# Question #65437

Jul 23, 2015

QUANTUM NUMBERS

The three main quantum numbers describe the shape, energy level, and orientation of the atomic orbitals.

There is a fourth (${m}_{s}$) which describes the spin of an electron in an orbital, and it is purely based on the properties of the electron.

• $n$ is the principal quantum number which describes the energy level.

$n = 1 , 2 , 3 , . . .$ and is always a positive integer.

• $l$ is the orbital angular momentum quantum number which describes the shape of the orbital. $l = 0 , 1 , 2 , 3 , . . . , n - 1$.

$l = 0 , 1 , 2 , 3 , \ldots$ for $s , p , d , f , \ldots$ orbitals, respectively.

• $m$ (more specifically, ${m}_{l}$) is the magnetic quantum number, which corresponds to each unique orbital orientation. ${m}_{l}$ takes on the set of integers from $- l$ to $l$, i.e.

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

• ${m}_{s}$ (see why we need to say ${m}_{l}$?) is the electron spin quantum number, which describes the spin of the electron. It is only $\pm \frac{1}{2}$, independent of the other quantum numbers.

2P ATOMIC ORBITAL

For example, a $2 p$ orbital is said to have:

• $n = 2$ since it's $\textcolor{h i g h l i g h t}{2} p$
• $l = 1$ since it's $2 \textcolor{h i g h l i g h t}{p}$
• ${m}_{l} = \left\{0 , \pm 1 , . . . , \pm l\right\} = \left\{- 1 , 0 , + 1\right\}$
(if $l = 2$, then ${m}_{l} = \left\{- 2 , - 1 , 0 , + 1 , + 2\right\}$)

If the orbital has an electron, the electron's ${m}_{s}$ is either $+ \text{1/2}$ or $- \text{1/2}$ for spin up and spin down, respectively. Technically, we just say $\pm \text{1/2}$ because it could take either configuration (neither is preferential over the other).

We cannot say which electron is which, because they are indistinguishable.

The $2 {p}_{z}$ orbital looks like this:

• While ${m}_{l} = 0$, that corresponds to the $2 {p}_{z}$ orbital.
• While ${m}_{l} = - 1$, that corresponds to the $2 {p}_{x}$ or $2 {p}_{y}$ orbital.
• While ${m}_{l} = + 1$, that corresponds to the $2 {p}_{y}$ or $2 {p}_{x}$ orbital. (The uniqueness that ${m}_{l} = \pm 1$ matters, but the choice of assigning either to ${p}_{x}$ or ${p}_{y}$ is arbitrary.)

Lastly, the observations you learn from General Chemistry describe the idea that electrons fill the orbitals:

• with the lowest energy first, in part due to $n$ (Aufbau Principle)
• one at a time
• in such a way that ${m}_{s}$ is not the same for any two electrons both in a doubly-occupied orbital (Pauli Exclusion Principle).