# How would you find how many orbitals a sublevel has?

May 17, 2018

You examine the range of allowed values of ${m}_{l}$ for a given $l$.

For an atomic orbital, the angular momentum quantum number $l$ corresponds to the shape of the orbital, and we have:

$\underline{l \text{ "" ""shape/subshell}}$
$0 \text{ "" } \textcolor{w h i t e}{.} s$
$1 \text{ "" } \textcolor{w h i t e}{.} p$
$2 \text{ "" } \textcolor{w h i t e}{.} d$
$3 \text{ "" } \textcolor{w h i t e}{.} f$
$4 \text{ "" } \textcolor{w h i t e}{.} g$
$5 \text{ "" } \textcolor{w h i t e}{.} h$
$6 \text{ "" } \textcolor{w h i t e}{.} i$
$7 \text{ "" } \textcolor{w h i t e}{.} k$
$\vdots \text{ "" } \vdots$

${m}_{l}$, the magnetic quantum number, is the z-projection of $l$ in integer steps:

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

Each ${m}_{l}$ value corresponds to an orbital in the subshell defined by $l$. There are $2 l + 1$ such values of ${m}_{l}$ for a given $l$, and thus, there are $\boldsymbol{2 l + 1}$ orbitals in each subshell.

$\underline{l \text{ "" ""shape/subshell"" "" ""Number of Orbitals}}$
$0 \text{ "" "color(white)(.)s" "" "" "" "" "" "" } 2 \left(0\right) + 1 = 1$
$1 \text{ "" "color(white)(.)p" "" "" "" "" "" "" } 2 \left(1\right) + 1 = 3$
$2 \text{ "" "color(white)(.)d" "" "" "" "" "" "" } 2 \left(2\right) + 1 = 5$
$3 \text{ "" "color(white)(.)f" "" "" "" "" "" "" } 2 \left(3\right) + 1 = 7$
$4 \text{ "" "color(white)(.)g" "" "" "" "" "" "" } 2 \left(4\right) + 1 = 9$
$5 \text{ "" "color(white)(.)h" "" "" "" "" "" } \textcolor{w h i t e}{. .} 2 \left(5\right) + 1 = 11$
$6 \text{ "" "color(white)(.)i" "" "" "" "" "" "" } 2 \left(6\right) + 1 = 13$
$7 \text{ "" "color(white)(.)k" "" "" "" "" "" } \textcolor{w h i t e}{. .} 2 \left(7\right) + 1 = 15$
$\vdots \text{ "" "vdots" "" "" "" "" "" } \textcolor{w h i t e}{.} \vdots$