AS V Schrödinger Model of of H Atom

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1 chem101/3, wi2010 pe 05 1 AS V Schrödinger Model of of H Atom Wavefunctions/ Orbitals chem101/3, wi2010 pe 05 2 General Bohr s Quantum Theory fails to explain why e s don t loose energy by constantly radiating, don t fall into nucleus etc.... Quantum Numbers Visualizations Ref 7: 2, 3 Prob FUP: 7: 6-8 EofC: 7: (odds) Adv Rdg 8: 1-3 New theory required: quantum mechanics = wave mechanics fundamental point: all matter ( esp. e s) has dual nature: 1) particle nature: mass, inertia, momentum,... & 2) wave nature: diffraction/interference behavior.. demonstrated by Pet. Fig. 8.16b (SB Fig. 7.14) chem101/3, wi2010 pe 05 3 Pet. Fig ( SB Fig. 7.14) X-rays diffracted Electrons diffracted by metal foil by metal foil chem101/3, wi2010 pe 05 4 Results of Quantum Mechanics 1) de Broglie Relationship relates wavelength to momentum of a piece of matter λ = h p = h m u (where m = mass, u = speed of particle) Fig. 8.16b is taken as proof that e s have wave properties it leads to the conclusion that e s in the nuclear environment behave as standing waves derivation for EMR: E = mc 2 (Einstein) & E = hν (Planck) mc 2 = hν with ν = c λ λ = M h mc in analogy, for matter waves: h λ = m u, where c has been replaced by the speed of the matter particle, u

2 chem101/3, wi2010 pe ) Heisenberg Uncertainty Principle Momentum (p = mu) of a moving particle and its position, X, cannot be known precisely at the same time. Mathematically, chem101/3, wi2010 pe ) Schrödinger Equation under the influence of the nucleus, e behaves as a standing wave, analogous to the behavior of a guitar string. see Pet. Fig (SB Fig. 7.13) ΔX Δp h 4π, where Δ are uncertainties Qualitative interpretation: location of e in atom is diffuse = blurry best described as e cloud, e density, or probability to find e at point X chem101/3, wi2010 pe 05 7 Schrödinger... Notes re: Pet. Fig. 8.18: several discrete states are possible (here the states are a, b, c) max. amplitudes occur at fixed positions nodes (where amplitude = 0) exist; their number range from 0, 1, 2... nodes are at symmetric locations for (a) at 1 2 L for (b) at 1 3 L and 2 3 L chem101/3, wi2010 pe 05 8 Schrödinger... math. symbolism to describe matter waves H (Ψ) = Ε Ψ where H, hamiltonian, complex math. operator Ψ, wavefunction E, energy complex differential equ n for e as a wave equ n has many solutions wavefunctions, Ψ also called orbitals each Ψ indicates - location of e (i.e., e density in 3D), - energy of e M - is associated w/ 3 different quantum # s

3 chem101/3, wi2010 pe 05 9 Three Quantum Numbers n, principal... l, angular momentum... m l, magnetic... Possible Values & Properties n = 1, 2, 3,... (main) indicator of size & energy of orbital l = 0, ( 1, 2, 3,...) n 1, also s, p, d, f,... chem101/3, wi2010 pe quantum # s... m l = - l, - l+1,..., 0, 1,..., +l e.g., if l = 3, possible m l values are: m l = - 3, - 2, -1, 0, +1, +2, +3 indicator of orientation in 3 D Note: for any n, can have n 2 orbitals e.g. n=1 1 orbital n=2 4 orbitals n=3 9 orbitals indicator of shape of orbital chem101/3, wi2010 pe Practice Identify all orbitals w/ n = 3 chem101/3, wi2010 pe Visualization of Orbitals Definition of Terms 1) Ψ, allowed wavefunction illustrated by the first three s orbitals (1s, 2s, 3s) in HT Fig. 5.1 shows value of Ψ vs. distance (r) from nucleus, has no direct physical meaning. Note: ψ for 1s has the same sign throughout, ψ for 2s crosses the zero line once, and ψ for 3s crosses the zero line twice.

4 chem101/3, wi2010 pe ) Ψ 2, electron probability density, also point probability, or PP probability to find e in an infinitesimally small volume e.g., for 1s e chem101/3, wi2010 pe ) 4πr 2 Ψ 2, radial probability distribution, also radial probability, or RP probability to find e in a particular, infinitesmal small spherical shell ( onion skin like ) or at a particular distance e.g., for 1s e chem101/3, wi2010 pe ) Electron Density Chart density of dots indicates the level of probability to find e ( PP ) at a particular location e.g., for 1s e chem101/3, wi2010 pe ) envelope or surface boundary sketch (most common representation) shows the space within which the e can be found at a specified probability (usually 90% or 95% level) e.g. 1s orbital: z x y also examine old Pet. Fig for Ψ 2 & electron chart representations

5 chem101/3, wi2010 pe chem101/3, wi2010 pe Old Pet. Fig s Orbitals (SB Fig.7.17) chem101/3, wi2010 pe chem101/3, wi2010 pe 05 20

6 chem101/3, wi2010 pe Pet. Fig (SB fig. 7.18) 2p orbitals chem101/3, wi2010 pe Nodes & Phases Nodes: surfaces where point probability, PP or Ψ 2 = 0 Pet. Fig (SB fig. 7.19) 3d orbitals can be planar (angular), spherical (radial) or conical # of nodes = n 1, therefore 1s s... 1 (spherical) 2p... 1 (planar)... 3d... 2 (planar), (3dz 2, conical, counts as 2 nodes) Phases: indicate relative sign of underlying Ψ often indicated by different colors different phases separated by nodes chem101/3, wi2010 pe chem101/3, wi2010 pe 05 24

7 chem101/3, wi2010 pe chem101/3, wi2010 pe Pet. Fig Radial Probability Functions chem101/3, wi2010 pe chem101/3, wi2010 pe 05 28

8 chem101/3, wi2010 pe chem101/3, wi2010 pe chem101/3, wi2010 pe chem101/3, wi2010 pe Some Guidelines for PP s and RP s PP s have x, y or z, RP s have r as horizontal axis PP s depend on direction, except s orbitals; (need to know shape & orientation of p & d orbitals) RP s do not; Nodes exist where PP curves touch 0; number of nodes as n PP s for s orbitals start at pos. values; PP s for all others start at 0 RP s always start at 0

9 chem101/3, wi2010 pe Example 2 p z orbital; RP's?; PP's z x chem101/3, wi2010 pe Comments on Energy & Radius (compare Bohr & Schrödinger) In H and other hydrogenic atoms each n is associated w/ different energy level (value is same as for Bohr model) RP different orbitals with same n (e.g., 3s, 3p, 3d) have same energy level, are degenerate PP PP r PP for 1s e : radius where RP is highest = Bohr atom radius, a o zero throughout zero throughout x y z chem101/3, wi2010 pe Summary of Lesson AS V principles of de Broglie, Heisenberg: matter waves, uncertainties of location/ momentum Schrödinger: e described as standing wave, with fixed wavefunction, occupying orbitals 3 quantum numbers: n = 1, 2, 3,... l = 0, 1, 2,..., n-1 m l = l,..., +l visualization of orbitals: Ψ, Ψ 2 (PP), 4πr 2 Ψ 2 (RP), e chart, surface boundary distinguish PP & RP nodes & phases of orbitals penetration phenomenon detailed description of 1s, 2s, 2p, 3s, 3p, 3d orbitals