# Why is the maximum number of (electrons, orbitals) related to each principal energy level equals 2n^2?

Nov 14, 2016

Because of several noticed patterns:

• There are $n$ orbitals in each electron "shell". For example, the $n = 2$ shell has two types of orbitals, $s$ and $p$.
• Of the $n$ orbital types (subshells), each type (which corresponds to each $l$) has $2 l + 1$ number of actual orbitals.
• ${l}_{\max} = n - 1$.

Because of that, at each energy level $n$, we have the following:

$n = 1$: One $1 s$ orbital, because $2 \left(0\right) + 1 = \boldsymbol{1}$

$n = 2$: One $2 s$, three $2 p$ orbitals, because $\left[2 \left(0\right) + 1\right] + \left[2 \left(1\right) + 1\right] = \boldsymbol{4}$.

$n = 3$: One $3 s$, three $3 p$, five $3 d$ orbitals, because $\left[2 \left(0\right) + 1\right] + \left[2 \left(1\right) + 1\right] + \left[2 \left(2\right) + 1\right] = \boldsymbol{9}$.

...etc.

As a general rule then, the total number of orbitals in an electron "shell" is:

$\boldsymbol{{n}_{\text{orbs}}} = {\sum}_{l = 0}^{{l}_{\max}} \left(2 l + 1\right) = \boldsymbol{{\sum}_{l = 0}^{n - 1} \left(2 l + 1\right)}$

If we work this out and turn it into a simpler form:

$\implies \left(2 \cdot 0 + 1\right) + \left(2 \cdot 1 + 1\right) + \left(2 \cdot 2 + 1\right) + . . . + \left(2 {l}_{\max} + 1\right)$

$= \left(2 \cdot 0\right) + \left(2 \cdot 1\right) + \left(2 \cdot 2\right) + . . . + \left(2 {l}_{\max}\right) + n$

$= 2 \left(0 + 1 + 2 + 3 + . . . + {l}_{\max}\right) + n$

Now if we realize that the sum of the natural numbers is the last number (${l}_{\max}$) plus the next number (${l}_{\max} + 1$), then the quantity divided by $2$, we have:

$\implies \cancel{2} \left(\frac{{l}_{\max} \cdot \left({l}_{\max} + 1\right)}{\cancel{2}}\right) + n$

$= {l}_{\max} \cdot \left({l}_{\max} + 1\right) + n$

Now substitute ${l}_{\max} = n - 1$ to get:

$= \left(n - 1\right) \cdot \left(\left(n - 1\right) + 1\right) + n$

$= {n}^{2} - n + n$

$\implies \textcolor{b l u e}{{n}_{\text{orbs}} = {n}^{2}}$

Therefore, the total number of orbitals in one quantum level is ${n}^{2}$.

Since the maximum number of electrons in each orbital is $2$, the maximum number of electrons in an entire quantum level is $\textcolor{b l u e}{2 {n}^{2}}$.