nLab Avogadro constant

In macroscopic thermodynamic systems one deals with a large number of microscopic particles. The number of particles is, for ‘human sized’ systems (e.g. engines, rooms), roughly on the order of the Avogadro constant (or ‘Avogadro's number’ in historical usage). This constant is used as a scaling factor between macroscopic and microscopic observations.

More precisely, Avogadro’s number is defined as the number of atoms in $0.012$ kg of the carbon isotope $\mathrm{C}^12$. This is roughly the real number

$$6.02214179(30) \times 10^{23} \;\in\; \mathbb{R} \,.$$

An amount of any chemical substance which contains precisely this number of molecules is called a mole of substance. So in particular, by definition, a mole of $\mathrm{C}^{12}$ is precisely of weight $12 \,\mathrm{g}$.

Attempts to define the SI unit of mass, i.e. the kilogram, in terms of invariant and basic physical concepts generally involve trying to measure Avogadro’s number at increasing levels of accuracy. As with the speed of light, once the measurement of this derived quantity (taking the kilogram as the mass of certain reference kilogram cylinders) reaches a certain threshold of accuracy, it will make sense to specify Avogadro’s number exactly and then derive the units of mass.

Of course the choice of $12 \,\mathrm{g}$ of $\mathrm{C}^{12}$ is entirely arbitrary and just serves as some example of a typical laboratory-scale macroscopic quantity.

Accordingly, later it was found useful to have different such macroscopic reference quantities, for instance the pound mole in addition to the (standard) gram mole. For exchanging between these it was found useful to introduce units and treat ‘amount of substance’ as a dimension?.

Then the Avogadro constant is taken to be

$$N_A = 6.02214179(30) \times 10^{23} \,mol^{-1} \,.$$

(Strictly speaking this is $N_A = 1 \in \mathbb{R}$ in natural units.)