# Each electron shell can hold up to 2n² electrons. Is there a geometric reason for this?

It seems such a simple formula, that each electron shell can hold in principle two times a square number. I imagine there must be a geometrical reason for this, that's to do with the way electron orbitals "fit" around the nucleus. If this is so, what is it?

• $2n^2$ rule is not always true; mind it! – user5764 Jul 16 '15 at 4:57
• When you say "geometrical reason" it makes me think "if only chemistry was so simple" haha. The reason is quantum mechanical in nature. – orthocresol Jul 16 '15 at 5:39
• @orthocresol ncbi.nlm.nih.gov/pmc/articles/PMC219917 – DavePhD May 19 '16 at 18:34
• @user5764 I don't think so. Can you back your claim up with some examples? – Gaurang Tandon Jul 3 '18 at 18:08

A good insight!

The number of electrons in each orbital are double the odd numbers. The s orbital can hold 2, p holds 6, d holds 10 and f has 14: double 1, 3, 5 and 7. If you add the odd numbers, you get a square number:

1 = 1

1 + 3 = 4

1 + 3 + 5 = 9

1 + 3 + 5 + 7 = 16

The doubling is due to the Pauli exclusion principle; in each orbital, an electron of a specific energy can be in one of two spin states.

It is interesting to see spectral lines split in a magnetic field (Zeeman effect) or an electrostatic field (Stark effect): one line may become 3, 5 or 7 lines. It clearly shows how orbital theory was derived.