Section 7.4 Periodic Table
We now have the basic tools to explain the periodic table of the elements. Electrons are fermions since they have a half-integral spin quantum number (\(s = 1/2\) for an electron). Consequently, electrons obey the Pauli Exclusion Principle (see Section 6.4): no two electrons can be in the same state. So, if there is more than one electron in an atom, no two of them can have the same combination of quantum numbers \(n, l, m_l\) and \(m_s\text{.}\)
Consider lithium, which has three protons in the nucleus and three electrons orbiting it. The first two electrons can go into the ground state, but Pauli demands that the third electron go into a new state. If the lithium atom is in its lowest possible energy state, that third electron must be in some \(n=2\) orbital. Most of the chemical behavior of atoms is due to the shape and size of their outer electron orbitals, so the Pauli exclusion principle is what makes lithium act very differently than helium.
This process repeats many times over to make up the periodic table. The more electrons there are, the higher the energy levels must be filled, similar to the five particle system in Chapter 6, Example 6.1. For some specific orbital, meaning some specific values of \(n\text{,}\) \(\ell\) and \(m_\ell\text{,}\) there are two possible single-electron states: \(|n\,\ell\,m_\ell\uparrow\rangle\) and \(|n\,\ell\,m_\ell\downarrow\rangle\text{,}\) and we can put two electrons into the antisymmetric combination of these two states. Any additional electrons will have to go into a new orbital.
Were it not for the Pauli Exclusion Principle (i.e., if it didn't apply to electrons in atoms), then all elements would have all of their electrons in the lowest energy state with \(n = 1\) (except when excited). Were that the case, then there would be no chemical diversity — each element would behave almost identically to each other, and there would be no chemical interactions. Chemistry would be really dull in this case. Even worse, we wouldn't be around to notice that chemistry had become dull.