# Question 10924

Aug 16, 2017

Here's what I got.

#### Explanation:

I'm guessing that you want to figure out a possible set of quantum numbers that can describe an electron located in the $5 f$ subshell.

For starters, you should know that we can use a total of $4$ quantum numbers to describe the location and spin of an electron inside an atom. Now, the subshell in which an electron resides is given by the angular momentum quantum number, $l$, which can take on of the following values

• $l = 0 \to$ designates the s subshell
• $l = 1 \to$ designates the p subshell
• $l = 2 \to$ designates the d subshell
• $l = 3 \to$ designates the f subshell
$\vdots$

and so on. So in your case, all the electrons that can reside in the $5 f$ subshell will have

$l = 3$

Notice that the value of the angular momentum quantum number depends on the value of the principal quantum number, $n$.

This means that in order for your electrons to have access to the $\textcolor{red}{5} f$ subshell, they need to be located on the fifth energy level, so

$n = \textcolor{red}{5}$

The $5 f$ subshell contains a total of $7$ orbitals, all described by a distinct value of the magnetic quantum number, ${m}_{l}$.

${m}_{l} = \left\{- 3 , - 2 , - 1 , 0 , 1 , 2 , 3\right\}$ The spin quantum number, ${m}_{s}$, which describes the spin of the electron inside its orbital, can only take two possible values

${m}_{s} = \left\{- \frac{1}{2} , + \frac{1}{2}\right\}$

Since each orbital can hold a maximum of $2$ electrons of opposite spins, i.e. of opposite ${m}_{2}$ values--think Pauli's Exclusion Principle here--you can say that the $5 f$ subshell can hold a maximum of

7 color(red)(cancel(color(black)("orbitals"))) * "2 e"^(-)/(1color(red)(cancel(color(black)("orbital")))) = "14 e"^(-)#

This implies that you can write a total of $14$ distinct quantum number sets to describe one of the $14$ electrons that can share the $5 f$ subshell.

For example, you can have

• $n = 5 , l = 3 , {m}_{l} = - 3 , {m}_{s} = + \frac{1}{2}$
• $n = 5 , l = 3 , {m}_{l} = 0 , {m}_{s} = - \frac{1}{2}$
• $n = 5 , l = 3 , {m}_{l} = 2 , {m}_{s} = + \frac{1}{2}$
• $n = 5 , l = 3 , {m}_{l} = - 1 , {m}_{s} = - \frac{1}{2}$

and so on. Notice that all the $14$ sets share the principal quantum number and the angular momentum quantum number, which is what you should expect for electrons that share a subshell--the $f$ subshell-- located on a specific energy level--the fifth energy level.