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Note on uncertainties resulting from proposed kilogram redefinitions

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IOP PUBLISHING

METROLOGIA

Metrologia 44 (2007) L4–L6

doi:10.1088/0026-1394/44/2/L02

LETTER TO THE EDITOR

Note on uncertainties resulting from proposed kilogram redefinitions B P Leonard Department of Mechanical Engineering, The University of Akron, Akron, OH 44325, USA

Received 12 February 2007 Published 14 March 2007 Online at stacks.iop.org/Met/44/L4 Abstract Redefinitions of the kilogram based on either the electron mass, the carbon-12 atomic mass or a pseudo-mass proportional to the Planck constant are examined, together with respective resulting uncertainties in a number of important constants. The kilogram-to-dalton mass ratio is shown to play a central role in coupling mass unit definitions with those of units for amount of substance, since the kg/Da mass ratio is equal to the kilomole-to-entity amount-of-substance ratio in a compatible formulation. Because of this coupling, a redefinition of the unit for amount of substance using a fixed exact value of the kmol/ent ratio implies an exact value of the dalton in terms of the kilogram (regardless of how the kilogram itself is defined), thereby eliminating any need for a ‘correction factor’ in expressions involving amount-specific mass. In particular, compatible definitions guarantee that NA Da = Da ent −1 = kg kmol−1 = g mol−1 , exactly.

1. Coupling of units for mass and amount of substance We consider proposed kilogram redefinitions of the general form kg = NX∗ ma (X), where NX∗ is an exactly specified dimensionless constant, chosen to provide seamless continuity with the current artefact-based definition, and ma (X) is either the electron mass, me , the carbon-12 atomic mass, ma (12 C), or a pseudo-mass, mh , proportional to the Planck constant [1]. In each case, the kilogram would thus be defined as an exact multiple of a physically invariant quantity. This note examines the uncertainties in a number of important constants resulting from each of these definitions. If the kilogram-to-dalton mass ratio is represented by a dimensionless constant, N , then we have NX∗ ma (X) = kg = N /NG = N Da,

(1)

where NG is the number of atomic-scale mass units (dalton) per unit of mass, analogous to the Avogadro constant, NA , the number of atomic-scale amount-of-substance units (entity) per unit of amount of substance. Note that NG = 1 Da−1 , exactly. Similarly, NA = 1 ent−1 , exactly, so that, for amount of substance, we have kmol = N /NA = N ent. 0026-1394/07/020004+03$30.00

(2)

In this case, the unit entity—the amount of substance consisting of exactly one (specified elementary) entity—is the physically invariant quantity [2]. Note how the kilogram-todalton mass ratio, N , is also the kilomole-to-entity amountof-substance ratio. This is because Mu = kg kmol−1 = g mol−1 = Da ent−1 = NA Da = NA /NG , exactly; and terms in equation (1) are exactly Mu times the corresponding terms in equation (2). The kg/Da or kmol/ent ratio, N , is clearly of fundamental importance in any redefinitions of the units for mass and amount of substance, inherently coupling equations (1) and (2). For example, if we choose to retain the current definition of the dalton, Da = ma (12 C)/12, then N = (12 kg)/ma (12 C) is an exactly specified constant only if ma (X) = ma (12 C). For any other mass or pseudo-mass used for defining the kilogram, N will not be known exactly using this definition of the dalton. In particular, the numerical value of the Avogadro constant, expressed as NA = N kmol−1 , will be inexact. Alternatively, if we choose to fix N at an exact value, N ∗ , then in order to assure seamless continuity, we require that Da = ma (12 C)0 /12, exactly, which means that N ∗ = (12 kg)/ma (12 C)0 , exactly, where ma (12 C)0 is the nominal mass of the carbon-12 atom in kg at the time of transition, t = 0. In this case, we have Da = (1/N ∗ ) kg,

© 2007 BIPM and IOP Publishing Ltd

Printed in the UK

L4

Letter to the Editor

Table 1. Uncertainties in the values of the relative atomic mass of carbon 12, the carbon-12 mass ratio, the Avogadro constant, the electron mass and the Planck constant for inexact (N ) and exact (N ∗ ) kg/Da mass ratios and the invariant atomic-scale masses used to define the kilogram. The arrows indicate the influence of RC on NA for kg/Da = N and on Ar (12 C) for kg/Da = N ∗ . kg/Da

ma (X)

ur [Ar (12 C)]

ur (RC )

