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The mole is defined to be exactly $\mathrm{mol}=6.02214076\cdot 10^{23}$ entities. Avogadro's constant is defined as $N_A=6.02214076\cdot 10^{23}\mathrm{mol}^{-1}$.

Then why is it not usual to write $$N_A=6.02214076\cdot 10^{23}\cdot (6.02214076\cdot 10^{23})^{-1}=1?$$

It doesn't make sense to me.

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    $\begingroup$ One mole of anything contains exactly 1.0 moles of that thing. Important as it is, this fact is somewhat too trivial to be stated more often. $\endgroup$
    – Ivan Neretin
    Commented Aug 2, 2021 at 11:07
  • $\begingroup$ @Ivan Neretin This is not the point of this question at all $\endgroup$
    – Nomas
    Commented Aug 2, 2021 at 11:10
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    $\begingroup$ It’s not a silly question, but what you’ve proven is precisely that: 1 mole of stuff has exactly 1 mole of stuff in it. In a way, we would probably be in trouble if it wasn’t 1. $\endgroup$
    – orthocresol
    Commented Aug 2, 2021 at 11:22

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You have mixed up Avogadro number $N_0$ with Avogadro constant $N_\mathrm{A}$, and used the constant to define itself, which is incorrect mathematically. Analytical form of your suggestion is as follows:

$$N_\mathrm{A} = N_0\cdot\pu{mol^-1} \label{eqn:1}\tag{1}$$ $$\implies \pu{1 mol^-1} = N_\mathrm{A}\cdot N_0^{-1} \label{eqn:2}\tag{2}$$

Plugging \eqref{eqn:2} in \eqref{eqn:1} (even though it makes no sense mathematically) one gets

$$N_\mathrm{A} = N_0\cdot N_\mathrm{A}\cdot N_0^{-1}\tag{3}$$

Or, since both $N_0$ and $N_\mathrm{A}$ are non-zero by definition,

$$1 = 1\tag{4}$$

This is correct, but useless.

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  • $\begingroup$ Please explain how I used the constant to define itself. The definitions I wrote were taken from Wikipedia (I changed "particles" to "entities", though). $\endgroup$
    – Nomas
    Commented Aug 2, 2021 at 11:35
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    $\begingroup$ There is a circular definition $\pu{mol}\to N_\mathrm{A}\to\pu{mol}.$ The quantities you are using in numerical form are, as I said, Avogadro number $N_0$ with Avogadro constant $N_\mathrm{A},$ correspondingly. You are "taking them together" against mathematical principles. $\endgroup$
    – andselisk ♦
    Commented Aug 2, 2021 at 13:17
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    $\begingroup$ Numerical value of $N_\mathrm{A}$ is used in the definition of the mole, $N_0$ and $N_\mathrm{A}$ are inseparable entities, at least in context of current definition. $\endgroup$
    – andselisk ♦
    Commented Aug 2, 2021 at 13:34
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    $\begingroup$ This is not true. The number you for some reason return to all the time is "is the fixed numerical value of the Avogadro constant" (SI Brochure). Look, let me ask you straight: don't you think it's strange that according to your proposition the same constant has two vastly different values, one which is dimensionless? $\endgroup$
    – andselisk ♦
    Commented Aug 2, 2021 at 13:54
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    $\begingroup$ The constant has one and only one value. $N_A=6.02214076\cdot 10^{23}\,\mathrm{mol}^{-1}=1$. Indeed, the equality is correct if $\mathrm{mol}=6.02214076\cdot 10^{23}$. Nevertheless, I managed to find an article resolving this "apparent issue (my question)". pubs.acs.org/doi/pdf/10.1021/acs.jchemed.9b00467 You may also find it interesting. $\endgroup$
    – Nomas
    Commented Aug 2, 2021 at 14:08

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