# How many orbitals can there be in an energy level?

May 24, 2015

The easiest way to determine how many orbitals an energy level can hold is to use that respective energy level's electron capacity, i.e. the maximum number of electrons it can hold.

If you know the maximum number of electrons an energy level can hold, you can use the fact that each individual orbital can hold no more than 2 electrons to determine how many orbitals you have.

The formula for electron capacity looks like this

$\text{max number of electrons} = 2 {n}^{2}$, wehre

$n$ - the principal quantum number - used to describes the energy level.

This means that you can determine how many orbitals an energy level has by using this equation

$\text{number of orbitals} = \frac{2 {n}^{2}}{2} = {n}^{2}$

So, here's how the calculations would look like

• First energy lelve, $n = 1$

The first energy level can hold a maximum of

$2 \cdot {1}^{2} = {\text{2 e}}^{-}$

This means that the number of orbitals found in the first energy level will be equal to

$\text{number of orbitals} = {1}^{2} = 1$ $\to$ $1 s$

• Second energy level, $n = 2$

The number of orbitals found in the second energy level will be equal to

$\text{nr. of orbitals} = {n}^{2} = {2}^{2} = 4$ $\to$ $1 s$, $2 {p}_{x}$, $2 {p}_{y}$, $2 {p}_{z}$

• Third energy level, $n = 3$

The third energy level will contain

$\text{nr. of orbitals} = {3}^{2} = 9$

More specifically, the third energy level will contain the following orbitals:

$3 s , 3 {p}_{x} , 3 {p}_{y} , 3 {p}_{z} , 3 {d}_{x y} , 3 {d}_{x z} , 3 {d}_{y z} , 3 {d}_{{x}^{2} - {y}^{2}} , 3 {d}_{{z}^{2}}$