Why does the arrangement of the Periodic Table make no sense?

[Bonkus Klunkit]

TL;DR Summary

Electron Shells and the periodic table

We're told that the electron shells give the relative reactivity/affinity to other atoms by their incompleteness, completeness or over-completeness, and arranged in the periodic table accordingly as columns. The shells are are concentrically arranged around the nucleus from smaller capacity to larger starting with the first a shell that fits two. Hence row 1 has two (H & He), one on the left and one on the right of the table. The one on the right (He) has a full shell and is stable and is placed with the column of "noble gasses". The next shells continue in their holding capacity to hold 8, 18 and 32 respectively and are placed into the "columns" they would fall into and Ne having a full 2 shell and a full 8 shell is stable and placed under He as another "noble gas". Shell 3 holds 18, but we see row 3 with only 8 elements and Ar placed under the "noble gasses" column as if it had a complete shell (18)--but it only has 8! Row 4 has 18, but what happened to row 3? And shouldn't row 4 have 32 (row 6 & 7 do)? I'm confused, this would make the concentric shells hold respectively 2, 8, 8, 18, 18, 32, 32. I know this BASIC physics, but I'm confused.

 

[jfizzix]

...Shell 3 holds 18, but we see row 3 with only 8 elements and Ar placed under the "noble gasses" column as if it had a complete shell (18)--but it only has 8! Row 4 has 18, but what happened to row 3? And shouldn't row 4 have 32 (row 6 & 7 do)? ...

As a general rule, the periodic table is arranged in the order that electrons would fill orbitals at higher energy.

Here, orbitals are particular quantum states of the electron orbiting a hydrogen-like atom with principal quantum number $n={1,2,3,...}$, angular momentum quantum number $\ell={0,1,2,...,n-1}$, magnetic quantum number $m_{\ell}={-\ell,-\ell+1,...,\ell-1,\ell}$, and spin quantum number $m_{s}={-1/2,1/2}$. The models describing the valence electrons around different elements are more sophisticated, but these quantum numbers are useful enough to organize the elements.

Each orbital corresponds to a unique value of $n$, $\ell$, and $m_{\ell}$

Each orbital can be occupied by up to 2 electrons, due to $m_{s}$ having two possible values, and the Pauli exclusion principle preventing two electrons having the same set of all 4 quantum numbers at the same time.

In order of increasing energy, the orbitals are:
1s (row 1, 2 elements)
2s, 2p ( row 2, 2 + 6 = 8 elements)
3s,3p ( row 3, 2 + 6 = 8 elements)
4s,3d,3p ( row 4, 2 + 10 + 6 = 18 elements)
5s,4d,4p ( row 5, 2 + 10 + 6 = 18 elements)
6s, 4f, 5d, 5p ( row 6, 2 + 14 + 10 + 6 = 32 elements)
7s, 5f, 6d, 6p ( row 7, 2 + 14 + 10 + 6 = 32 elements)

Here as short-hand $\ell={0,1,2,3,4}\rightarrow{s,p,d,f,g}$, so that the 6d orbital corresponds to $n=6$ and $\ell=2$, which means $m_{\ell}={-2,-1,0,1,2}$, and so there are five, 6d orbitals, and it takes 10 electrons to fill it.

All electrons in an orbital with a particular value of $n$ is in that electron shell. Electron shells don't precisely relate to how the periodic table is organized.

I don't have a good explanation for why the 3d orbital has a lower energy than the 3p orbital and a higher energy than the 4s orbital, or why the noble gases below neon are noble in spite of having incompletely filled electron shells, but I hope this helps :)

 

[mitochan]

I don't have a good explanation for why the 3d orbital has a lower energy than the 3p orbital and a higher energy than the 4s orbital, or why the noble gases below neon are noble in spite of having incompletely filled electron shells, but I hope this helps :)

Let me explain my understandings on this matter.

The two rules are sure;

As for different prime numbers and the same orbital number x,
nx < (n+1)x for x=s,p,d,.. where "nx" is energy level of nx orbit.
e.g.
1s < 2s < 3s < ...
1p < 2p < 3p < ...
1d < 2d < 3d < ...

As for different orbit number n and the same prime numbers x,
ns < np < nd < nf < ...
e.g.
1s < 1p < 1d <...
2s < 2p < 2d <...
3s < 3p < 3d <...

Satisfying these basic rules something like "crossover" could take place between
"nx" and "my". It is like that : a lower story of a building built on a higher land can become higher than the higher story of the building built on the lower land. For nx n is land height and x-th is the number of story.