# Question #673da

Sep 13, 2016

Two electrons.

#### Explanation:

As you know, we can use four quantum numbers to describe the position and spin of an electron inside an atom.

Now, notice that the problem provides you with the energy level on which the electron is located, i.e. the principal quantum number, $n = 3$, and the orbital in which it resides, i.e. the magnetic quantum number, ${m}_{l} = 2$.

This means that the answer to the question is $2$ electrons.

That is the case because every energy level has its distinct orbitals. In this case, the ${m}_{l} = 2$ value designates one of the five d-orbitals located on the third energy level.

This particular orbital is unique because you can only have one set of d-orbitals on the third energy level, one set of d-orbitals on the fourth energy level, and so on.

Now, each orbital can hold a maximum of $2$ electrons, one having spin-up, or ${m}_{s} = + \frac{1}{2}$, and the other having spin-down, or ${m}_{s} = - \frac{1}{2}$, as given by the Pauli Exclusion Principle.

This basically means that when you're given a specific orbital, i.e. the value of $n$ and the value of ${m}_{l}$, the answer can only be $2$ electrons.

In this particular case, you have

• $n = 3 , l = 2 , {m}_{l} = 2 , {m}_{s} = + \frac{1}{2}$

This descrribes an electron located on the third energy level, in the d-subshell, in one of the five 3d-orbitals, that has spin-up

• $n = 3 , l = 2 , {m}_{l} = 2 , {m}_{s} = - \frac{1}{2}$

This descrribes an electron located on the third energy level, in the d-subshell, in one of the five 3d-orbitals, that has spin-down