N N N∗ N∗ N∗

me mh ma (12 C) me mh

0 0 0 U1 U3

U1 U3 0 U1 U3

← ←

→ →

exactly, kmol = N ∗ ent, exactly, and NA = N ∗ kmol−1 , exactly. Clearly, the inexact N is related to the exact N ∗ by N = N ∗/RC , where RC is the carbon-12 mass ratio, RC = ma (12 C)/ma (12 C)0 . The uncertainty in N is thus that of RC , and this will depend on the particular form of ma (X) chosen for the mass unit definition. Another way of examining this follows from the identity Ar (12 C)/N ≡ 12RC /N ∗ , for cases in which ur (RC ) = 0. For the amount-of-substance unit, we can either choose Ar (12 C) = 12, exactly (i.e. the current definition), in which case N = N ∗/RC is inexact; or we can choose N = N ∗ , exactly (i.e. the so-called ‘fixed-Avogadro-constant’ definition), in which case Ar (12 C) = 12RC is inexact. Strictly speaking, the latter should be called the ‘exact kilomole’ definition: kmol = N ∗ /NA = N ∗ ent, exactly—since NA is always fixed exactly in terms of the physically invariant unit, entity: NA = ent−1 , exactly. In this interpretation, the reciprocal Avogadro constant, 1/NA ≡ ent, could be considered to be a fundamental physical invariant.

For the fixed-electron-mass definition of the kilogram, we have (3)

thereby fixing the value of me /kg exactly. The numerical values used for this and the other redefinitions are for illustrative purposes only; clearly, values with the lowest uncertainty at the time of transition would be used in order to assure seamless continuity. Equation (3) would also be used for realizing the kilogram. The relative standard uncertainty of ma (12 C)/kg— and thus of the carbon-12 mass ratio—in this case is ur (RC ) = ur [ma (12 C)/me ] = U1. Currently, U1 ≈ 4.4 × 10−10 [3]. The corresponding uncertainty in the value of the Planck constant is obtained from the identity (c0 me α 2 )/(2R∞ h) ≡ 1, giving ur [h/(J s)] = ur [R∞ /(m−1 )] + 2ur (α) = U2. Currently, ur [R∞ /(m−1 )] ≈ 6.6×10−12 [3] and ur (α) ≈ 7.0×10−10 [4]. So that U2 is approximately 14.1 × 10−10 . For the fixed-Planck-constant kilogram definition, we first write (4) kg = (6.022 1415 × 1026 )mh , exactly, where the numerical factor is N ∗ and mh is a pseudo-mass equal to exactly one dalton at the time of transition: mh = ν ∗ (h/c02 ) = ma (12 C)0 /12 = (1/N ∗ ) kg, exactly. This gives mh = (2.252 342 737 773 × 1023 Hz)(h/c02 ), exactly,

(5)

thereby fixing the value of h. Note that the pseudo-de Broglie– Compton frequency, ν ∗ , is comparable to that of the proton Metrologia, 44 (2007) L4–L6

ur (me /kg)

ur [h/(J s)]

U1 U3 0 0 0

0 U2 U1 0 U2

U2 0 U3 U2 0

mass, νp = 2.268 7318 . . . × 1023 Hz, as would be expected. Consequently, this two-part fixed-h definition does not involve any physically impossible quantities, and should be easily comprehended. Instruments that measure h or h/c02 would be used for realizing the kilogram from these equations. The relative standard uncertainty of me /kg = (me /mh )/N ∗ is ur (me /mh ) = U2. For ur (RC ), we have ma (12 C)/kg = [ma (12 C)/me ](me /mh )/N ∗ , so that ur (RC ) = U1 + U2 = U3 ≈ 18.5 × 10−10 . Finally, for the fixed-carbon-12-atomic-mass definition, we have kg = (5.018 451 25 × 1025 )ma (12 C),

(6)

noting that the numerical factor is N ∗/12, exactly. Here we have RC ≡ 1, so that ur (RC ) = 0; and the exact numerical value of NA (in number per kilomole) is N ∗ . For the electron mass, ur (me /kg) = U1. And for the Planck constant, ur [h/(J s)] = U1 + U2 = U3.

