4 - The Orbital Primer pt. 1

This comic gave me a good laugh, despite not being directly related to quantum mechanical orbitals…     Image by Randall Munroe via      xkcd

This comic gave me a good laugh, despite not being directly related to quantum mechanical orbitals…

Image by Randall Munroe via xkcd

I know how much you were looking forward (or perhaps dreading) this part. Well, if not looking forward to, perhaps still trying to reason how all of this is going on within every element in your body and facing a slight existential crisis. All I have to say, in that case, is to try not to worry about it, and, of course, prepare for another dive into the boundary of quantum mechanics and chemistry.

I, again, recommend that you check out the previous parts of the Elemental Flow series, especially Part 2. Familiarizing yourself with the quantum mechanics of electrons before this lesson will make this a smooth experience.

Quantum Numbers

What do the letters “s”, “p”, “d”, “f”, Aufbau, spin, and shapes have in common?

You might be scratching your head and wondering, “Why is he asking me this when I know he’s going to tell me.” And you’re right.  Let’s break out our handy dandy…periodic table. And notebook, if you have one (I got you, 90s kids).

Image via      Science Notes

Image via Science Notes

Orbitals are organized according to their energy levels. Naturally, every element on the periodic table must exist at a separate energy state in nature than each other (otherwise, what you learned in Part 1 and Part 2 would be meaningless – electrons have no need to configure themselves in different ways if every atom exists at the same energy level). Therefore, there must be a different orbital for each element. And, indeed, there is. The periodic table shows that, at the bottom of each element, there is a different electron configuration for each element. This exclusivity in configuration is called the Pauli Exclusion Principle.

Attentive readers might have already noticed the letters at the bottom of the elements on the periodic table – “s”, “p”, “d”, and “f”.

These labels are only descriptors of one of four quantum numbers.

If you are lamenting the word “quantum” after Part 2 of the Elemental Flow series, don’t fret. A quantum number is a quantity within a quantum mechanical system. Since electrons are quantum mechanical systems, they hold specific quantities. That’s all.

Electrons have four such quantities. We call them the principal quantum number, or n, the orbital quantum number, or l, the magnetic quantum number, or m, and the spin quantum number, or s.

The Quantum Dive – Breaking the Shells

1) Principal Quantum Number (n)

The first, n, is the number that labels the energy level of an electron. Since we know, from Part 1, that the electrons with the lowest energy are the ones closest to the nucleus, we can tell that the lower the value of n, the lower the energy level. In chemistry, the energy level of an electron is described as its electron shell.

These shells are labeled from K to O starting from their innermost shell to their outermost shell, according to their principal quantum number. To be clear, an n of 1 is referred to as the K shell, n = 2 is L, n = 3 is M, n = 4 is N and n = 5 is O. While they do have this alphabetical label, when describing shells, more people recognize the numeric value of n. However, they can help you organize when we talk about the subsequent quantum numbers.

2) Orbital Quantum Number (l)

Next up, l. This quantum number tells about the orbital’s angular momentum, or the momentum brought about by rotation of electrons around some center – the nucleus.

Before I move on from here, I did want to explain this number more carefully.

For those familiar with astronomy, this quantum number is also called the azimuthal quantum number, as an azimuth is a certain angular measurement within a sphere.

Image via      UOregon

Image via UOregon

Within our three-dimensional world, we can describe the position of anything in relation to anything else. As you can see, the azimuth and altitude give the direction of a star in this spherical system. North is usually used as the starting point for an angular calculation when comparing the horizontal position of a star and the altitude is given as angle from the observer in the center to the position on the sphere that the star is located.

But how does this relate to an electron? Take a look at this:

Image by tonfilm via      vvvv

Image by tonfilm via vvvv

The observer is the nucleus, or, in this image, where the x, y, and z axes (representing each of our three dimensions) connect. The star is the point P, or the electron, with some x, y, and z coordinate. The angle between north, or the x-axis, and the point is defined as the azimuth. Lastly, the altitude angle would be the angle between the z-axis and the position of the point.

If the electron was just a point, or particle, then this would be a simple case of calculating the ordinary angular momentum.

But it isn't just a particle, is it?

Image by Yuta Aoki via      Wikimedia

Image by Yuta Aoki via Wikimedia

The angular momentum is harder to calculate in this system than it is in ordinary physics, due to the Wave-Particle Duality of the electron (discussed in Part 2). The electron is traveling quickly on a wave-like pattern. At any given time, we are unsure of where the electron may be. Instead of treating the electron like a particle, it is treated like a standing wave. More simply put, its wave functionality, constant and uninterrupted as it is, is used to determine its angular momentum. Therefore, the orbital angular momentum, is calculated completely differently than the classical angular momentum, which will be discussed when we reach the spin quantum number.

Perhaps the actual mathematics can be shown in a separate lesson if the interest is great enough? For now, recognize that, by solving this equation for the angular momentum, you receive non-negative integer values, or 0, 1, 2, 3, etc.

