Avogadro's Number

Avogadro's number, $N_A = 6.022 \times 10^{23}\ \text{mol}^{-1}$, is defined as the number of elementary entities (atoms, molecules, ions, or any other specified particle) contained in exactly one mole of a substance -- historically fixed so that one mole of a substance's mass in grams equals its molar mass (e.g. one mole of oxygen molecules, $32\ \text{g}$, contains exactly $N_A$ oxygen molecules).

Its significance throughout this chapter, and throughout kinetic theory generally, is that it is the single conversion factor bridging two otherwise completely separate scales of description: the MICROSCOPIC scale of individual molecules (with their individual masses $m$, individual kinetic energies $\tfrac{1}{2}m\overline{v^2}$, individual degrees of freedom), which are far too small and far too numerous to ever observe or count directly, and the MACROSCOPIC, MOLAR scale of laboratory quantities (moles $n$, molar mass $M$, the universal gas constant $R$), which are what can actually be measured and controlled in an experiment. Every microscopic-to-macroscopic conversion used earlier in this chapter passes through $N_A$: molar mass relates to molecular mass by $M = N_Am$; Boltzmann's constant relates to the universal gas constant by $k_B = R/N_A$; and the total number of molecules in $n$ moles is $N = nN_A$. …