# How many orbitals are found in the d sublevel?

Dec 3, 2015

We can derive this from knowing that the atomic orbitals each exist in accordance with angular momentum quantum number $l$.

Recall how $l$ tells you the shape of the atomic orbital. That just means:

$l = 0 \to s$ orbital
$l = 1 \to p$ orbital
$l = 2 \to d$ orbital
etc.

Additionally, what we have associated with $l$ is the magnetic quantum number ${m}_{l}$, the projection of $l$ in the negative, unsigned, and positive directions. In other words...

${m}_{l} = 0 , \pm 1 , \pm 2 , . . . \pm \left(l - 1\right) , \pm l$

If $l = 2$, then:

$\textcolor{b l u e}{{m}_{l} = - 2 , - 1 , 0 , + 1 , + 2}$

Each individual ${m}_{l}$ value corresponds to a unique orbital.

That indicates that five orbitals are available in the $d$ orbitals. Specifically, the ${d}_{{z}^{2}}$, ${d}_{{x}^{2} - {y}^{2}}$, ${d}_{x y}$, ${d}_{x z}$, and ${d}_{y z}$ orbitals.

Furthermore, from knowing that the spin quantum number ${m}_{s}$ for an electron is $\pm \text{1/2}$ (two spin states), and recalling the Pauli exclusion principle (two electrons in one orbital must be opposite spins), there can be a max of two electrons per orbital.

Therefore, the total number of electrons in five orbitals can be a maximum of 10.