# How many quantum number sets can be written for an electron located in the 4d subshell?

Dec 28, 2017

Here's what I got.

#### Explanation:

The first thing to note here is that the coefficient added in front of the name of the subshell tells you the energy shell in which the electron resides.

In your case, an electron located in the $4 d$ subshell will have a principal quantum number, $n$, equal to $4$.

$n = 4 \to$ the fourth energy shell

Now, the identity of the energy subshell is given by the angular momentum quantum number, $l$, which can take the following values

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

and so on. In your case, the electron is located in the $d$ subshell present on the fourth energy level, so it will have

$l = 2$

Now, in order to describe an orbital, you need to have subscripts added to the identity of the energy subshell. The magnetic quantum number, ${m}_{l}$, which tells you the orientation of an orbital present in a given subshell, can take the following values for a $d$ subshell

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

This means that the $d$ subshell contains a total of $5$ orbitals, each designated using a subscript.

For example, you can have

• ${m}_{l} = - 2 \to$ the $4 {d}_{{x}^{2} - {y}^{2}}$ orbital
• ${m}_{l} = - 1 \to$ the $4 {d}_{y z}$ orbital
• ${m}_{l} = \textcolor{w h i t e}{+} 0 \to$ the $4 {d}_{{z}^{2}}$ orbital
• ${m}_{l} = + 1 \to$ the $4 {d}_{x z}$ orbital
• ${m}_{l} = + 2 \to$ the $4 {d}_{x y}$ orbital

Finally, the spin quantum number, ${m}_{s}$, which tells you the spin of the electron inside the orbital, can take one of two possible values.

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

So, one set of quantum numbers that can describe an electron located in the $d# subshell would be $n = 4 , l = 2 , {m}_{l} = 0 , {m}_{s} = + \frac{1}{2}$This set describes an electron that is located on the fourth energy level, in the $4 d$subshell, in the $4 {d}_{{z}^{2}}$orbital, and that has spin-up. Another example would be $n = 4 , l = 2 , {m}_{l} = + 2 , {m}_{s} = - \frac{1}{2}$This set describes an electron that is located on the fourth energy level, in the $4 d$subshell, in the $4 {d}_{x y}$orbital, and that has spin-down. Since the $4 d$subshell contains a total of $5$orbitals, and since each orbital can hold a maximum of $2$electrons of opposite spins, you can write a total of $10$quantum number sets for the electrons that can reside in the $4 d$subshell. You can find all $10\$ quantum number sets in this Socratic answer.