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I have asked my teacher, as she was introducing the concept of mole to us, why that number was chosen, instead of more convenient one. She told me that it came from the definition of the mole, that is the number of atoms in 12g of Carbon 12. When I asked why that definition was chosen, she answered me that my question wasn't really a question, and I understood from her stare that asking why? twice in a row was not appreciated.

However, I certainly hope that this definition wasn't chosen randomly and that there is a reason for it being it.

I gave it some thought and I came to believe that it may be for convenience after all. There are 12 nucleons in a carbon atom and if we say that the electrons mass can be neglected, then we get that a mole is the number of nucleon it takes to get one gram of them. However, can we really neglect the mass of electrons, when the atoms get bigger? And if so, why do the element's mass per mole do very rarely end up being integers?

This makes me think there may be another reason, or some things I do not understand. And if there are no reason, then I guess I might just as well be doing religious studies..!

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Why is the definition of the moles as it is?

It is a rather arbitary definition that the mole is the number of atoms in 12g of carbon 12. This has not always been the definition. For example, prior to 1960, the definition was based upon oxygen rather than carbon-12.

The first standard was based upon 1 gram of hydrogen. Later, the standard was changed to 16 grams of oxygen being a mole, for convenience because oxygen formed compounds with many other elements. Eventually it was realized that elements including oxygen have different isotopes of different mass, and the mass of a particular isotope is a more specific and measurable standard than the mass of natural abundance oxygen. 12 grams of carbon-12 matches the old standard of 16 grams of natural abundance oxygen more closely than choosing 16 grams of oxygen 16.

can we really neglect the mass of electrons, when the atoms get bigger?

The mass of a proton or neutron is about 1836 times that of an electron. So depending upon the ratio of neutrons to protons, electrons are at most 1/1836 of the mass of a neutral atom. If accuracy of more than 1 part in 1836 is desired, it is important to consider the electrons, regardless of whether atoms are big or small.

why do the element's mass per mole do very rarely end up being integers?

No element other than carbon-12 will exactly be an interger. This is because masses of atoms depend upon number of protons, neutrons, electrons and binding energy.

In other words, considering that carbon-12 has an equal number of protons and neutrons (and electrons), other isotopes with a 1:1 proton/neutron ratio would have essentially integral atomic masses except for binding energy. For example cadmium 112 is 111.90 instead of exactly 112 because binding energy is slightly more than 0.1 amu, partially offest by a higher number of neutrons than protons, the neutron mass being slightly greater than the combined mass of an electron and proton.

As pointed out by Matt Black, the natural abundance atomic masses further deviate because they are weighted averages of the masses of isotopes of a given element, weighted by their abundance on Earth.

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    $\begingroup$ That's a good answer - I would also add that all standards are chosen for two reasons: consistency and convenience, and C-12 meets both requirements better than any other element. $\endgroup$
    – thomij
    May 29, 2014 at 18:21
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    $\begingroup$ @thomij If you wanted to expand on that a little, it would be a great second answer. $\endgroup$
    – jonsca
    May 29, 2014 at 19:23
  • $\begingroup$ +1 nice answer however I think hydrogen is not related to the concept of mole... $\endgroup$
    – G M
    May 29, 2014 at 22:40
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    $\begingroup$ The major reason why elements masses are not integers or even close to them is that most quoted numbers in periodic tables are based on a mixture of isotopes. The errors from binding energy etc. are usually much smaller (parts per thousand so affecting the 2nd or 3rd decimal place). $\endgroup$
    – matt_black
    May 29, 2014 at 23:14
  • $\begingroup$ @matt_black that is right however $H^{1}$ =1.007 825 032 07(10) and hydrogen standard atomic weight is 1.008(1) $\endgroup$
    – G M
    May 30, 2014 at 8:40
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Why is the definition of the moles as it is?

Here what say The International System of Units (SI)

“Atomic weights” were originally referred to the atomic weight of oxygen, by general agreement taken as 16. But whereas physicists separated the isotopes in a mass spectrometer and attributed the value 16 to one of the isotopes of oxygen, chemists attributed the same value to the (slightly variable) mixture of isotopes 16, 17 and 18, which was for them the naturally occurring element oxygen. Finally an agreement between the International Union of Pure and Applied Physics (IUPAP) and the International Union of Pure and Applied Chemistry (IUPAC) brought this duality to an end in 1959/60. Physicists and chemists have ever since agreed to assign the value 12, exactly, to the so-called atomic weight of the isotope of carbon with mass number 12 (carbon 12, 12C), correctly called the relative atomic mass Ar( 12C). The unified scale thus obtained gives the relative atomic and molecular masses, also known as the atomic and molecular weights, respectively

Atomic Weight' - The Name, Its History, Definition, And Units Prepared For Publication By P. De Bievre' And H. S. Peiser

The unit could be fixed arbitrarily. By choice, it was linked to 12C and the macroscopic mass unit, the kilogram of the SI. Nevertheless, one should note from the definition below that the magnitude of that link factor of NA does not affect the amount of substance of any entity.

