# What is the maximum number of orbitals in a p sub-level?

Feb 4, 2018

There can only be three $p$-orbitals for any value of $n$ (in any shell).

#### Explanation:

This result comes from the manner in which the orbitals are determined for particular orbitals.

First, the principal quantum number $n$ is determined. This decides to which shell the orbital belongs. $n$ can have any positive integer value starting with 1.

Next, the angular momentum quantum number, $l$ must be specified. $l$ can be any value from zero up to $n - 1$.

An orbital is a $p$=orbital if it has an angular momentum quantum number, $l$ equal to 1 (which implies that these orbitals first exist for quantum level $n = 2$, and are found for every value of $n$ after that).

Finally, for determining orbitals, the one remaining quantum number to be specified is the magnetic quantum number, ${m}_{l}$. Like each quantum number, there are restrictions on the values ${m}_{l}$ can possess. In this case it is $- l , - l + 1 , - l + 2 , \ldots , 0 , 1 , 2 , \ldots l - 1$.

Therefore, putting all this together: if $l = 1$, (so we are referring to a $p$-orbital, the possible value for ${m}_{l}$ are only $- 1 , 0 , \mathmr{and} + 1$. These three possible values create the orbitals known as ${p}_{x} , {p}_{y} , \mathmr{and} {p}_{z}$ as the only possibilities, regardless on the value of $n$.