Why Is Binding Energy Subtracted in Nuclear Mass Calculations?

[Alex_physics]

The mass of the nucleus is calculated as follows:

mass (nucleus) = [Number of protons * Mass of proton] + [Number of Neutrons * mass of neutron] - [Binding energy/c^2]

Why is the contribution to mass from the binding energy subtracted from the mass of the nucleus rather than being added to the nuclear mass ?

 

[Orodruin]

Because that is the definition of binding energy. If you had to spend energy in order to make a system bound, that system would be unstable and decay to the original constituents.

 

[PlancksApprentice]

Two hydrogen atoms come together to form a stable helium atom. Surely the mass of the helium atom's nucleus should be the sum of the masses of the two protons and neutrons i.e.

m_nucleus = (m_proton * 2) + (m_neutron * 2)

But, this does not add up to the actual mass of a helium atom. There is a loss of ≈0.0303 amu which is the mass defect. Using Einstein's equation relating mass and energy, E = mc², we find that the nuclear binding energy is 4.53 x 10^-12 J.

Thus in the equation which you stated above, the mass defect is calculated by dividing by c², thus getting mass back. And to obtain the actual mass of the nucleus it has to be subtracted.
NOTE: By switching the positions of 'binding energy' and 'mass(nucleus)' we get the mass defect equation.
Δm = Zm_proton + (A - Z)m_neutron - M
Where, Δm = mass defect
Z = number of protons
A = number of nucleons
M = mass of the nucleus.