# What is the dimension of Avogadro's constant?

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?

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: $$N_0 = (\pu{1 mol})\times N_\mathrm{A} = \pu{6.02214076E23}$$
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 $$\pu{1/mol}$$.