How did Rayleigh compute the Avogadro number? Knowing that the oil, oleic acid, had spread evenly over the water, Rayleigh calculated that the thickness of the film was 1.6 nm by knowing the volume of oil dropped and the area of coverage. In addition, he used these calculations to prove the existence of the Avogadro number.
Could someone explain how Rayleigh computed the Avogadro number from the thickness of the film? This is one of the exercises in the "Exercises for the Feynman Lectures on Physics" book.
 A: In 1890 Lord Rayleigh published "Measurements of the Amount of Oil Necessary in Order to Check the Motions of Camphor upon Water" in which he estimated the size of oil molecules but did not actually use that estimation to calculate Avogadro's number. 
His experiment consisted of coating a pool of water (surface area ~$0.6\ m^2$) with the thinnest possible layer of olive oil such that the layer was only one molecule thick. Placing bits of a natural tree wax called camphor on the water allowed him to determine precisely when he had added enough oil because the camphor bits jiggle wildly only while dissolving in contact with the water. By weighing the oil and knowing its density he could calculate the volume of oil, and knowing the surface area of the pool allowed him to calculate the height of the monomolecular oil layer at about $1\ \text{nm}$. 
To get from the height of an oil molecule to Avogadro's number, you would have to make an assumption. For example: that the oil molecule is a cube so the volume of one molecule would be $1\ \text{nm}^3$. Dividing the measured volume of oil by $1\ \text{nm}^3$ would give the number of oil molecules in the measured mass of oil. This value, in molecules of oil per gram of oil should be compared to Avogadro's number divided by the known molecular weight of the oleic acid (main component of olive oil):
$$\frac{6.02\ \times 10^{23}\ \text{molecules}}{1\ \text{mol}}\times\frac{\text{1 mol of oil}}{282\ \text{g}} =\ \frac{2.1\ \times 10^{22}\ \text{molecules of oil}}{1\ \text{gram}}$$
You will find that using the assumption of the oil molecule cube and Rayleigh's data will bring you very close to the accepted value of Avogadro's number!
