Avogadro's number is defined as the amount of molecules in 12 grams of Carbon-12. Just as we know there are 12 donuts in a dozen, we know there are 6.02223 donuts in a mole of donuts. In this way, Amedeo Avogadro can't be wrong as he chose to define the number as such.
However, it was Jean Perrin who was awarded the Nobel Prize in Physics by rigorously determining/measuring the quantity of carbon atoms in 12 grams of carbon via several different methods. One is coulometry (see:
others can be learned here
While I am not entirely certain how Avogadro got that number, I think we can be reasonably confident that the number you see in your textbooks is correct, even if Avogadro's number itself was a little off. There are derivations of equations of state that tell us how many atoms or molecules there are in a sample. Similarly, the value of the electric charge is known, as is the mass of the electron (because when an electron meets with an antiproton, they produce gamma rays, and the frequency of the gamma rays tells us the energy, and thus the mass). It's a fairly simple matter of extrapolating back from the known masses of subatomic particles using matter-antimatter annihilation energies to recover Avogadro's constant. There is always uncertainty in the value, but it's close enough.
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