Quantum Numbers, Part 2: No More Lies

Hello, everyone! I’m back! Are you surprised? I am…

Anyway, I just wrote a blog post on neutral lipids (and I mean just wrote), but I need to get some more studying done, so now I’m going to teach you some inorganic! /throws glitter/

If you read my earlier posts that I wrote while in Gen Chem, you’ll notice that, at various points, I stated that I knew I was being lied to. Now that I’m in Inorganic I, which is basically Gen Chem III, because it’s taught by the same professor, I’m learning the theory behind some of the lies we were told in freshmen year.

Although I don’t think I wrote a blog post on quantum numbers, they were most definitely a central component of my Gen Chem I coursework. Now that I’m back for Gen Chem III, I figure I should actually bother to put in a word or so about them. Since. You know. They’re central to pretty much all of chemistry. All of it.

All right! Quantum mechanics, let’s goooo~!


Electrons can be described by a set of four numbers known as quantum numbers. Each of these numbers tells you something different about the electron, which is super handy dandy. The four quantum numbers are as follows:

The principle quantum number, n, describes the energy level of an electron, and can take on any positive integer value.

The angular quantum number, l, describes the type of orbital in which an electron resides, and can take on any positive integer value n-1.

The magnetic quantum number, mldescribes the specific orbital in which an electron resides, and can take on any integer value between -l and l.

The spin quantum number, ms, describes the spin of an electron, and can take on the value of ±½.

In Gen Chem I, this was essentially left as granted. Although our professor tried to explain the theory behind it using the Schrödinger equation (believe me, he did), we didn’t get into the detail that we would have really needed to understand the dependence of different electronic characteristics on the different quantum numbers.

That’s where inorganic came in.

The first thing we did this semester was revisit the dreaded beast that was the Schröndinger equation. This perfectly innocuous little equation is shown below:

EΨ = ΨĤ

Where Ĥ is the Hamiltonian operator, beautiful little projection operator that it is, and Ψ is the wave function of the quantum system.

Yeah, those are words.

As it turns out, what you really need to know is that Ψ, the wave function of the system, is defined in polar coordinates, and can be broken down into two separate equations: the radial equation, and the angular momentum equation.

Firstly, let’s look a bit at the radial portion. In case you hadn’t deduced already, the radial portion of the wave function of an electron tells you how far it is from the nucleus.

Sounding familiar? Remind you of a quantum number or so?

The radial portion of Ψ is defined by a long, complicated equation that I don’t feel like formatting at the moment, but it’s dependent on two different quantum values: one we designate as n, and one we designate as l. This tells you a very important piece of information: the distance that an electron is from the nucleus is dependent on its energy level (n) and the subshell that it resides in (l).

Next, let’s look at the angular portion. This equation tells us the direction of the vector of the wave function. This is another complicated equation that would hurt to format, but it depends on two quantum values: l and ml.This tells you that the angular momentum of an electron is dependent upon its subshell and the specific orbital in which it resides.

Okay, okay. So what does that mean, really?

It means you can determine a lot of cool stuff from your quantum numbers!

(Well, if you’re dealing with one-electron species, that is…)

Energy

For a single-electron species (think hydrogen), the energy of an electron increases with increasing n. For example, let’s take the following set of quantum numbers:

n = 2, l = 1, ml = -1

n = 3, l = 0, ml = 0

Which describes a function of high energy? The n = 3, of course! That’s all there is to it!

Size

For any sets of quantum numbers with the same l quantum number, size of the orbital increases in radius with increasing n. For example:

n = 2, l = 0, ml = 0

n = 3, l = 0, ml = 0

For the following set of quantum numbers, the second one describes an electron in an orbital with a larger radius than the first. Note, again, that it is important that the l numbers are equivalent.

Nodes

The number of nodes (areas of zero electron density) for a given wave function is given by n – l – 1. For example:

n = 4, l = 2, ml = -2

The number of nodes present in this function is equal to n – l – 1 = 4 – 2 – 1 = 1.

Lobes

The number of lobes of electron density for any given set of quantum numbers is equal to 2l. For example, for the set of quantum numbers used above, the number of lobes equals 2l =22 = 4. This makes sense, since we know that d orbitals (l = 2) have four lobes.

Number of Orbitals With Same n

Finally, the quantum numbers of a wave function can tell you how many possible orbitals exist with the same n value. This boils down to being the number of values that ml can take on at any given n and l. Since ml can take on any value between -l and +l, this can be reduced into the general equation # = 2l + 1. For example, for the set of quantum numbers given in the “nodes” section, the number of orbitals = 2l +1 = 4 + 1 = 5.

That about wraps it up for our discussion of quantum numbers. Now, you may be wondering, “What about things that aren’t hydrogen?” Well, that’s a little bit more complicated, and it involves determining nuclear shielding using a set of rules known as Slater’s rules. First, though, we need to go back and revisit our polar lipids, because they are literally impossible to live without.


Questions? Comments? You know what to do. 😉

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