How many electrons can go into the third quantum level?

A quick way to tackle this question is by using the equation

#color(blue)(ul(color(black)("no. of e"^(-) = 2n^2)))#

Here

• #n# is the principal quantum number

In your case, you will have

#color(darkgreen)(ul(color(black)("no. of e"^(-) = 2 * 3^2 = "18 e"^(-))))#

Now, let's double-check this result by counting the orbitals that are present in the third energy shell.

You know that we can use a total of four quantum numbers to describe the location and spin of an electron in an atom.

![figures.boundless.com]()

To find the number of electrons that can be located in the third energy shell, which is the energy level that corresponds to #n=3#, we must find the number of orbitals that can exist in this particular energy shell.

For starters, we know that

#n = 3 implies l = {0, 1, 2}#

This means that the third energy shell holds #3# energy subshells, each corresponding to one value of the angular momentum quantum number, #l#.

Now look at the values that correspond to the magnetic quantum number, #m_l#, because these values give you the actual orbitals present in each subshell

• #ul(l = 0 implies m_l = 0) -># the #l=0# subshell can only hold #1# orbital

• #ul(l=1 implies m_l = {-1, 0 ,1}) -># the #l=1# subshell can hold #3# orbitals

• #ul(l=2 implies m_l = {-2, -1, 0 ,1, 2}) -># the #l=2# subshell can hold #5# orbitals

Therefore, you can say that the third energy shell holds a total of

#1 + 3 + 5 = "9 orbitals"#

According to the Pauli Exclusion Principle, each orbital can hold a maximum of #2# electrons*, which means that the maximum number of electrons that can be placed in the third energy shell is

#color(darkgreen)(ul(color(black)("no. of e"^(-) = 2 * 9 = "18 e"^(-))))#