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There is something I don't understand about the units of unified atomic mass.

I have seen on Khan Academy that $1$ unified atomic mass ($u$) is equal to $1.6605\times 10^{-27}$ kg.

But after 2 videos he writes that the unit of $u$ is $\frac{\mathrm{g}}{\mathrm{mol}}$. Why is it true?
Why I can say that according to table of elements, for hydrogen, $u=1.01 \frac{\mathrm{g}}{\mathrm{mol}}$

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    $\begingroup$ Suppose some eggs are 500 g/dozen. How much would they be per egg? $\endgroup$
    – The Photon
    Commented Oct 3, 2020 at 16:09

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In SI units a mole is simply a number, a measure of quantity. It is defined as the number of atoms in $12$ grams of carbon $12$. Its value is approximately $6.022 \times 10^{23}$. So a mass of $1$ gram/mole in kg is

$\displaystyle \frac {10^-3}{6.022 \times 10^{23}} \approx 1.66 \times 10^{-27}$ kg

If an atom (or a molecule) has a mass of $n$ atomic mass units then one mole of that atom/molecule will have a mass of $n$ grams. One mole of carbon $12$ has a mass of exactly $12$ grams so one atom of carbon $12$ has a mass of exactly $12$ atomic mass units.

The reason that one atom of hydrogen has a mass slightly greater than $1$ atomic mass unit is because an atom of carbon $12$ is slightly less than $12$ times as massive as an atom of hydrogen, due to binding energy.

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