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To my understanding the mole is the unit used to translate between mass on the atomic level and mass in the macro level, defined as the number of atoms in $12$ grams of Carbon-$12$, which apparently turns out to be $\pu{6.0221409e23}$ atoms, and this was defined as Avogadro's Number.

How was this number derived? Is it something I can derive myself? If I had $12$ grams of Carbon-$12$, how would I be able to deduce that it had $\pu{6.0221409e23}$ atoms?

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    $\begingroup$ It would be hard, very hard. I'll point out that there is a gigantic different in stating that $N_A \approx 6.022 \times 10^{23}$ and stating that $N_A = 6.0221409\times 10^{23}$. Determining a value to eight significant figures requires extremely good experimental technique. $\endgroup$ – MaxW Jul 13 '18 at 3:29
  • $\begingroup$ @MaxW How did Avogadro do it in his time, to whatever degree of precision that he did? $\endgroup$ – user51819 Jul 13 '18 at 3:48
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    $\begingroup$ This information is readily available on the web in sources such as Wikipedia. en.wikipedia.org/wiki/Avogadro_constant, "By dividing the charge on a mole of electrons by the charge on a single electron..." Look up Millikan oild rop experiment for charge of e-, Faraday and electrolysis for atomic mass/charge ratio etc. $\endgroup$ – DrMoishe Pippik Jul 13 '18 at 4:36
  • $\begingroup$ @DrMoishePippik What is interesting to me is that apparently the oil drop experiment gave an answer that was technically wrong / inaccurate, but we still used it anyway to derive all these other numbers? $\endgroup$ – user51819 Jul 13 '18 at 4:43
  • $\begingroup$ It's been refined over time. As @MaxW states, original values had a very large margin of error. It's like Galileo using a chandelier as a clock... instrumentation has improved since then. $\endgroup$ – DrMoishe Pippik Jul 13 '18 at 5:09
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To determine Avogadro's number you have to measure the same unit at the atomic and macroscopic scales.

This was first achieved by Millikan who measured the charge of an electron. The charge of one mole of electrons was already known and is a Faraday. Dividing both, you get Avogadro's number.

Before that, Josef Loschmidt was the first one to calculate the number of particles in a cubic meter of gas, using the kinetic theory of gases. This is, of course, also related to Avogadro's number. Avogadro himself did not give any number. He just stated that equal volumes of different gases at the same pressure and temperature have equal amount of particles.

A more modern way of doing it is determining the density of an ultra pure element and then determine the number of atoms and their distances in a unit cell with X-ray diffraction.

So yes, you could, in principle, derive Avogadro's number with pure monocrystalline carbon 12. You will, however, need very good instrumentation.

As per the answer to this question on Chemistry SE it might become the other way round. The Avogadro number will be absolute (defined, with 0 error) and then the kilogram will be redefined as a function of the number of atoms in a monocrystalline perfect $\ce{^{28}Si}$ sphere.

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