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5.111 Lecture Summary #7 Readings for today: Section 1.10 (1.9 in 3rd ed) – Electron Spin, Section 1.11 (1.10 in 3rd ed) – The Electronic Structure of Hydrogen. Read for Lecture #8: Section 1.12 (1.11 in 3rd ed) – Orbital Energies (of many-electron atoms), Section 1.13 (1.12 in 3rd ed) – The Building-Up Principle. _______________________________________________________________________________ Topics: I. RPD for s-orbitals (continued from Lecture #6) II. p-orbitals A. The shape of p-orbitals B. Radial probability distributions III. Electron spin and the fourth quantum number IV. Wavefunctions for multielectron atoms ________________________________________________________________________________ I. RADIAL PROBABILITY DISTRIBUTIONS (RPD) FOR S-ORBITALS Radial Probability Distribution: The probability of finding an electron in a spherical shell of thickness dr at a distance r from the origin. Knowing only probability is one of main consequences of Quantum Mechanics. Unlike CM, QM is non-deterministic. The uncertainty principle forbids us from knowing r exactly. II. P ORBITALS • For any subshell of l = 1, there are three p-orbitals, _____, _____, and _____. • m = ±1 states combine to give px and py orbitals. m = 0 is called the pz orbital. 1 A. THE SHAPE OF P-ORBITALS • Unlike s orbitals, p-orbitals have θ, φ dependence. P-orbitals _____________ spherically symmetrical. • p orbitals consist of two lobes (of opposite sign) separated by a _____________ plane on which Ψ = 0 (and Ψ2 = 0). • There is zero probability of finding a p-electron in a nodal plane. Thus, there is ________ probability of finding a p-electron at the nucleus. Probability density maps of p orbitals: ! 22pz ! 22px !2 2p y Z Z X Y X Z Y X Y Figures by MIT OpenCourseWare. HIGHEST PROB Along ___________ Along ____ axis Along ____ axis POSITIVE Ψ where z is __________ where x is positive where y is positive NODAL PLANE _______________ θ = 90° ______ plane φ = 90° ______ plane φ = 0° Nodal planes (planes that have no electron density) arise from angular nodes in the wavefunction. • Angular nodes: values of ______ or ______ at which Ψ (and Ψ2) = 0. • Recall from Lecture #6: Radial nodes are values of r at which Ψ (and Ψ2) = 0. In general, an orbital has: n–1 total nodes ___________ angular nodes ___________ radial nodes For 2s: _________ = ____ total nodes _________ = ____ angular nodes For 2p: _________ = ____ total nodes _________ = ____ angular nodes 2 _________ = ____ radial nodes _________ = ____ radial nodes B. RADIAL PROBABILITY DISTRIBUTIONS 3 RPD rmp 2s 2 RPD 3 0 0 0 5a0 10a0 n = 3, l = 0 3s rmp 1 n = 2, l = 0 1 2 0 5a0 10a0 15a0 r 20a0 r rmp 2p 2 n = 2, l = 1 1 0 0 5a0 10a0 2 n = 3, l = 1 3p rmp 1 0 0 5a0 10a0 15a0 20a0 r r rmp(2p)_______rmp(2s) 3 RPD RPD 3 RPD 3 2 rmp 1 0 0 5a0 10a0 n = 3, l = 2 15a0 20a0 3d r rmp(___) < rmp(___) < rmp(___) Figures by MIT OpenCourseWare. Interpreting RPD plots • As n increases (from 1 to 2 to 3), the orbital rmp “size” ______________. • As l increases (from s to p to d) for a given n, the orbital rmp “size”___________. • Only electrons in s states have a substantial probability of being very close to nucleus. This means that although the “size” of s orbitals is larger than p or d orbitals, s-electrons are the ___________ shielded. III. ELECTRON SPIN AND THE FOURTH QUANTUM NUMBER From quantum mechanics, a fourth quantum number appears that describes the spin of an electron within an orbital. Spin magnetic quantum number, ______ 3 There is no classical analogy to spin. • An electron can have two spin states: ms = ________ (spin up) or ms = _______ (spin down). • Similar to the angular momentum quantum number, l, ms describes the magnitude of an angular momentum. However, ms completes the description of an __________________ and is NOT dependent on the orbital. The property of electron spin was first proposed by S. Goudsmit and G. Uhlenbeck in 1925 to explain tiny deviations from the expected frequencies of spectral lines: _________________ expected ν _______________ observed ν So we can describe a given orbital using three quantum numbers (n, l, ml) and a given electron using 4 quantum numbers (n, l, ml, ms). !nlml describes an ___________ !nlmlms describes an ___________ PAULI EXCLUSION PRINCIPLE No two electrons in the same atom can have the same four quantum numbers. In other words, no two electrons can be in the same orbital and have the same spin. The Pauli exclusion principle limits an atom to ________ electrons per orbital. ie. How many electrons in a single atom can have the following two quantum numbers: n = 4 and ml = -2? _______ 4 MIT OpenCourseWare http://ocw.mit.edu 5.111 Principles of Chemical Science Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.