# How many electrons can be described by the quantum numbers n=5, l=2 in a particular atom ?

Jun 12, 2017

${\text{10 e}}^{-}$

#### Explanation:

The key here is actually the value of the angular momentum quantum number, $l$, which tells you the energy subshell in which an electron resides inside an atom.

This value is important because you can use it to determine the number of orbitals, and consequently the number of electrons, that can reside in this particular energy subshell.

So, you know that you're working on the fifth energy level of the atom, hence the value $n = 5$ for the principal quantum number.

Now, the value $l = 2$, which corresponds to the $d$ subshell, allows for $5$ individual orbitals as given by the possible values of the magnetic quantum number, ${m}_{l}$.

$l = 2 \implies {m}_{l} = \left\{- 2 , - 1 , \textcolor{w h i t e}{-} 0 , + 1 , + 2\right\}$

As you know, each orbital can hold a maximum of $2$ electrons, one having spin-up and the other having spin-down.

This means that the maximum number of electrons that can reside in a $d$ subshell is equal to

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

So, you can say that a maximum of $10$ electrons can share the quantum numbers

$n = 5 , l = 2$

These electrons are located on the fifth energy level, in the $5 d$ subshell