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I have the following question:

A stationary nucleus of uranium-$238$ undergoes alpha decay to form thorium-$234.$ The following data are available. Energy released in decay $4.27 MeV$, Binding energy per nucleon for helium $7.07 MeV$ and binding energy per nucleon for thorium $7.60 MeV$.Calculate the binding energy per nucleon for uranium-238

From my understanding the reaction is

$Uranium \rightarrow Thorium +Helium+Energy$

Since energy is conserved, the sum of energies on the R.H.S should give the binding energy of Uranium.

Therefore, binding energy per nucleon of uranium $= \frac{(234 \cdot7.6) + (4 \cdot 7.07) + 4.27}{238}$. However, this turns out to be the wrong answer. The answer specifies that energy released $(4.27)$ must be subtracted. Why should energy released be subtracted and not added to binding energy?

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Binding energy tends to lower the total energy of the system, making it more stable. Thus, a nucleus has less energy than the particles it's made up of, and since the nucleus arrangement has a lower total energy, it's a more stable configuration of the system. Thus, binding energy is how much the nucleus is below the energy where particles could break free from themselves, or in other words, the amount of energy you need to put in the system to make the particles be free.

This means higher binding energies are actually lower system total energies, corresponding to more stable systems. Since nuclei tend to decay into more stable isotopes, we can figure out what is happening now.

The extra energy given out by the nuclei is actually it lowering its energy down to get to a more stable arrangement, which actually increases its total binding energy.

The more radical the reaction, the more the system's total energy decreases, making the binding energy increase even further. Remember, binding energy is simply how much energy we'd need to kick in for the energy to be high enough for the system to fly apart.

Since more binding energy decreases the system's total energy, it is then reasonable to understand that, if our products have a certain binding energy, and the reaction gives off whatever amount of energy, the binding energy of the initial nucleus should have been the binding energy of the resulting system minus the energy given off, because remember, higher binding energies mean less energetic systems. The system lost energy, which means it's now in a more stable state, with more binding energy per nucleon, the previous system had to have less binding energy per nucleon, not more, as it was more unstable.

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