I think that "number of particles" should be a dimensional quantity, with the same dimension as "amount of substance", because they are only scaled by Avogadro's constant, which then should be dimensionless.

For instance, an electron in an hydrogen atom has an energy of $-2,18 \times 10^{-18} \:\text{J}$. Then, the ionization energy should be $2,18 \times 10^{-18} \:\text{J atom}^{-1} = 1312 \:\text{kJ mol}^{-1} $. Nevertheless, the standard is to consider the first one as plain joules, without the "amount" dimension.

Is there any reason behind this, and by consequence the dimensional character of $N_A$?

  • 1
    $\begingroup$ Possible duplicate: Why is the mole/“amount of substance” a dimensional quantity? $\endgroup$
    – lemon
    Nov 13, 2016 at 16:28
  • $\begingroup$ I have already read that before. It didn't answer my doubt, because I understand why moles are dimensional, I just think that "atoms"/"molecules"/etc. should be considered dimensional too, and by consequence $N_A$ would be dimensionless. $\endgroup$
    – J. C.
    Nov 13, 2016 at 16:39

1 Answer 1


Avogadros constant is not dimensionless. It is the number of atoms/molecules per mole. The mole is a substance unit which was introduced by chemists before the number of atoms/molecules per mole was actually known. The situation is similar to the arbitrary choice of the coulomb as a unit of charge which disregards the number of elementary charges it is composed of.

The number of particles is, indeed, dimensionless as long as you don't define it by the equivalent number of moles.

  • $\begingroup$ Sorry my question was not very clear. What matters here is not the Avogadro constant itself, but the "number of particles". If a mole of atoms has dimension N (amount), then $\frac{1}{6.022 \times 10^{23}} \text{mol}$, which corresponds to a single particle, should also have dimension N. The number of particles is a multiple of this, so it should be of dimension N too, from my point of view. $\endgroup$
    – J. C.
    Nov 13, 2016 at 18:13
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    $\begingroup$ @Joao - You are right, if you describe the number of particles via the mole, it has the unit of mole. The number of particles actually has the implicit unit "particle" which you just write in plain text. Maybe the SI should introduce this as a new unit. $\endgroup$
    – freecharly
    Nov 13, 2016 at 18:32
  • $\begingroup$ Then, how can you describe the number of particles if not via the mole? Also, if we consider the way I described it, I think that the most logical is to consider that "particle" is a submultiple of moles, like grams to kilograms, isn't it? $\endgroup$
    – J. C.
    Nov 13, 2016 at 18:48
  • $\begingroup$ @Joa0 - The number of particles is only secondary to the mole. This number was later added to the mole. You can, of course, define particle number via the mole. The comparison is not like between $kg$ and $g$, it is more like between the unit of capacitance farad ($F$) in the SI system as compared to the unit $cm$ (for capacitance) in the Gauss system. $\endgroup$
    – freecharly
    Nov 13, 2016 at 19:12
  • $\begingroup$ I know that at the beginning we didn't know how many particles there were in a mole, but now we do know, so why can't we change our approach to these concepts? Furthermore, could the "dozen" be considered a unit of amount of substance? If it could, I don't understand why "units" cannot be it too. $\endgroup$
    – J. C.
    Nov 15, 2016 at 18:35

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