The link to inertia mass

What is the link between unified atomic mass unit and inertia mass?

Jean Perrin’s and other scientists’ experiments in the early 1900s to determine the Avogadro constant were based on a gram-molecule, which is the mass of a gas that occupies the same volume as two grams of hydrogen gas at the same temperature and pressure. Their experiments produced a range of values for the constant when different gases are used. This is because different real gases, though having the same amount of particles, have different volumes.

A better definition of the mole is therefore needed and was chosen in 1967 to be:

The amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon-12.

The choice to peg the Avogadro constant to carbon-12 seems arbitrary. However, it is due to this definition that we are able to link the unified atomic mass unit scale to the inertia mass scale. So,

$1^{12}C=12u$

$\frac{0.012\; kgmol^{-1}}{N_{A}\; mol^{-1}}=12u\; \; \; \; \; \; \; (1)$

$1u=1.660539\times 10^{-27}\; kg\; \; \; \; \; \; \; (2)$

We can rewrite eq1 as

$1u=\frac{M_{u}}{N_{A}}g$

where M_uis the molar mass constant and is equal to 1 gmol^-1.

Even though the definition of the mole was changed to ‘a mole is the amount of substance of a system that contains exactly 6.02214076 x 10²³ elementary entities’ in Nov 2018, the peg of 1¹²C to 12 u remains. However, the exact value of the Avogadro constant leads to the molar mass of carbon-12 having a relative uncertainty in the order of 10^-10. It is no longer exactly 0.012 kgmol^-1 and is given by the formula (see the article on ‘Bohr model‘ for derivation):

$M(^{12}C)=\frac{24hR_{\infty }N_{A}}{c\alpha^{2}A_{r}(e)}\; \; \; \; \; \; \; \; (3)$

where

$h$ is the Planck constant

$R_{\infty }$ is the Rydberg constant

$c$ is the speed of light

$\alpha$ is the fine-structure constant

$A_{r}(e)$ is the ‘relative atomic mass’ of an electron

The uncertainty in the value of 0.012 kgmol^-1, which will be determined in future experiments, is primarily due to the uncertainty in the fine-structure constant.

Similarly, the molar mass constant , which was a constant with the value 1 gmol^-1, is now described by the formula:

$M_{u}=\frac{2hR_{\infty}N_{A}}{c\alpha ^{2}A_{r}(e)}$

By the same logic, the unified atomic mass unit has the formula:

$u=\frac{2hR_{\infty }}{c\alpha^{2}A_{r}(e)}$

The link to inertia mass

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