Estimate of molecular size: a more formal method

Imagine molecules in a gas; dots spaced far apart. Add arrows to show the random motion, not all speeds (arrows) the same but speeds around the average. You could say:

'Here is a snapshot of air molecules in this room with the camera focused at one distance. To find how one molecule would move through this vast array of moving neighbours is too difficult a business. Instead, pretend that we freeze all the molecules except one and watch that one molecule go hurtling through the crowd.'

Redraw the picture showing each molecule as one round blob without any indication of velocity. Draw the path of the chosen molecule, as it moves to collide with another, as a cylinder swept out between the two molecules. The diameter of the cylinder is equal to the diameter of the molecule and its length is equal to the mean free path. Bend the path at the collision and another cylinder is swept out as shown this diagram:

The mean free path is many times longer than the separation between molecules and so the cylinder should pass many other molecules on the way to a collision.

Now move off to a separate preparatory discussion looking at such a collision in detail. Draw a large round molecule bouncing against another molecule.

'How far apart are the molecules, centre to centre, at the collision? One diameter.'

'I am now going to show you a trick for finding out how far a molecule goes before hitting another. This trick has been invented by scientists and is not what really happens but gives good results. When two molecules collide they must be 2 radii, or 1 diameter apart. Instead of drawing the collision like that, I could pretend that the molecule flying along to make the collision is much bigger, and any other molecule that it hits is much smaller. We get the same result as long as we have the centres of the two molecules 1 diameter apart at the collision. I am now going to push this to the limit and make the flying molecule have double the radius, equal to 1 diameter, and the molecule it hits have no radius at all.'

'Now we start this story all over again. Here is the artificial molecule flying along with radius equal to one molecular diameter. It sweeps out a cylinder of 1 molecular diameter in radius and collides with the artificial point sized molecule where it bends its path.'

'Think about the path swept out by this flying molecule which is possessively patrolling its "share" of the volume of the box. This volume is equal to d2 x 10-7 m.'

[For justification of mean free path being 10-7 m, see Guidance note Further discussion of mean free path

'We need to know the volume of space that belongs to one molecule of air in this room. The volume change from liquid air to air is about 1:750. If for liquid air each molecule of diameter, d, occupies a cubical box of side d, then the volume occupied is d3 on the average.'

750d3 = d2 x 10-7

d = 4 x 10-10 m

'We have found the diameter of a typical molecule of air. An atom is probably about half that size. This is certainly a rough estimate because our measurements were difficult and we made all kinds of risky moves carrying out our calculations. Yet this is a very good estimate for many working purposes. It is the right order of magnitude.

All we are really measuring here is an order-of-magnitude distance of approach at which inter-molecular forces grow large enough to have a noticeable effect. Air of course is a mixture of different gases, mainly nitrogen (about 78%) and oxygen (about 21%).

Careful measurements for particular molecules give different diameters according to the experiment chosen and the method of interpretation used. After all, the diameter of a molecule is not as definite a thing as the diameter of a steel ball. Both nitrogen and oxygen are diatomic molecules. Not only are diatomic molecules 'oblong' but they behave as if squashy, so more violent collisions are likely to reveal a smaller effective diameter. Nitrogen molecules are very slightly larger than oxygen molecules; in their gaseous state both have effective diameters of about 3 x 10-10 m.