# Question #28c50

Oct 22, 2017

To designate a specific orbital within a subshell, you need $n , l ,$ and ${m}_{l}$.

#### Explanation:

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

• For example, the $3 d$ subshell is in the $n = 3$ shell, the $2 s$ subshell is in the $n = 2$ shell, etc.

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.

• For example, the $n = 3$ shell has subshells of $l = 0 , 1 , 2$, which means the $n = 3$ shell contains $s$, $p$, and $d$ subshells (each containing their respective orbitals). 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 $2 p$ orbital with $n = 2$ and $l = 1$, we can have ${m}_{1} = - 1 , 0 , 1$. This tells us that the $p$ orbital has $3$ possible orientations in space.

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

Therefore, to describe any specific orbital within a subshell, where we care about the specific orientation of the orbital, we would need three quantum numbers, as described above.

If we had all four quantum numbers, we could then begin to describe the electrons "within" the orbitals. The fourth quantum number is the electron spin quantum number, ${m}_{s}$, and 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 (this is is because electrons are a type of fermion, which have half integer spins).

• Remember that there are only two electrons to every orbital, and that they should have opposite spins (again, this is because electrons are fermions $\to$ 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 $- \frac{1}{2}$.

In summary, $n$ describes the shell, both $n$ and $l$ describe a subshell, $n$, $l$, and ${m}_{l}$ describe an orbital, and all four quantum numbers ($n$, $l$, ${m}_{l}$, ${m}_{s}$) describe an electron.