Question

What values of `l`, `m_l`, and `m_s` are possible for `n = 3`?

Answer 1

Well, your set of quantum numbers is not "allowed" for a particular electron because of the value you have for `"l"`, the angular momentum quantum number.

The values the angular momentum quantum number is allowed to take go from zero to `"n-1"`, `"n"` being the principal quantum number.

So, in your case, if `"n"` is equal to 3, the values `"l"` must take are 0, 1, and 2. Since `"l"` is listed as having the value 3, this puts it outside the allowed range.

The value for `m_l` can exist, since `m_l`, the **magnetic quantum number, ranges from `-"l"`, to `"+l"`.

Likewise, `m_s`, the spin quantum number, has an acceptable value, since it can only be `-"1/2"` or `+"1/2"`.

Therefore, the only value in your set that is not allowed for a quantum number is `"l"=3`.

Answer 2

There are 4 quantum numbers which describe an electron in an atom.
These are:

`n` the principal quantum number. This tells you which energy level the electron is in. `n` can take integral values 1, 2, 3, 4, etc

`l` the angular momentum quantum number. This tells you the type of sub - shell or orbital the electron is in. It takes integral values ranging from 0, 1, 2, up to `(n-1)`.

If `l` = 0 you have an s orbital.
`l=1` gives the p orbitals
`l=2` gives the d orbitals

`m` is the magnetic quantum number. For directional orbitals such as p and d it tells you how they are arranged in space. `m` can take integral values of `-l ............. 0.............+l`.

`s` is the spin quantum number. Put simply the electron can be considered to be spinning on its axis. For clockwise spin `s`= +1/2. For anticlockwise `s` = -1/2. This is often shown as `uarr` and `darr`.

In your question `n=3`. Let's use those rules to see what values the other quantum numbers can take:

`l=0, 1 and 2`, but not 3.This gives us s, p and d orbitals.

If `l` = 0 `m` = 0. This is an s orbital
If `l` = 1, `m` = -1, 0, +1. This gives the three p orbitals. So `m` = 0 is ok.
If `l` = 2 `m` = -2, -1, 0, 1, 2. This gives the five d orbitals.

`s` can be +1/2 or -1/2.

These are all the allowed values for `n=3`

Note that in an atom, no electron can have all 4 quantum numbers the same. This is how atoms are built up and is known as The Pauli Exclusion Principle.