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- Homework Statement
- x

- Relevant Equations
- E = mc^2

If we consider a nuclear decay in which element X decays into Y and an alpha particle, we can see that the total energy released is ##c^2(-m(\alpha) -m(y) + m(x))##.

If we replace ##m(alpha)## with ##s(alpha) - be(\alpha)/c^2##, and similarly for the other particles, X and Y, we can see that the energy released is equal to## c^2( s(x) - be(x)/c^2 - s(\alpha) + be(alpha)/c^2 - s(y) + be(y)/c^2)##

Tidying up gives ##c^2(s(\alpha) +s(y) -s(x)) +be(y) + be(alpha) - be(x)##, we can see that ##s(\alpha) +s(y) -s(x)## is equal to ##0## (imagine breaking up X, Y and Z, Y and Z are clearly made of the same stuff as X, hence same mass), therefore the energy released is equal to the change in binding energies, ##be(y) + be(\alpha) – be(x)##.

Now imagine having X in front of you, you put ##be(x)J## of energy in, and form Y and alpha, which releases ##be(y) + be(\alpha) J##, leaving a "net" energy release of ##be(\alpha) + be(y) - be(x)##. Is this identical to what happens in decay? I don't think so, as the entire nucleus has no need to break down for alpha decay.

So where does the energy required to break off the alpha particle come from?

If we replace ##m(alpha)## with ##s(alpha) - be(\alpha)/c^2##, and similarly for the other particles, X and Y, we can see that the energy released is equal to## c^2( s(x) - be(x)/c^2 - s(\alpha) + be(alpha)/c^2 - s(y) + be(y)/c^2)##

Tidying up gives ##c^2(s(\alpha) +s(y) -s(x)) +be(y) + be(alpha) - be(x)##, we can see that ##s(\alpha) +s(y) -s(x)## is equal to ##0## (imagine breaking up X, Y and Z, Y and Z are clearly made of the same stuff as X, hence same mass), therefore the energy released is equal to the change in binding energies, ##be(y) + be(\alpha) – be(x)##.

Now imagine having X in front of you, you put ##be(x)J## of energy in, and form Y and alpha, which releases ##be(y) + be(\alpha) J##, leaving a "net" energy release of ##be(\alpha) + be(y) - be(x)##. Is this identical to what happens in decay? I don't think so, as the entire nucleus has no need to break down for alpha decay.

So where does the energy required to break off the alpha particle come from?

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