What values of m are permitted for an electron with l = 2?

Jan 2, 2017

Possible ${m}_{l}$ values are $- 2 , - 1 , 0 , 1 , 2$. See below.

Explanation:

There are four quantum numbers: the principle quantum number, $n$, the angular momentum quantum number, $l$, the magnetic quantum number, ${m}_{l}$, and the electron spin quantum number, ${m}_{s}$. For this question we are concerned with $l$ and ${m}_{l}$.

The angular momentum quantum number, $l$, describes the shape of the subshell and its orbitals, where $l = 0 , 1 , 2 , 3. . .$ corresponds to $s , p , d ,$ and $f$ subshells (containing $s , p , d , f$ orbitals), respectively. Each shell has up to $n - 1$ types of subshells/orbitals.

An angular momentum quantum number of $l = 2$ describes a $d$ subshell.

The magnetic quantum number, ${m}_{l}$, describes the orientation of the orbitals (within the subshells) in space. The possible values for ${m}_{l}$ of any type of orbital ($s , p , d , f \ldots$) is given by any integer value from $- l$ to $l$.

Therefore, given $l = 2$, the possible ${m}_{l}$ values are $- 2 , - 1 , 0 , 1 , 2$. This tells us that the $d$ subshell contains five $d$ orbitals, each with a different orientation (${d}_{y z}$, ${d}_{x y}$, ${d}_{x z}$, ${d}_{{x}^{2} - {y}^{2}}$, and ${d}_{{z}^{2}}$).