Ten reasons NOT to fix the numerical value of the Avogadro constant

In September 2010, I circulated a draft paper to several authors who had publically proposed redefining the mole in terms of a fixed numerical value of the Avogadro constant, describing why I thought such a redefinition would be a bad idea. A revised version [1] of that paper has now been published in Metrologia. A second paper [2], describing a problem with the current “conventional definition” of amount of substance, has also been accepted for publication by Metrologia. A discussion of the didactical problems relating to amount of substance has been submitted to Journal of Chemical Education [3].

1. There would be no metrological benefit from fixing the numerical value of N A . It is widely accepted that fixing the numerical values of the Planck constant h and the elementary charge e would allow the results of many measurements to be expressed more precisely in SI units, so-called "quantum metrology". No such claim can be made for fixing the numerical value of N A . There are two nontrivial physical constants whose uncertainties would depend on the uncertainty in N A : the Faraday constant F and the molar gas constant R (assuming the kelvin is also to be redefined in terms of a fixed numerical value of the Boltzmann constant k B ). 1 The residual uncertainties in F and R would still be almost three orders of magnitude lower than the relative measurement uncertainties of the best direct determinations of these constants, and so of the best measurements of the corresponding physical quantities. (see also Appendix 1)

No new techniques have become available for the general measurement of amount of substance.
The metre was redefined in 1983 in part because laser interferomery had arisen as an extremely precise method of measuring length. The current push towards redefining the kilogram in terms of a fixed {h} and the ampere in terms of a fixed {e} comes in part because of the measurement of electrical quantities by the Josephson effect and the quantum Hall effect. No comparable development in the measurement of n has occurred that might justify a redefinition of the base unit. 3. Amount of substance is not a count of entities, any more than electric charge is a count of elementary charges. The unit of amount of substance wasn't introduced because chemists didn't want to deal with big numbers: it was in use (as the Mol. or mol.) even before Raoult needed to describe colligative properties, such as boiling-point elevation and melting-point depression, in 1882, and before the Avogadro constant had been estimated.
Despite the title of their most recent paper [9], the International Avogadro Coordination (IAC) do not "count atoms of silicon": they compare a measurement of a microscopic quantity (the {220} lattice distance of silicon) with a measurement of a macroscopic quantity (the volume of a one-kilogram silicon sphere), as is the basis for almost all practical measurements of amount of substance. 4. Amount of substance is a useful quantity dimension at the macroscopic scale, and should not be confused with the dimension 1: any unit of amount of substance based on a fixed {N A } would risk being confused with a unit of dimension 1, as already happens despite the current definition. A statement such as "the standard molar entropy of chlorine gas is 223.081(10) JK -1 mol -1 " obviously refers to an ensemble of chlorine molecules, and cannot be reduced to the scale of an individual molecule. While convenient in contexts such as industrial chemistry and chemical engineering, the use of specific quantities (per unit mass) represents a loss of information, as can be illustrated by Clausius' failure to realize that all ideal gases share the same molar gas constant. 5. The Avogadro constant is irrelevant for the measurement of amount of substance. Chemists were measuring stoichiometric quantities for more than a hundred years before Perrin determined N A . The vast majority of modern measurements of n would be recognizable to Dalton and Gay-Lussac. 6. The mole is not "thought of by chemists as an Avogadro number of entities" as the IUPAC ICTNS would have us believe. I contend that never in the field of chemical science has anyone ever thought "I've got to weigh out 6.02 × 10 21 molecules of benzoic acid" or "I'm going to need 1.5 × 10 21 hydrogen ions to neutralize that solution." 7. The current confusion surrounding amount of substance does not stem from the current definition of the mole. In my experience, it stems from an undue emphasis on the numerical value of the Avogadro constant when introducing the subject. Educators have great fun telling kids that there are 6 × 10 23 atoms in 12 grams of carbon 12, and that that's a really big number, far too large to count: is it any surprise that students are confused as to how they're supposed to measure it? The standard approach to teaching amount of substance misses the more fundamental point entirely: that one doesn't need to know the numerical value of N A to measure n, one merely needs to know the relative masses (or other relative quantities) of the entities in question.