Can we really neglect the mass of electrons when the atoms get bigger?

I would say yes, Neutrons mass is 1.00866491588(49) u for a proton is 1.007276466879(91) u while the electrons at rest have a mass of 0.000548579909 u so even if we take the heaviest atom Oganesson that has 118 electrons we will have as a very approximate weight of the electrons 0.06473242927025999 u.

Why do the element's mass per mole do very rarely end up being integers?

At the root, there is the fact that neither the atomic mass unit u is an integer actually it's 1.660539040(20)×10−27 kg. The kilogram was defined long before the studies of sub-atomic particles so it wasn't designed to be a perfect multiple of the atomic mass unit. However, even if we decide to adjust the kilogram so that the atomic mass unit is 1.00000000000×10−27 kg we will eventually end up with non-integer masses, this is due mainly to two factors:

Nuclear binding energy

The "mass defect" that is related to the fact that: $$E=mc^{2}$$ So the mass is linked to the energy, different elements have different energy due to different Nuclear binding energy and so different mass!

Neutrons heavier than protons

Neutrons weight 1.00866491588(49) u while protons 1.007276466879(91) u so elements with more protons are heavier.

Isotopic abundance? Not really...

Natural abundance (NA) is not at the root of the problem, your observation is valid not only for Standard atomic weight (Ar, standard,) that take into account the isotopic abundance but also for the weight of single isotopes. The periodic table below shows you that also element without isotopes (blue background) have non integer mass. Of course, if you consider the standard atomic weight of an element ( that is the result of a weighted average so you might already expect that is a non integer... ) the isotopic abundance have a greater impact on the result, but if you consider the atomic mass as defined by IUPAC so you are analyzing a single isotope the root of the problem are the reasons described above.

enter image description here

Holden, N., Coplen, T., Böhlke, J., et al. (2018). IUPAC Periodic Table of the Elements and Isotopes (IPTEI) for the Education Community (IUPAC Technical Report). Pure and Applied Chemistry, 90(12), pp. 1833-2092. Retrieved 18 Jan. 2019, from doi:10.1515/pac-2015-0703

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  • $\begingroup$ The biggest reason why the elemental atomic masses quoted in periodic tables are not integers isn't the binding energy: it is the existence of multiple isotopes. For example: chlorine is 35.45, far from an integer but that is because it consists of two major isotopes ~75% chlorine-35 and 25% chlorine-37. The atomic mass of Cl-35 is ~34.969 and this is a result of nuclear binding energy causing deviation from an integer value. $\endgroup$
    – matt_black
    Jan 17, 2019 at 15:10
  • $\begingroup$ @matt_black thanks for your comment, but the atomic masses are actually of an element and in the periodic table are quoted the standard atomic weight not the atomic mass most of the times. Besides that, the user question is the element mass per mole and not the value reported in the periodic tables. $\endgroup$
    – G M
    Jan 18, 2019 at 15:29
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    $\begingroup$ I used standard atomic mass as a synonym for standard atomic weight. Those are, in effect, the mass per mole of a standardised sample of the element with its natural atomic abundance. This is what even the table you reproduce shows as it is what chemists work with. Your assertion that the non-integer nature isn't (mostly) caused by isotopic abundance is just wrong: binding energies affect mostly the second decimal place. But chlorine, for example, has a quoted mass/weight of 35.45. That's isotopic abundance not nuclear binding energy. $\endgroup$
    – matt_black
    Jan 18, 2019 at 15:43
  • $\begingroup$ @matt_black I am not saying that it doesn't affect the standard atomic weight that are reported in the periodic table, but I am saying that is not the root of the fact that the weights are not integers and in fact also if we determine the weight of a single isotope we will have a non integer value. $\endgroup$
    – G M
    Jan 18, 2019 at 15:48
  • $\begingroup$ Yes but the binding energy for most elements doesn't affect more than about 5 points in the second decimal place of the atomic weight. No mono isotopic element has a mass that doesn't start with xx.9 or xx.0. But plenty of standard atomic weights do. Isotopic composition is by far the biggest contributor to the non-integer character of atomic weights. $\endgroup$
    – matt_black
    Jan 18, 2019 at 15:56

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