Atomic Orbitals — Part 1
Introduction
It’s been almost six months since my last article. I’ve been very busy and so am sorry for my unofficial hiatus
Anyway with that being said, I wanted to talk about something that fascinates me quite a lot — atomic orbitals. Atomic orbitals are more of a chemistry thing rather than physics, but in fact they’re just a manifestation of quantum wavefunctions (who knows maybe I’ll do an article on that soon).
“I’m confused.”
~ Literally every single chem student studying orbitals
Bohr’s model of the hydrogen atom— a prelude
There have been countless models for the atom from Thompson’s plum pudding model (electrons are essentially embedded as tiny plums in a large positively charged sponge pudding) to Rutherford’s model (electrons orbit the nucleus kind of like how planets orbit the solar system)
In fact, if you ask someone to draw an atom, they’ll likely draw Rutherford’s model. However, there are flaws with each model of the atom. For example, Thompson’s model forgot to account for the fact that there would be a repulsion between the electrons (they are all negatively charged) and Rutherford’s model’s main flaw was that if electrons orbited the nucleus, they would start to radiate energy and fall into the nucleus. However, we know that this does not take place as atoms are stable (if not we wouldn’t have our world working).
One man proposed an even better model, in fact it was a genius scientific contribution. This man was Niels Bohr, and his significance in science cannot be understated. In fact, his model for an atom is still relevant to this day.
Bohr proposed that the electrons actually occupy energy levels around an atom.
This looks more like a solar system doesn’t it? The rings are referred to as n and start from n=1 pretty much until infinity. The ring closest to the nucleus has the least energy (we’ll get into why in a bit).
To go to n=2, energy has to be supplied to the electron. This is because the electron in n=1 has an energy of -13.6 eV (eV stands for electron volts, it is used to measure small amounts of energy, 1 eV is approximately 1.6 x 10^-19 J). If the electron was at n=2, it would have -3.40 eV. Therefore, energy has to be added to the electron to increase its energy from -13.6 eV to -3.40 eV.
When the electron is at n=1, we say that it is in the ground state. When an electron transitions to another energy level, we call it an excited state. When an electron transitions to an excited state, the electron is kind of unstable because it has all this additional energy, so it immediately releases this energy as a photon and transitions back to the ground state.
Also, electrons can only occupy the energy levels, and cannot exist in between them. Thus, an electron cannot be anywhere in the gaps. Where’s the experimental proof for this? The answers lie within absorption and emission spectras.
Absorption and emission spectra
We saw that in order to excite an electron, we must add energy. One way we can do this is by shining light (photons also have energy!)
The electron absorbs this photon and transitions to a higher energy level. But, how does this prove that electrons exist only at the energy levels?
Well, if we were to analyse just plain white light and split it up into the colours we can see, we’d get something like this:
If we were to merge all these colours together, we’d just get white. Now, let’s suppose I shine the light on an atom and then analyse the light after I shine it on the atom and split it up into its various colours. I’d end up getting something like this:
These black bands represent the colour of light that was absorbed by the atom in order to excite the electron. Now, we also saw that an electron releases the energy it absorbs as a photon when it transitions back to the ground state, analysing that light would get something like this:
Notice how the colours in the emission spectrum line up exactly with the colours in the absorption spectrum? This tells us that the electron releases the exact amount of energy it absorbs.
So, to answer the question. The atom only absorbs certain wavelengths of light in order to excite the electron, notice how the emission spectrum shows only certain colours and not a continuous spectrum? This tells us that all the other colours not present in the spectrum did not have enough energy to excite the electron to a full energy level, it probably only had enough energy to excite it half-way or three-quarters.
How do we measure the energy of a colour? Well, all colours have a certain wavelength which can be ascertained from the electromagnetic spectrum
Notice that as you go from violet to red, the wavelength increases? Also, all electromagnetic waves travel at the speed of light. So, we can set up an equation like so:
Where c is the speed of light, f is the frequency of light and λ is the wavelength. Solving for f gives us
The energy of a particle is also given by
Where h is planck’s constant which is about 6.62 x 10^-34 Js
Substituting in for f gives us:
Notice that a larger wavelength also gives a smaller energy
So, if we had a more detailed emission spectrum like the one below:
We can actually calculate the energy required to completely transition from one energy level to another.
In fact, the emission spectrum is very useful because it is unique to each element, no two elements have the emission spectrum. Hence, the emission spectrum acts almost as a barcode (looks like one too) for an element. If you had an unknown element and if you were to get the emission spectrum, you could compare it to already known emission spectrums and identify the element. If you get a completely knew emission spectrum (and there were no glaring errors in your experiment) then it is very likely that you have discovered a new element!
Schrödinger’s model
It turns out that Bohr’s model is actually kind of flawed. First of all, the model is only valid for hydrogen. For other elements, you have more than one electron, and as a result there is a force of repulsion between the electrons (all electrons are negatively charged). Furthermore, Bohr’s model assumes that the electron is a particle (we’ll see cases where electrons exhibit both wave-like and particle-like behaviour), and the model also violates Heisenberg’s uncertainty principle (you can’t know the position and momentum of a quantum object simultaneously)
The Bohr model assumes you know the position (which energy level the electron is at) and the momentum (which energy level the electron transitions to), when in fact you can’t know both. This is because by looking at an electron (remember we see things because light reflects off of it), the photon sort of moves it in a random direction.
There are also cases where electrons exhibit both particle-like and wave-like behaviour. I basically touched upon this in my post on superposition so you should check it out!
Now, Erwin Schrödinger proposed his theory of the atom which basically seemed to correct the flaws present in Bohr’s model
Schrödinger’s model involves probabilities. Due to the weird nature of electrons, we can’t say for certain where an electron actually is at a given point in time.
Therefore, we come up with the idea of orbitals which is basically just an area around the nucleus in which there is a 90% probability of finding an electron.
There are four types of orbitals: s, p, d, and f. Each orbital can hold a maximum of 2 electrons.
Each energy level can be split up into different subshells. So, energy level n=1 can be split up into one s orbital. n=2 can be split up into one s orbital and one p orbital and so on.
The s orbital
The s orbital looks like a sphere:
It turns out that every single energy level contains one s orbital. Therefore, the s orbital at n=1 is called the 1s orbital, the s orbital at n=2 is the 2s orbital and so on. Any orbital with the number in front is also called a subshell (this becomes more apparent for the other orbitals)
The p orbital
The p orbitals are dumbell shaped, and it turns out that there are three p orbitals in each energy level that contains a p subshell. So for example, the 2p subshell (the p orbitals at n=2) contains three p orbitals, each oriented around the x, y, and z axis. Hence, the p orbital oriented around the x axis is also called the p_x orbital. Notice that the p subshell entirely can hold six electrons (two in each orbital).
The d orbital
The d orbitals are even weirder, with their orientations following mathematical equations. Note that orbitals aren’t actually a physical thing, they’re just a mathematical description of where can potentially find an electron.
Each d orbital can hold 2 electrons, and there are a total of five d orbitals in a given d subshell, hence the entire d subshell can hold ten electrons.
The f orbital
If you thought the d orbital was weird, the f orbital is way worse. Each one can hold two electrons and there a total of 7 f orbitals in a given subshell, so each f subshell can hold 14 electrons.
Electron configuration — Aufbau’s principle
The electron configuration of an element basically tells us how electrons arrange themselves within these orbitals. Aufbau’s principle basically tells us that electrons start arranging themselves in the lowest energy subshells first. It turns out that the 1s subshell has the lowest energy and so electrons fill the 1s subshell first, and eventually occupy higher energy orbitals.
Now, how do I know what the level of energy actually is? Not to worry, the periodic table helps you with this
The block an element is in basically tells you which orbital contains the valence electron or the last electron in an element. Hence, Lithium for example is in the s block. Thus, its valence electron is in an s orbital. Let’s take a look at the electron configuration of Hydrogen. Hydrogen is clearly in the s block and starts with the 1s subshell, as it only has 1 electron, we can write the electron configuration of H as 1s¹. The coefficient tells you the energy level, the letter tells you the orbital while the “exponent” tells you how many electrons are in that subshell. Helium for example has one more electron than H, and it just so happens that the 1s subshell can occupy one more electron as the s orbital can hold two electrons and there is only one s orbital in an subshell, so we write the electron configuration of He as 1s².
What about a more complicated case like Nitrogen? Nitrogen has 7 electrons. Initially, two electrons fill themselves in the 1s subshell. There are now 5 electrons left to fill, two more fill themselves in the 2s subshell. The remaining three electrons fill themselves in the 2p subshell. So, the electron configuration of Nitrogen is 1s²2s²2p³.
Hund’s rule
Hund’s rule basically states that electrons are first placed in empty orbitals in order to minimise the repulsion between them. If there are no more empty orbitals, electrons are placed in orbitals that can occupy more electrons. In the case of Nitrogen, the electrons in the 2p subshell will originally be arranged like this:
1 electron goes in the 2p_x orbital, 1 electron goes in the 2p_y orbital and 1 electron goes in the 2p_z orbital.
A diagram for the 2p subshell would look like that. What about oxygen? Oxygen has the electron configuration 1s²2s²2p⁴, the initial 3 electrons in the 2p subshell are arranged like the one above, each in empty orbitals. The fourth electron does not have a free p orbital to go to, so it occupies the p_x orbital again.
Pauli’s exclusion principle
Electrons have a property called spin. It is really hard to explain and there is no classical analogy for spin, it is a type of angular momentum. Pauli’s exclusion principle states that no two electrons can be placed in the same orbital if they have the same spin. That’s why, in the diagram above. I drew the fourth electron with a downwards arrow, this shows that the other three electrons are spin-up while the fourth electron is spin-down. It doesn’t really matter which one you do, you could draw the three electrons spin-down and the fourth one spin-up.
Exceptions to electron configurations
It turns out that there are two exceptions to the rules above for atomic orbitals and electron configuration. The two are copper and chromium. Let’s start with chromium, chromium has 24 electrons. If we were to follow the general rule of filling electrons we’d get the following electron configuration: 1s²2s²2p⁶3s²3p⁶4s²3d⁴ (You can rewrite it as 1s²2s²2p⁶3s² 3p⁶3d 4s² as well. It turns out the 4s subshell has lesser energy than the 3d subshell, so the 4s subshell is filled first according to the Aufbau principle. However, when the 3d subshell has electrons, the 4s subshell is pushed to a higher energy level.)
Now, it turns out that the electron configuration above is wrong and it might seem apparent if we draw the orbital diagram for 3d and 4s
Subshells are basically stable if they are fully filled, half-filled or empty. Notice that the 4s subshell is stable because it is fully filled but the 3d subshell is not because four orbitals contain one electron while the fifth orbital contains no electrons. So, the 4s subshell donates one electron in order to stabilise the 3d subshell. The 4s subshell is still stable as it becomes half-filled. The explanation is basically the same for copper, you just have to draw out the orbital diagrams.
This article has already become very long. Part 2 will be a contination and will go into quantum numbers!
