Principle Quantum Number, Energy Levels and Nodes

The principal quantum number determines the number of nodes in the electron standing wave.

Let’s pull that apart a little bit.

We know that Schrödinger’s Theory supposes that electrons are standing waves centered on the nucleus. We also know that as the energy of a wave increases, so does the number of nodes. Lastly, we know that the lowest energy version of a standing wave is one without any node.

That means that a level 1 electron has no node, that a level 2 electron has 1 node and so on. In other words, we can say that

(n-1)=number of nodes

The Angular Momentum Quantum Number determines the number of angular nodes that exist in an orbital (details are here), but there is another possible type of node - a spherical node (also called a radial node). 

A spherical node is a node defined only by the distance from the nucleus - a given radius or "r" value. Remembering that at node is a place where the wave has no measurable value, a spherical node is an infinitely thin shell around the nucleus at a distance r where the electron has no measurable presence. An s orbital with a single spherical shell looks like this:

Energy Level and Orbital Size

It must be noted that energy level affects the size of the orbital. This is a complicated matter, since orbitals don't actually have edges. A good way to think about this is to go back to our analogy of sound waves. If you haven't read this analogy yet, you need to in order to make sense of what follows. A note played softly (with low energy) will have a fairly small sphere of sound, while a note played with more energy will make a larger sphere of sound. The same is true of orbitals. A 2s orbital is larger than a 1s orbital and a 3s is larger still.

Some complications

As stated above, (n-1) is the number of nodes in an orbital and as stated elsewhere ℓ is the number of angular nodes. Putting those two ideas together, we can deduce that

(n-1)-ℓ = the number of spherical nodes in an orbital

To understand this idea, we will look at two different applications

First, let's look at the difference between a 1s orbital, a 2s orbital and a 3s orbital. All three have ℓ = 0, and therefore have no angular nodes (that's what makes them s orbitals. So, the difference between them is only the number of spherical nodes.

This means that all three have the same spherical shape of an s orbital, but they differ in size and spherical nodes.

Now let's look at the s, p and d orbitals on level 3.

This leads to the following pictures: