# Question #3ccc1

##### 1 Answer

Because there are

#### Explanation:

For starters, you should keep in mind that the maximum number of electrons that can be added on the **fourth energy level** is equal to *an atom* can hold up and including the fourth energy level.

Now, the number of **subshells** present on the fourth energy level, which is described by *angular momentum quantum number* can take.

You know that

#l = {0, 1, ..., n-1}#

In your case, the angular momentum quantum number can take four possible values

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

Now, the number of **orbitals** present in each subshell is given by the *magnetic quantum number*,

#m_l = {-l, -(l-1), ..., -1, 0, 1, ..., (l-1), l}#

This means that you have

#l = 0 implies m_l = {0}# #l = 1 implies m_l = {-1, 0, 1}# #l = 2 implies m_l = {-2, -1, 0, 1, 2}# #l = 3 implies m_l = {-3, -2, -1, 0, 1, 2, 3}#

The orbitals look like this--the single orbital present in the **subshell** is at the top and the **orbitals** present in the **subshell** are at the bottom.

If you add up the number of orbitals present in each subshell, you will end up with

#overbrace("1 orbital")^(color(blue)("in the s subshell")) + overbrace("3 orbitals")^(color(blue)("in the p subshell")) + overbrace("5 orbitals")^(color(blue)("in the d subshell")) + overbrace("7 orbitals")^(color(blue)("in the f subshell")) = "16 orbitals"#

Now, you should that **each orbital** can hold a maximum of **electrons** of opposite spins, i.e. one having spin-up and the other having spin-down **Pauli's Exclusion Principle** here.

This means that the maximum number of electrons that the fourth energy level can hold is equal to

#16 color(red)(cancel(color(black)("orbitals"))) * "2 e"^(-)/(1color(red)(cancel(color(black)("orbital")))) = "32 e"^(-)#

The equation that gives you the maximum number of electrons that can be added on a given energy level

#color(darkgreen)(ul(color(black)("max no. of electrons" = 2 * n^2)))#

In your case,

#"max no. of electrons" = 2 * 4^2 = "32 e"^(-)#

Notice that **number of orbitals** that are present on a given energy level