Question

In the ground state `"Kr"` atom, how many electrons have the quantum number `m_l = -2` ?

Answer 1

Two electrons.

Explanation:

First of all, you should know that the magnetic quantum number, `m_l`, describes the orientation of the orbital in which an electron is located in an atom.

The values that the magnetic quantum number can take depend on the values of the angular momentum quantum number, `l`, which tells you the energy subshell in which an electron resides.

Now, we know that

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

This implies that in order to be able to have

`m_l = -2`

you need to have

`l >=2`

As you know, the angular momentum quantum number can take the following values

• `l = 0 ->` describes the `s` subshell
• `l = 1 ->` describes the `p` subshell
• `l = 2 ->` describes the `d` subshell
  `vdots`

and so on. For the `d` subshell, you have

`m_l = { -2, -1, 0, 1, 2}`

This means that you're looking for

1. the nubmer of `d` subshells that contain electrons in a neutral atom of krypton in its ground state
2. the number of electrons located in one the five `d` orbitals present for every `d` subshell that contains electrons

Now, krypton, which is located in period `4`, group `18` of the Periodic Table, has the following electron configuration

`"K: " 1s^2 2s^2 2p^6 3s^2 3p^6 ul(3d^10) 4s^2 4p^6`

As you can see, a neutral atom of krypton has `10` electrons in the `3d` subshell, the only `d` subshell that contains electrons in this atom.

Since the `3d` subshell is completely filled, i.e. it contains `10` electrons, you know for a fact that each of the five `3d` orbitals contains `2` electrons of opposite spins.

This implies that a total of `2` electrons will have `m_l = -2`, i.e. two electrons of opposite spins can share the orbital described by `m_l = -2`.

Depending on what naming convention you use, you can say that these two electrons will be located, for example, in the `3d_(x^2-y^2)` orbital.