# Why are the masses of atoms less than the sum of their subatomic particles?

The mass of carbon-12 is $$\pu{12 u}$$ by definition. However, one carbon-12 atom comprises 6 neutrons (each weighing $$\pu{1.0087 u}$$), 6 protons (each weighing $$\pu{1.0072 u}$$), and 6 electrons (each weighing $$\pu{0.0005 u}$$), which all add up to $$\pu{12.0894 u}$$.

Where does the $$0.7\%$$ difference in mass come from?

• In very simple and short terms: In bound states, the mass of the involved particles changes, i.e. mass defect. The loss of bonding energy translates to a loss of mass, basically demonstrated by $E=m\cdot\mathcal{c}^2$. – Martin - マーチン Mar 17 '15 at 8:37
• Exactly that mass difference is used in nuclear reactions to "generate" energy: for light nuclei (smaller than iron), energy is released by fusing them to heavier ones; for large nuclei (larger than iron), it is released by splitting them. – ahemmetter Mar 17 '15 at 10:07

The binding energy of carbon-12 is quoted on Wikipedia as $$\pu{92.162 MeV}$$. Therefore we can estimate the mass defect of a carbon-12 atom, $$\Delta m$$, using $$E = (\Delta m)c^2$$:
$$\Delta m = \frac{(\pu{92.162 \times 10^6 eV})(\pu{1.6022 \times 10^-19 J eV-1})}{(\pu{2.9979 \times 10^8 m s-1})^2} = \pu{1.6430 \times 10^-28 kg} = \pu{0.098943 u}$$
The difference in the mass of carbon-12 to the mass of its constituent particles is $$\pu{0.08940 u}$$, so we can see that our calculation is a reasonable estimate of the mass defect. The slight difference is due to other more complicated factors that I have not taken into account, but it still illustrates the main reason for the mass defect.