Why is Avagadro's number independent of substance? [closed]








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In the book of Irey, *Thermodynamics*, in vol1, IA, page 4, it is given that


Molecular (atomic) weight $M$:



> 
> The ratio of the mass of a molecule (atom) of a substance to one twelfth of the mass of an atom of the most common isotope of carbon $^{12}\mathrm C$.
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Mole:



> 
> A gram mole is the amount of a substance whose mass is equal to its molecular weight in grams. The mass of the one gram mole equals $M$ grams ($M$: molar mass).
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Avogadro's number: 



> 
> Since the mass of a mole is proportional to molecular weight, there are a definite number of molecules in any molar unit, called $N\_\mathrm A$.
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And in the remark, the author mentioned that *$N\_\mathrm A$ is independent of molar units*, and we know that it is even independent of substance.


However, with these definitions, I cannot understand why $N\_\mathrm A$ should be independent of substance. 


**Question:**


With these definitions, why is the constant $N\_\mathrm A$ independent of substance that is used to measure $N\_\mathrm A$? 




It is the same reason that a dozen doesn't depend on whether you are counting grapes or elephants. That is how the mole is defined: it is a number, nothing else.


The confusion, I suspect, is because of how we *measure* that number (or, strictly, how we originally measured it as the definition changed recently). The intention of the unit was always to define a number of things independent of the thing being measured. But to measure that number chemists originally defined it as the count of atoms in a particular substance (originally on 16O and later 12C). Since November 2018 it is now defined as an exact number.


Presumably something similar happened as humanity learned to count beyond 10. To illustrate the concept of 12 they might have used 12 rocks or 12 wildebeest. But the idea of a dozen was independent of the things being counted: the things were merely a concrete illustration or example of the idea of a dozen. Same with the mole: the thing being counted is irrelevant even if you had to use a concrete example of counting the atoms in a real substance to illustrate the number. 


 13

Let's look at the $\ce{^9\_4Be}$ isotope and apply the definitions:


**Relative atomic mass**


The relative atomic mass of $\ce{^{12}C}$ is, by definition, 12. Looking at the periodic table, we find that the relative atomic mass of beryllium is 9.0121831(5). That makes sense because $\ce{^{12}C}$ has a mass number of 12 (6 protons and 6 neutrons) while beryllium has a mass number of only 9 (4 protons and 5 neutrons).


**Mass of one mole of substance**


According to the definition, one mole of beryllium has a mass of 9.0121831(5) g, just as one mole of $\ce{^{12}C}$ has a mass of 12 g (exactly, until May 20th 2019). These two samples have same number of particles because their mass ratios are equal to their relative atomic mass ratios (look back at the definition of the relative atomic mass, it relates one atom of beryllium to one atom of carbon-12). 


**Avogadro constant $N\_\mathrm A$**


A mole of anything contains the same number of particles as a mole of $\ce{^{12}C}$. This follows from combining the definitions of relative atomic mass and of the unit mole (pre May 20th, 2019). The Avogadro constant is the ratio of number of particles to amount of substance:


$$ N\_\mathrm A = \frac{N\_\text{particles}}{n\_\text{sample}}$$


Before May 20th, 2019, the Avogadro constant was experimentally determined. You would have to count atoms in a sample of known chemical amount, typically indirectly. For example, if you know the mass, the radius, the atomic distances of a silicon sphere and the average molar mass of the silicon sample, you could figure out the Avogadro constant.


Since May 20th, 2019, the Avogadro constant is set to a constant value:


$$ N\_\mathrm A = \pu{6.02214076e23 mol-1}$$


Your textbook is out of date, just like most textbooks were when you asked your question, which does not make sense anymore given the current definition of the Avogadro constant.


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