Is 12 amu for Carbon-12 exact or rounded?

Question

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!

Answer 1

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.

Answer 2

Reading your comments to the other answer, I feel like something should be clarified here.

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.