# How many orbitals make up the 4d subshell?

Jun 7, 2018

The same number that makes up any individual $d$ subshell... $5$. What are these five ${m}_{l}$ values, specifically, for $d$ orbitals?

$d$ orbitals have an angular momentum quantum number $l$ of $2$:

$\textcolor{w h i t e}{.} \underline{l \text{ "" ""shape}}$
$\textcolor{w h i t e}{.} 0 \text{ "" } s$
$\textcolor{w h i t e}{.} 1 \text{ "" } p$
$\textcolor{w h i t e}{.} 2 \text{ "" } d$
$\textcolor{w h i t e}{.} 3 \text{ "" } f$
$\textcolor{w h i t e}{.} 4 \text{ "" } g$
$\vdots \text{ "" } \vdots$

$l$ has a range of $0 , 1 , 2 , 3 , . . . , n - 1$. Its projection in the $z$ direction is ${m}_{l}$, the magnetic quantum number, and $| {m}_{l} | \le l$, i.e.

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

You can see that there is an odd number of ${m}_{l}$ values because this tells you that ${m}_{l} = 0 , \pm 1 , \pm 2 , . . . , \pm l$. Hence, there are $2 l + 1$ values of ${m}_{l}$ for a GIVEN $l$.

Since $l = 2$...

$2 \left(2\right) + 1 = \boldsymbol{\text{five}}$ $d$ orbitals exist in one $d$ subshell of any $n$.

And these orbitals are each given one value of ${m}_{l}$ in the set of ${m}_{l} = \left\{- 2 , - 1 , 0 , + 1 , + 2\right\}$.