# What is the maximum number of electrons that can occupy the 3d orbitals?

Dec 16, 2016

There are five $3 d$ orbitals, each of which can hold up to $2$ electrons, for $10$ total electrons.

#### Explanation:

An orbital is described by the principle quantum number, $n$, the angular momentum quantum number, $l$, and the magnetic quantum number, ${m}_{l}$. An electron is described by each of these quantum numbers, with the addition of the electron spin quantum number, ${m}_{s}$.

The principle quantum number , $n$, is the energy and distance from the nucleus, and represents the shell.

The $3 d$ orbital is in the $n = 3$ shell, just like the $2 p$ and $2 s$ orbitals are in the $n = 2$ shell.

The angular momentum quantum number , $l$, describes the shape of the orbital or subshell, where $l = 0 , 1 , 2 , 3. . .$ corresponds to $s , p , d ,$ and $f$ orbitals, respectively.

Therefore, a $d$ orbital has an $l$ value of $2$. It is worth noting that each shell has up to $n - 1$ types of orbitals.

For example, the $n = 3$ shell has orbitals of $l = 0 , 1 , 2$, which means the $n = 3$ shell contains $s$, $p$, and $d$ subshells. The $n = 2$ shell has $l = 0 , 1$, so it contains only $s$ and $p$ subshells.

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 $l$.

So, for a $3 d$ orbital with $n = 3$ and $l = 2$, we can have ${m}_{1} = - 2 , - 1 , 0 , 1 , 2$. This tells us that the $d$ orbital has $5$ possible orientations in space.

If you've learned anything about group theory and symmetry in chemistry, for example, you might vaguely remember having to deal with various orientations of orbitals. For the $d$ orbital, those are ${d}_{y z}$, ${d}_{x y}$, ${d}_{x z}$, ${d}_{{x}^{2} - {y}^{2}}$, and ${d}_{{z}^{2}}$. So, we would say that the $3 d$ subshell contains $5$ $3 d$ orbitals (shown below).

Finally, the electron spin quantum number, ${m}_{s}$, has only two possible values, $+ \frac{1}{2} \mathmr{and} - \frac{1}{2}$. As the name implies, these values describe the spin of each electron in the orbital.

Remember that there are only two electrons to every orbital, and that they should have opposite spins (think Pauli exclusion principle). This tells us that there are two electrons per orbital, or per ${m}_{l}$ value, one with an ${m}_{s}$ value of $+ \frac{1}{2}$ and one with an ${m}_{s}$ value of $- \frac{1}{2}$.

(Tl;dr) Thus, as stated above, each individual $3 d$ orbital can hold $2$ electrons. Because there are five $3 d$ orbitals in the $3 d$ subshell, the $3 d$ subshell can hold $10$ electrons total ($5 \cdot 2$).