3. Summary

2. Redefinitions of the kilogram

kg = (1.097 769 2395 × 1030 )me , exactly,

ur [NA ·(kmol)]

These results are summarized in table 1 for the inexact N (= N ∗/RC ) for ma (X) = me and mh , and the exact N ∗ for each of the proposed kilogram redefinitions. In the first two rows, we see that choosing Ar (12 C) = 12, exactly, implies an inexact value of NA ·(kmol) = NA /(kmol−1 )—and this could equally be interpreted as kmol/ent, which, of course, is N itself. In this case, the dalton is defined in the traditional way, Da = ma (12 C)/12. All relative atomic-scale masses retain their current values and uncertainties expressed in terms of the inexact dalton—although they would, of course, be updated periodically. For ma (X) = ma (12 C), we have an exact Ar (12 C), RC ≡ 1 and the kilomole is defined as an exact number, N ∗ , of entities. The dalton is now (1/N ∗ ) kg, exactly, so that the absolute entity masses expressed in Da now have the same numerical values (and uncertainties) as the corresponding relative atomic-scale masses. When the exact kg/Da or kmol/ent ratio, N ∗ , is chosen and ma (X) = ma (12 C), Ar (12 C) is no longer exact; instead, Ar (12 C) = 12RC . In fact, it should be clear that, for any substance, Y, Ar (Y)(t=0+ ) = RC Ar (Y)(t=0− ) . At t = 0+ , the nominal value of RC is unity, with an uncertainty of U1 (for the fixed-me kilogram definition) or U3 (for the fixedh definition). But at subsequent times, as more precise measurements of ma (12 C)/kg become available, the nominal value of RC will deviate (very slightly) from unity, as its uncertainty decreases. And corresponding changes will occur in all atomic-scale masses expressed in terms of Da. This has caused some concern among researchers involved with L5

Letter to the Editor

proposals for SI unit redefinitions. For example, Mills et al, in their study of the fixed-h kilogram redefinition [5], recommended (albeit with different terminology) using the exact N ∗ for the kmol/ent amount-of-substance ratio together with an incompatible inexact N for the kg/Da mass ratio, in order to avoid disturbing relative atomic-scale mass values and their uncertainties expressed in terms of the inexact atomic mass unit, u. The incompatibility is then corrected, where necessary, by a multiplicative correction factor, (1 + κ)—which is equal to the carbon-12 mass ratio, RC , as defined here. Specifically, with Da = ma (12 C)0 /12 = (1/N ∗ ) kg, let u = ma (12 C)/12 = RC Da = (1/N ) kg; with Ar (Y) = ma (Y)/Da, let ATM r (Y) = ma (Y)/u. Then M(Y) = Ar (Y)Mu = RC ATM (Y)M u . Note that, in this r case, uNA = [(1/N ) kg][N ∗ kmol−1 ] = RC Mu = Mu . The correction factor is then set equal to unity for practical chemical calculations. This is an awkward strategy and might possibly be confusing, especially for beginning chemistry students. It would seem that the straightforward compatible approach of using the exact N ∗ for the kg/Da mass ratio as well as for the kmol/ent amount-of-substance ratio, i.e. using the exact dalton as the atomic-scale absolute mass unit, would be better in the long run—and much more easily comprehended. Since we are then dealing with absolute atomic-scale masses in terms of the exactly defined dalton, Da = (1/N ∗ ) kg, with expected decreases in U1 and U2 (and thus U3)— especially in ur (α)—in the near future, uncertainties in

L6

ma (Y)/Da are likely to soon be negligible compared with current uncertainties in Ar (Y). In the meantime, selected mass ratios of particularly small uncertainty could be appended to the main compilation of absolute masses represented by ma (Y)/Da. Periodic updates would be appropriate as the measured atomic-scale absolute mass values became more precise. We then have NA Da = Da ent−1 = Mu , exactly. And, of course, the well-known formula, M(Y) = Ar (Y)Mu , is then satisfied exactly, without the need for any correction factor.

References [1] Becker P, De Bi`evre P, Fujii K, Glaeser M, Inglis B, Luebbig H and Mann G 2007 Considerations on future redefinitions of the kilogram, the mole and other units Metrologia 44 1–14 [2] Leonard B P 2007 On the role of the Avogadro constant in redefining SI units for mass and amount of substance Metrologia 44 82–6 [3] Mohr P J and Taylor B N 2005 CODATA recommended values of the fundamental constants 2002 Rev. Mod. Phys. 77 1–107 [4] Gabrielse G, Hanneke D, Kinoshita T, Nio M and Odom B 2006 New determination of the fine structure constant from the electron g value and QED Phys. Rev. Lett. 97 030802/1–4 [5] Mills I M, Mohr P J, Quinn T J, Taylor B N and Williams E R 2006 Redefinition of the kilogram, ampere, kelvin and mole: a proposed approach to implementing CIPM recommendation 1 (CI-2005) Metrologia 43 227–46

Metrologia, 44 (2007) L4–L6