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Understandably, the relative atomic masses of isotopes are often not close to whole numbers as they're adjusted for isotope abudandancy eg. $A_r(Cl)\approx35.45$. However, wouldn't one expect the relative atomic masses of individual isotopes to be rather close to whole numbers, given that they're merely the sum of the relative atomic masses of the protons, neutrons, and electrons within the atom and that the masses of protons and neutrons $\approx1$?

I.E, we have $Rb-85$, which has a literature $u=84.91$. Attempting to calculate $u$ given proton, neutron and electron count: $$ A_r(p^+) = 37\times 1.0073\approx 37.2701\space Da $$ $$ A_r(n^0)=48\times 1.0087\approx 48.4176\space Da $$ $$ A_r(e^-)=37\times 5.4858\times 10^{-4}\approx 0.02030\space Da$$ $$37.2701+48.4176+0.02030=85.708$$ The answer I get is $85.708$, whereas something close to $84.91$ is expected.

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    $\begingroup$ Do you consider the total nucleus mass is less than sum of masses of protons and neutrons by the nuclein bound energy? $\endgroup$
    – Poutnik
    Dec 5, 2021 at 5:32
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    $\begingroup$ E.g. Mass of 4He is less than mass of 2p + 2n by about 0.7%, AFAIK. And as average bonding energy per differ from the one for 12C, nuclei masses in Da differ from integer values. Another minor factor is various n/p ratio. $\endgroup$
    – Poutnik
    Dec 5, 2021 at 6:41
  • $\begingroup$ Please note that the dalton (unity symbol $\mathrm{Da}$) is not a unit of the quantity relative atomic mass (quantity symbol $A_\mathrm r$). Relative atomic mass is a quantity of dimension one (a so-called dimensionless quantity). Also please note that $u=84.91$ doesn't make any sense at all. $\endgroup$
    – Loong
    Dec 5, 2021 at 9:26
  • $\begingroup$ Interesting facts: The lowest mass per nucleon has 56Fe, while the highest binding energy per nucleon has 62Ni. $\endgroup$
    – Poutnik
    Dec 5, 2021 at 9:37

2 Answers 2

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In the world of atomic masses, everything is "weighed" with reference to carbon-12. Unfortunately, this is a circular approach, in the sense that carbon-12 is arbitrarily set to 12 as an exact integer. Today, there is a very small change due to the refining of the definitions of mass/ length in terms of fundamental constants.

There is no reason as to why the mass of a single isotope be an integer. Recall, carbon-12, it is arbitrary. Yes, the count of protons and neutrons is an integer but not their masses. Two centuries ago, atomic weights were all different in the time of Berzelius because they had other standards.

To add insult to injury in the nuclear world, 1+1 is not equal to 2. It is always slightly less. This is one of the fascinating mysteries of nature. The missing mass is accounted by the binding energy of nucleons (protons+neutrons) in the nucleus.

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    $\begingroup$ One can even state that each proton and each neutron loses a small fraction of its mass when joining to make a nucleus. This mass loss is transformed into energy by Einstein's law : $\pu{E = mc^2}$ $\endgroup$
    – Maurice
    Dec 5, 2021 at 11:20
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Because the energy involved in nuclear forces holding nuclei together are so large they affect the net mass of the nucleus

The underlying reason is Einstein's famous $E = mc^2$.

The forces that hold nuclei together are very strong, as they have to be to stop charged protons from flying apart because of their electrostatic repulsion. And those really strong nuclear forces involve a lot of energy, which has to come from somewhere.

When multiple protons and neutrons are combined into a stable nucleus (which is stable because the binding forces are strong enough to outweigh the electrostatic repulsion) the result has a much lower energy than the isolated protons and neutrons. And this can be a lot of energy (enough to make a star shine brightly or to create an enormously destructive fusion bomb).

That energy, using Einstein's famous equation, constitutes a significant proportion of the mass of the resulting nucleus so a combination of neutrons and protons into a stable nucleus will have less mass than the sum of the masses of isolated protons and neutrons. In the case you calculate that difference is a little under 1% of the isolated masses of protons and neutrons in the Rb nucleus.

The exact mass defect depends on the specific nucleus which complicates things for chemists who tend to reference atomic masses versus specific isotopes where it is easy to measure (once $\ce{^16O}$, more recently $\ce{^12C}$). But, if you understand the principle, the non-integer masses of isotopes is easy to understand.

And that history of how chemists measure relative nuclear mass also explains why the mass of isolated nuclei are not exact integers (they are measured relative to a specific isotope which has a lower mass than the sum of its isolated components).

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  • $\begingroup$ This explanation better suits why 2+2 is not equal to 4 for atomic masses, however there is no fundamental explanation as to why the mass of a single proton or a neutron is not an integer. In short, we do not know why atomic masses are non-integers. The Einstein's equations is perhaps only a small part of the story. The reason is more or less circular- C-12 has been historically set to an integer. I know it has been changed very marginally but... $\endgroup$
    – AChem
    Oct 18, 2022 at 12:41
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    $\begingroup$ @achem That is simply an obvious result of measuring the mass relative to a particular isotope. $\endgroup$
    – matt_black
    Oct 18, 2022 at 12:44
  • $\begingroup$ @AChem The title of the question mentions integers, but the body of the question is more about the non-additivity. Trivially, 1.0073 looks more like an integer than e.g. 1.0073 * 100 = 100.73. BTW the molar mass of 12C is 11.999 999 9958(36) g/mol according to CODATA2018. $\endgroup$
    – Karsten
    Oct 18, 2022 at 13:17
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    $\begingroup$ Carbon -12 was chosen to keep the atomic weight of the natural abundance of oxygen as close to 16. as possible and choosing an isotope for the physical scale. This prevented previous chemical data from becoming not so good. The improvements in mass spectrometry meant all the physical masses were being redone anyway. The chemical scale was based on the mix of oxygen set to 16. There is nothing circular about defining a standard mass. $\endgroup$
    – jimchmst
    Oct 18, 2022 at 15:07
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    $\begingroup$ @jimchmst This is correct. The underlying reason why the numbers we get for isotope and nucleon masses is the mass deficit caused by nuclear binding energy. The "circular definition" isn't circular: it is simple about choosing a point (specific isotope) to which other masses are referenced. The fact we need to choose a reference point is a side effect of the mass deficit issue, not the cause of non-integer masses. $\endgroup$
    – matt_black
    Oct 18, 2022 at 16:10

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