# What is the maximum number of electrons allowed in n = 2?

Mar 11, 2016

THE MAIN QUANTUM NUMBERS OVERALL DEFINE AVAILABLE ORBITAL SUBSHELLS

$n$ is the principal quantum number, indicating the quantized energy level, and corresponds to the period/row on the periodic table. $n = 1 , 2 , 3 , . . . , N$ where $N$ is some large arbitrary integer in the real numbers.

$l$ is the angular momentum quantum number, which corresponds to the shape of the orbital ($s$, $p$, $d$, $f$, $g$, $h$, etc). It is equal to $0 , 1 , . . . , n - 1$.

It defines a unique subshell.

With $n = 2$, we have access to the $2 s$ and $2 p$ orbitals, since we assign $l = 0$ for the $s$ orbitals and $l = 1$ for the $p$ orbitals. (Similarly, we assign $l = 2$ for the $d$ orbitals, $l = 3$ for the $f$ orbitals, and so on alphabetically: $g$, $h$, etc.)

Note that with $n = 2$ we do not get $l = 2$ because $n - 1 = 2 - 1 = 1$ for a max $l$ of $1$.

THE MAGNETIC QUANTUM NUMBERS FURTHER DEFINE A SET OF (ORTHOGONAL) ORBITALS

With $l = 0 , 1$, we introduce the magnetic quantum number ${m}_{l}$, which takes on the values $0 , \pm 1 , . . . , \pm l$.

Therefore:

• $\setminus m a t h b f \left({m}_{l} = 0\right)$ for $l = 0$. For each ${m}_{l}$ value, we have one orbital that corresponds.
• $\setminus m a t h b f \left({m}_{l} = - 1 , 0 , + 1\right)$ for $l = 1$. For each ${m}_{l}$ value, we have one orbital that corresponds.

Furthermore, we have the spin quantum number ${m}_{s}$ for each unique electron, which can be $\pm \text{1/2}$ and cannot be the same for two electrons with the same ${m}_{l}$.

That is, since all electrons in the same orbital have the same $n$, $l$, and ${m}_{l}$, they can only have ${m}_{s} = \text{+1/2}$ show up once and ${m}_{s} = \text{-1/2}$ show up once.

That means two electrons per orbital only.

TOTAL NUMBER OF ELECTRONS POSSIBLE

Finally, with the $s$ and $p$ subshells combined, we have:

$\left(1 + 3\right) \cancel{\text{orbitals") xx "2 electrons"/cancel("orbital}}$

$= \textcolor{b l u e}{8}$ $\textcolor{b l u e}{\text{electrons allowed}}$

which is what we expect for the octet rule for $n = 2$.