Molar mass constant

The molar mass constant, usually denoted by M_u, is a physical constant defined as one twelfth of the molar mass of carbon-12: M_u = M(¹²C)/12.^[1] The molar mass of an element or compound is its relative atomic mass (atomic weight) or relative molecular mass (molecular weight or formula weight) multiplied by the molar mass constant.

The mole and the atomic mass unit (dalton) were originally defined in the International System of Units (SI) in such a way that the constant was exactly 1 g/mol, which made the numerical value of the molar mass of a substance, in grams per mole, equal to the average mass of its constituent particles (atoms, molecules, or formula units) relative to the atomic mass constant, m_u. Thus, for example, the average molecular mass of water is approximately 18.015 daltons, making the mass of one mole of water approximately 18.015 grams.

On 20 May 2019, the SI definition of mole changed in such a way that the molar mass constant remains nearly but no longer exactly 1 g/mol. However, the difference is insignificant for all practical purposes. According to the SI, the value of M_u now depends on the mass of one atom of carbon-12, which must be determined experimentally. The 2022 CODATA recommended value of M_u is 1.00000000105(31)×10⁻³ kg⋅mol⁻¹.^[2]^[3]

The molar mass constant is important in writing dimensionally correct equations.^[4] While one may informally say "the molar mass M of an element 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 M_u.

Prior to 2019 revision

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,^[5] the molar mass of carbon-12 was exactly 12 g/mol. From the definition of relative atomic mass,^[6] 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

M_{\text{u}}={{\text{molar mass }}[M(^{12}\mathrm {C} )] \over {\text{relative atomic mass }}[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:

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).)^[7]

Post-2019 revision

Because the 2019 revision of the SI 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 exact. 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 ⋅ $N A$ .

The wording of the 9th SI Brochure^{[Note 1]} implies that the first statement remains valid, which means the second is no longer exactly 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 $N A h$ at the time this Resolution was adopted, namely 4.5×10⁻¹⁰, and that in the future its value will be determined experimentally",^[8]^[9] which makes no reference to the dalton and is consistent with either statement.

Notes

^ 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."