# Question fb349

Jul 18, 2017

Here's what I got.

#### Explanation:

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

I assume that you're fairly familiar with how they work, so I won't go into too much detail here.

So, you know that you must find the sets of quantum numbers that can be used to describe an electron located in the $4 d$ subshell, i.e. in one of the $5$ possible $4 d$ orbitals.

The number added in front of the subshell tells you the energy level on which the subshell is located, i.e. the value of the principal quantum number, $n$.

In your case, you have the $4 d$ subshell, so you can say that

$n = 4 \to$ the fourth energy level

The $d$ subshell is described by an angular momentum quantum number, $l$, that takes the value

$l = 2 \to$ the d subshell

For the $d$ subshell, the magnetic quantum number, ${m}_{l}$, can take one of five possible values

${m}_{l} = \left\{- 2 , - 1 , 0 , 1 , 2\right\} \to$ five orbitals in the d subshell

Each value of the magnetic quantum number describes one of the five orbitals present in the $d$ subshell.

Finally, each orbital can hold a maximum of $2$ electrons of opposite spins, which implies that the spin quantum number, ${m}_{s}$, can only take two possible values

${m}_{s} = \left\{- \frac{1}{2} , \frac{1}{2}\right\} \to$ two electrons of opposite spins

So, you can say that the $4 d$ subshell can hold a maximum of

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

that can be described using the following sets of quantum numbers--each individual orbital is shown in a different color

• $n = 4 , l = 2 , \textcolor{b l u e}{{m}_{l} = - 2} , {m}_{s} = - \frac{1}{2}$
• $n = 4 , l = 2 , \textcolor{b l u e}{{m}_{l} = - 2} , {m}_{s} = + \frac{1}{2}$
• $n = 4 , l = 2 , \textcolor{p u r p \le}{{m}_{l} = - 1} , {m}_{s} = - \frac{1}{2}$
• $n = 4 , l = 2 , \textcolor{p u r p \le}{{m}_{l} = - 1} , {m}_{s} = + \frac{1}{2}$
• $n = 4 , l = 2 , \textcolor{red}{{m}_{l} = 0} , {m}_{s} = - \frac{1}{2}$
• $n = 4 , l = 2 , \textcolor{red}{{m}_{l} = 0} , {m}_{s} = + \frac{1}{2}$
• $n = 4 , l = 2 , \textcolor{\mathrm{da} r k g r e e n}{{m}_{l} = 1} , {m}_{s} = - \frac{1}{2}$
• $n = 4 , l = 2 , \textcolor{\mathrm{da} r k g r e e n}{{m}_{l} = 1} , {m}_{s} = + \frac{1}{2}$
• $n = 4 , l = 2 , \textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{{m}_{l} = 2} , {m}_{s} = - \frac{1}{2}$
• $n = 4 , l = 2 , \textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{{m}_{l} = 2} , {m}_{s} = + \frac{1}{2}$