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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… Bohr’s Model of the H Atom e- occupies only certain quantized energy states e- orbits the nucleus in a fixed radius circular path nucleus Ee- in the nth state En = -2.18 x 10-18 1 J( 2 n ) n = 1,2,3,… depends on Coulombic attraction of nucleus(+) and e-(-) always negative First Four e- Energy Levels in Bohr Model n=4 n=3 nucleus excited states n=2 E n=1 n=2 n=3 n=1 ground state n=4 E Levels are spaced increasingly closer together as n Energy of H atom e- in n=1 state? En = -2.18 x 1 J( 2 n ) n = 1,2,3,… In J/atom: 10-18 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 First Four e- Energy Levels in Bohr Model n=4 n=3 -1.36 x 10-19 J/atom -2.42 x 10-19 J/atom n=2 -5.45 x 10-19 J/atom E n=1 n=2 n=3 n=4 n=1 -2.18 x 10-18 J/atom the more - , the lower the En What is DE for e- transition from n=4 to n=1? (Problem 1) n=4 n=3 -1.36 x 10-19 J/atom -2.42 x 10-19 J/atom n=2 -5.45 x 10-19 J/atom E n=1 n=2 n=3 n=1 -2.18 x 10-18 J/atom n=4 DE = En=1 - En=4 = -2.18 x 10-18J/atom - (-1.36 x 10-19J/atom) = -2.04 x 10-18J/atom What is l of photon released when emoves 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. 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. Wave Mechanics Quantum, Part II Wave Mechanics Incorporates Planck’s quantum theory But very different from Bohr Model Important ideas Wave-particle duality Heisenberg’s uncertainty principle Wave-Particle Duality e- can have both particle and wave properties Particle: e- has mass Wave: e- can be diffracted like light waves e- or light wave wave split into pattern slit Wave-Particle Duality Mathematical expression (deBroglie) l = h/mu u = velocity m = mass Any particle has a l but wavelike properties are observed only for very small mass particles 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 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 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 )2 gives probability of finding e- in a particular space 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 Quantum Numbers Q# = conditions under which Yn can 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 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) Orbital Shapes l = 0 (s orbitals) l = 1 (p orbitals) these have different ml values Orbital Shapes l = 2 (d orbitals) these have different ml values Energy of orbitals in a 1 e- atom l=0 l=1 l=2 orbital n=3 3s 3p 3d n=2 E 2s 2p n=1 1s Three quantum numbers (n, l, ml) fully describe each orbital. The ml distinguishes orbitals of the same type. 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 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). Energy of orbitals in a 1 e- atom orbital E 3s 3p 2s 2p 1s 3d Filling Order of Orbitals in Multielectron Atoms The Quantum Periodic Table s block l =0 n =1 p lblock =2 dl block 1 2 3 4 5 6 7 6 7 f lblock =3 More About Orbitals and Quantum Numbers 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 1st shell; n = 2: e- in 2nd shell; ... Defines orbital E and diameter n=1 n=2 n=3 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.’ 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 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 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) ml l = 0, s orbital l = 1, p orbital l = 2, d orbital 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 Problem: What orbitals are in the n=4 level? Solution One s orbital Three p orbitals Five d orbitals Seven f orbitals