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As far as I get it, we did weigh one carbon (12,6-isotope) atom. Then we said that electrons have nearly zero weight and protons weight nearly the same as neutrons. Hence we took the weight of that one atom and divided it by twelve and this way we got the mass of a proton/neutron:

$w_\mathrm {n,p} = 1.6726 \cdot 10^{-24}\ \mathrm g$

Because that isn't easy to calculate with, we introduce a number which multiplied with the weight of a proton/neutron gives us one gram:

$N_\mathrm A = w_\mathrm {n,p}^{-1} = 1\ \mathrm{mol}$

So far so good. Now hydrogen weights 1 g/mol and carbon weights 12 g/mol. Two questions to that:

  1. How did we actually measure the weight of one carbon atom in the first place?
  2. Why is the molar mass not precisely 1 and 12?
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    $\begingroup$ 1.) did you check wikipedia? Its explained thoroughly there. 2.) nuclear binding energy, $E=mc^2$ $\endgroup$ – Karl Jan 9 at 0:40
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    $\begingroup$ Your equation for the Avogadro constant is not correct. $\endgroup$ – Loong Jan 9 at 7:14
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    $\begingroup$ Why is this question being voted to be closed? It is a very good focused question which has deep historical roots and it is the Mr. Wikipedia that doesn't have decent and focused answers on this topic. $\endgroup$ – M. Farooq Jan 9 at 15:06
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This is a question worth debating and discussion. First of let us realize the fact that we are used to looking at relative atomic weights or masses (if someone wants to be too accurate). No Table lists the absolute atomic masses of atoms in the periodic table, i.e., through out the history we have to assumed either the weight of hydrogen atom is exactly 1.0000, or oxygen-16 had a weight of 16.0000 or lately the weight of carbon-12 atom is exactly 12.0000 units. So let us forget the picture that someone actually made a measurement, took a single atom of hydrogen or carbon and weighed by some fancy means. These are all man made references and numbers. We could have taken the weight of C-12 as 100, and all the atomic masses would change proportionally.

So far so good. Now hydrogen weights 1 g/mol and carbon weights 12 g/mol. Two questions to that: How did we actually measure the weight of one carbon atom in the first place? Why is the molar mass not precisely 1 and 12?

See above. You have to choose a reference. Fix hydrogen to 1 or carbon as 12 exactly but not both. Now you may ask how these numbers came up? You have to look at the question historically, when Avogadro's number was not used. If you understand German, have a look at Handbuch der Anorganischen Chemie by Abegg. There is long history how the number 12 got associated with carbon.

https://archive.org/details/a2handbuchderano03abeguoft/page/12;

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You must realize that one proton free, lonely, has not the same mass as when included in a nucleus. A proton looses a small part of its mass when entering a nucleus. This mass loss is transformed into energy through the Einstein law : E = mc2.

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