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Molar mass constant

## Summary

The molar mass constant, usually denoted by Mu, is a physical constant defined as the ratio of the molar mass of an element (or a compound) and its relative mass.

The mole and the relative atomic mass were originally defined in the International System of Units (SI) in such a way that the constant was exactly g/mol. That is, the numerical value of the molar mass of an element, in grams per mole of atoms, was equal to its atomic mass relative to the atomic mass constant, mu. Thus, for example the average atomic mass of chlorine is approximately 35.446 daltons, and the mass of one mole of chlorine atoms was approximately 35.446 grams.

On 20 May 2019, the SI definition of mole changed in such a way that the molar mass constant is no longer exactly 1 g/mol. However, the difference is insignificant for all practical purposes. According to the SI, the value of Mu now depends on the mass of one atom of carbon-12, which must be determined experimentally. As of that date, the 2018 CODATA recommended value of Mu is 0.99999999965(30)×10−3 kg⋅mol−1.[1][2]

The molar mass constant is important in writing dimensionally correct equations.[3] While one may informally say "the molar mass of an element M is the same as its atomic weight A", the atomic weight (relative atomic mass) A is a dimensionless quantity, whereas the molar mass M has the units of mass per mole. Formally, M is A times the molar mass constant Mu.

## Prior to 2019 redefinition

The molar mass constant was unusual (but not unique) among physical constants by having an exactly defined value rather than being measured experimentally. From the old definition of the mole,[4] the molar mass of carbon 12 was exactly 12 g/mol. From the definition of relative atomic mass,[5] the relative atomic mass of carbon 12, that is the atomic weight of a sample of pure carbon 12, is exactly 12. The molar mass constant was thus given by

${\displaystyle M_{\text{u}}={{\text{molar mass }}[M(^{12}\mathrm {C} )] \over {\text{relative atomic weight }}[A_{\text{r}}(^{12}\mathrm {C} )]}={{12\ {\rm {g/mol}}} \over 12}=1\ {\rm {g/mol}}}$

The molar mass constant is related to the mass of a carbon-12 atom in grams:

${\displaystyle m({}^{12}{\text{C}})={\frac {12\times M_{\text{u}}}{N_{\text{A}}}}}$

The Avogadro constant being a fixed value, the mass of a carbon-12 atom depends on the accuracy and precision of the molar mass constant.

(The speed of light is another example of a physical constant whose value is fixed by the definitions of the International System of Units (SI).)[6]

## Post-2019 redefinition

Because the 2019 redefinition of SI base units gave the Avogadro constant an exact numerical value, the value of the molar mass constant is no longer exact, and will be subject to increasing precision with future experimentations.

One consequence of this change is that the previously defined relationship between the mass of the 12C atom, the dalton, the kilogram, and the Avogadro number is no longer valid. One of the following had to change:

• The mass of a 12C atom is exactly 12 daltons.
• The number of daltons in a gram is exactly equal to the numerical value of the Avogadro number: i.e. 1 g/Da = 1 mol ⋅ NA.

The wording of the 9th SI Brochure[Note 1] implies that the first statement remains valid, which means the second is no longer true. The molar mass constant is still very close to 1 g/mol, but no longer exactly equal to it. Appendix 2 to the 9th SI Brochure states that "the molar mass of carbon 12, M(12C), is equal to 0.012 kg⋅mol−1 within a relative standard uncertainty equal to that of the recommended value of NAh at the time this Resolution was adopted, namely 4.5×10−10, and that in the future its value will be determined experimentally",[7][8] which makes no reference to the dalton and is consistent with either statement.

## Notes

1. ^ A footnote in Table 8 on non-SI units states: "The dalton (Da) and the unified atomic mass unit (u) are alternative names (and symbols) for the same unit, equal to 1/12 of the mass of a free carbon 12 atom, at rest and in its ground state."

## References

1. ^ "2018 CODATA Value: molar mass constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
2. ^ Mohr, Peter J.; Taylor, Barry N. (2005). "CODATA recommended values of the fundamental physical constants: 2002". Rev. Mod. Phys. 77 (1): 1–107. arXiv:1507.07956. Bibcode:2005RvMP...77....1M. doi:10.1103/RevModPhys.77.1.
3. ^ de Bièvre, Paul; Peiser, H. Steffen (1992). "'Atomic Weight' — The Name, Its History, Definition, and Units" (PDF). Pure and Applied Chemistry. 64 (10): 1535–43. doi:10.1351/pac199264101535.
4. ^ International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), pp. 114–15, ISBN 92-822-2213-6, archived from the original (PDF) on 2017-08-14
5. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "relative atomic mass (atomic weight)". doi:10.1351/goldbook.R05258
6. ^ Penrose, r (2004). The Road to Reality: A Complete Guide to the Laws of the Universe. Vintage Books. pp. 410, 411. ISBN 978-0-679-77631-4. "... the most accurate standard for the metre is conveniently defined so that there are exactly 299,792,458 of them to the distance travelled by light in a standard second, giving a value for the metre that very accurately matches the now inadequately precise standard metre rule in Paris."
7. ^ "Resolutions adopted" (PDF). Bureau international des poids et mesures. November 2018. Archived from the original (PDF) on 2020-02-04. Retrieved 2020-02-04.
8. ^ Nawrocki, Waldemar (2019-05-30). Introduction to Quantum Metrology: The Revised SI System and Quantum Standards. Springer. p. 54. ISBN 978-3-030-19677-6.