Quantum Numbers

There are chemists who have spent the last hundred years or so developing a language to describe how an atom behaves and how electrons orbit the nucleus. Neils Bohr first thought that electrons orbited the nucleus in a way similar to planets orbiting the sun, but that wasn't quite right. The theory seemed to work with Hydrogen, but after that there seemed to be something missing...What was missing was "The Numbers."

The four quantum numbers most students are taught are n, l, m_l, and m_s.

n: This is the principle quantum number. These numbers are positive whole numbers from 1 to whatever. They are related to the different energy levels, or shells of orbitals. The lowest energy level has n = 1 and the numbers increase from there. As the n's get bigger, so do the size of orbitals.

1: This is the azimuthal quantum number often referred to as a subshell or orbital. It is also a positive integer, but l's range goes from 0 to n_1. Notice that this is a range. If you're asked what the l values are for n = 3, the answer is not l = 2; it's l = 0, 1 or 2. Unless you’re given other information, i.e., the letter that goes with that number, you cannot say which it is. This quantum number is related to an orbital's shape. Unfortunately, the values are usually represented as letters instead of a number — they are in no particular order!

1=	1	2	3	4	5	6	...
	s	p	d	f	g	h	etc.

Actually, the first four letters actually stood for something once and are pretty much the only ones you ever really use later on. After the f's, it's just alphabetical.

m_l: Just like l was based on n, m_l is based on l. It's usually called the magnetic quantum number and can be any integer from -l to +l (including 0) and determines the orientation of the orbital.
If l = 3 then m_l can be -3, -2,-1, 0, 1, 2, or 3.

m_s: This is the spin quantum number and describes how the electrons in an orbital are spinning. m_s can be +1/2 or -1/2. The most electrons you can have in any orbital is two. One will spin one way and will have m_s = +1/2, and the other will spin in the opposite direction and will have m_s = -1/2. It's not directly related to the other three and is mainly used when discussing box orbital notation.

Putting them together:

For the n = 1 shell, the rules say we can only have one value for l, l = 0 which is the number for an s subshell or orbital. So we only have one orbital in the first energy level, which we call a 1s orbital. The number tells you the energy level and the letter tells you what type you have. And since there’s only one l we can only have one m_l, m_l=0.

For n = 2, l can be 0 (an s orbital) or 1 (a p orbital). There are two types of orbitals in the second shell. When you have an s orbital, m_l = 0 (just like for n = 1) and when there’s a p orbital, m_l = -1, 0, or 1. This doesn't mean there is one p orbital that can be in three different directions, but we have three different p's which are each orientated differently. So the total number of orbitals in the second shell is 4, a 2s and three 2p's.

For n = 3, l can be 0 (s orbital), 1 (p orbital), or 2 (d orbital). Just as before, there is one s orbital and 3 p's. But here, we also have a d orbital, which has m_l = -2, -1, 0, 1, or 2; five m_l's means we have five d orbitals. So we have a grand total of nine orbitals in shell three: one 3s, three 3p's and five 3d's. And it goes on from there. And if you haven't noticed, for every n shell there are a total of n² orbitals.

Some basic shapes

Let's take a minute to talk about what these orbitals actually look like. There are really, Really, REALLY, complicated math equations to find out where an electron will probably appear around a nucleus. When they do this a couple of zillion times with a couple of zillion sets of numbers, there are a couple of zillion points around the nucleus. And what they found was that for s orbitals (l=0), you get the shape of a sphere.

Now remember, m_l tells the orientation of the orbital. There is only have one m_l, and there's really only one way to orientate a sphere in 3D space, so it makes sense that this is the shape for l=0.

And when you do the same for p orbitals (l=0, m_l = -1,0, or +1), you get a dumbbell or a 3D figure eight. And since we have three possible m_l's, we have three possible orientations of this dumbbell on the x-y-z axis. One along the x axis, one along the y, and one along the z.

After the p's, it gets a little more complicated, but it's basically the same idea. And remember, no matter what the value for n is, if the above rules allow it, you will always have an s orbital, three p orbital, etc. These orbitals will all have the same shape, but they get larger as the value for n gets bigger.

Adapted from http:// www.louisville.edu/~immatt01/chemtutor.html