Why fission pays off

Problem

A nucleus of $^{235}\text{U}$ splits into two medium-small fragments. The sum of the masses of the two fragments turns out to be less than the mass of the starting uranium. Explain where the released energy comes from and why it justifies the industrial use of fission.

Solution

The energy comes from the missing mass $\Delta m$: the difference between the mass of the initial uranium and the total mass of the fragments does not vanish, but is converted into energy according to Einstein’s equivalence $E = \Delta m\, c^2$. Since $c^2$ is enormous, even a tiny fraction of mass produces a significant release of energy.

The reason the fragments weigh less lies in the binding energy per nucleon curve $B/A$. Uranium sits on the branch of heavy nuclei, where $B/A$ is lower; the medium-mass fragments sit closer to the maximum of the curve (around iron). A higher $B/A$ means more strongly bound nucleons and hence lower rest mass: the difference is exactly the energy released.

Each fission releases about $200\ \text{MeV}$, millions of times more than the energy of a chemical reaction. A single gram of $^{235}\text{U}$ releases energy comparable to about $20\ \text{t}$ of TNT: this extraordinary energy density is what makes fission worthwhile for industrial use in nuclear reactors.

$\ev{E=\Delta m\,c^2;\ \text{fragments with a higher }B/A\text{ than uranium}}$

Connections

Topics: Nuclear physics Concepts: Nuclear fission · Nuclear binding energy · Mass-energy equivalence