In high-school chemistry classes, we learn about the mol or "Avogadro constant", a number around 6.02×1023. For example, 18 grams of water have one mol molecules, since 18 is the atomic mass of H2O — 16 from the oxygen and 2 from both hydrogen atoms. (BTW, is there an "Imperial" definition of mol? :)
At this point, the student asks "how is this going to change my life?"
Setting aside the fact having a bit of culture and curiosity can't hurt, the Avogadro constant has direct influence in our everyday lives. For example, have you ever prepared 2-part epoxy?
Here in Brazil, there is very popular product from Loctite called Durepox; it is a household name like J.B.Weld. (I didn't find an exact counterpart for Durepox in Loctite international website; the nearest I found was Repair Putty.) It is a very thick putty that comes in two parts, one dark gray, another off-white, and should be mixed by kneading before use.
The directions say "knead very well", but how well is well enough? Does it suffice to reach an uniform color, or should we keep kneading by half an hour? I did a test: get a Durepox bar, and keep mixing the two components, taking a small sample from time to time, to see how long we should be mixing to get the desired result.
The chosen mixing method is the simplest possible: fold the putty and press, changing the folding direction every other fold. It is not the most efficient method, but it can be defined objectively and this will be useful later. As you can infer by the figure above, I took a sample every 5 folds, going up to 80, when too little putty was left. Left curing for 24 hours, as say by the directions.
For the epoxy to cure, the parts must be mixed down to molecular levels. It seems impossible to get such homogeneity just by folding, but folding is an exponential process. In 30 folds, we have divided the putty in 1 billion parts. With 60 folds, we reach 1018 parts. With 80 folds, we surpass the Avogadro number and it is probably pointless to go on.
By the way, here is one major practical consequence of the Avogadro constant. If, for example, the mol was around 10100, it would take five times as much effort to mix well. We would still get there, thanks to the exponential character of mixing processes. But everything in life would take much more work, be it mixing epoxy or kneading bread.
Before I looked into the empirical results, I tried to estimate what would happen, based on the mol. I had mixed 50g of epoxy. The molecular weight of epoxy monomer is around 393g per mol, so I had prepared around 1/8 of mol, or 7.5×1022 molecules. This number is near 276, so in theory I get a perfect epoxy putty mix with 75 folds.
But there is another factor: I took a bit of the putty every five folds. Being smaller and smaller, the remaining putty mass mixes faster and faster. For example, the 50-fold sample is actually equivalent to 55 folds, because it is as well-mixed as if I had folded 55 times the whole 50g mass.
Well, time to see if theory meets empirical results. The epoxy balls evaluation was a bit subjective, since I hadn't taken care of making them the same size and shape. I used my bare hands and cutting pliers to check their strength. It was possible to divide the samples in three major groups.
The samples mixed at least 60 folds (68 effective folds) were really good, very tough, as the product usually sets in. It would take something more sophisticated than cutting pliers to determine strength differences.
The samples between 30 and 55 folds had hardened, but they were less tough. The 55-fold sample released a fragment just by hand pressure. The 40-fold sample had a soft spot, it was certainly a badly-mixed piece. In general, the hardened putties would still fulfill less demanding roles, like covering a seam.
At 25 folds and below, the putty was still sticky to the touch and crumbly when forced. It didn't harden at all below 15 folds. 25 was also the point of visual homogeneity. It seems there is an actual relationship between a visually homogeneous putty and minimum strength.
The theoretical estimation got pretty close, especially taking the precariousness of the experience into account. There are many unknowns as well: Durepox is not pure epoxy; the hardened part is of unknown composition and molecular weight; and so on. The error was around 7 folds (68 folds were enough to get a tough result, while the estimation was 75). If we had made this experience to estimate the Avogadro constant, we would have underestimated it by a factor of 100.
The concept of mol comes from the kinectic theory of gases. In his time, Amedeo Avogadro hypothesized that, if two gas samples had the same volume, same presure and same temperature, they would have the same number of particles, even if constituted of dissimilar substances.
Avogadro never tried to estimate how many particules were there. The constant's name was a posthumous homage. It had been estimated by indirect methods, with increasing precision, since the XIX century. The current value has 9 digits of precision.
On the other hand, the weight of a gas sample depends solely on the atomic mass. For example, a sample of 22.4 liters of gas at 0ºC and 1 atm has one mol of molecules, but such a sample of SF6 weighs 70 times more than a sample of hydrogen.