0.4 Quantum energy levels in atoms

Foundation

The atomic molecular theory provides us a particulate understanding of matter. Each element is characterizedas consisting of identical, indestructible atoms with atomic weights which have been determined. Compounds consist of identicalmolecules, each made up from a specific number of atoms of each of the component elements. We also know that atoms have a nuclearstructure, meaning that all of the positive charge and virtually all of the mass of the atom are concentrated in a nucleus which isa very small fraction of the volume of the atom. Finally, we know that the electrons in the atom are arranged in "shells" about the nucleus, with each shell farther from the nucleus thatthe previous. The electrons in outer shells are more weakly attached to the atom than the electrons in the inner shells, andonly a limited number of electrons can fit in each shell.

Goals

The shell model of the atom is a good start in understanding the differences in the chemical properties of theatoms of different elements. For example, we can understand the periodicity of chemical and physical properties from our model,since elements in the same group have the same number of electrons in the valence shell.

However, there are many details missing from our description. Other than a very crude calculation of"distance" of the shells from the nucleus, we have no description of what the differences are between the electrons indifferent shells. What precisely is a "shell?"

Most importantly, the arrangement of elements into groups and the periodicity of chemical properties both dependon the concept that a shell is "filled" by a certain number of electrons. Looking at the number of elements in eachperiod, the number of electrons which fills a shell depends on which shell is being filled. In some cases, a shell is filled byeight electrons, in others, it appears to be 18 electrons. What determines how many electrons can "fit" in a shell? Whyis there a limit at all?

Finally, a closer look at the ionization energies here reveals that our shell model must be incomplete. Our model implies that theelements of the second period from Lithium to Neon have their valence electrons in the second shell. With increasing nuclearcharge, the ionization energy of these atoms should increase from Lithium to Neon. As a general trend, this is true, but there arevariations. Note that the ionization energy of Oxygen atoms is less than that of Nitrogen atoms. We need to pursue additional detail inour model of the structure of the atom.

Observation 1: the spectrum of hydrogen

To begin, we need to know a little about light. All forms of electromagnetic radiation travel as anoscillating wave, with an electric field component perpendicular to a magnetic field component. As a wave, the radiation can becharacterized by its "wavelength", symbolized as $\lambda$ , which is the distance between adjacent peaks in the wave.Different wavelengths correspond to different forms of electromagnetic radiation. For example, microwave radiation haswavelength in the range of $10^{-2}$ to $10^{-3}$ meters, whereas x-ray radiation has wavelength in the range $10^{-9}$ to $10^{-10}$ meters. Radiation which is visible to the human eye has wavelength in thevery narrow range from $3.8E-7$ to $7.8E-7$ meters.