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What is the dimension of Avogadro's constant ($N_\mathrm{A}$).

On Wikipedia it says it is dimensionless, but in Nigel Wheatley's article (pdf) On the dimensionality of the Avogadro constant and the definition of the mole it says it is $\mathsf{N}^{-1}$.

\begin{align} \text{Number of particles} &= N_\mathrm{A}\times \text{Amount of substance}\\ [\text{Amount of substance}] &= \mathsf{N}\\ [\text{Number of particles}] &= \mathsf{1} \end{align}

If $[N_\mathrm{A}] = \mathsf{N}^{-1}$ then the above equation follows, otherwise it doesn't. Is this the right value?

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The dimension/unit of the Avogadro constant $N_\mathrm{A}$ is actually $1/\mathsf{N}$ or $\pu{mol-1}$. It is shown that way in your first reference as well, i.e. $$ N_\mathrm{A} = \pu{6.02214076E23 mol-1}. $$

What is "dimensionless" is the Avogadro number or the number of atoms / molecules in a single $\pu{mol}$ of that substance, sometimes written as $N$ or $N_0$.

The numerical value is the same, but the concept is different. Basically the relationship between the two is: $N_0 = (\pu{1 mol})\times N_\mathrm{A} = \pu{6.02214076E23}$

See also the answer discussed in the comment by Ed V, where the new definition of the constant is explained: Effects of Changing Avogadro's Constant.

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The dimensions of Avogadro's Constant is $1/\mathsf{N}$. In SI units, this is $\pu{1/mol}$.

Your unit analysis is correct. I have seen this type of question before, I believe the confusion is caused by people normally saying particles per mole, and "particles" is a "phantom" unit (similar to radian). But either way, the SI units are definitely $\pu{1/mol}$.

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