An Exact Value for Avogadro's Number
Untangling this constant from Le Gran K could provide a new definition of the gram
Farewell to Le Gran K
A similar solution can solve the dilemma of the current time-dependent definition of Avogadro's number. The idea is simply to define N A , once and for all, as was done for the speed of light. Unlike that case, however, the range of known possible values for N A is astronomical. Three desirable basic properties for a reasonable value for N A help narrow the search.
First, since Avogadro's number purports to count the number of atoms in some theoretical specimen, its value should be an integer, as any schoolchild would expect. This would avoid having to interpret one-third of an atom, or worse yet, 1/p of an atom.
Second, the value chosen should be within the currently accepted range, (6.0221415 ± 0.0000010) × 1023.
Third, the value chosen for Avogadro's number should ideally have some inherent physical significance. Since volumes of objects are measured cubically, as in cubic centimeters and cubic yards, and not spherically (for example, via volumes of spheres with unit radii or diameters), and since the current definition of Avogadro's number counts the number of atoms in a solid specimen,it is reasonable to imagine the object as being a perfect geometrical cube. That implies that the value chosen should be a perfect numerical cube.
The range of acceptable integers in the current estimate of N A is two hundred quadrillion (2 × 1017), but within that huge range of values there are only 10 perfect cubes—from 84,446,8843 to 84,446,8933. For our purposes, any one of those 10 may be used, but the one closest to the best current estimate of Avogadro's number, and the only one accurate to within one unit in the eighth significant digit of the current best estimate, is
N A * = 602,214,141,070,409,084,099,072 = 84,446,8883.
Our proposal is simply to define Avogadro's number, permanently, as was done with the speed of light and with the second, and to set it equal to this specific integer. If the sides of the cube of atoms were only six atoms shorter or longer, the number of atoms it contains would no longer be within the currently accepted range for Avogadro's number, since 84,446,8833 = (6.02214034+) × 1023 and 84,446,8943 = (6.02214269+) × 1023.
Since the shape of a volume certainly affects the numbers of molecules it can contain—extremely long, thin cylinders can contain none—it seems natural to ask that the shape of the defining volume be a cube. Of course any other solid shape could also suffice as the defining object, but using a rectangular solid or parallelpiped would require specification of three numbers: the length, width and height. Using a sphere precludes choosing an integer at all, because of the irrationality of p.
At first glance, another possible candidate for the exact value of Avogadro's number might be 602,214,150, 000,000,000,000,000, which is dead center in the current range of values. This value, however, has little physical significance. It is neither a perfect cube nor a perfect square, so no perfect geometrical cube or square of atoms could be constructed which has that exact volume or area.
Moreover, the method of simply using the most recent best estimate of N A is not robust, unlike the methods that were used for defining fixed values for the speed of light and the second. If current experimental estimates of Avogadro's number increase the known number of significant digits by four or five places, for example, the "current best estimate" method of fixing the value for Avogadro's number would presumably also change by those same four or five digits.
The fixed values for the meter in terms of the speed of light and for the second in terms of vibrations of a cesium atom, however, were nearest-integer solutions, insensitive to further fractional refinements of the exact measurements. In exactly that same spirit, the definition of N A * above is also a nearest-integer solution—the nearest integral side length of a cube containing Avogadro's number of atoms. As such, the value chosen is also insensitive, within one atom either way, to improved experimental estimates of N A . The choice of an integer value for N A * seems essential, whereas the requirement that it be a perfect cube is largely esthetic, but with practical and intuitive physical significance as well.