Hydrogen atom family albumCH101 General Chemistry, Fall 2013, Boston UniversityWe have learned for the one-dimensional examples of standing waves that the electron wave for the lowest possible energy has one loop, that the electron wave for the second lowest possible energy has two loops, and one loop is added for each higher energy.Here is how to extend this idea of the number of loops reflecting relative energy to the three dimensional electron waves in atoms. First, because atoms are spherical, there will be separate sets of loops in the radial direction (going out from the nucleus) and in the angular directions (going around the nucleus at fixed distance from it). Second, because the angular electron wave at f=0 is the same as f=2 p, electron waves meet smoothly after one full cycle in f. This means that the lowest energy angular variation must have one full wavelength (that is, two loops rather than one loop), and each increase in angular excitation must be accompanied by the addition of another full wavelength (two additional angular loops). Angular loops appear in the electron density as nodal planes through the nucleus.We call the number of radial loops j and the number of angular wavelengths (the number of nodal planes) { .With this background, let's sort out atomic electron waves in families according to the number, {, of wavelengths moving around the nucleus at fixed distance from it, and then, within a family, according to the number, j, of loops moving radially outward from the nucleus.In the displays below are shown the probability densities in the xy plane of the simplest members for the families of hydrogen atom electron waves with zero, one and two angular wavelengths. The displays consist of cross sections through the electron "cloud" of several members of each family. The nucleus is at the center of each cross section. Each cross section is displayed at five different scales (magnifications) . From left to right, the width and height are 1, 10, 30, 60 and 90 Þ, that is, 0.1, 1, 3, 6, and 9 nm. The brightness of the display is proportional to the probability density. Since all of the displays at a given magnification are on the same scale, comparing displays of different orbitals makes it easy to compare their relative sizes.Copyright © 2012 Dan Dill ([email protected]). All rights reservedà Spherical (s) familyThe members of the family of hydrogen atom electron waves with zero angular wavelengths (and with spherical symmetry) are call s orbitals. The lowest energy member, 1s, has one radial loop (moving out from the nucleus), the second lowest energy member, 2s, has two radial loops, etc.Probability density in the xy plane of the 1s hydrogen atom electron wave (wave function). The nucleus is at the center. From left to right, the width and height are 0.1, 1, 3, 6, and 9 nm. The brightness of the display is proportional to the probability density.Probability density in the xy plane of the 2s hydrogen atom electron wave (wave function). The nucleus is at the center. From left to right, the width and height are 0.1, 1, 3, 6, and 9 nm. The brightness of the display is proportional to the probability density.Probability density in the xy plane of the 3s hydrogen atom electron wave (wave function). The nucleus is at the center. From left to right, the width and height are 0.1, 1, 3, 6, and 9 nm. The brightness of the display is proportional to the probability density.Probability density in the xy plane of the 4s hydrogen atom electron wave (wave function). The nucleus is at the center. From left to right, the width and height are 0.1, 1, 3, 6, and 9 nm. The brightness of the display is proportional to the probability density.Hydrogen atom family album, CH101 General Chemistry, Fall 2013, Boston University 2Copyright © 2012 Dan Dill ([email protected]). All rights reservedà Dumbbell (p) familyThe members of the family of hydrogen atom electron waves with one angular wavelengths (and so one nodal plane through the nucleus) are call p orbitals. There are three different kinds p orbitals, differing by the axis along which the dumbbell shape is aligned. The three different kinds of p orbitals are called px, py, and pz. The lowest energy members, the three 2p orbitals, each have one radial loop (moving out from the nucleus), the second lowest energy members, 3p, each have two radial loops, etc.Probability density in the xy plane of the 2px hydrogen atom electron wave (wave function). The nucleus is at the center. From left to right, the width and height are 0.1, 1, 3, 6, and 9 nm. The brightness of the display is proportional to the probability density.Probability density in the xy plane of the 3px hydrogen atom electron wave (wave function). The nucleus is at the center. From left to right, the width and height are 0.1, 1, 3, 6, and 9 nm. The brightness of the display is proportional to the probability density.Probability density in the xy plane of the 4px hydrogen atom electron wave (wave function). The nucleus is at the center. From left to right, the width and height are 0.1, 1, 3, 6, and 9 nm. The brightness of the display is proportional to the probability density.Hydrogen atom family album, CH101 General Chemistry, Fall 2013, Boston University 3Copyright © 2012 Dan Dill ([email protected]). All rights reservedà Cloverleaf (d) familyThe members of the family of hydrogen atom electron waves with two angular wavelengths (and so two nodal planes through the nucleus) are called d orbitals. There are five different kinds of d orbitals, differing by the orientation of the cloverleaf shape. The five different kinds of d orbitals are called dz2, dyz, dzx, dxy, and dx2-y2. The lowest energy members, the five 3d, each have one radial loop (moving out from the nucleus), the second lowest energy members, 4d, each have two radial loops, etc.Probability density in the xy plane of the 3dxy hydrogen atom electron wave (wave function). The nucleus is at the center. From left to right, the width and height are 0.1, 1, 3, 6, and 9 nm. The brightness of the display is proportional to the probability density.Probability density in the xy plane of the 4dxy hydrogen atom electron wave (wave function). The nucleus is at the center. From left to right, the width and height are 0.1, 1, 3, 6, and 9 nm. The brightness of the display is proportional to the probability density.Hydrogen atom family album, CH101 General Chemistry, Fall 2013, Boston University 4Copyright © 2012 Dan Dill ([email protected]). All