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Atoms that have eight electrons in their outer shell are extremely stable. It can't be because both the $s$ and the $p$ orbitals are full, because then an atom with 13 or 18 valence electrons would be extremely stable. ($d$ has 10, and 5 is also stable).

Why is it that atoms with eight electrons in the outer shell are extremely stable?

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First, this isn't quite true. It is true for the first row of the periodic chart (from lithium to neon). It is almost true for the second row (from sodium to argon. But there are exceptions there. Beyond that it really isn't true at all for the elements beyond the first two columns.

The reason for the increased stability for the first two rows lies in quantum mechanics. Classically we can note that there are no $d$ electrons there. Another ways of looking at it from a classical point of view is that the early elements are too small to allow too many other atoms or groups of atoms around them. That tends to go away as you go down the periodic chart and the atoms get "fatter". A typical example is chloroplatinic acid which has six chlorines around it.

Most transition metals also can have more than four groups around them as well.

I suspect that this isn't an exceptionally useful explanation. As I said, the answer really lies in quantum mechanics. In looking up "molecular orbital theory", one reference can be found here.

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This video nicely explains your question.

The essence is that paulis exclusion principle states that two electrons(or generally fermions) cannot occupy the same quantum state.

This is because electrons are fundamentally indistinguishable,but their wavefunctions are not. So there must be some quantum factors which allows to distinguish between their quantum states. In general we say that quantum properties of electrons can be conveyed by some numbers which we call the quantum numbers of that electron.

The distinguishablity in the first shell electrons is assured by spin quantum number. They can have $+1/2$ or $-1/2$ spin. So, there are 2 electrons in the first shell. But in the second shell,there are additional quantum numbers like azimuthal quantum number and magnetic quantum number, that can vary which allows for upto 8 electons in a single shell.

So,why are 8 electrons stable? because the protons and electrons attract each other.so,the atom in which all electrons are maximum filled is the most stable system.

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