Question

How many electrons are in `n=3`, `l= 2`?

Answer 1

`"10 e"^(-)`

Explanation:

The idea here is that you must use the value of the angular momentum quantum number, `l`, which tells you the energy subshell in which an electron resides, to find the possible values of the magnetic quantum number, `m_l`.

The number of values that the magnetic quantum number can take tells you the number of orbitals that are present in a given subshell.

So, you know that the magnetic quantum number depends on the value of the angular momentum quantum number

`m_l = {-l, -(l-1), ..., -1, color(white)(-)0, +1, ..., +(l-1), +l}`

In your case, you have `l=2`, which is an accepted value for the angular momentum quantum number given the fact that the principal quantum number, `n`, is equal to `3`, so you can say that

`m_l = {-2, -1, color(white)(-)0, +1, +2}`

This tells you that the `d` subshell, which is denoted by `l=2`, holds a total of `5` orbitals.

Since each orbital can hold a maximum of `2` electrons, one having spin-up and one having spin-down, you can say that you have

`5 color(red)(cancel(color(black)("orbitals"))) * "2 e"^(-)/(1color(red)(cancel(color(black)("orbital")))) = "10 e"^(-)`

Therefore, a maximum number of `10` electrons can share these two quantum numbers in an atom.

`n=3, l=2`

These electrons are located on the third energy level, in the `3d` subshell.