Questions: Binding Energy and the Nuclear Stability Curve

Question 1

Both nuclear fusion (combining hydrogen into helium) and nuclear fission (splitting uranium) release energy, even though one combines nuclei and the other splits them. The unifying explanation is:

• A. Both reactions produce lighter nuclei, and lighter nuclei always have more binding energy
• B. Both reactions destroy neutrons, releasing the energy stored in neutron mass
• C. Both reactions move product nuclei toward the peak of the binding energy per nucleon curve near iron, producing more tightly bound products
• D. Fission releases energy because uranium is radioactive, while fusion releases energy because hydrogen has too few neutrons

Answer: (C)

The binding energy per nucleon curve peaks near iron (A ≈ 56). Light nuclei like hydrogen are on the left side of the peak; heavy nuclei like uranium are on the right. Fusion moves light nuclei up the left slope toward iron (more tightly bound products), releasing energy. Fission moves heavy nuclei down the right slope toward iron (more tightly bound fragments), also releasing energy. Both processes converge on iron from opposite sides. The unifying principle is that both reactions produce products with higher binding energy per nucleon than the starting material.

Question 2

A massive star fuses hydrogen into helium, then helium into carbon, and so on up the periodic table until it builds up a core of iron. Why does fusion stop producing energy at iron?

• A. Iron nuclei are too large for the strong nuclear force to bind any additional nucleons
• B. Iron is chemically inert and its electrons prevent nuclei from getting close enough to fuse
• C. Iron sits at the peak of the binding energy per nucleon curve; fusing or fissioning iron would produce less tightly bound products, requiring energy input rather than releasing it
• D. Iron has too many neutrons relative to protons to undergo further fusion reactions

Answer: (C)

Iron (A ≈ 56) is at the bottom of the nuclear energy valley — the most tightly bound configuration of nucleons. Any reaction that moves nuclei away from iron (either toward lighter or heavier products) requires energy; any reaction toward iron releases energy. Stars 'fall downhill' toward iron during nucleosynthesis. Once the core is iron, there is no further energy to extract from nuclear reactions, and the core collapses under gravity — triggering a supernova.

Question 3

Heavy stable nuclei (like lead or bismuth) have roughly equal numbers of protons and neutrons, just as light nuclei like helium-4 do.

• T. True
• F. False

Answer: False

For light nuclei, N ≈ Z is stable. But for heavier nuclei, the stability curve curves above the N = Z line: heavy stable nuclei have more neutrons than protons. The reason is electrostatic repulsion: as Z increases, the cumulative proton-proton repulsion becomes substantial and destabilizes the nucleus. Extra neutrons provide additional strong-force binding without adding to the Coulomb repulsion, diluting the proton density. Bismuth-209 (the heaviest stable nucleus) has 83 protons and 126 neutrons. Nuclei on the N = Z line at high mass number are unstable and undergo beta-plus decay or proton emission.

Question 4

The binding energy of a nucleus represents the energy that must be supplied to completely disassemble it into free, separated protons and neutrons.

• T. True
• F. False

Answer: True

This is the definition: BE = (Z·m_p + N·m_n − M_nucleus)·c². The mass defect — the 'missing' mass of the assembled nucleus compared to its free constituents — has been converted into binding energy that holds the nucleus together. To reverse the process (disassemble the nucleus into free nucleons), you must supply exactly this energy. A larger binding energy means more tightly bound, more stable nucleus. The binding energy per nucleon (BE/A) allows comparison across nuclei of different sizes and is the quantity plotted on the stability curve.

Question 5

Using the binding energy per nucleon curve, explain why a nuclear power plant (fissioning uranium) and the sun (fusing hydrogen) both extract energy from nuclear reactions, even though one splits nuclei and the other combines them.

Model answer: The binding energy per nucleon curve peaks near iron (A ≈ 56) at about 8.8 MeV/nucleon. Uranium (A ≈ 235) sits on the right side of the peak at a lower BE/A. When uranium fissions into two medium-weight fragments (near A ≈ 90-140), those fragments sit closer to the peak and have higher BE/A — the products are more tightly bound than the reactant. The difference in binding energy per nucleon is released as kinetic energy and radiation. Hydrogen (A = 1) sits far to the left of the peak at nearly zero BE/A. When hydrogen fuses to helium-4 (BE/A ≈ 7 MeV), the helium is much more tightly bound, releasing about 7 MeV per nucleon. Both reactions extract energy by moving toward the iron peak from opposite sides.

The key conceptual step is recognizing that 'more tightly bound' = lower mass (via mass defect = binding energy / c²). Reactions that produce more tightly bound products release the mass difference as energy. The curve is a map of nuclear stability, and both fission and fusion navigate downhill on this map toward iron.