In my textbook, it says that the maximum number of electrons that can fit in any given shell is given by 2n². This would mean 2 electrons could fit in the first shell, 8 could fit in the second shell, 18 in the third shell, and 32 in the fourth shell.

However, I was previously taught that the maximum number of electrons in the first orbital is 2, 8 in the second orbital, 8 in the third shell, 18 in the fourth orbital, 18 in the fifth orbital, 32 in the sixth orbital. I am fairly sure that orbitals and shells are the same thing.

Which of these two methods is correct and should be used to find the number of electrons in an orbital?

I am in high school so please try to simplify your answer and use fairly basic terms.

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Shells and orbitals are not the same. In terms of quantum numbers, electrons in different shells will have different values of principal quantum number n.

To answer your question...

In the first shell (n=1), we have:

The 1s orbital

In the second shell (n=2), we have:

The 2s orbitalThe 2p orbitals

In the third shell (n=3), we have:

The 3s orbitalThe 3p orbitalsThe 3d orbitals

In the fourth shell (n=4), we have:

The 4s orbitalThe 4p orbitalsThe 4d orbitalsThe 4f orbitals

So another kind of orbitals (s, p, d, f) becomes available as we go to a shell with higher n. The number in front of the letter signifies which shell the orbital(s) are in. So the 7s orbital will be in the 7th shell.

Now for the different kinds of orbitalsEach kind of orbital has a different "shape", as you can see on the picture below. You can also see that:

The s-kind has only one orbitalThe p-kind has three orbitalsThe d-kind has five orbitalsThe f-kind has seven orbitals

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Each orbital can hold two electrons. One spin-up and one spin-down. This means that the 1s, 2s, 3s, 4s, etc., can each hold two electrons because they each have only one orbital.

The 2p, 3p, 4p, etc., can each hold six electrons because they each have three orbitals, that can hold two electrons each (3*2=6).

The 3d, 4d etc., can each hold ten electrons, because they each have five orbitals, and each orbital can hold two electrons (5*2=10).

Thus, to find the number of electrons possible per shell

First, we look at the n=1 shell (the first shell). It has:

The 1s orbital

An s-orbital holds 2 electrons. Thus n=1 shell can hold two electrons.

The n=2 (second) shell has:

The 2s orbitalThe 2p orbitals

s-orbitals can hold 2 electrons, the p-orbitals can hold 6 electrons. Thus, the second shell can have 8 electrons.

The n=3 (third) shell has:

The 3s orbitalThe 3p orbitalsThe 3d orbitals

s-orbitals can hold 2 electrons, p-orbitals can hold 6, and d-orbitals can hold 10, for a total of 18 electrons.

Therefore, the formula $2n^2$ holds! What is the difference between your two methods?

There"s an important distinction between "the number of electrons possible in a shell" and "the number of valence electrons possible for a period of elements".

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There"s space for $18 exte^-$ in the 3rd shell: $3s + 3p + 3d = 2 + 6 + 10 = 18$, however, elements in the 3rd period only have up to 8 valence electrons. This is because the $3d$-orbitals aren"t filled until we get to elements from the 4th period - ie. elements from the 3rd period don"t fill the 3rd shell.

The orbitals are filled so that the ones of lowest energy are filled first. The energy is roughly like this: