Science of Symmetry II

This is the second of a two-part blog post. If you missed last week’s post, it’s here. So welcome back – this week we get into gritty mathematical concepts and more pretty pictures. We are going to talk about; Symmetrical Isomerism, which is a nice bit of related chemistry, and symmetry groups, which is a taste of group theory.

Symmetrical Isomerism

Symmetrical Isomerism, sometimes known as Optical Isomerism, relies on rotational symmetry of molecules around a certain carbon. An isomer is a pair of molecules that are made of the same stuff but look different in relation to our skeletal formulae. So, for example let’s take lactic acid, the chemical that makes your muscles hurt when you run.

Skeletal formulae of Lactic acid

This type of isomerism only occurs in certain molecules, similar to how only certain shapes have rotational symmetry. The condition for this type of symmetry in molecules is the occurrence of something called a ‘chiral carbon’. A chiral carbon must have four different sub-molecules attached to it. That must sound confusing so let us take an example:

Here we have a molecule with some other sub-molecules coming off it: OH, NH2, H3C and an invisible H, due to skeletal formulae ignoring hydrogens. We can see that these are all different. The carbon which is circled in green is chiral.

Where is our chiral carbon in Lactic Acid? See whether you can find it.

After some fiddling and looking, you should be able to see that the carbon with the red diamond on it is the chiral carbon. It has a H3C, OH, H and a COOH attached to it, again four different sub-molecules. Furthermore, looking below at the symmetric pair, we can see a link to reflective symmetry we saw in the previous blog post,

(Note that I have been alerted that the O and OH are not symmetrical. I am fixing this.)

These are the same molecule but reflected in an invisible mirror between the two molecules. Note that the wedges show whether the sub-molecules are above or below the plane of the page. For example, the solid wedge on OH means that it is pointing upwards out of your screen, however the stripy wedge shows that the sub-molecule is below the screen.

This is cool and all, but you haven’t heard the best bit! Molecules like these can polarise light. That’s right, if you have a pure enough sample of lactic acid and you shine light through it, it should polarise it!

Polarisation

When I talk about polarised light, I mean light that oscillates in one direction. Light is a wave… ish, light oscillates up and down, but in random directions. A good analogy I could use is if you had a pack of spaghetti and dropped it on the floor, the pieces would all be pointing in different directions. The polariser comes along and takes all the spaghetti pointing in one direction and lets it through, leaving the others behind.

Symmetry Groups

Back to Maths and more specifically symmetry groups. In the previous post we showed that molecules are basically the same things as shapes, now we are going to use this and apply group theory to shapes and hence, to molecules.

Groups? What are they and why?

What is a group? Well, technically it’s an algebraic structure with axioms that constrain the contents. However, that definition is dry and hard to grasp. We shall call a group a box, with things in it. The things in the box have properties. One thing does nothing, like a useless piece of tat you picked up at a conference, a pen that doesn’t work. Then you have objects that do things, like a pencil and an unsolved Rubiks cube. However, every object must have an inverse, something that undoes what it does, like an eraser for the pencil. The inverse of an object can be itself, like the Rubiks cube, which once solved can be unsolved again.

Although this analogy is brilliant for describing what a group is, it isn’t so good for describing wider group theory. I fear flooding you with symbols so I will explain as we go.

Let us use our hexagon, as that’s how I was taught about this and I love hexagons #hexagonlover. Firstly, we will go back to an illustration I put in last week’s post:

An image containing a hexagon and square. They have lines of reflective symmetry on them.

The hexagon’s angle of rotational symmetry is 60∘, as it has order of symmetry 6. In terms of a symmetry group, that means there are 7 elements, one for each rotation and an extra for the 0∘ or 360∘ rotation. We can introduce some notation for the elements of the group:

Rotation060120180240300
$g^n$$e$$g$$g^2$$g^3$$g^4$$g^5$

Basically, this notation accentuates the point that if your hexagon is at the 240$\circ$ (g4) position and you rotate it through another 180$^\circ$ (g3), then you would have rotated basically 60$^\circ$ (g). I should also note that we don’t use the notation of g0 as it can be seen as ambiguous, instead we use e. We define this type of structure as cyclic, as we are doing the same action again and again, making it possible to go from g2 to g3 by only doing the same thing you used to get from g to g2. If you aren’t convinced, try it out for yourself:

I think we have answered the question, what.

Why do groups exist? I propose a question back, why not? To an extent this is pure maths and it has no rhyme, no reason, just fun. However, if you are a chemist, you can make certain groups of molecules and then all the elements of that group have certain properties. For instance, all the molecules that have chiral carbons form a symmetry group. Another group is all of the molecules that have triangular pyramidal shape, as we saw in the diagram above when defining chiral carbons. It’s a way of grouping or classifying using Maths.

So, we can now apply this to molecules and get molecular symmetry groups! I may come back at a later date and talk about the symmetry matrices that we use in group theory. That may be next week’s post… I’m unsure. If you are wondering where I learn my appalling knowledge of group theory, then it was all from my A-level Further Maths. No rigour, just pictures. I shall leave you with my favourite group, the Ih molecular group, which includes C60 also known as Bucky Balls.

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