Aha! A new possibility reveals itself. If you were to believe that electrons were like their classical systems, you would probably say that it’s impossible for an object that’s moving to have 0 momentum. After all, if you’re walking down the street, you have momentum; how could an electron which is constantly revolving around the nucleus have 0 momentum? But quantum mechanically, a system with 0 angular momentum or, in other words, l = 0, simply orbits the nucleus in its wave-like pattern along the xy plane without any z altitude. The electron only has an azimuth, no altitude.

When I said in the last part that you had already learned about orbitals, this is what I meant. Because an electron orbiting the nucleus on an xy plane looks oddly familiar…

The Bohr Model of Hydrogen     Image via      Wikispaces

The Bohr Model of Hydrogen

Image via Wikispaces

Is that…? Why, yes it is! The Bohr Model! He did come up with the initial idea of orbitals. was that dastardly scientist was right!?

Not so fast.

First, let’s clean up this image. Since we’re not concerned with the electron’s decrease in energy state (like we were in part 2 when this was being explained), let’s look at the hydrogen atom at its lowest energy state.

The electron is still on the n = 1 orbital and orbiting the nucleus like above. Only the n = 2 and n = 3 orbitals have been removed.     Image via      Chemistry LibreTexts

The electron is still on the n = 1 orbital and orbiting the nucleus like above. Only the n = 2 and n = 3 orbitals have been removed.

Image via Chemistry LibreTexts

Bohr was only correct when it comes to things with one electron and an angular momentum of 0. This is because Bohr’s elementary model only observes the energy between the nucleus and the electrons only, not the energy created between the repulsion of electrons. The image above shows the only case in which the Bohr model is correct - with the hydrogen atom. Hydrogen only has one electron and its nucleus, meaning this fits Bohr’s model. Furthermore, as per the aforementioned Pauli Exclusion Principle, there are no other elements like hydrogen in nature.

To wrap up the second and most complicated quantum number, we make note of the fact that we can’t really know what the xy-axis is as it electrons see it.

To conceptualize what I’m saying, think of the image above as the wheels on a gyroscope.

Of course, in a quantum mechanical system, there is more discrete and fast motion, but this is a fantastic example. Now, if the revolving electron in the Bohr model was taking any of the wheels in the above gyroscope, you can understand why it would be hard to tell exactly where the electron would be, let alone where its x and y axes are.

But what you do know is that the electron is somewhere within the sphere made by the rotating gyroscope wheels.

This might sound familiar to you who remember or re-read Part 2 because this is the exact conclusion you derive from the Electron Cloud model that  Schrodinger developed!

I love it when science comes together…You gotta admit, that’s pretty cool.     Image via      Ask A Mathemtician/Physicist

I love it when science comes together…You gotta admit, that’s pretty cool.

Image via Ask A Mathemtician/Physicist

With all of that said, the purpose of the orbital quantum number is to tell you the shape of the electron subshell. Like the principal quantum number, these have an alphabetical naming system too…and it’s one you’ve already seen! That is: s, p, d and f. l = 0 to l = 3 represents s to f. Unlike the principal quantum number, however, it is conventional to use s, p, d and f rather than the numeric orbital quantum number value.

With that said, if the principal quantum number, n, is equal to 1 and the orbital quantum number, l, is equal to 0, you have a 1s system, which is the electron configuration of the hydrogen atom (check the bottom of the hydrogen element on your periodic table).

Now, I think after all of that, it’s good to keep ourselves organized.

To recap, the principal atomic number tells us the electron shell or energy level of the element, and the orbital quantum number tells us the electron subshell. There can be multiple subshells in one shell. And each individual electron configuration is called an orbital.

Image by thomji via      Chemistry Stack Exchange

Image by thomji via Chemistry Stack Exchange

3) Magnetic Quantum Number (m)

Before we wrap up with this image, there’s something important in it that will help you understand the magnetic quantum number.

You might have noticed above that there are three different p orbitals in the above image. That is due to the different ways an orbital can orient itself within space as the amount of electrons in the system increases. This is what Bohr did not account for in his model; when more electrons enter the system, in order to keep energy as low as possible, the system must orient itself differently within the three dimensions.

The magnetic quantum number is a handy way of telling us how many ways a subshell can orient itself. I say it’s handy because there’s a simple equation to tell you how.

For every value of l, there is a range of values of m from -l to l, including zero.

So, if our system is a d orbital, which is l = 2, there would be 5 ways the subshells could orient themselves, given by the m values -2, -1, 0, 1 and 2.

That’s all there is to that. See? Not so bad.

Intermission

Okay, okay. Break time. We don’t want to overcook your brains with chemistry and quantum theory. Fortunately, what we are going over is, theoretically, all there is to orbitals (without the droll mathematics, of course). And, as you already know, this leads to some amazing science. After all, after learning about electron shells, the next step can only be how elements combine to reduce energy even further!

Are you enjoying learning about how these microscopic particles work and seeing how something so complex can create something as simple as a drop of water and/or something as ubiquitous as air?

Let me know. I would love to talk science with you guys.

As always, thank you for joining me, and I will see you again very soon.