# Can why electrons exist in shells be explained by the Pauli exclusion principle?

Do you know the Pauli exclusion principle?-'No two particles could be in the same quantum state at once'. Well can you use that principle to explain why electrons stay in shells and electrons in separate shells can never get closer than a certain length to electrons in another shell. I learned somewhere that what the Pauli exclusion principle is really saying is that you should be able to indistinguishably identify two separate fermions or something very similar to that. Well I'd get why that happens in the first shell (because there are only 2 electrons and they have opposite spins) but how could that apply to shells with higher electrons like for example the 2nd shell which has 4 electrons? I also think that there's an equation describing this and I'd love to know what that equation is. Any help would be greatly appreciated.

• Have you read the wikipedia page on the aufbau principle? – jacob1729 Sep 30 '20 at 12:14
• I did but wikipedia wasn't all that helpful – alienare 4422 Sep 30 '20 at 12:20
• Note that the exclusion principle prohibits two fermions occupying states with identical quantum numbers, the 2nd energy shell has 4 orbitals, $s^x$, $p^x$, $p^y$ and $p^z$ each of these can separately hold 2 electrons since their $l$ and $m_l$ quantum numbers are different. – Charlie Sep 30 '20 at 12:34
• *I shouldn't have attached an $x$ to the s-orbital, can't edit it now, my mistake. – Charlie Sep 30 '20 at 12:50
• @Charlie, You can't edit a comment after five minutes, but you can copy it's text into a new comment and then delete the old one. – Solomon Slow Sep 30 '20 at 14:51

The state of a bound electron in an atom is described by four quantum numbers:

1. The principal quantum number $$n$$ which takes integer values $$1,2,3, \dots$$ and determines which shell the electron is in.
2. The azimuthal quantum number $$l$$ which takes integer values from $$0$$ to $$n-1$$ and determines the angular momentum of the electron.
3. The magnetic quantum number $$m_l$$ which takes integer values from $$-l$$ to $$l$$.
4. The spin quantum number $$s$$ which takes values $$\pm \frac 1 2$$.

The Pauli exclusion principle then prevents two bound electrons having exactly the same set of values for these four quantum numbers.

In shell $$1$$ we have $$n=1$$, $$l=0$$, $$m_l=0$$ and $$s=\pm \frac 1 2$$. So there are at most two electrons in shell $$1$$.

In shell $$2$$ we have $$n=2$$ and $$l=0, 1$$. When $$l=0$$ then $$m_l=0$$ and $$s=\pm \frac 1 2$$, which allows up to $$2$$ electrons. When $$l=1$$ then $$m_l=-1, 0, 1$$ and $$s=\pm \frac 1 2$$, which allows up to $$6$$ electrons. So there are at most $$8$$ electrons in shell $$2$$.

And. in general, there can be at most $$2n^2$$ electrons in shell $$n$$.