# Is 12 amu for Carbon-12 exact or rounded?

Dive 07/26/2017. 2 answers, 128 views

I'm having a difficult time understanding how Carbon-12 has an atomic mass of 12. Each proton and neutron has an atomic mass that is just slightly above 1 amu, so wouldn't Carbon-12 also be above 12? Neutron = 1.0034 amu Proton = 1.0073 amu Carbon-12 = 6 protons and 6 neutrons. AMU of 6 protons (6.0438amu) and 6 neutrons (6.0204amu) = 12.0642amu I understand atomic weight, being the average of mass of isotopes on earth. But I do not understand how Carbon-12 = 12 amu, unless is is just rounded down to a whole number for simplicity purposes. If someone could please shed some light upon this I'd be most grateful!

Qmechanic♦ 07/28/2017

Martin Beckett 07/26/2017.

Carbon12 is 12 unified amu by definition (see David's comment below).

The masses you quote for free protons and neutrons are not the same as their mass when bound in a nucleus. The binding energy of C12 is around 92MeV which accounts for the missing 0.064amu in your example.

1 David Hammen 07/26/2017
Re "Carbon12 is 12 amu by definition": That should be "The mass of one $^{12}C$ atom in its rest state is exactly 12 unified atomic mass units, by definition (currently)." Being a bit pedantic, amu refers to the old $^{16}O$ standard. It's better to be precise and use either unified atomic mass units or Daltons, or their abbreviations u or Da. Regarding "currently": This may change with the proposed redefinition of the SI.
Dive 07/26/2017
Martin Becket, thank you for your response. So what you're saying is that when protons and neutrons become binded together they lose 0.064amu? If I interpret your response correctly, where then does this mass go?
Dive 07/26/2017
David Hammen, reading your response about Daltons helps me understand, I think. Daltons are essentially equivalent the the number of protons and/or neutrons? Each particle equating to 1 dalton. Whereas the amu in other situations (the amu of a single proton being 1.0073amu) its somewhat of a different unit, pertaining more to mass rather than of a quantity? I hope I don't come off supremely ignorant, I'm new to the studies of Chemistry and am self educating prior to the beginning of the semester. Thanks for your help everyone!
Martin Beckett 07/26/2017
@Dive the extra mass goes to energy (e=mc^2). Dalton and amu are (essentially) the average mass of a proton+neutron. A Dalton or unified amu is 1/12 the mass of a C12 nucleus. In chemistry you would give the mass of a molecule in Daltons, in physics it gets a bit complicated because the mass of a particle depends on its state - you wouldn't normally state the mass of a proton or neutron in Daltons
Dive 07/26/2017
I see. So my question was reaching out into physics which deviates from the chemistry definition of amu, correct? To assure my understanding, an example; O18 would have an amu of 18, correct? And I127 has an amu of 127. Yes?

Noiralef 07/28/2017.

In an atomic nucleus, a number of protons and neutrons are in a bound state -- just like an electron and a proton in a Hydrogen atom. Such a bound state of particles has a binding energy. Imagine you begin with all the parts of the nucleus (the partons) infinitely far apart, at this point their potential energy is zero. If you move them together, once they are close enough to feel the strong nuclear force, they will attract each other / "fall in a potential well". The potential energy they are losing during this process is the binding energy $E_b$.

Now, the total energy of the system is $$E = \sum_{\text{parton } k} m_k\, c^2 - E_b .$$ This is the energy of the bound state, the nucleus, at rest and therefore corresponds to its rest mass: $$M = E/c^2 = \sum_k m_k - \frac{E_b}{c^2} .$$ You can see there is a mass deficit: The mass of the nucleus is smaller than the sum of the masses of its components.

It is very hard to calculate the mass deficit of a nucleus; in fact, this is a many-body problem which can not be analytically solved. You might be interested in the semi-empirical mass formula of Bethe and Weizsäcker or in this diagram of binding energy per nucleus for various isotopes.

Finally, C12 has 6 pairs of protons and neutrons, O16 has 8 pairs, but the mass of O16 is not exactly 8/6 times the mass of C12 but again slightly less. One u is now simply defined to be 1/12 the mass of a C12 atom, hence all smaller nuclei tend to have slightly "too high" masses and all larger nuclei tend to have "too small" masses.