## The separation of molecules in a gas

Teaching Guidance for 14-16

As a crude picture that will lead to a rough estimate, assume that each molecule in a liquid occupies a cubical box of side, *d* , the diameter of the molecule. (Of course real molecules are not hard lumps like billiard balls and certainly not spherical.)

At first glance, this may seem to place the molecules too close together for liquid behaviour. However, the volume of space occupied, *d*^{ 3} , is almost twice the volume of a sphere with diameter *d* , so the assumed cubical spacing would have liquid behaviour.

In the closely packed array we imagine for a liquid, the spacing for molecules, neighbour to neighbour, is therefore *d* , one molecule diameter.

'How much greater is the spacing in a gas such as air?'

If the spacing in the gas is *D* , then a volume of *D*^{ 3} is needed for each gaseous molecule.

The ratio *d*^{ 3} / *D*^{ 3} = volume occupied by a liquid gas/volume occupied by the same mass of the gas.

So, the average separation of the gas molecules is the (volume of gas/volume of liquid)
^{
1/3
}
x the molecular diameter.

The change in volume from the liquid to the gaseous phase for:

- air (nitrogen) is about 1: 750
- carbon dioxide about 1: 850
- petrol at 90°C about 1: 800
- water at 100°C about 1: 1650 (water is unusually high)

If we assume that the volume change is about 1: 1000, then we can conclude that **in a gas, the molecules are about 10 molecular widths apart, on average.** Nine diameters apart would give a volume change of 729 and 12 would give 1728.

This shows the powerful effect of a cube root in estimating molecular separations.