# Energy Released in a Fission Reaction

I've been told the following is incorrect, but I can't really see how.

Consider the fission event described by the equation $$\rm ^{235}U+n\rightarrow{}^{93}Rb+{}^{140}Cs+3n$$ The energy released is given by \begin{align} Q&=\Delta mc^2 \\&=(m_\mathrm U+m_\mathrm n-m_\mathrm{Rb}-m_\mathrm{Cs}-3m_\mathrm n)c^2 \\&=(m_\mathrm U-m_\mathrm {Rb}-m_\mathrm {Cs}-2m_\mathrm n)c^2 \end{align} and writing the nuclear masses in terms of the masses of their constituent nucleons less than nuclear binding energy we find $$Q=B_\mathrm{Rb}+B_\mathrm{Cs}-B_\mathrm{U}$$ You could then estimate these binding energies using the SEMF giving a value of roughly $$Q=145\rm\,MeV$$ (in reality it should be something like $200\rm\,MeV$). However I'm just concerned about whether or not the general method is correct or not. Thanks in advance.

• Your first equation for Q is okay. By "general method" do you mean using the SEMF to find the binding energies for your final calculation? If so, why do that when the masses are tabulated and easily looked up? The SEMF is a gross approximation and doesn't fit individual nuclides well enough for better than a 10-20% uncertainty in Q calculations. – Bill N Apr 2 '16 at 16:36
• By general method I mean obtaining Q in terms of the Bs. Could you direct me to such a source? – Watw Apr 2 '16 at 17:15
• Instead of some estimate, one could get the 2012 Atomic Mass Evaluation file from the Atomic Mass Data Center, linked from the National Nuclear Data Center at www.nndc.bnl.gov. It lists every evaluated isotope, with binding energy and lots of other information. – Jon Custer Apr 2 '16 at 18:06
• we get about 200 MeV from fission of U-235 but the products are Ba-141 and Kr-92...but here you are quoting Rb-93 but 93 is Niobium and Cs-140 but we have Cerium-140... so i think its not a possible fission event on which you are trying to check. – drvrm Apr 2 '16 at 18:24
• chart of the nuclides. Click on the nuclide you're interested in. Atomic mass is one of the items. atom.kaeri.re.kr/nuchart – Bill N Apr 2 '16 at 21:32

The Nuclear Wallet Cards make this process a little easier by listing mass excesses rather than binding energies. For your reaction we have $$\begin{array}{cr} \text{nuclide} & \Delta\,\rm (MeV) \\\hline \rm n & +8.1 \\ \rm^{140}Cs & -77.1 \\ \rm^{93}Rb & -72.6 \\ \rm^{235}U & +40.9 \end{array}$$ So your initial state has mass excess $\rm \Delta n + \Delta U = +49.0\,MeV$, your final state has mass excess $\rm \Delta Rb + \Delta Cs + 3 \Delta n = -125.4\,MeV$, and the $Q$-value for this particular fission is apparently about $\rm 175\,MeV$. I would say that your estimate from the semi-empirical mass formula is not that far off, all things considered.
You can convince yourself that my method and your method are equivalent by using the mass excess to find the binding energy of each nucleus. For instance, the binding energy per nucleon for U-235 is $7.59\rm\,MeV$, which corresponds to a total binding energy of $\rm 1783\,MeV$. (This is almost two nucleons' worth of binding energy!) If I were to break my U-235 nucleus into 92 protons and 143 neutrons my mass excess would go from 40.9 MeV (above) to $$\rm 92\Delta p + 143 \Delta n = 92 \times 7.29\,MeV + 143 \times 8.07\,MeV = 1824\,MeV$$ and $\rm 1824\,MeV - 40.9\,MeV = 1784\,MeV$, as we expected. You're welcome to check the others if you are so inclined.