# Simple molecular model of different states of matter

##### Demonstration

Using models to support a discussion of the mean-free-path of molecules in a gas, and, perhaps, to lead to a value for the size of a molecule.

Coins, 20 to 30

#### Procedure

a Scatter coins upon the table, spaced so that they are well apart from one another. Discuss the possible mean-free path.

b Use a ruler to sweep them closer together and, again, discuss the possible mean-free paths.

#### Teaching notes

A typical discussion might be as follows.

Let's pretend that the coins represent air molecules. If they are close together we must have liquid air. If you can guess how crowded the molecules would look in that liquid, you will be able to find the size of a single molecule. This is where you will have to make a guess.

Imagine the molecules are round balls – not true but it will get us going. Experiments show that the volume change from gaseous air to liquid air is about 750 to 1. Imagine that compression has pushed the molecules together until they are touching.

Do you think the molecules could be that close? Could molecules arranged like that behave as a liquid which can be poured easily? No; this would be a solid. In a liquid which can be poured and move around easily, the molecules are probably a little further apart than that.

If we give the molecules twice as much space, does that look like a liquid?

It might, but the spaces do look large. It looks as if a molecule could move quite a long way among its neighbours. Diffusion would be fairly fast, but actually diffusion is fairly slow in liquids. (Try putting copper sulphate crystals at the bottom of a tall jar of water, wait, and see how slowly the blue solution diffuses.) Molecules with this spacing would be a gas.

Think of a crowd of people when it is behaving as a liquid; a crowd that can flow through streets to a railway station or a football match but is too dense to allow individual people to move far among their neighbours. An intermediate guess somewhere between the extremes of no distance between molecules, which would lock them tight like a solid, and spaces as big as one whole diameter which would make them behave as a cool gas.

Look at one molecule and guess its mean-free-path. Draw an arrow to show how far it can move in any direction. Start the arrow at the surface of the molecule and continue it until you meet the surface of another molecule.

Do that in many different directions, measure the length of the arrows and find the average. (This is only a 2-dimensional plot but we will use it to make guesses in 3-dimensions.) The average length is about 1/3 of a diameter, its mean-free-path.

(If students do this as well, the range of their guesses is likely to be from 1/10 to 9/10 of a diameter. This whole range covers only one order of magnitude; an uncertainty of half an order of magnitude is not a problem. Accept whatever they come up with otherwise you might as well tell them the answer!)

All right, we agree on 1/3 of a diameter for the mean-free-path at liquid crowding. Then, we have squeezed the mean-free-path down by 750 from that in ordinary air until it is only 1/3 of a diameter, d/3.

Therefore, the mean-free-path in the gas, 10-7 m, is squeezed down to 10-7/750 m = d/3. So, d = 4 x 10-10 m and we have found the ‘diameter’ of a single molecule of air! (An atom, or diatomic molecule, is probably half that size.) It is a rough guess but it is a good estimate; it is the right order of magnitude. We have made an atomic measurement.

The difference between a solid and liquid is not just the change in molecular separation. Intermolecular forces and kinetic energies also differ.

This experiment was safety-checked in July 2006.