4. Discussion

The Avogadro constant is in essence a conversion factor between atom-scale and macroscopic units. Its scale size is therefore specified to many significant figures historically, by the relationship between macroscopic and atomic units of mass. If the new-SI allows us to select an exact value for this constant, there may be scientific and pedagogical value for using a physically-meaningful model. The scientific value, for example, might be found in our ability to construct and maintain precise molar/mass standards by simply counting the number of atoms (e.g. with an electron microscope) to make sure all atoms are still there before putting the standard to use. Carbon and its graphite/graphene structures are already among the most intensely-studied by nano-technologists, and the most stable at least in the absence of hot oxygen or molten iron. Thus a machine able to ``turn-out" a nanotube of arbitrary length might be able to dispense masses precisely defined by that length, since for instance a one-meter length of the 1 nm diameter tube shown in the experimental image (Unrau2009) would contain only ~200 femto-moles of carbon. 

Independent of our ability to manufacture and maintain 12-gram (or much smaller) graphite standards which contain a precise number of moles of 12C, on the pedagogical front one can certainly build graphite and ``pretend-graphite" models of the hexagonal prism. Figure 4 illustrates with an m = 4 hex-prism having form-factor similar to that of the m = 51,150,060 version. This can be constructed in the rescaled size of the Avogadro's number prism using today's 3D printing technology. 

If the prism is made of graphite, or a substance with density comparable to graphite, then it will give folks a concrete idea of the space taken up by a mole of condensed matter as well as the weight of a mole of carbon. The weight of a mole of other atoms will be proportionally heavier or lighter according to the ratio between atomic weights.

The space taken up by a mole of matter for most solids and liquids will be a bit larger than this because of the high atomic density of graphite (≈1.13×1023/cc), but not by much since e.g. the number of atoms per cubic centimeter in elemental solids and liquids is around (4.70±2.56)×1022. Equivalent sphere radii (which go inversely as the cube root of density) show even less variability, with a standard deviation less than 20% of the mean. 

Acknowledgements: I would like to thank colleagues across the regional nanoalliance for decades of interesting specimens to examine and think about, which investigations in turn provided background information for this note.

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