Slide 14 of 50

CLASS EXAMPLE – PROBABILITIES: For a H atom in the ground electronic state

find the total probability that the electron and the nucleus (proton) occupy the same volume if the radius of the nucleus is 1.0 x 10-15 m. How can (a) numerical integration and (b) approximations be used to simplify the calculation?

CLASS EXAMPLE – THE 1s H WAVE FUNCTION: 

H ATOM S ORBITALS – GRAPHS: 

H ATOM S ORBITALS – GRAPHS: The corresponding radial part of the 2s

orbital wave function, R2,0 (r) looks very similar (see text). Class example: At what point does the probability density for an electron in the 2s orbital of a H atom have its maximum value?

H ATOM S ORBITALS – GRAPHS: Graphs will be constructed/drawn in class

showing two routes to the desired information: Route 1: Plot either ψ2,0,0 (r) or R2,0 (r) vs r Route 2: Plot either ψ22,0,0 (r) or R22,0 (r) vs r Aside: One might wish to multiply both ψ2,0,0 (r) and R2,0 (r)! Is “Route 2” better?

H ATOM WAVE FUNCTIONS: Plots of Rn,l (r) vs r are much less regular

than we saw earlier for the one dimensional PIAB. For the H atom, for example, nodes are not equally spaced and Rn,l (r) amplitudes do not show the “regular” variations seen for the PIAB (where all ψn(x) maxima and minima had the same absolute value).

S ORBITALS ARE SPHERICAL!! Although hardly news, the form of all s

orbital wave functions – having no angular dependence - mandates that the smallest volume containing a given % of the atom’s electron density is a sphere.

NODAL SURFACES: While not immediately obvious for the H

atom we can easily/often calculate/describe the position of a nodal surface. The simplest example of a nodal surface occurs for the H atom with the single electron in a 2s orbital (an electronically excited state!).

NODAL SURFACES – THE H 2s ORBITAL: 

NODAL SURFACES – OTHER 1 e- SPECIES: One electron “atoms” with higher nuclear

charges (He+, Li2+…) have a spherical nodal surface (for the 2s orbital) with a smaller radius (and a much smaller associated volume). This follows from the form of the relevant wave functions and/or coulombic considerations.

Rn,l and R2n,l PLOTS FOR s ORBITALS: Class examples: Use the Rn,o (r) functions to

draw Rn,o (r) vs r plots and R2n,o (r) plots for the 1s, 2s and 3s H atoms orbitals. Can both sets of plots be used, for example, to locate radial nodes?

THE RADIAL DISTRIBUTION FUNCTION: 

THE RADIAL DISTRIBUTION FUNCTION: 

 FIGURE 8-35

Radial probability distributions

GENERAL CHEMISTRY: CHAPTER 8

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H ATOM – RADIAL AND ANGULAR NODES: 

H ATOM – RADIAL AND ANGULAR NODES: 