Question

If both the number of protons and the neutrons are conserved in each nuclear reaction, in what way is mass converted into energy (or vice versa) in a nuclear reaction? Explain.

Hint:

Nucleon number conserved ≠ mass conserved. Mass defect accounts for nuclear energy via \( E = mc^2 \).

Answer 1

Mass-Energy Conversion in Nuclear Reactions
In a nuclear reaction, even if the number of protons and neutrons is conserved, the total mass of the nucleus before and after the reaction may not be exactly the same. This difference in mass, called the mass defect, is converted into energy according to Einstein’s mass-energy equivalence principle:
\[ E = \Delta m \, c^2 \]
where \( \Delta m \) is the mass defect and \( c \) is the speed of light.
Explanation:
- When nucleons (protons and neutrons) bind together to form a nucleus, the total mass of the bound nucleus is slightly less than the sum of the individual masses of the separate nucleons. The “lost” mass appears as binding energy that holds the nucleus together.
- Conversely, in nuclear fission or fusion, a small fraction of mass is converted into energy, which is released as kinetic energy of the products and radiation.
- Thus, even though the number of protons and neutrons remains the same, the conversion of mass into energy occurs due to changes in nuclear binding energy.
Example:
- In the fusion of hydrogen nuclei to form helium, the helium nucleus has slightly less mass than the combined mass of four hydrogen nuclei. The missing mass is released as energy, which powers stars.
Conclusion:
Mass is converted into energy (or vice versa) in nuclear reactions through changes in nuclear binding energy, despite the conservation of the number of protons and neutrons. This principle underlies the enormous energy output of both nuclear fission and fusion reactions.

Answer 2

A more visual way to see this is through the binding-energy-per-nucleon curve, the graph of \( E_b / A \) plotted against mass number \( A \) for all known nuclei.

This curve rises steeply for light nuclei, peaks around \( A \approx 56 \) near iron, and then falls off slowly for heavier nuclei. Nucleons in a nucleus near the peak are the most tightly bound of all, in the sense of having the highest binding energy per nucleon.

Why the mass changes while nucleon count doesn't:
The mass of any nucleus equals the sum of the masses of its free, separated nucleons minus \( E_b / c^2 \) for that nucleus. A nucleus with a higher position on the binding-energy-per-nucleon curve has converted more of its constituent mass into binding energy, so it weighs less per nucleon than one lower on the curve, purely because of how tightly it is bound, not because it contains different nucleons.

Fission (heavy nuclei):
A heavy nucleus, sitting on the falling right-hand side of the curve, splits into two medium-mass nuclei that sit closer to the peak, at a higher binding energy per nucleon. Since the same total number of nucleons is now more tightly bound than before, some of the original mass has converted into the released energy of fission.

Fusion (light nuclei):
Light nuclei, sitting on the steeply rising left-hand side of the curve, combine into a single heavier nucleus closer to the peak, again moving to a higher binding energy per nucleon. The nucleons involved are the same before and after, but the resulting nucleus is more tightly bound, and the corresponding mass difference is released as energy, this is what powers the Sun and other stars.

In both cases, protons and neutrons are simply rearranged into a configuration with different binding energy; it is this change in binding energy, not any change in the number of nucleons, that shows up as the mass converted to or from energy.