Maths In Chemistry

Familiarity with Maths is very useful to anybody studying any of the Sciences, and Chemistry is no exception. In this short article I will show how a knowledge of Vector Geometry can help to make a seemingly difficult calculation - finding out the bond angle of the Methane molecule - surprisingly simple.

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Chemistry textbooks will tell you that the Methane molecule (CH₄) is a regular Tetrahedron, with a bond angle of approximately 109.5º, but few will go on to explain the maths behind this. In fact, the procedure used to derive this bond angle is an excellent demonstration of the power of Vector Geometry.

We will start by considering a regular Cube, with sides of length 2, centred on the Origin of a simple 3-D space. Why do we start talking about a cube, when the shape under discussion is a Tetrahedron? Well, that will become clear shortly.

When we say that this cube is “centred” on the Origin, this is a short-hand way of stating that the distance from the origin to each of the cube`s 8 corners (or “vertices”) is exactly the same. This is an important point.

The cube we have described can be written in Vector Geometry in terms of 8 vectors, each of which is the vector from the origin to the corner.

Since the cube has sides 2 long, we can write these vectors (in x,y,z format) as follows:

OA (1,1,1)

OB (1,-1,1)

OC (-1,-1,1)

OD (-1,1,1)

OE (-1,1,-1)

OF (1,1,-1)

OG (1,-1,-1)

OH (-1,-1,-1)

As is obvious by inspecting the vectors, they are all the same length. So, we can take any 4 of these points, and they will form a shape where all 4 vertices are equidistant from centre, i.e. the Origin.

If we take alternate points, we will make a regular Tetrahedron whose centre is the origin.

For example, we can take OA, OC, OE, and OG.

How does this help us to find the bond angles of the Methane molecule? Remember that the Methane molecule consists of a central Carbon atom, surrounded by 4 Hydrogen atoms. Electrons repel each other, which is the basis of the Valence Shell Electron Pair Repulsion (VSEPR) Theory of molecular shape. So, with four identical atoms, each forming an identical s-p bond, (sp³ hybridisation), and without any lone electron pairs to distort it, we would expect the molecule to be perfectly symmetrical, and indeed it is.

Another way of saying this is that the Hydrogen atoms are arranged so that:

1. They are equidistant from the Carbon atom

2. They are equidistant from each other.

Both of these requirements are satisfied: the first, because we know that all 4 vertices were chosen from the set of 8 vertices of the cube, which, by definition, are all the same distance from the centre, which is where we are placing the Carbon atom. The second requirement is also met, because the Hydrogen-Hydrogen bond lengths are four diagonals of the cube`s faces, and these are also all equal.

The bond angle we are talking about is the angle formed by H―C―H, and this is equivalent to asking – what is the angle between OA and OC ?– or indeed between any any other pair selected from the 4 vectors we chose from the list of 8 above.

This is where the Vector Geometry really comes in handy. Remember that the Vector “DOT” Product is related to the cosine of the angle between two vectors as follows:

X • Y = |X| * |Y| cos(θ)

Where |X| is the length of X and θ is the angle between the vectors, and the • symbol refers to the Dot Product, which is the sum of products the individual vector terms.

So, using OA and OC we have:

OA = (1,1,1)

OC = (-1,-1,1)

OA • OC = ם ם +1 = ם

|OA| * |OC| = |OA| * |OA| = 1² + 1² + 1² = 3

giving:

cos(θ) = -1/3

and θ ≈ 109.5º