How on Earth did chemists settle on such a seemingly arbitrary figure for Avogadro's number? To understand how it was derived, we have to first tackle the concept of the atomic mass unit (amu). The atomic mass unit is defined as 1/12 of the mass of one atom of carbon-12 (the most common isotope of carbon). Here's why that's neat: Carbon-12 has six protons, six electrons and six neutrons, and because electrons have very little mass, 1/12 of the mass of one carbon-12 atom is very close to the mass of a single proton or a single neutron. The atomic weights of elements (those numbers you see below the elements on the periodic table) are expressed in terms of atomic mass units as well. For instance, hydrogen has, on average, an atomic weight of 1.00794 amu.
Unfortunately, chemists don’t have a scale that can measure atomic mass units, and they certainly don’t have the ability to measure a single atom or molecule at a time to carry out a reaction. Since different atoms weigh different amounts, chemists had to find a way to bridge the gap between the invisible world of atoms and molecules and the practical world of chemistry laboratories filled with scales that measure in grams. In order to do this, they created a relationship between the atomic mass unit and the gram, and that relationship looks like this:
1 amu = 1/6.0221415 x 1023 grams
This relationship means that if we had Avogadro's number, or one mole, of carbon-12 atoms (which has an atomic weight of 12 amu by definition), that sample of carbon-12 would weigh exactly 12 grams. Chemists use this relationship to easily convert between the measurable unit of a gram and the invisible unit of moles, of atoms or molecules.
Now that we know how Avogadro's number comes in handy, we need to examine one last question: How did chemists determine how many atoms are in a mole in the first place? The first rough estimate came courtesy of physicist Robert Millikan, who measured the charge of an electron. The charge of a mole of electrons, called a Faraday, was already known by the time Millikan made his discovery.
Dividing a Faraday by the charge of an electron, then, gives us Avogadro's number. Over time, scientists have found new and more accurate ways of estimating Avogadro's number, most recently using advanced techniques like using X-rays to examine the geometry of a 1 kilogram sphere of silicon and extrapolating the number of atoms it contained from that data. And while the kilogram is the basis for all units of mass, some scientists want to begin using Avogadro's number instead, much the way we now define the length of a meter based on the speed of light instead of the other way around.