitals) • Orbitals labeled 2s and 2p n(l) 1s one of these 2s one 2p three

  33. Problem:What orbitals are present in n=3 level? • If n = 3 • l = 0, 1, or 2 (three types of orbitals, s, p,and d) • l = 0, s orbital • l = 1, p orbital • l = 2, d orbital • ml • for l = 0, ml = 0 (one s orbital) • for l = 1, ml = -1, 0, +1 (three p orbitals) • for l = 2, ml = -2, -1, 0, +1, +2 (five d orbitals) • Orbitals labeled 3s, 3p, and 3d n(l) 3s one of these 3p three 3d five

  34. Problem:What orbitals are in the n=4 level? • Solution • One s orbital • Three p orbitals • Five d orbitals • Seven f orbitals

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