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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 $^{12}\mathrm C$.

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).

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 $N_\mathrm A$.

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 $N_\mathrm A$?

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    $\begingroup$ It's not related to your actual question, but your book is more than 40 years old. In particular, the concept of "gram mole" is obsolete. You might want to consider getting a new book. $\endgroup$
    – user7951
    May 29, 2019 at 9:55
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    $\begingroup$ In short, the constant $N_A$ is independent of substance precisely because it is defined to be independent of substance. $\endgroup$
    – Ivan Neretin
    May 29, 2019 at 10:08
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    $\begingroup$ @Loong That is one of the reasons why I particularly have chosen this book, apart from that is a great book; I cannot understand newly written books, regardless of their content. $\endgroup$
    – Our
    May 29, 2019 at 10:46
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    $\begingroup$ If you cannot see such a statement in the definition then you are looking at the wrong definition $\endgroup$
    – Waylander
    May 29, 2019 at 10:53
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    $\begingroup$ Irey's definitions of mole and molar mass inevitably lead to $N_A$ being a constant. The definition is a bit cumbersome, because only a few weeks ago, $N_A$ has once and forever been defined to be one specific integer number. $\endgroup$
    – Karl
    May 29, 2019 at 20:45

2 Answers 2

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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.

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  • $\begingroup$ So, If understand correctly, we experimentally found that this number was independent of the substance, and then defined it in a way which also described how to measure that number ? and this is the main confusion; there is nothing that forces that quantity be independent of substance, but it is an experimental fact that it is the case ? $\endgroup$
    – Our
    May 29, 2019 at 11:06
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    $\begingroup$ @onurcanbektas No. The definition is not empirical. Chemists observed that things combine in fixed ratios long before they even had a concept of the atom. As the concept of the atom emerged, chemists wanted to count them. Then they developed the concept of the mole and needed to measure how many atoms were in it. Don't confuse the number (independent of what is being measured) with how it was first counted. A dozen is not an empirical concept depending on what thing we are counting: it is an abstract number. $\endgroup$
    – matt_black
    May 29, 2019 at 11:12
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    $\begingroup$ @onurcanbektas You are missing the point. A mole has always been a number and is currently defined as a precise number. Initially chemists used specific substances to measure it but the definition didn't change because of the difficulty of measuring it. in fact, if a measurement of a different substance looked off, what chemists would change is the atomic mass not the count of moles. $\endgroup$
    – matt_black
    May 29, 2019 at 11:47
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    $\begingroup$ @onurcanbektas a dozen is exactly the same sort of concept as the mole. The only issue is that counting a mole using your fingers is a lot harder. We don't need to check empirically that a dozen elephants had the same number of things in it as a dozen grapes. Replace dozen with moles and the grapes and elephants with different atoms and the sentence still works exactly the same way. $\endgroup$
    – matt_black
    May 29, 2019 at 11:55
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    $\begingroup$ @onurcanbektas You are veering off into persistent ignorance and rudeness. A mole is a certain number of things, there is probably not a mole of stars in the sky, there is a little bit more than a mole of water in a half full glass, there is exactly a mole of electrons in half a mole of hydrogen gas. Same as there are usually not a dozen pups to the dog, usually more than a dozen children in a regular sized 5th grade class and exactly a dozen in a 12 pack of eggs. a mole is a mole by definition, exactly 1 per atom in a dozen grammes of carbon atoms. what-if.xkcd.com/4 $\endgroup$
    – Stian
    May 29, 2019 at 20:13
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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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    $\begingroup$ I understood why their mass ratios are equal to their relative atomic mass ratios; however, how does that imply that both samples have the same number of particles ? $\endgroup$
    – Our
    May 29, 2019 at 19:29
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    $\begingroup$ The mass ratio of one atom Be to one atom C is 9 to 12. If I compare a million atoms of Be to a million atoms of C, the mass ratio will be 9 to 12. Conversely, if the mass ratio is 9 to 12, I will have the same number of atoms of Be and C, no matter what the absolute mass is. $\endgroup$
    – Karsten ♦
    May 29, 2019 at 20:05
  • $\begingroup$ You mean the same number of atoms of Be relative to C ? $\endgroup$
    – Our
    May 29, 2019 at 20:07
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    $\begingroup$ Yes, in order to get the same number of atoms of Be relative to the number of atoms in a C sample, their masses must have the ratio of 9 to 12. $\endgroup$
    – Karsten ♦
    May 29, 2019 at 20:08

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