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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 $m_{proton}<m_{neutron}$ (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?

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    $\begingroup$ mass defect, see e.g. chem.purdue.edu/gchelp/howtosolveit/Nuclear/… $\endgroup$ Sep 15, 2021 at 17:32
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    $\begingroup$ 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. $\endgroup$
    – Paul
    Sep 15, 2021 at 17:43
  • $\begingroup$ 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. $\endgroup$
    – AChem
    Sep 15, 2021 at 20:10
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    $\begingroup$ 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. $\endgroup$ Sep 16, 2021 at 8:03
  • $\begingroup$ 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. $\endgroup$
    – Alchimista
    Sep 16, 2021 at 9:46

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Your textbook is wrong and there are several confusions in your argument

The first issue is the textbook claim. Which is wrong for two possible reasons. One is that the masses quotes for atoms ni most periodic tables are for the natural abundance of the element given multiple possible isotopes (hydrogen has three: one with a single proton, one with an extra neutron and one with two extra neutrons but both are very rare). Chlorine has two common isotopes (one with 18 neutrons and one with 20; about 3/4 of chlorine atoms have 18 neutrons giving an average mass of ~35.5). Are you sure you consistently quote the atomic mass of specific isotopes?

The other is that the atomic nucleus has a binding energy (which from E=MC2 makes a difference to the net mass which isn't just the combined mass of naked protons plus naked neutrons). In other words the energy it takes to hold a nucleus together actually makes a notable difference to the net mass of the nucleus.

The second effect is at least as important as the difference in mass of naked nuclei. So any claim that the "deviation" is caused by that is just wrong.

By the way, these small details are rarely important in normal chemistry and only matter to nuclear chemists or physicists.

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