Orbitals and how many electrons each holds is central to the process of chemical bonding, and from a physics perspective, orbitals are closely tied to the energy levels of the electrons in the atom in question. If you’ve been asked to find orbitals for a specific energy level, understanding how these two are linked will both deepen your understanding of the topic and give you the answer you’re looking for.

#### TL;DR (Too Long; Didn't Read)

The principal quantum number, *n*, determines the energy level of the electron in an atom. There are *n*^{2} orbitals for each energy level. So for *n* = 3 there are nine orbitals, and for *n* = 4 there are 16 orbitals.

## Understanding Quantum Numbers

When discussing electron configurations, “quantum numbers” are widely used. These are numbers that define the specific state an electron is in for its “orbit” around the nucleus of the atom. The main quantum number you’ll need, to work out the number of orbitals for each energy level, is the principal quantum number, which is given the symbol *n*. This tells you the energy level of the electron, and a bigger principal quantum number means the electron is farther away from the nucleus.

The other two quantum numbers that explain orbitals and sublevels are the angular momentum quantum number (*l*) and the magnetic quantum number (*m _{l}*). Like ordinary angular momentum, the angular momentum quantum number tells you how quickly the electron is orbiting, and it determines the shape of the orbital. The magnetic quantum number specifies one orbital out of those available.

The principal quantum number *n* takes whole number (integer) values such as 1, 2, 3, 4 and so on. The angular momentum quantum number *l* takes whole number values starting from 0 and up to *n* − 1, so for *n* = 3, *l* could take values 0, 1 or 2 (if *n* = 3, then *n* – 1 = 2). Finally, the magnetic quantum number *m _{l}* takes whole number values from –

*to +*

*l**l*, so for

*l*= 2, it can be −2, −1, 0, +1 or +2.

#### Tips

In chemistry in particular, the

*l*numbers are each given a letter. So*s*is used for*l*= 0,*p*is used for*l*= 1,*d*is used for*l*=2 and*f*is used for*l*= 3. From this point onwards, the letters increase alphabetically. So an electron in the 2_p_ shell has*n*= 2 and*l*= 1. This notation is often used to specify electron configurations. For example, 2_p_^{2}would mean there were two electrons occupying this subshell.

## How Many Orbitals in Each Energy Level? The Simple Method

The easiest way to work out how many orbitals in each energy level is to use the information above and simply count the orbitals and sublevels. The energy level is determined by *n*, so you only need to consider one fixed value for *n*. Using *n* = 3 as an example, we know from the above that *l* can be any number from 0 to *n* – 1. This means *l* could be 0, 1 or 2. And for each value of *l*, *m _{l}* can be anything from –

*to +*

*l**l*. Each combination of

*l*and

*m*is a specific orbital, so you can work it out by going through the options and counting them.

_{l}For *n* = 3, you can work through the values of *l* in turn. For *l* = 0, there is only one possibility, *m _{l}* = 0. For

*l*= 1, there are three values (

*m*= −1, 0 or +1). For

_{l}*l*= 2, there are five possible values (

*m*= −2, −1, 0, +1 or +2). So adding the possibilities gives 1 + 3 + 5 = 9 orbitals in total.

_{l}For *n* = 4, you can go through the same process, but in this case *l* goes up to 3 instead of just two. So you’ll have the nine orbitals from before, and for *l* = 3, *m _{l}* = −3, −2, −1, 0, +1, +2 or +3. This gives seven extra orbitals, so for

*n*= 4 there are 9 + 7 = 16 orbitals. This is a bit of a labor-intensive way of working out the number of orbitals, but it’s reliable and simple.

## How Many Orbitals in Each Energy Level? A Quicker Method

If you’re comfortable with taking the square of a number, there is a much quicker way to find orbitals for an energy level. You may have noticed above that the examples followed the formula number of orbitals = *n*^{2}. For *n* = 3, there were nine, and for *n* = 4, there were 16. This turns out to be a general rule, so for *n* = 2, there are 2^{2} = 4 orbitals, and for *n* = 5 there are 5^{2} = 25 orbitals. You can check these answers with the simple method if you like, but it does work out in any case.

## How Many Electrons in Each Energy Level?

There is also an easy way to work out how many electrons are in each energy level. Each orbital holds two electrons, because they also have one extra quantum number: *m _{s}*, the spin quantum number. This can only take two values for electrons: −1/2 or +1/2. So for every orbital, there are a maximum of two electrons. This means that: maximum number of electrons in an energy level = 2_n_

^{2}. In this expression,

*n*is the principal quantum number. Note that not

*all*of the available spots will be full in every case, so you have to combine this with a bit more information, such as the number of electrons in the atom in question, to find orbitals that will be fully occupied by electrons.