# Question #a4966

Oct 5, 2017

Two electrons.

#### Explanation:

As you know, the principal quantum number, $n$, tells you the energy shell in which an electron is located inside an atom.

$n = 3$

which means that your electrons are going to be located in the third energy shell, i.e. on the third energy level.

Now, notice that the problem provides you the value of the magnetic quantum number, ${m}_{l}$, which tells you the orbital in which an electron is located.

${m}_{l} = + 1$

so your goal now is to figure out how many subshells located on the third energy level can hold an orbital designated by that value of the magnetic quantum number.

As you know, the identity of the energy subshell in which an electron is located is given by the angular momentum quantum number, $l$, which depends on the principal quantum number as follows

$l = \left\{0 , 1 , \ldots , n - 1\right\}$

$l = \left\{0 , 1 , 2\right\}$

So the third energy shell can hold a total of $3$ energy subshells.

• $l = 0 \to$ the s subshell
• $l = 1 \to$ the p subshell
• $l = 2 \to$ the d subshell

Now, the magnetic quantum number depends on the angular momentum quantum number as follows

${m}_{l} = \left\{- l , - \left(l - 1\right) , \ldots , - 1 , 0 , 1 , \ldots , \left(l - 1\right) , l\right\}$

This means that an orbital described by ${m}_{l} = + 1$ can be found for

• $l = 1 \implies {m}_{l} = \left\{- 1 , 0 , + 1\right\}$
• $l = 2 \implies {m}_{l} = \left\{- 2 , - 1 , 0 , + 1 , + 2\right\}$

This means that a total of $2$ orbitals, one located in the $3 p$ subshell and the other located in the $3 d$ subshell, match this description.

Finally, the problem gives you a value for the spin quantum number, ${m}_{s}$.

As you know, an orbital can hold a maximum of two electrons of opposite spins, i.e. one spin-up electron and one spin-down electron.

This means that a total of $2$ electrons, one for each orbital, can share the quantum numbers given to you.

$n = 3 , \textcolor{b l u e}{l = 1} , {m}_{l} = + 1 , {m}_{s} = + \frac{1}{2}$

This set describes an electron located on the third energy level, in the $3 \textcolor{b l u e}{p}$ subshell, let's say in the $3 {p}_{y}$ orbital, that has spin-up

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

This set describes an electron located on the third energy level, in the $3 \textcolor{red}{d}$ subshell, let's say in the $3 {d}_{y z}$ orbital, that has spin-up