The third row of the periodic table is similar to the second. The two $3s$ states are filled in sodium and magnesium. The two columns on the left of the periodic table represent the two electrons that can go into an $s$ subshell. Then the six $3p$ states are filled, one by one, in aluminum through argon. The six columns on the right represent the six electrons of the $p$ subshell. Argon $(Z=18\text{,}1s^{2}2s^{2}2p^{6}3s^{2}3p^6)$ is another inert gas, although this may seem surprising because the $3d$ subshell is still open.

The fourth row is where the periodic table begins to get complicated. You might expect the closure of the $3p$ subshell in argon to be followed, starting with potassium $(Z=19)$, by filling the $3d$ subshell. But if you look back at Figure 41.14, where the energies of the different nl states are shown, you will see that the $3d$ state is slightly higher in energy than the $4s$ state. Because the ground state is the lowest energy state consistent with the Pauli exclusion principle, potassium finds it more favorable to fill a $4s$ state than to fill a $3d$ state. Thus the ground-state configuration of potassium is $1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}4s$ rather than the expected $1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}3d$.

At this point, we begin to see a competition between increasing $n$ and decreasing $l$. The highly elliptical characteristic of the $4s$ state brings part of its orbit in so close to the nucleus that its energy is less than that of the more circular $3d$ state. The $4p$ state, though, reverts to the “expected” pattern. We find that

so the states across the fourth row are filled in the order $4s$, then $3d$, and finally $4p$.

Because there had been no previous $d$ states, the $3d$ subshell “splits open” the periodic table to form the 10-element-wide group of transition elements. Most commonly occurring metals are transition elements, and their metallic properties are determined by their partially filled $d$ subshell. The $3d$ subshell closes with zinc, at $Z=30$, then the next six elements fill the $4p$ subshell up to krypton, at $Z=36$.

Things get even more complex starting in the sixth row, but the ideas are familiar. The $l=3$ subshell ($f$ electrons) becomes a possibility with $n=4$, but it turns out that the $5s$, $5p$, and $6s$ states are all lower in energy than $4f$. Not until barium $(Z=56)$ fills the $6s$ subshell (and lanthanum $(Z=57)$ adds a $5d$ electron) is it energetically favorable to add a $4f$ electron. Immediately after barium you have to switch down to the lanthanides at the bottom of the table. The lanthanides fill in the $4f$ states.

The $4f$ subshell is complete with $Z=70$ ytterbium. Then $Z=71$ lutetium through $Z=80$ mercury complete the transition-element $5d$ subshell, followed by the $6p$ subshell in the six elements thallium through radon at the end of the sixth row. Radon, the last inert gas, has $Z=86$ electrons and the ground-state configuration

This is frightening to behold, but we can now understand it!

The white lettering on the periodic table of Figure 41.19 summarizes the results, showing the subshells as they are filled. It is especially important to note how the electron’s spin is absolutely essential for understanding the periodic table. Explaining the periodic table of the elements is a remarkable success of the quantum model of the atom.