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

##### 1 Answer

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#2l + 1# number of actual orbitals. #l_max = n - 1# .

Because of that, at each energy level

...etc.

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

#bb(n_"orbs") = sum_(l = 0)^(l_max) (2l + 1) = bb(sum_(l = 0)^(n - 1) (2l + 1))#

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

#=> (2*0 + 1) + (2*1 + 1) + (2*2 + 1) + . . . + (2l_max + 1)#

#= (2*0) + (2*1) + (2*2) + . . . + (2l_max) + n#

#= 2(0 + 1 + 2 + 3 + . . . + l_max) + n#

Now if we realize that the sum of the natural numbers is the last number (

#=> cancel(2)([l_max*(l_max + 1)]/cancel(2)) + n#

#= l_max*(l_max + 1) + n#

Now substitute

#= (n - 1)*((n - 1) + 1) + n#

#= n^2 - n + n#

#=> color(blue)(n_"orbs" = n^2)#

Therefore, the total number of orbitals in **one** quantum level is

Since the maximum number of electrons in each orbital is **maximum number of electrons in an entire quantum level** is