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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 $N_A$?

J. C.
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1 Answers1

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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.

freecharly
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  • 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. – J. C. Nov 13 '16 at 18:13
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    @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. – freecharly Nov 13 '16 at 18:32
  • 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? – J. C. Nov 13 '16 at 18:48
  • @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. – freecharly Nov 13 '16 at 19:12
  • 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. – J. C. Nov 15 '16 at 18:35
  • @Joao - You are, in principle, right. Very often different units for the same thing are chosen for convenience. But you would not like to teach children math specifying 7+3 apples and 3 +7 eggs in moles units instead of 7 and 3. – freecharly Nov 15 '16 at 18:44
  • My point is not that. Of course you use different units for different contexts; I think nobody would measure the distance between Earth and the Sun in $mm$, nor would they measure the height of the Eiffel Tower in light-years. I'm asking if these "units" do or do not have the same dimensions, in this case, N (amount of substance). – J. C. Nov 15 '16 at 18:49
  • @Joao - Yes, you are right. The dimension for length is the same for all different length units. The amount of substance you could consider similarly. But you should not forget that in different unit systems the same thing can also have different dimensions, or no dimension in one and dimension in the other, depending how you define the quantity. I gave you the example that the capacitance of a capacitor has the dimension [capacitance]=[charge]/[voltage] with unit $F=As/V$ in the Si system, and in the Gauss system the dimension [length] with unit $cm$. – freecharly Nov 15 '16 at 19:08