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# lecture_20 - 109 One more thing: radial probability...

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109 One more thing: radial probability distributions There are two different ways to view the radial probability of the electron in the hydrogen atom. In the first, we view the distribution in one dimension . Let’s take the 1s function as an example: ψ 1s (r) 2 = 0 a / r 2 3 0 e a 1 π This function has a maximum at the origin (r = 0), and decreases exponentially with increasing distance. In one dimension , the probability of finding the electron in an element of length dr at distance r from the origin is given as ψ 1s (r) 2 dr In three dimensions , the element of length dr sweeps out a spherical shell as θ sweeps out its full range from 0 to π , and φ varies from 0 to 2 π . The probability of finding the electron in that shell , with volume dV = 4 π r 2 dr is just ψ 1s (r) 2 dV = ψ 1s (r) 2 4 π r 2 dr For the one-dimensional case, dr does not vary as a function of r. However, dV = 4 π r 2 dr does vary with r. The shell volume goes to zero as r 0. Angular functions: When the l quantum number is non-zero, the orbitals have an angular dependence. We will learn that this angular dependence ultimately determines the shapes of molecules 1-D 3-D

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110 l = 1: these are p-orbitals. Possible values of m l are –1, 0, +1 The p-orbitals look like the following: Notice that the three p-orbitals only differ in spatial orientation. These orbitals point along the coordinate axes. Alternate lobes have opposite signs. Each orbital has one nodal plane. l = 2: these are d-orbitals . Possible values of m l are –2, –1, 0, +1, +2 Notice that the five d-orbitals (almost) only differ in spatial orientation. Alternate lobes have opposite signs. Each orbital has two nodal planes. Finally, it is possible to have orbitals with l = 3: these are f-orbitals . Possible values of m l are –3, –2, –1, 0, +1, +2, +3
111 Notice that the seven f-orbitals (almost) only differ in spatial orientation. Alternate lobes have opposite signs. Each orbital has three nodal planes. Remember that in the one-electron atom, there is no dependence

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## This note was uploaded on 04/16/2008 for the course CHM 132 taught by Professor Farrar during the Spring '08 term at Rochester.

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lecture_20 - 109 One more thing: radial probability...

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