Binding energies of $Be$ Just a quick question. 
How does Beryllium 8 decay into 2 alpha particles?
Beryllium 8 has a binding energy of 56.499508 Mev
An alpha particle has a binding energy of 28.3 so two of these would have 56.6 Mev
So how does this reaction happen? 
Beryllium doesn't have enough energy.
 A: I always get confused when I try to use binding energies directly, partly because of the sign issue, and partly because differences in binding energies show up in the third or fourth significant figure. My favorite reference instead lists for each isotope a "mass excess," which is the difference (in energy units) between an observed mass and the naïve mass of $A$ atomic mass units.  For beryllium and helium we have
\begin{align}
\rm\Delta({}^8Be) &= \rm 4.9416\,MeV \\
\rm\Delta({}^4He) &= \rm 2.4249\,MeV \\
2\cdot\rm\Delta({}^4He) &= \rm 4.8498\,MeV 
\end{align}
This makes it clear that a $^8\rm Be$ is heavier than two $^4\rm He$ by about $0.1\,\rm MeV$ — it has more excess mass.
So what was wrong with your binding energy approach? You have
\begin{align}
\rm BE({}^8Be) &= 56.49955\,\rm MeV \\
\rm BE({}^4He) &= 28.3\rm\,MeV &\text{(careful with precision here)}\\
2\cdot\rm BE({}^4He) &= 56.6\rm\,MeV
\end{align}
Here you have the same energy difference, about $0.1\rm\,MeV$, but it appears to go the other way, as you point out in your question.  That's because helium is more tightly bound than beryllium; when beryllium-8 fissions, the extra binding energy is released.
A: A higher binding energy translates into less mass. Here the mass of two alphas combined is less than that of Be8. The mass difference would give you the kinetic energies of the two alpha when they are infinite apart.
A: You maybe didn't understand the point that dmckee made: so, here is a detailed explanation.
The binding energy is the energy that would be released if creating a nucleus from its nucleon components. If a nucleus X created from nucleons has a lower binding energy than another nucleus Y created from the same nucleons, then in creating X less energy was eliminated than in creating Y. So, inside X there remained more energy than in Y.
As the total binding energy of $^8Be$ is less than of two $\alpha$, it means that in creating $^8Be$ there remained inside it more energy than remains in the $\alpha$ particles if we would create them. Thus, inside $^8Be$ there is some additional energy than in the two $\alpha$, and it can decay into 2 $\alpha$. The excess of energy in the $^8Be$ will transform into kinetic energy of the $\alpha$ nuclei.