A definition of the mole in terms of a fixed numerical value of N A would not be "conceptually simpler" than the current definition.
It is only conceptually simpler in terms of a misconception of amount of substance as a count of entities. The present definition allows the theory behind the calculation of molar mass to be explained to elementary students: the proposed redefinition would incorporate all the high-level theory of the Rydberg constant into the justification of everyday molar-mass calculations. 9. The current measurement uncertainty in N A is conceptually meaningful.
The Avogadro constant relates quantities measured at the macroscopic scale with similar quantities measured at the atomic scale. The uncertainty in the measured value of N A is a simple statement of the uncertainty in our ability to compare the two scales of measurement. This uncertainty would still be present in any redefined system of units, it would merely be transferred to the molar mass constant M u and so retained in the atomic mass constant m u . There is no benefit to be gained from simply transferring the measurement uncertainty from a well-known physical constant onto an (unduly) obscure one. 10. There is no reason to remove the implicit link to the kilogram from the definition of the mole, especially as amount of substance is most commonly measured through measurements of mass. The ampere is currently defined in terms of the newton because, for nearly 100 years, the most accurate way of measuring electric current was through the Biot-Savart law. The metre is defined in terms of the second because the most accurate way of measuring length (at the laboratory scale) is by laser interferometry. Under the "explicitconstant" definitions proposed by the CCU [8], not one of the SI base units would be defined by a constant of the same kind as the unit! Leonard's contention that the uncertainty in the realization of the kilogram is transmitted to the mole [10] is incorrect: the current definitions of the mole and the kilogram are exact, although any practical realization of the mole will, of course, have a measurement uncertainty. The contention of May et al. [11] that users consider the mole to be a unit of mass because of the present definition is simply illogical: users do not consider the ampere to be a unit of force, after all!
The conditions for redefinition of the kilogram [12,13] have not been met [9]  I look forward to the proposed Mise en pratique for the redefined mole, upon which the chemical metrology community has yet to have a chance to debate. If it suggests that chemists should undertake precision dimensional and density measurements on single crystals in order to measure n at the highest level, it would be a very eloquent argument against the proposed redefinition; if it merely repeats the principles of the current Mise en pratique [16], it would be an admission that the proposed redefinition is, in fact, impractical and unnecessary.
For these reasons, I urge the CCQM to reconsider its stated "preference" for a redefinition of the mole in terms of a fixed value of the Avogadro constant. Users are served perfectly well by a definition based on the molar mass of carbon 12, and the admitted shortcomings in the present wording of the definition could easily be resolved by an "explicit-constant" definition of the type: The mole, mol, is the unit of amount of substance; its magnitude is set by fixing the numerical value of the molar mass of unbound carbon 12 atoms, at rest and in the ground state, to be equal to exactly 0.012 when it is expressed in the unit kg mol -1 . In this International Year of Chemistry, our time and energy would be better spent explaining why we don't measure amount of substance by counting elementary entities, rather than by pretending that somehow we do.

Appendix 2: Molar Planck constant
An oversight in the preparation of manuscript [1] led to the omission of a discussion of the molar Planck constant N A h. The discussion is important in this context, as May et al. [11] include the molar Planck constant in their list of physical constants whose value would be fixed by fixing the numerical value of the Avogadro constant.
Under the current CODATA adjustment [17], the molar Planck constant is not dependent on the values of either N A or h: instead it is given by (1) where α, A r (e) and R ∞ are independently refined constants while c and M u have defined values. As such, the uncertainty in the value of N A h (u r = 1.4 × 10 -9 ) is much lower than the uncertainties in the values of N A or h on their own, and is essentially due to the uncertainty in α 2 .
Under proposals to redefine the kilogram and the mole in terms of fixed numerical values of h and N A respectively, (1) would have to be reinterpreted as a definition of M u (2), which would have the same measurement uncertainty as N A h at present.
There has been some interest in the utility of the molar Planck constant as a stringent test of Einstein's mass-energy equivalence [18,19]. The principle behind such tests is the examination of a neutron-capture reaction A X(n,γ) A+1 X (more than one γ photon is usually emitted), where (3) must hold for mass-energy equivalence. ∆E = c 2 ∆m (3) The importance of the molar Planck constant in this method is that ∆m can only be measured in daltons, but must be compared to γ wavelengths measured in metres. The conversion factor (CODATA 2006 values) is (1 m -1 )h/c = 1.331 025 0394(19) × 10 -15 Da (4) with u r = 1.4 × 10 -9 , more than an order of magnitude better than the u r = 5.0 × 10 -8 in the value of the atomic mass constant [17].
As is discussed in [1], the uncertainty in the value of m u would be much reduced (effectively to the uncertainty in α 2 ) by redefining the kilogram in terms of a fixed {h}, but no further reduction in the uncertainty would be achieved by redefining the mole in terms of a fixed {N A } (because of the consequential uncertainty in M u ). For the same reason, the conversion factor in (4) would still have the same measurement uncertainty with fixed {N A }. Hence a redefinition of the mole in terms of fixed {N A } would not allow an improvement in tests of E = mc 2 by this method. It is noted in passing that the current best measurements of nuclear binding energies (expressed in terms of measured wavelengths) have uncertainties more than two orders of magnitude higher than u r (α 2 ) [19,20], so this is not currently the limiting factor in our ability to test mass-energy equivalence: the Avogadro constant remains as irrelevant to practical quantum metrology as it is to practical amount-of-substance measurements.
• Replace point four: "although the existing definition of the candela is not linked to a fundamental constant, it may be viewed as being linked to an exact value of an invariant of nature," with "although the existing definitions of the mole and the candela are not linked to fundamental constants, they may be viewed as being linked to exact values of invariants of nature," Section "the International Committee will also propose" • In the introductory paragraph, insert ", mole" after "metre" • Insert new point after second point: "the mole, mol, is the unit of amount of substance; its magnitude is set by fixing the numerical value of the molar mass of unbound carbon 12 atoms, at rest and in the ground state, to be equal to exactly 0.012 when it is expressed in the unit kg mol -1 ," Section "In consequence" • Delete fifth point "the definition of the mole in force since 1971 (14th CGPM, 1971, Resolution 3) based on a definition whereby the molar mass of carbon 12 had the exact value of 0.012 kg mol -1 will be abrogated," • In sixth point: insert ", mole" after "second"; insert ", the 14th CGPM (1971, Resolution 3)" after "the 13th CGPM (1967/68, Resolution 1)" Section "further notes that on the same date" • Delete fourth point "that the molar mass of carbon 12 M( 12 C) will be exactly 0.012 kg mol -1 but with a relative uncertainty equal to that of the recommended value of N A just before the redefinition and that subsequently its value will be determined experimentally." Section "encourages" • Delete "and N A ,"; insert "and" before "k" Section "invites" • In second point: delete "and mole,"; insert "and" before "kelvin"