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Mathematics for Physics

186 Nuclear Energy

The energy released in nuclear reactions such as fission or fusion when part of the reactants’ rest mass converts to kinetic energy.

Mass-to-kinetic energy transmutation. In fission and fusion, the products have less rest mass than the reactants, and the missing rest mass appears as kinetic energy of the fragments. Rest-mass deficit is that difference of rest masses. This principle is used to compute the kinetic energy released in a nuclear reaction.

The nuclear energy release is

\[ Q = (\Delta m)\,c^{2} \]

where

  • \(Q\) is the energy released.
  • \(\Delta m\) is the rest-mass deficit.
  • \(c\) is the speed of light.

Conservation of total relativistic energy. Individual rest masses are not conserved, while the total relativistic energy of an isolated system is conserved. Total relativistic energy is rest energy plus kinetic energy. This principle is used to show that the missing rest energy reappears as kinetic energy of the fragments.

Nuclear binding energy as a mass deficit. A bound nucleus is lighter than its free protons and neutrons because energy must be added to pull it apart. Binding energy is the energy equivalent of that mass difference. This principle is used to decide whether a reaction releases energy.

Locked rest-mass energy. Rest energy stored in nucleons can be unlocked by rearranging them into a more tightly bound nucleus. Rest energy is the energy \(E_{0}=mc^{2}\) of a particle at rest. This principle is used to model stellar fusion and terrestrial fusion reactors.

Note: Also called mass defect.

186.1 References

  1. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — source for the heading explanation.
  2. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — fission and fusion; rest-mass deficit; binding energy.
  3. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — rest mass into kinetic energy; conservation of total energy.
  4. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — rest-mass deficit; rest energy \(E_{0}=mc^{2}\).
  5. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — binding energy and nuclear stability.
  6. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — rest energy as a reservoir.