Let's say that we have a nucleus with 7 units of mass. And let's say that the combined mass of its constituents is 10 units. The 3 units contribute toward the "binding energy." However, we do know that when (for example) hydrogen fuses to create helium, energy is also released along with that reaction. Shouldn't we say that: Energy (Mass) of constituents = Energy (Mass) of resulting nucleus + Binding Energy+ Energy Released (as radiation) instead of: Energy (Mass) of constituents = Energy (Mass) of resulting nucleus + Binding Energy?
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The released energy is exactly that binding energy. Binding things together lowers the total energy in the system (neglecting kinetic energy for the moment). The excess energy present in the system before must go somewhere (because of the conservation of energy), so it is released in some way (mostly emitting by photons or as kinetic energy).
So in your example, the system first has an energy equivalent of 10 mass units, and after binding only 7. When the binding reaction occurs, suddenly there is an energy excess equivalent to 3 mass units in the system, so this energy (the binding energy) is released in some way.
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$\begingroup$ How can we say that binding things lower the total energy of the system? We're binding protons in the nucleus. At infinity, the energy is zero. However, when we bring protons close together, they do have positive energy (by storing it in the form of potential energy). So, shouldn't "binding" (i.e. sticking) things together means increasing their energy? $\endgroup$ Mar 25, 2019 at 13:49
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$\begingroup$ If binding would increase the energy, it would not occur. That's for the simple reason that nature tries to minimize energy wherever possible. For protons the energy increases from coulomb repulsion until the strong force takes over, which radically lowers the energy. That is exactly why fusing nuclei is hard. $\endgroup$– noahMar 25, 2019 at 13:52
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$\begingroup$ @noah That's for the simple reason that nature tries to minimize energy wherever possible. I don't believe such a principle exists in physics. You're right that If binding would increase the energy, it would not occur. But this is simply because it wouldn't be possible to conserve energy. Unless initial particles do have some kinetic energy to match energy balance. $\endgroup$ Mar 25, 2019 at 16:03
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$\begingroup$ @ElioFabri Such a principle most certainly exists. The argument that it does not occur only because it won't conserve energy is not valid. We could just add the "missing" energy via a photon or, as you state yourself, kinetic energy. But even then, it won't occur. $\endgroup$– noahMar 25, 2019 at 16:29