# Why atomic masses aren't integers?

Most of the elements have isotopes, so the atomic masses are calculated depending on the percentage of the existing isotopes. That is clear. However, what about elements that have only one isotope (monoisotopic) - like fluorine? Shouldn't the atomic mass for it be a whole number and not 18.9984?

The short answer is nuclear binding energy, which is the energy needed to disassemble an atom into its subatomic parts (or in some cases the energy released when this happens). The binding energy is a consequence of the strong and weak nuclear forces that hold atoms together.

Where does this energy come from? It comes from the mass of the nucleons! What? It is true; most atoms have less mass than the sum of their parts, and that mass defect is converted into the energy that holds them together. If you do not believe me, let us look at one atom of carbon-12, which is used as the definition of the atomic mass unit:

$$\mathrm{1\ u}=\dfrac{1}{12}\text{ of the mass of one }\ce{^12C}\text{ atom}$$

Thus, the atomic mass of an atom of $\ce{^12C}$ is by definition $\mathrm{12\ u}$. The atom is constructed from 6 protons, 6 neutrons, and 6 electrons, and the masses of those particles are:

particle   mass                     number    total mass
p          1.00727646681290     u   6         6.0436588008774    u
n          1.0086649160043      u   6         6.0519894960258    u
e          5.485799094622×10^−4 u   6         0.0032914794567732 u
TOTAL:  12.0989397763600    u


The sum of the parts of the $\ce{^12C}$ atom are more than the mass of the atom! This mass defect, $\mathrm{0.0989397763600\ u}$ in this case, is the value of the nuclear binding energy when converted to energy using $E=mc^2$.

This handy graph from the Wikipedia article linked above shows that the binding energy per nucleon is not the same for all nuclei.

If it were, then we would have integer values for monoisotopic nominal masses. However, the consequences of this change in reality could be as drastic as the heavier elements (like iron) not having enough energy to hold themselves together. An additional consequence is that there would be no productive nuclear reactions (fission, fusion, or radioactivity). All nuclear transmutations are driven by conversion of nuclei with lower binding energy per nucleon to nuclei with higher binding energy per nucleon, resulting in a net conversion of mass to energy. Without a difference in binding energy per nucleon, we would not have nuclear bombs (and probably we would be better off as a civilization), but we would also not have nuclear power and radioisotopes for compound labeling, cancer therapy, and medical diagnostics. We would also not have a ready supply of deuterium for NMR solvents, and we might not be able to use variations in isotopic distributions of $\ce{H}$, $\ce{C}$, and $\ce{O}$ to determine all sorts of things like did that "locally grown" food from that restaurant you like actually come for your geographic region?

• Just curious... since most people read atomic masses from periodic table; is it not average atomic mass? Hence shall relative isotopic abundance be also not included in answer for sake of completion? – ipcamit Jul 28 '15 at 14:10
• @ipcamit : The posted question explicitly excludes such considerations. – Eric Towers Jul 28 '15 at 15:16
• oops! missed that line! – ipcamit Jul 28 '15 at 15:51
• "...the binding energy per nucleon is not the same for all nuclei. If it were, then we would have integer values for monoisotopic nominal masses." Because of the proton/neutron mass difference, we still wouldn't have integer values. Carbon 12 has a 1:1 ratio of protons to neutrons. In the absence of any difference in binding energy per nucleon, any isotope with a 1:1 proton to neutron ratio would have an integer atomic mass, but an isotope with any other ratio would not. – Will Orrick Jul 28 '15 at 20:58
• Calculated from the mass defect, the average binding energy per nucleon for $\ce{^{12}C}$ is $7 680 151.46~\mathrm{eV}$ which matches exactly the given $7.68~\mathrm{keV}$. – DHMO Jan 14 '17 at 13:16

## protected by orthocresol♦Oct 13 '18 at 16:50

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