|
|
| Fig. 1: Comparison of atomic and nuclear length scales. Electrons in atoms occupy regions on the order of 10-10 m (1 Angstrom), while protons and neutrons are confined within nuclei of characteristic size about 10-15 m (1 Fermi). (Source: Wikimedia Commons) |
Nuclear fuel differs from chemical fuel in the amount of energy that can be released per unit mass. A frequently cited comparison is that one gram of uranium can release approximately the same amount of energy as a metric ton of oil, a ratio on the order of one million to one. [1]
Experiments show that nuclear reactions release energies on the order of mega-electron-volts (MeV) per reaction, whereas chemical reactions typically release only a few electron-volts (eV). For example, the fission of a U-235 nucleus releases approximately 200 MeV of energy. [2] By contrast, the combustion of methane releases approximately 9.2 eV per molecule. [2] Since
| 200 × 106 eV 9.2 eV |
= | 2.17 × 1017 |
this means that the energy released in a typical nuclear reaction is roughly twenty million times greater than that released in a typical chemical reaction.
Understanding the difference between chemical and nuclear energy first requires examining the characteristic size of atoms and atomic nuclei, as these determine the forces involved in the reactions. These sizes have been determined experimentally.
The typical size of atoms is on the order of 10-10 m. Atomic dimensions can be determined using techniques such as X-ray diffraction in crystalline materials. More detailed discussions of X-ray diffraction and related techniques can be found in Adams. [3] These measurements show that electrons in atoms occupy regions with characteristic dimensions of roughly one Ångstrom (10-10 m). [4]
Atomic nuclei are vastly smaller. Experiments in which high-energy electrons are scattered from nuclei reveal the spatial distribution of nuclear charge and allow nuclear radii to be determined. More detailed discussions can be found in Hofstadter. [5] These measurements establish that nucleons are confined within regions on the order of 10-15 m. Atomic nuclei are therefore roughly one hundred thousand times smaller than atoms, whose characteristic size is about 10-10 m (Fig. 1).
This enormous difference in spatial scale plays a central role in determining the vastly different energy scales of chemical and nuclear reactions.
Chemical energy is released when atoms rearrange their electrons to form new molecules. It is the energy that is released or stored during a reaction. In a chemical reaction the atomic nuclei remain unchanged and only the electronic structure of the participating atoms is altered. As a result, the amount of energy that can be released in any chemical process is determined entirely by the behavior of electrons.
The energy released by chemical reactions is determined experimentally using calorimetry. [6] In a bomb calorimeter a fuel is burned in a sealed chamber and the temperature rise of the surrounding water is measured. The heat released by the reaction is calculated from
where C is the heat capacity of the calorimeter and ΔT is the temperature increase. Calorimetry measurements show that typical chemical reactions release energies of a few eV per molecule, for example 9.2 eV for the combustion of methane. [2,6]
Independent evidence for this energy scale comes from atomic spectroscopy experiments, which measure the binding energies of electrons in atoms. These measurements show that electrons are bound to atoms with energies of order 1 - 10 eV. Because chemical reactions involve rearrangements of electrons, the characteristic energy scale of chemical processes is therefore limited to this range.
Nuclear reactions involve changes within the atomic nucleus itself. In nuclear reactions the identities and configurations of atomic nuclei are altered. [4] In these processes protons and neutrons are rearranged or transformed into different nuclei. Unlike chemical reactions, which involve only electrons, nuclear reactions involve the much more tightly bound nucleons inside the nucleus. As a result, the energy released in nuclear reactions is determined by the interactions between nucleons. The energy released in nuclear reactions is determined experimentally by measuring the kinetic energies of the particles produced in the reaction. [2] In nuclear fission experiments detectors measure the energies of the emitted fission fragments, neutrons, and gamma radiation. By summing the energies carried away by these particles, the total energy released in the reaction can be determined. [2]
Understanding why nuclear energies are so much larger than chemical energies requires examining the forces that act inside atomic nuclei.
 |
| Fig. 2: Average nuclear binding energy as a function of atomic number. [7,11] (Source: Wikimedia Commons) |
As shown in Fig. 2, the energy stored in a nuclear bond is on the order of 1 MeV, while a typical chemical bond stores energy on the order of 1 eV, approximately one million times smaller. [1,2] This enormous difference arises from the combination of fundamental forces and the quantum mechanical consequences of confinement at different length scales. The different mass scales involved in chemical and nuclear interactions also affect the total energy released.
Chemical reactions are governed by the electromagnetic force, which binds electrons to atomic nuclei. This interaction is strong at distances comparable to the size of an atom, roughly 10−10 m. Because electrons are relatively light particles and the electromagnetic interaction at this scale is moderate, the energy released when chemical bonds form or break is typically only a few electron-volts.
