### Electrons and Orbitals

- ayubdova
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An atom is (a little bit) like an Indian bus.

This is how they work.

You get in and find an empty seat. The seats fill up.

Then people stand in the aisle. The aisle fills up.

Then people get on the roof. The roof fills up. (Or not).

Then you’re ready to go.

The seats are the most comfortable place to ride. That’s why they fill up first. Ever been on a bus where everyone is standing in the aisle and nobody is sitting in the seats? Of course not. We’re far too lazy to allow that to happen.

Let’s say the bus stops at a village and someone leaves their seat to get off. Another person from the aisle will immediately take their place. Then there will be room for someone to go from the roof to the aisle. Then there will be room on the roof for a passenger from the village to get on. [yes, I can understand that maybe the roof could be more comfortable than the aisle in a hot crowded bus. Bear with me.

Maybe we can draw a comfort diagram here.

If you look around you’ll see lots of examples of this behavior. Queues. Seating arrangements in the living room at Super Bowl parties. Open-admission concerts. **Whenever a preferred place vacates, there is a rush to fill it.**

Electrons in an atom are just like passengers in a bus: they each need their** own** seat, and the seats fill up in a very specific order. The “one butt per seat” principle is in effect – just as you can’t have 2/3, 0.17 or another fraction of a human, you can’t have a fraction of an electron either. This is what we mean by **quantization**. The fact that they fill in a specific order is called the aufbau principle or building-up principle.

The “seats” for the electrons are areas of space we call orbitals. Orbitals are regions of space which represent 3-dimensional graphs of an electron’s likely location. Whereas seats correspond to the shape of the human form, the shapes of orbitals are described by an equation developed in the 1920’s by Erwin Schrödinger that we call the Schrödinger wave equation. (Schrödinger figured out that the energies of electrons can be calculated quite effectively by treating them as waves).

If you take upper level chemistry courses, you learn how to play around with the Schrödinger wave equation. If not, here’s the boiled down version.

Just as people seek the most comfortable seat on a bus, electrons seek the orbital that is closest to the nucleus – where the force of attraction between the positively charged nucleus and the negatively charged electron is greatest.

- Electrons behave as both particles and waves. Treating electrons as waves, orbitals are shapes that describe areas of space which allow for constructive wave overlap. [note – orbitals are the areas where there’s a 95% probability of finding the electron]
- Every electron on an atom has a unique “identification code”, or quantum number. [Triva: this property that the quantum numbers not be the same is a general property of a class of particles called fermions, to which electrons belong; these are distinguished from bosons, which can occupy the same quantum states (photons are a prominent example, but you may have heard of the Higgs boson).] The four quantum numbers are:

- the principal quantum number n. This defines the energy level of the electron which is ultimately a reflection of how close it is to the nucleus. This also defines the number of “nodes” (areas of zero electron density) in an orbital. If we go back to our bus example, a rough analogy would be n=1 (seats), n=2 (aisle), n=3 (roof). As we increase the number we are decreasing the overall comfort level.
- The azimuthal quantum number
*l*. This defines the shape of the orbital, and it ranges from 0 to (n-1). - The magnetic quantum number
*m*. This defines the number of different orientations of a given shape, and is defined by (-l, 0, +l). - The spin quantum number
*s*, which can be either +1/2 or –1/2.

These are the only values that provide solutions for the Schrodinger wave equation.

So for n=1, we get the quantum numbers, n=1, l=0, m = 0, and spin = +1/2 or –1/2 or more succinctly, [1 0 0 +1/2] and [1 0 0 –1/2]

That’s it. Two electrons. If we want to add more electrons we have to increase n. This is like filling up the seats on our bus and then adding the next group of passengers to the aisle.

The next set of electron “passengers” have to go into the aisle, where n=2.

For n = 2, *l*=0, we get 2 0 0 +1/2 and 2 0 0 –1/2 for a total of two electrons.

When n=2 we can also let *l* = 1. Watch this:

So for n=2 we have a total of 8 electrons: 2 for *l*=0, and 6 for *l*=1. If we want to add more electrons, we have to increase n again (to 3) which is like filling the aisle of our bus and sending the next group of passengers up to the roof.

Let’s stop here for a second. the different numbers *l*=0 and *l*=1 correspond to different shapes of orbitals. In practice we have names for them. They are the S and the P orbitals.

s orbitals look like spheres. How do we know this? From the wave equation. The fact that m=0 is a clue – it means there is only one orientation possible across all the axes, which can only be possible for a sphere.

p orbitals look like teardrops. Again, we know this from the wave equation. What’s interesting here is that we can have 3 different values for m. In practice what this means is that there are 3 different ways to orient the orbitals – along the three different axes x, y, and z.

**In Org 1 and Org 2 we will deal exclusively with s and p orbitals. The shapes have tremendous consequences for chemical bonding, as we shall soon see.**

What about the higher levels? There are also different shapes of orbitals for *l*=2 and *l*=3. For *l*=2, these are called the d orbitals, found in the transition metal series, and there are 5 different orientations of these orbitals (m=2,1,0,-1,-2). For *l*=3, there are the f orbitals, which are found in the lanthanides and actinides, and there are 7 different orientations of these orbitals.

If you take upper year courses you’ll learn more about the d orbital series, since transition metals have many important applications in organic chemsitry. Chemistry of the d orbitals is pretty fascinating. In contrast, the chemistry of atoms involving the f-block tends to be pretty similar, since at this point the electrons are really far apart from the nucleus and thus are far less differentiated. That’s why the lanthanides are often found as mixtures – often the trouble is separating their minerals from each other.

The higher orbitals than that (g, h, etc.) dont really appear in chemistry because the very heavy nuclei you’d need (>120) are extremely unstable. Someone has come up with an extended periodic table which incorporates g-block elements. Beautiful! – but more of theoretical rather than practical interest. Compare it to our Indian bus – you could keep adding platforms on top of the bus to fit successive levels of passengers, but there comes a point where the whole thing is just going to collapse under its own weight.