What is the mass of a proton?

My textbook explains that the deviation from integer atomic mass is caused by mass difference between proton and neutron, which are $$1.67262·10^{-27}$$ kg and $$1.67493 ·10^{-27}$$ kg, respectively.

If SI system currently define $$1$$ mole as the number of entities in exactly $$12$$ g of $$C^{12}$$, which is Avogadro's number $$N_A = 6.02214\times 10^{23}$$, then $$6.02214\times10^{23} ( C^{12}) = 12g$$ Likewise atomic mass unit (u) for relative mass is also defined by $$C^{12}$$ isotope as $$12 u = C^{12}$$ atom. So $$1 g = 6.02214\times 10^{23}u$$ Then if $$12 u = 6(m_{proton}+m_{neutron}+m_{electron})=1C^{12}$$, and $$m_{proton} (as stated by the text), $$6\times\frac{m_{proton}}{m_{total}}<\frac1{2}$$proton mass ratio in carbon-$$12$$ is less than half, and one proton mass will be $$m_{proton}$$ $$1 u$$ = $$1.66054 ·10^{-27}$$ kg

This proton mass is less than the value in the book $$1.67262·10^{-27}$$ kg and would produce a much lower atomic mass for $$H^1$$ than $$1.00783$$ u. Catastrophic! Why is this and what am I missing here?

• mass defect, see e.g. chem.purdue.edu/gchelp/howtosolveit/Nuclear/… Sep 15, 2021 at 17:32
• The 1.00783 u is the average atomic mass of hydrogen and also includes a small fraction of deuterons. Note also that the definition of the Avogadro constant has been changed and is an exact number since 2019. Therefore, the mass of a carbon atom now carries uncertainty.
– Paul
Sep 15, 2021 at 17:43
• The key question is why the atomic mass be integers and integers in which units. I don't think there is such a fundamental requirement. Sep 15, 2021 at 20:10
• I apologise for the wording of my original post which may have caused some confusion. All the atomic mass values I quote (from Clayden's Organic Chemistry, mass spectrometry, p.51) are not average isotope abundance but the exact mass of a single isotope. @KarstenTheis and matt_black pointed out that this difference between calculated value and actual mass of atoms is contributed by mass defect and formation energy, therefore it's observed that calculating atomic mass from the sum of its parts would result in a value higher than the actual exact mass. Sep 16, 2021 at 8:03
• The definition of Mole has been changed (tough this does not change the issue that you have). Just for sake of precision as this is a chemistry site. Similarly a proton does not have atomic mass. Sep 16, 2021 at 9:46