Nuclear reactions occur at a much smaller spatial scale. As discussed earlier, the radius of an atomic nucleus is approximately 10−15 m, about 100,000 times smaller than the size of an atom. At these distances the dominant interaction is the strong nuclear force, which binds protons and neutrons together within the nucleus. The strong force is extremely powerful at very short distances and has been estimated to be roughly 137 times stronger than the electromagnetic force at the scale of nucleons. [7,8] As a result, much larger amounts of energy can be stored in nuclear systems than in chemical bonds between electrons and atoms. Interested readers can find further discussion in Griffiths and Krane. [7,8]
The strong nuclear force determines the very small size of the nucleus by tightly binding nucleons together. At these extremely small length scales, quantum mechanical effects become essential to understanding nuclear structure. The behavior of particles at this scale is governed by the Schrödinger equation,
| - | ℏ2 2m |
∇2ψ + V ψ | = | E ψ |
where ℏ is the reduced Planck constant, m is the mass of the nucleon, V is the potential energy function, and ψ is the wavefunction describing the particle.
For illustration, the nucleus can be approximated as a potential well confining a single nucleon within a region of size L. [9] Taking the ground state (n = 1), the corresponding energy is
| E1 | = | π2ℏ2 2 m L2 |
According to the principles of quantum mechanics, confining particles to a very small region of space leads to large uncertainties in their momentum and therefore higher characteristic energies. [10] Because nucleons are confined within a region on the order of 10−15 m, the energy scale associated with their motion is naturally much larger than that of electrons in atoms. This quantum confinement provides a key explanation for why nuclear systems exhibit energy scales far greater than those of chemical systems. [9,10]
The energy stored within the nucleus is described by the nuclear binding energy, which represents the energy required to separate a nucleus into its individual protons and neutrons. [2,11] This energy originates from the mass defect that occurs when nucleons bind together to form a nucleus. According to Einstein's relation
a small decrease in mass corresponds to a large release of energy. Because nucleons have much more mass than electrons, changes in nuclear structure involve much larger energy differences than those associated with electronic rearrangements in chemical reactions.
The average nuclear binding energy as a function of atomic number is illustrated by the nuclear binding energy curve shown in Fig. 2. Heavy nuclei such as U-235 lie lower on this curve and can release energy when they split into smaller nuclei with higher binding energy per nucleon. The difference in binding energy between the original nucleus and the resulting fission products appears as the kinetic energy of the fragments, neutrons, and radiation.
Because nuclear forces act at distances roughly 100,000 times smaller than atomic dimensions, they involve the strong nuclear force rather than the electromagnetic force. They also involve much heavier particles than electrons. As a result, nuclear reactions naturally involve energy scales millions of times larger than those observed in chemical reactions.
The enormous difference between chemical and nuclear energy ultimately arises from the fundamental structure of matter. Chemical reactions involve the rearrangement of electrons bound by the electromagnetic force at atomic length scales, producing energies on the order of a few electron-volts. Nuclear reactions, in contrast, occur within the atomic nucleus, where protons and neutrons interact through the much stronger nuclear force at distances roughly one hundred thousand times smaller.
© Tanmay Prakash. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] J. H. Ausubel, "Power Density and the Nuclear Opportunity," Program for the Human Environment, Rockefeller University, December 2015.
[2] M. Ripani, "Energy from Nuclear Fission," EPJ Web Conf. 98, 05001 (2015).
[3] F. C. Adams, "X-Ray Absorption and Diffraction -= Overview," in Encyclopedia of Analytical Science, 3rd Ed., edited by P. Worsfold et al. (Elsevier, 2019).
[4] S. Şahin, Y. Wu, and H. M. Şahin, "Nuclear Energy Production: Fission Energy," in Comprehensive Energy Systems, 2nd Ed., ed. by I. Dincer (Elsevier, 2025).
[5] R. Hofstadter, "Electron Scattering and Nuclear Structure," Rev. Mod. Phys. 28, 214 (1956).
[6] A. Dale al., "High Precision Calorimetry to Determine the Enthalpy of Combustion of Methane," Thermochim. Acta 382, 47 (2002).
[7] S. Dull, "Chemical Versus Nuclear Reactions," Physics 241, Stanford University, Winter 2018.
[8] E. M. A. Hussein, Radiation Mechanics: Principles and Practice (Elsevier, 2007).
[9] K. S. Krane, Introductory Nuclear Physics (Wiley, 1987).
[10] D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed. (Cambridge University Press, Cambridge, 2018).
[11] M. F. L'Annunziata, "The Atomic Nucleus," in Radioactivity, 2nd ed. (Elsevier, 2016).
